DIAGONAL CYCLES AND EULER SYSTEMS II: THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR HASSE-WEIL-ARTIN L-FUNCTIONS

HENRI DARMON AND VICTOR ROTGER

Abstract. This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over ℚ viewed over the fields cut out by certain self-dual Artin representations of dimension at most 4. When the associated L-function vanishes (to even order ≥ 2) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida p-adic L-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain p-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.

Contents

Introduction 1 1. Global Λ-adic cohomology classes 8 2. Local Λ-adic cohomology classes 21 3. The syntomic Abel-Jacobi map on products of semistable curves 31 4. Bloch-Kato logarithms of weight two specializations 35 5. The explicit reciprocity law 43 6. Application to the Birch and Swinnerton-Dyer conjecture 46 References 52

Introduction

Let E be an elliptic curve over ℚ and let

ϱ : G_{ℚ} = Gal (ℚ̄/ℚ) ⟶ Aut_{L}(V_{ϱ}) ≃ GL_{n}(L)

be an Artin representation with coefficients in a finite extension L ⊂ ℚ̄ ⊂ ℂ of ℚ, factoring though the Galois group of a finite extension H/ℚ. The Hasse-Weil-Artin L-series L(E, ϱ, s) of E twisted by ϱ, defined on the right half-plane Re(s) > 3/2 by an absolutely convergent Euler product of degree 2n, is expected to admit an analytic continuation to the whole complex plane.

Let E(H)^{ϱ} := Hom_{G_{ℚ}}(V_{ϱ}, E(H) ⊗ L) denote the ϱ-isotypical component of the Mordell-Weil group E(H), and define the analytic and algebraic rank of the twist of E by ϱ as

r_{an}(E, ϱ) = ord_{s=1}L(E, ϱ, s), r(E, ϱ) = dim_{L} E(H)^{ϱ}.

A Galois-equivariant refinement of the Birch and Swinnerton-Dyer conjecture predicts that

(1)  r_{an}(E, ϱ) =^{?} r(E, ϱ).

The main object of this article is to study conjecture (1) when ϱ = ϱ_{1} ⊗ ϱ_{2}, where

ϱ_{1}, ϱ_{2} : G_{ℚ} ⟶ GL_{2}(L)

are odd, irreducible two-dimensional Artin representations satisfying det(ϱ_{1}) = det(ϱ_{2})^{−1}. This last condition implies that ϱ has real traces, i.e., is isomorphic to its dual, or contragredient, representation.

Thanks to the major advances achieved in the last two decades, the elliptic curve E and the Artin representations ϱ_{1} and ϱ_{2} are known to be associated to cuspidal newforms

f ∈ S_{2}(N_{f}),  g ∈ S_{1}(N_{g}, χ),  h ∈ S_{1}(N_{h}, χ^{−1})

of weight 2, 1, 1, respectively. The nebentype character χ = det(ϱ_{1}) of g is an odd Dirichlet character whose conductor divides both N_{g} and N_{h}.

The L-function L(E, ϱ, s) can be identified with the triple-product convolution L(f, g, h, s), whose analytic continuation and functional equation follow from the work of Garrett, Piatetski-Shapiro and Rallis. In particular, the analytic rank r_{an}(E, ϱ) is defined unconditionally, suggesting that (1) might repay closer scrutiny in this setting.

It is assumed throughout this work that the level N_{f} of f is relatively prime to N_{g}N_{h}. This implies (cf. [Pr, thm. 1.4]) that all the local signs in the functional equation of L(f, g, h, s) = L(E, ϱ, s) are +1, hence that the same is true for the global sign, implying that r_{an}(E, ϱ) is even. Subject to these hypotheses, the main results of this article are Theorems A and B below. (Cf. Theorem 6.7 and Theorem 6.13 for somewhat more general statements.)

Theorem A. If L(E, ϱ, 1) ≠ 0 then E(H)^{ϱ} = 0.

Theorem A has broad implications for the arithmetic of E over ring class fields of quadratic fields, both imaginary and real. For any ring class character ψ of a quadratic field K (of conductor relatively prime to N_{f} =: N_{E}), let H/K denote the ring class field cut out by it, and define

E(H)^{ψ} = {P ∈ E(H) ⊗ L such that P^{σ} = ψ(σ)P for all σ ∈ Gal (H/K)}.

When K is a real quadratic field, assume further that the pair (E, K) satisfies the non-vanishing hypothesis of Definition 6.8. This mild hypothesis is expected to always hold, and is satisfied, for example, as soon as E has quadratic twists E′ of both possible signs for which L(E′/K, 1) ≠ 0.

Corollary A1. If L(E/K, ψ, 1) ≠ 0 then E(H)^{ψ} = 0.

When K is imaginary quadratic, theorems of this sort were already established by a variety of approaches (cf. [Ko89], [BD97, Th.B], [BD99, Thm.1.2.], [BD05, Cor.4] and [Ne3, Th.A’], for example) using Heegner points on modular or Shimura curves, combined with p-adic techniques and/or congruences between modular forms. Their common feature is that they ultimately rely on Kolyvagin’s Euler system of CM points and on some variant (either classical, or p-adic) of the Gross-Zagier formula [GZ86]. The approach followed in this article makes no use of CM points, which accounts for why it extends unconditionally to more general settings, including the case where K is a real quadratic field.

The intimate connection between Corollary A1 for K real quadratic and the theory of Stark-Heegner points initiated in [Da] is suggested by the articles [BDD07] and [LRV13], which give a conditional proof of Corollary A1 resting on the algebraicity of Stark-Heegner points, essentially by replacing CM points with their conjectural real quadratic counterparts in the proof of [BD99, Thm.1.2]. Stark-Heegner points are still poorly understood, but the hope that the structures developed in the proof of Corollary A1 might shed light on their algebraicity was an important motivation for the present work.

Specialising Theorem A to the case where the representation ϱ = ϱ_{1} ⊗ ϱ_{2} is irreducible establishes the Birch and Swinnerton-Dyer conjecture in analytic rank zero, in scenarios lying beyond the scope of both Heegner and Stark-Heegner points. For instance, in the case where the projective representations attached to ϱ_{1} and ϱ_{2} cut out the same A_{5}-extension of ℚ, but where ϱ_{1} and ϱ_{2} are not twists of each other, Theorem A leads to the following instance of the Birch and Swinnerton-Dyer over quintic fields, which is spelled out in Theorem 6.11.

Corollary A2. Let E be an elliptic curve over ℚ, let K be a non-totally real quintic field with Galois group A_{5}, and assume that the discriminants of E and K are coprime. Then

ord_{s=1}L(E/K, s) = ord_{s=1}L(E/ℚ, s) ⇒ rank E(K) = rank E(ℚ).

The second main theorem of this article is concerned with the case where L(E, ϱ, s) vanishes at s = 1 (and hence to order at least 2). Note that ϱ_{1} and ϱ_{2} are both regular, i.e., there exists σ ∈ G_{ℚ} acting on each of these Galois representations with distinct eigenvalues.

Fix an odd prime p not dividing N := lcm(N_{f}, N_{g}, N_{h}) at which f is not Eisenstein and fix throughout an embedding ℚ̄ ↪ ℚ̄_{p}. Let Frob_{p} ∈ G_{ℚ} denote the Frobenius element at p induced by this embedding. Let L_{p} denote the completion of L in ℚ̄_{p}. Let (α_{g}, β_{g}) and (α_{h}, β_{h}) be the pairs of eigenvalues of ϱ_{g}(Frob_{p}) and ϱ_{h}(Frob_{p}), respectively. It shall be assumed throughout this article that

E is ordinary at p, and α_{g} ≠ β_{g}, α_{h} ≠ β_{h}.

Such a prime exists by the Chebotarev density theorem.

Theorem B below relates the p-Selmer group attached to E and ϱ to the behaviour of the Garrett-Hida p-adic L-functions attached to f, g, and h. In order to describe these p-adic L-functions more precisely, let Sord Λ̃(M, ψ) denote the space of ordinary Λ-adic modular forms of tame level M ≥ 1 and tame character ψ with coefficients in a finite flat extension Λ̃ of the Iwasawa algebra Λ = ℤ_{p}[[1 + pℤ_{p}]]. These spaces are locally free Λ̃-modules of finite rank, and are equipped with a natural Λ̃-linear action of the Hecke operators.

A Hida family \underline{ϕ} can be viewed concretely as a formal q-series with coefficients in Λ̃ whose specialisations give rise to a p-adically coherent collection {ϕ_{x}} of overconvergent ordinary modular forms indexed by the points x in the rigid analytic space Ω̃ := Hom_{cts}(Λ̃, ℂ_{p}). This space is equipped with a finite flat morphism w : Ω̃ ⟶ Ω to the weight space Ω := Hom_{cts}(Λ, ℂ_{p}), which is also equipped with a natural embedding ℤ ⊂ Ω. When k = w(x) ∈ ℤ, the form ϕ_{x} belongs to the space S_{k}^{oc}(M, ψω^{1−k}) of overconvergent modular forms of weight k, and is classical when k ≥ 2 by [Hi93, Ch. 7.3, Thm. 3]. When k = 1, the form ϕ_{x} may be classical or not.

If \underline{ϕ} is a Hida family of eigenforms of tame level M and tame character ψ, let

(2)  Sord Λ̃(M, ψ)[\underline{ϕ}] := \{\underline{φ} ∈ Sord Λ̃(M, ψ) s.t. | \begin{matrix}T_{ℓ}\underline{φ} = a_{ℓ}(\underline{ϕ})\underline{φ}, & ∀ℓ ∤ Mp, \\ U_{p}\underline{φ} = a_{p}(\underline{ϕ})\underline{φ}, \end{matrix} \} ,

(3) Sord Λ̃(M, ψ)^{∨}[\underline{ϕ}] = \{\underline{γ} : Sord Λ̃(M, ψ) → Λ̃ s.t. | \begin{matrix}\underline{γ}(T_{ℓ}^{∗}\underline{φ}) = a_{ℓ}(\underline{ϕ})\underline{γ}(\underline{φ}), & ∀ℓ ∤ Mp, \\ \underline{γ}(U_{p}^{∗}\underline{φ}) = a_{p}(\underline{ϕ})\underline{γ}(\underline{φ}) & \underline{φ} ∈ Sord Λ̃(M, ψ) \end{matrix} \}

denote the \underline{ϕ}-isotypic subspaces of Sord Λ̃(M, ψ) and of its Λ̃-linear dual, in which the latter is endowed with the adjoint action of the Hecke operators, as described in (24) below, the adjoints being taken relative to Poincaré duality.

Let f_{α} ∈ S_{2}(N_{f}p) denote the unique ordinary p-stabilisation of f, and write g_{α}, g_{β} ∈ S_{1}(N_{g}p, χ) (resp. h_{α}, h_{β} ∈ S_{1}(N_{h}p, χ^{−1})) for the pair of ordinary stabilisations of g (resp. h). By [Hi93, Ch. 7.3, Thm. 3] combined with a recent result of Bellaïche and Dimitrov [BeDi], there is a unique triple of Hida families, denoted by (\underline{f}, \underline{g}, \underline{h}), which specialises to (f_{α}, g_{α}, h_{α}) in weights (2, 1, 1). These Hida families are normalised eigenforms

\underline{f} ∈ S_{Λ_{f}}^{ord}(N_{f}),  \underline{g} ∈ S_{Λ_{g}}^{ord}(N_{g}, χ),  \underline{h} ∈ S_{Λ_{h}}^{ord}(N_{h}, χ^{−1}),

with coefficients in suitable finite flat extensions Λ_{f}, Λ_{g} and Λ_{h} of Λ.

The Garrett-Hida p-adic L-functions denoted

ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h}),  ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h}),  ℒ_{p} ^{h_{α}}(˘\underline{f}, ˘ \underline{g}, ˘ \underline{h}^{∗})

depend on the choice of a triple (˘\underline{f}, ˘ \underline{g}, ˘ \underline{h}) of Λ-adic test vectors

(4)  ˘  \underline{f} ∈ S_{Λ_{f}}^{ord}(N)[\underline{f}],  ˘  \underline{g} ∈ S_{Λ_{g}}^{ord}(N, χ)[\underline{g}],  ˘  \underline{h} ∈ S_{Λ_{h}}^{ord}(N, χ^{−1})[\underline{h}],

as well as on a triple of dual test vectors

(5)  ˘  \underline{f}^{∗} ∈ S_{Λ_{f}}^{ord}(N)^{∨}[\underline{f}]  ˘  \underline{g}^{∗} ∈ S_{Λ_{g}}^{ord}(N, χ^{−1})^{∨}[\underline{g}],  ˘  \underline{h}^{∗} ∈ S_{Λ_{h}}^{ord}(N, χ)^{∨}[\underline{h}].

The specialisations f̆_{x}, ğ_{y} and h̆_{z} at points of weight k, ℓ, m ∈ ℤ belong to the space of overconvergent oldforms of tame level N and weights attached to f_{x}, g_{y} and h_{z} respectively, while f̆_{x}^{∗}, ğ_{y}^{∗} and h̆_{z}^{∗} belong to the ℂ_{p}-linear duals S_{k}^{oc}(N, ω^{1−k})^{∨}[f_{x}^{∗}], S_{ℓ}^{oc}(N, χ^{−1}ω^{1−ℓ})^{∨}[g_{y}^{∗}] and S_{m}^{oc}(N, χω^{1−m})^{∨}[h_{z}^{∗}].

The space S_{k}^{n-oc}(N, χ) of nearly overconvergent p-adic modular forms is described for instance in [DR13, §2.4]. Lemma 2.7 of loc.cit. asserts that the image of Hida’s ordinary projector e_{ord} := lim U_{p}^{n!} on S_{k}^{n-oc}(N, χ) is contained in the ordinary part of S_{k}^{oc}(N, χ). Let d = q \frac{d}{dq} be the Atkin-Serre operator on p-adic modular forms, and let ϕ^{[p]} denote the p-depletion of a classical or p-adic modular form ϕ of weight k, level N and character χ, so that d^{t}ϕ^{[p]} is determined by the fourier expansion

d^{t}ϕ^{[p]}(q) = ∑_{p∤n} n^{t}a_{n}(ϕ)q^{n}.

For t ≥ 0, this modular form belongs to the space S_{k+2t}^{n-oc}(N, χ). In particular, the modular form d^{t}ğ_{y}^{[p]} × h̆_{z} is a nearly overconvergent modular form of weight 2t + ℓ + m.

With these basic notions and the Λ-adic test vectors in hand, the Garrett-Hida p-adic L-function attached to (˘\underline{f}^{∗}, ğ, ˘ \underline{h}) is defined (as described in [DR13, Def. 4.4], but adopting a somewhat more flexible point of view) to be the p-adic rigid-analytic function on Ω_{f} × Ω_{g} × Ω_{h} satisfying

(6)  ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h})(x, y, z) := f̆_{x}^{∗}(e_{ord}(d^{t}ğ_{y}^{[p]} × h̆_{z})),  t := (k − ℓ − m)/2,

as (x, y, z) ranges over the dense set of points in Ω_{f} × Ω_{g} × Ω_{h} of integral weights (k, ℓ, m), with k ≡ ℓ + m (mod 2). Viewing this as a p-adic L-function attached to the triple convolution of the Hida families \underline{f}, \underline{g}, and \underline{h} is justified by fundamental results of Garrett, Harris-Kudla and Ichino relating the quantity in (6) to the central critical values

L(f_{x}, g_{y}, h_{z}, c),  c := (k + ℓ + m − 2)/2, when k, ℓ, m ≥ 1,  k ≥ ℓ + m,

up to a product of local factors which are non-zero for a suitable choice of test vectors. Identical definitions apply to ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h}) and ℒ_{p} ^{h_{α}}(˘\underline{f}, ˘ \underline{g}, ˘ \underline{h}^{∗}), with the difference that their regions of classical interpolation are defined by the inequalities ℓ ≥ k + m and m ≥ k + ℓ respectively.

The point (x_{0}, y_{0}, z_{0}) ∈ Ω_{f} × Ω_{g} × Ω_{h} of weight (2, 1, 1) for which (f_{x_{0}}, g_{y_{0}}, h_{z_{0}}) = (f_{α}, g_{α}, h_{α}) thus lies within the region of interpolation defining ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h}), and its value at this point is given by the formula

(7)  ℒ_{p} ^{f_{α}}(f̆^{∗}, ğ_{α}, h̆_{α}) := ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h})(x_{0}, y_{0}, z_{0}) = f̆_{x_{0}}^{∗}(e_{ord}(ğ_{α}^{[p]} × h̆_{α}))

arising from (6). Ichino’s explicit form of Garret’s formula asserts that the square of the expression on the right is a simple (non-zero, for a judicious choice of test vectors) multiple of the central critical value L(E, ϱ, 1) = L(f, g, h, 1). In particular, although ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h}) depends crucially on the Hida families \underline{f}, \underline{g} and \underline{h}, and hence on the p-stabilisations of g and h that were chosen to define these Hida families, the resulting values at the classical point (x_{0}, y_{0}, z_{0}) do not depend in an essential way on the choices α_{g} and α_{h} of U_{p}-eigenvalues, or on the concomitant choice of test vectors. It is therefore not too egregious an abuse of notations to set

ℒ_{p} ^{f_{α}}(f, g, h) := ℒ_{p} ^{f_{α}}(f̆^{∗}, ğ_{α}, h̆_{α}) ∼ ℒ_{p} ^{f_{α}}(f̆^{∗}, ğ_{α}, h̆_{β}) ∼ ℒ_{p} ^{f_{α}}(f̆^{∗}, ğ_{β}, h̆_{α}) ∼ ℒ_{p} ^{f_{α}}(f̆^{∗}, ğ_{β}, h̆_{β}),

where ∼ denotes an equality up to an algebraic factor (in the field generated by the Fourier coefficients of f_{α}, g_{α}, and h_{α}) which can be made non-zero with a good choice of test vectors.

The value

(8)  ℒ_{p} ^{g_{α}}(f̆, ğ_{α}^{∗}, h̆_{α}) := ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h})(x_{0}, y_{0}, z_{0}) = ğ_{α}^{∗}(e_{ord}(d^{−1}f̆^{[p]} × h̆_{α}))

of ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h}) at the point (x_{0}, y_{0}, z_{0}), which now lies outside the region of classical interpolation, is somewhat more subtle. It depends critically on the choice of stabilisation of g, but [DR13, Lemma 2.17] shows that ℒ_{p} ^{g_{α}}(f̆, ğ_{α}^{∗}, h̆_{α}) is left unchanged upon replacing h̆_{α} with the other p-stabilisation h̆_{β} of h. Furthermore, a change in the choice of test vector (belonging to the same (f, g_{α}, h)-isotypic subspaces for the action of the Hecke operators) has the effect of multiplying ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) by explicit local zeta integrals at the places dividing N, of exactly the same type as occur in Ichino’s formula for the Garret-Rankin triple product. One is hence justified in adopting the slightly abusive but simpler notations

(9)  ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) := ℒ_{p} ^{g_{α}}(f̆, ğ_{α}^{∗}, h̆_{α}) = ℒ_{p} ^{g_{α}}(f̆, ğ_{α}^{∗}, h̆_{β}),

and likewise for the other p-stabilisations of g and of h. The four non-classical p-adic L-values

(10)  ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆), ℒ_{p} ^{g_{β}}(f̆, ğ^{∗}, h̆), ℒ_{p} ^{h_{α}}(f̆, ğ, h̆^{∗}), ℒ_{p} ^{h_{β}}(f̆, ğ, h̆^{∗})

ostensibly bear no simple direct relation to L(f, g, h, 1), or to each other. When L(f, g, h, 1) = 0, they should rather be viewed as different p-adic avatars of L′′(f, g, h, 1).

Let Sel_{p}(E, ϱ) := H_{fin}^{1}(ℚ, V_{p}(E) ⊗ V_{ϱ} ⊗_{L} L_{p}) denote the ϱ-isotypic component of the Bloch-Kato Selmer group of E/H; cf. (154) for the precise definition of this group. The following theorem shows that the above p-adic values convey non-trivial information about Sel_{p}(E, ϱ).

Theorem B. If L(E, ϱ, 1) = 0 and ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) ≠ 0 for some choice of test vectors, then

dim_{L_{p}} Sel_{p}(E, ϱ) ≥ 2.

The same conclusion applies, of course, when any of the four p-adic L-values in (10) is non-zero. Recent joint work with Alan Lauder [DLR] describes an algorithm for calculating the expressions in (8), and hence ℒ_{p} ^{g_{α}}(f̆, ğ_{α}^{∗}, h̆_{α}), and uses this to verify their non-vanishing in many instances of analytic rank two, showing that Theorem B is not vacuous.

Theorem D below formulates a more precise version of Theorem B, in which two canonical elements in Sel_{p}(E, ϱ) are associated to the data (f, g_{α}, h), and shown to be linearly independent under the hypotheses of Theorem B. One of the main contributions of [DLR] and [DR15] is the formulation of a conjecture expressing these two elements as explicit p-adic linear combinations of global points in the Mordell-Weil group attached to E(H)^{ϱ}. (Cf. loc.cit. for the precise statement of this conjecture and a summary of the experimental evidence that has been gathered in its support.)

Theorem B can be compared to the Theorem B of Skinner and Urban [SU], which asserts that the rank of the Selmer group of an elliptic curve is ≥ 2 when the associated L-function vanishes to even order ≥ 2. There is little overlap between both theorems, whose methods of proof are very different. The result of [SU] has the virtue of requiring no non-vanishing hypotheses on an auxiliary p-adic Lfunction, while Theorem B applies to a different class of Artin representations, typified by the settings described in Corollaries A1 and A2.

The proofs of Theorems A and B rest on a system of global cohomology classes for the Rankin convolution of three modular forms of weights (2, 1, 1), whose construction is one of the main contributions of the present work. These classes arise from generalised Gross-Kudla-Schoen cycles in the product of three Kuga-Sato varieties fibered over a classical modular curve, and (crucially) from their variation in Hida families.

The prequel [DR13] to the present work initiated the authors’ study of generalised Gross-Kudla-Schoen cycles, relating their images under the p-adic Abel-Jacobi map to special values of Garrett-Hida p-adic L-series attached to the triple convolution of three (Hida families of) cusp forms. The p-adic Abel-Jacobi map has the feature—not present in its complex analogue, or in related complex and p-adic heights—of factoring through the restriction to G_{ℚ_{p}} of its p-adic étale counterpart. This turns out to be crucial in arithmetic applications relying on the theory of Euler systems. More precisely, the étale Abel-Jacobi images of generalised diagonal cycles give rise, after applying suitable projections, to distinguished global Galois cohomology classes

(11)  κ(f_{x}, g_{y}, h_{z}) ∈ H^{1}(ℚ, V_{f_{x}g_{y}h_{z}}(N)),

where

• f_{x}, g_{y} and h_{z} are classical specialisations of weights k, ℓ, and m which are balanced in the sense that each weight is strictly smaller than the sum of the other two;

• V_{f_{x}g_{y}h_{z}}(N) is the Kummer self-dual twist of the direct sum of several copies of the tensor product of the p-adic representations V_{f_{x}}, V_{g_{y}}, V_{h_{z}} of G_{ℚ} attached by Eichler-Shimura and Deligne to these forms, occuring in the middle cohomology of a Kuga-Sato variety E_{k} (resp. E_{ℓ}, E_{m}) of dimension k − 1 (resp. ℓ − 1, m − 1).

The extensions of p-adic Galois representations associated to the classes κ(f_{x}, g_{y}, h_{z}) arise from geometry; namely, they are realised in the p-adic étale cohomology of an open subvariety of the product of Kuga-Sato varieties E_{k,ℓ,m} := E_{k} × E_{ℓ} × E_{m}, which have good reduction at p. In particular, thanks to the work of Saito their restrictions to a decomposition group at p are known to be crystalline.

The main theorem of [DR13] can be interpreted as a direct relationship between the Bloch-Kato p-adic logarithms of the class κ(f_{x}, g_{y}, h_{z}) and the special values of Garrett-Hida p-adic L-functions at points which lie outside their range of classical interpolation. The concluding paragraphs of the introduction to [DR13] stressed the desirability of deforming this result along p-adic families and, in particular, of p-adically interpolating the classes κ(f_{x}, g_{y}, h_{z}) themselves as the triple (f_{x}, g_{y}, h_{z}) is made to vary over the classical, balanced specialisations of ordinary Hida families \underline{f}, \underline{g} and \underline{h}. Such a p-adic interpolation would be described by a global cohomology class

(12)  κ(\underline{f}, \underline{g}, \underline{h}) ∈^{?} H^{1}(ℚ, 𝕍_{\underline{f} \underline{g} \underline{h}}(N)),

where 𝕍_{\underline{f} \underline{g} \underline{h}}(N) is the direct sum of several copies of the tensor product of Hida’s Λ-adic representations 𝕍\underline{f}, 𝕍\underline{g}, 𝕍\underline{h} attached to \underline{f}, \underline{g}, and \underline{h}, twisted in such a way that for each triple (f_{x}, g_{y}, h_{z}) of classical (balanced or unbalanced) specialisations of (\underline{f}, \underline{g}, \underline{h}) there are G_{ℚ}-equivariant specialisation homomorphisms

ν_{xyz} : 𝕍_{\underline{f} \underline{g} \underline{h}}(N) ⟶ V_{f_{x}g_{y}h_{z}}(N).

The putative class κ(\underline{f}, \underline{g}, \underline{h}) would be uniquely determined by the requirement that its specialisations at balanced triples (f_{x}, g_{y}, h_{z}) should agree, up to normalisation by elementary fudge factors, with the classes κ(f_{x}, g_{y}, h_{z}) of (11). Guided by this ideal goal but falling somewhat short of it, Chapter 1 attaches a one-variable cohomology class to a triple (f, \underline{g}, \underline{h}) formed by an eigenform f of weight two and trivial nebentype and two Hida families \underline{g} and \underline{h} of tame character χ and χ^{−1}, respectively. This Λ-adic class is described by:

• a finite flat algebra Λ_{f\underline{gh}} over the Iwasawa algebra Λ = ℤ_{p}[[1 + pℤ_{p}]];

• a Λ_{f\underline{gh}}[G_{ℚ}]-module 𝕍_{f\underline{gh}}(N) whose classical specialisations are of the form V_{fg_{y}h_{z}}(N) where the pairs (g_{y}, h_{z}) range over the specializations of \underline{g} and \underline{h} at classical points of the same weight ℓ and nebentype character at p;

• a cohomology class κ(f, \underline{gh}) ∈ H^{1}(ℚ, 𝕍_{f\underline{gh}}(N)).

The class κ(f, \underline{gh}) gives rise to a collection of global classes in H^{1}(ℚ, V_{fg_{y}h_{z}}(N)), varying p-adic analytically as (g_{y}, h_{z}) varies over pairs of specialisations of \underline{g} and \underline{h} with common weight and nebenytpe character at p. In order to construct κ(f, \underline{gh}), Section 1.3 introduces a family \{Δ_{s}\}_{s≥1} of null-homologous twisted diagonal cycles on certain twists of finite quotients of the three-folds X_{0}(Np) × X_{1}(Np^{s}) × X_{1}(Np^{s}), satisfying an explicit compatibility under the natural push-forward maps as s varies. The class κ(f, \underline{gh}) is then manufactured in §1.4-1.6 as the inverse limit of the cohomology classes one obtains as the image of Δ_{s} under a suitable projection of the p-adic étale Abel-Jacobi map.

Although κ(f, \underline{gh}) is constructed by interpolating geometric constructions attached only to the weight two specialisations of \underline{g} and \underline{h}, it makes sense to consider its specialisations

(13)  κ(f, g_{y}, h_{z}) := κ(f, \underline{gh})_{y,z} ∈ H^{1}(ℚ, V_{fg_{y}h_{z}}(N))

at any point (y, z) attached to a pair (g_{y}, h_{z}) of classical modular forms of arbitrary weight ℓ ≥ 1 with inverse tame nebentype characters. The relevance of this class to the proof of Theorems B and C arises when f is the weight 2 newform associated to the elliptic curve E, and g_{y} = g_{α} and h_{z} = h_{α} are (ordinary) p-stabilisations of the forms g and h associated to the Artin representations ϱ_{1} and ϱ_{2}.

Since g_{α} ≠ g_{β} and h_{α} ≠ h_{β}, the construction described above yields four relevant cohomology classes, denoted

(14)  κ(f, g_{α}, h_{α}), κ(f, g_{α}, h_{β}), κ(f, g_{β}, h_{α}), κ(f, g_{β}, h_{β}) ∈ H^{1}(ℚ, V_{fgh}(N)),

where again V_{fgh}(N) is just a direct sum of several copies of V_{p}(E) ⊗ V_{ϱ} ⊗_{L} L_{p}. These classes are called the generalised Kato classes attached to the triple (f, g, h) of modular forms of weights (2, 1, 1). Justifying this terminology is the fact that the classes of (14) were first constructed and studied by Kato in the special case where both g and h are weight one Eisenstein series, where they form the basis for his remarkable proof of the Birch and Swinnerton-Dyer conjecture in analytic rank zero for L-functions of elliptic curves twisted by Dirichlet characters.

The Kummer-self-dual representation V_{fgh} is still crystalline, but there is no a priori reason for the classes in (14) to be crystalline at p because the weights (2, 1, 1) of the triple (f, g, h) are unbalanced, with f being the eigenform of dominant weight in the sense of [DR13] and thus the points in weight space attached to the classes in (14) lie outside the range of “geometric interpolation” defining κ(f, \underline{gh}).

Theorem 6.4 of §5.2 establishes a reciprocity law relating the image of the classes in (14) under the Bloch-Kato dual exponential map to the central critical value L(E, ϱ, 1). It is the main ingredient in the proof of Theorem A.

Theorem C. The generalised Kato class κ(f, g_{α}, h_{α}) is crystalline at p if and only if L(E, ϱ, 1) = 0.

When L(E, ϱ, 1) ≠ 0, Theorem C combined with the regularity of V_{g} and V_{h} at p implies that the cohomology classes in (14) are linearly independent, more precisely, that their natural images generate the “singular quotient” of the local cohomology at p. A standard argument involving the global Poitou-Tate exact sequence then implies that the image of the Selmer group of V_{fgh} maps to zero in the local cohomology H^{1}(ℚ_{p}, V_{fgh}) at p. The fact that the Mordell-Weil group of global points always injects into the group of local points then shows the triviality of the ϱ-isotypical component of the Mordell-Weil group of E, thereby proving Theorem A.

One also expects (as a consequence of the Shafarevich-Tate conjecture) that Sel_{p}(E, ϱ) is trivial when E(H)_{L}^{ϱ} is, but the proof of Theorem A described above does not yield the finiteness of the associated Selmer group or Shafarevich-Tate group. The Gross-Schoen diagonal cycles can be viewed as a special instance of a broader class of examples, namely the “Euler systems of Garrett-Rankin-Selberg type” described in [BCDDPR], which also encompass Kato’s Euler system of Beilinson elements and the Euler system of Beilinson-Flach elements. In both these settings, finiteness results for the Shafarevich-Tate groups have been obtained by Kato [Ka98] and Lei, Loeffler, and Zerbes [LLZ1] respectively. It would be of great interest to extend these authors’ refinements to the setting of diagonal cycles. Recent work of Yifeng Liu makes striking progress in a closely related setting, where diagonal cycles are replaced by twisted variants on the product of a modular curve and a Hilbert modular surface [Liu1] and on a Hilbert modular threefold [Liu2], and the Selmer groups that are studied are those attached to Galois representations of weights (2, 2, 2) (in the region of “balanced weights”, which can thus be treated without recourse to the p-adic deformation of Abel-Jacobi images of cycles).

When L(E, ϱ, 1) = 0, and hence r_{an}(E, ϱ) ≥ 2, Theorem C implies that the four classes in (14) belong to the Selmer group Sel_{p}(E, ϱ). The submodule of Sel_{p}(E, ϱ) generated by these four classes is expected to be non-trivial precisely when r_{an}(E, ϱ) = 2, i.e., when L′′(E, ϱ, 1) ≠ 0. The following result, whose proof is described in Theorem 6.13 of Section 6.4, involves ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) rather than L′′(f, g, h, 1), and directly implies Theorem B above.

Theorem D. If L(E, ϱ, 1) = 0 and ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) ≠ 0 for some choice of test vectors, there exist G_{ℚ}-equivariant projections j_{α}, j_{β} : V_{fgh}(N) → V_{p}(E) ⊗ V_{ϱ} such that the classes κ_{αα} = j_{α}(κ(f, g_{α}, h_{α})) and κ_{αβ} = j_{β}(κ(f, g_{α}, h_{β})) are linearly independent in Sel_{p}(E, ϱ).

The canonical nature of the elements in Sel_{p}(E, ϱ) described in Theorem D is key to the formulation of the conjectures of [DLR] and [DR15], which seem to admit no direct counterpart for the Selmer classes constructed in [SU, Thm. B]. To further compare Theorem D with the results of [SU], note that generalised Kato classes can only contribute to the pro-p Selmer group Sel_{p}(E/ℚ) of E over ℚ when ϱ contains the trivial representation as a constituent, which occurs precisely when g and h = ḡ are dual to each other, so that ϱ = 1 ⊕ Ad^{0}(ϱ_{g}), where Ad^{0}(ϱ_{g}) denotes the three dimensional adjoint of ϱ_{g}, consisting of the trace zero endomorphisms of ϱ_{g}. In the scenario where L(E/ℚ, s) admits a double zero at the center, the elliptic Stark conjectures of [DLR] predict that all of the p-adic L-values in (10) vanish and that the generalised Kato classes attached to (f, g, ḡ) never generate the Selmer group or Mordell-Weil group of E. However, they are expected to generate a one-dimensional subspace of Sel_{p}(E/ℚ) whenever L(E, Ad^{0}(ϱ_{g}), 1) ≠ 0, consisting of the classes whose restriction to ℚ_{p} lies in the kernel of the logarithm map log_{p} : E(ℚ_{p}) ⟶ ℚ_{p}. The non-triviality of such a class only implies that dim Sel_{p}(E/ℚ) ≥ 1, but it does imply that rank(E(ℚ)) ≥ 2 under the further (highly non-trivial) assumption that LLI(E/ℚ) is finite. See [DR15, §4.5] for a more detailed discussion of this intriguing scenario.

Underlying the crucial reciprocity law relating the local behaviour at p of the global classes in (14) to p-adic L-values is a precise relationship between the p-adic families κ(f, \underline{gh}) of global cohomology classes (more precisely, their restriction to a decomposition group at p) and the Garrett-Hida p-adic L-functions alluded to above. See Theorem 5.3 for the precise statement. Such relationships are part of a long tradition of explicit reciprocity laws going back to the seminal work of Coates and Wiles, and even further to the work of Kummer, Iwasawa and Kubota-Leopoldt on the arithmetic of cyclotomic fields. For more on how the approach of this paper fits into the larger perspective of Euler systems and explicit reciprocity laws, notably with Kato’s work on Beilinson elements and the work of Bertolini and the authors on Beilinson-Flach elements, the reader is invited to consult the survey [BCDDPR].

Acknowledgements: The authors benefitted from many stimulating exchanges with Massimo Bertolini, Antonio Lei, David Loeffler and Sarah Zerbes on the topics of this paper. They are also grateful to the two anonymous referees for their thorough proofreading and valuable feedback which led to significant improvements in the exposition. (Of course, any flaws that remain in this regard are to be blamed on the authors alone.) The first author was supported by an NSERC Discovery grant, and the second by Grant MTM2012-34611.

1. Global Λ-adic cohomology classes

1.1. Cycles, Chow groups, and correspondences. This section collects a few general facts and notations concerning algebraic cycles and correspondences for future reference.

Given a smooth proper irreducible variety W of dimension d over a field F, and any integer c ≥ 0, let C^{c}(W)(F) denote the group of codimension c cycles in W defined over F with coefficients in ℤ_{p}, and let

CH^{c}(W)(F) = C^{c}(W)_{/∼}

denote the Chow group of rational equivalence classes of such cycles. The field F is sometimes supressed from the notation when this leads to no ambiguity.

The standard conventions are employed for Tate twists: if H is a ℤ_{p}[G_{ℚ}]-module and k is an integer, then H(k) = H ⊗ ϵ_{cyc}^{k}, where

(15)  ϵ_{cyc} : G_{ℚ} ⟶ ℤ_{p}^{×}

denotes the p-adic cyclotomic character.

If Δ is a cycle in C^{c}(W), the same symbol Δ will often be used as well to denote the class of this cycle in the Chow group CH^{c}(W). There is a fundamental p-adic étale cycle class map over F

(16)  cl_{F} : CH^{c}(W)(F) ⟶ H_{et}^{2c}(W, ℤ_{p}(c)),

which is described in [Mi, §23] when F is algebraically closed. For more general fields F (such as number fields), the cycle class map is described in [Ja, eqn. (0.3)], and its target is to be interpreted as the continuous étale cohomology of loc.cit. After choosing an embedding of F into an algebraic closure F̄, the symbol \overline{W} is used to denote the variety over F̄ deduced from W by extension of scalars, and

(17)  cl : CH^{c}(W)(F) ⟶ H_{et}^{2c}(\overline{W} , ℤ_{p}(c))

denotes the natural map that is deduced from (16). A cycle in CH^{c}(W) which is in the kernel of cl is said to be null-homologous, and the subgroup of null-homologous cycles defined over F is denoted CH^{c}(W)_{0}(F). The Hochschild-Serre spectral sequence

H^{i}(F, H_{et}^{j}(\overline{W} , ℤ_{p}(c)) ⇒ H_{et}^{i+j}(W, ℤ_{p}(c))

of [Ja] relating continuous étale cohomology with the continuous group cohomology of G_{F} = Gal (F̄/F) gives rise to the p-adic étale Abel-Jacobi map

(18)  AJ_{et} : CH^{c}(W)_{0}(F) ⟶ H^{1}(F, H_{et}^{2c−1}(W̄ , ℤ_{p})(c)),

as described, for instance, in the penultimate displayed equation of [Ja]. This map plays a key role in the constructions of the global cohomology classes alluded to in the introduction.

A correspondence between two varieties W_{1}, W_{2} of dimension d is an element T ∈ Corr(W_{1}, W_{2}) := C^{d}(W_{1} × W_{2}). As indicated, the symbol T is used to denote both the cycle and its class in CH^{d}(W_{1} × W_{2}). Let T^{*} ∈ C^{d}(W_{2} × W_{1}) denote the transpose correspondence from W_{2} to W_{1}. An element T ∈ Corr(W_{1}, W_{2}) gives rise to a homomorphism from CH^{c}(W_{1}) to CH^{c}(W_{2}) for any c ≥ 0 by the rule T (Δ) := π_{W_{2},∗}(T · π_{W_{1}}^{∗}(Δ)), where · denotes the intersection product on W_{1} × W_{2}, the maps π_{W_{1}} and π_{W_{2}} denote the natural projections of W_{1} × W_{2} to its first and second factors, and π_{W_{1}}^{∗} and π_{W_{2},∗} denote the induced pull-back and proper pushforward maps associated to π_{W_{1}} and π_{W_{2}} respectively.

When W = W_{1} = W_{2}, set Corr(W) := CH^{d}(W × W). There is a natural identification

H^{2d}(\overline{W} × \overline{W} , ℚ_{p})(d) = ⊕_{i=0}^{2d}End(H^{i}(\overline{W} , ℚ_{p}))

arising from the Künneth decomposition combined with Poincaré duality, and the image of a correspondence T ∈ Corr(W) under the map cl of (17) gives rise to a degree 0 endomorphism of the cohomology of W, which is compatible with the action on Chow groups via the cycle class map. Given a correspondence T on W, the same symbol is routinely used to describe this induced endomorphism on Chow groups and on cohomology. The transpose correspondence T^{*} gives rise to the adjoint endomorphisms on cohomology relative to Poincaré duality.

1.2. Modular curves. Let M ≥ 1 be a positive integer and X_{0}(M) and X_{1}(M) denote the classical modular curves over ℚ attached to the Hecke congruence subgroups Γ_{0}(M) and Γ_{1}(M), respectively.

Several distinct models over ℚ of X_{1}(M) occur in the literature; in what follows, the affine curve Y_{1}(M) obtained by removing the cusps from X_{1}(M) is taken to be the coarse moduli space of pairs (A, i) where A is an elliptic curve and i : μ_{M} → A is an isomorphism between the group scheme μ_{M} and a closed finite flat subgroup scheme of A. This model of X_{1}(M) is characterized as being the one whose field of rational functions ℚ(X_{1}(M)) is the subfield of ℂ(X_{1}(M)) given by those functions whose q-expansion at the cusp i∞ have rational coefficients.

Fix an integer N ≥ 3 and an odd prime p ∤ N. For any s ≥ 1, let X_{s} := X_{1}(Np^{s}). It will be convenient to view X_{s} as classifying triples (A, i_{N}, i_{p}) where i_{N} : μ_{N} → A and i_{p} : μ_{p^{s}} → A are embeddings of finite group schemes. It will also be useful to consider the curve X_{s}^{†}/ℚ which arises as the coarse moduli space associated to the problem of classifying triples (A, i_{N}, P) where (A, i_{N}) is as above and P ∈ A is a point of exact order p^{s}. The curves X_{s} and X_{s}^{†} are not naturally isomorphic over ℚ, but are twists of each over the cyclotomic field of p^{s}-th roots of unity. See below for more details.

The curves X_{s} are endowed with the following additional structures:

Diamond operators. Let

(19)  j_{s} : X_{s} ⟶ X_{0}(Np^{s})

be the forgetful map sending a triple (A, i_{N}, i_{p}) to (A, C), where C = ⟨Im(i_{N}), Im(i_{p})⟩ is the subgroup generated by the images of i_{N} and i_{p}. It is a Galois cover, whose Galois group is identified with the group

G_{s}^{(N)} = {⟨d⟩, d ∈ (ℤ/Np^{s}ℤ)^{×}} ≃ (ℤ/Np^{s}ℤ)^{×}/⟨±1⟩

of diamond operators, acting on X_{s} by the rule ⟨d⟩(A, i_{N}, i_{p}) := (A, d · i_{N}, d · i_{p}). Let

Γ_{s} ⊂ G_{s} ⊂ G_{s}^{(N)}, Γ_{s} = 1 + p(ℤ/p^{s}ℤ), G_{s} = (ℤ/p^{s}ℤ)^{×}

be the natural subgroups, and write

(20)  Λ_{s} := ℤ_{p}[Γ_{s}],  Λ̃_{s} := ℤ_{p}[G_{s}],  Λ̃_{s}^{(N)} := ℤ_{p}[G_{s}^{(N)}]

for the associated group rings with ℤ_{p}-coefficients. The corresponding completed group rings are

(21)  Λ := lim_{←} Λ_{s},  Λ̃ := lim_{←} Λ̃_{s},  Λ̃^{(N)} := lim_{←} Λ̃_{s}^{(N)}.

All of the group rings in (20) and (21) act naturally on spaces of modular forms on Γ_{1}(Np^{s}) as well as on the various cohomology groups attached to X_{s}.

Given a ∈ (ℤ/Nℤ)^{×} and b ∈ (ℤ/p^{s}ℤ)^{×}, it will sometimes be convenient to denote by ⟨a; b⟩ the diamond operator ⟨d⟩, where d ∈ (ℤ/Np^{s}ℤ)^{×} is congruent to a modulo N and to b modulo p^{s}. This automorphism acts on X_{s} by the rule ⟨a; b⟩(A, i_{N}, i_{p}) = (A, a · i_{N}, b · i_{p}).

Let ζ_{s} be a primitive p^{s}-th root of unity and let ϵ_{s} : G_{ℚ} ⟶ (ℤ/p^{s}ℤ)^{×} denote the mod p^{s} cyclotomic character, factoring through Gal (ℚ(ζ_{s})/ℚ). The curve X_{s}^{†} introduced above is isomorphic to the twist of X_{s} by the cocycle

(22)  ⟨ϵ_{s}^{−1}⟩ : G_{ℚ} ⟶ Aut(X_{s}), ⟨ϵ_{s}^{−1}⟩(σ) := ⟨1; ϵ_{s}^{−1}(σ)⟩,

and therefore X_{s} is isomorphic to X_{s}^{†} over ℚ(ζ_{s}).

Duality. The (étale and de Rham) cohomology groups of X_{s} are endowed with perfect pairings

(23) ⟨ , ⟩_{s} : H_{et}^{1}(X̄_{s}, ℤ_{p}) × H_{et}^{1}(X̄_{s}, ℤ_{p}) ⟶ ℤ_{p}(−1),  H_{dR}^{1}(X_{s}/ℚ) × H_{dR}^{1}(X_{s}/ℚ) ⟶ ℚ

arising from Poincaré duality. The symbol ⟨ , ⟩_{s} is used to denote both the étale and de Rham Poincaré pairing, relying on the context will make it clear which is being used. The de Rham pairing is defined on classes of differentials of the second kind by the usual formula

⟨ω_{1}, ω_{2}⟩_{s} := ∑_{P ∈X_{s}} res_{P} (F_{ω_{1},P} · ω_{2}),

where the sum is taken over the closed points P of X_{s}, and F_{ω_{1},P} denotes a local primitive of ω_{1} at P.

Hecke operators. There is a commuting algebra of Hecke correspondences T_{n} indexed by the integers prime to Np (defined for instance as in [DS05, Ch. 5]). For all s ≥ 1, write 𝕋_{0}(Np^{s}) and 𝕋_{1}(Np^{s}) for the ℚ-algebra generated by these Hecke operators, together with the diamond operators in the latter case, in the ring of correspondences of the curves X_{0}(Np^{s}) and X_{s} respectively. These “good” Hecke operators commute with their adjoints relative to the pairings ⟨ , ⟩_{s} and, more generally, relative to the Poincaré duality on any modular curve. More precisely,

(24)  T_{n}^{*} = ⟨n⟩T_{n},  for all n with gcd(n, Np) = 1.

Atkin-Lehner automorphisms. Fix a norm-compatible collection {ζ_{s}}_{s≥1} of primitive roots of unity of order p^{s}. Associated to ζ_{s} is the Atkin-Lehner automorphism denoted w_{s} which is defined on triples (A, i_{N}, i_{p}) by the rule

w_{s}(A, i_{N}, i_{p}) = (A/C_{p}, i_{N}, i′_{p}), C_{p} := ⟨i_{p}(ζ_{s})⟩,

where i′_{p}(ζ_{s}) = x′ (mod C_{p}) and x′ is a point on A of order p^{s} satisfying ⟨i(ζ_{s}), x′⟩_{Weil} = ζ_{s}. Likewise, the choice of a primitive N-th root of unity ζ_{N} gives rise to a similar Atkin-Lehner operation at N, which shall be denoted w, and is defined on any of the curves X_{s}. The automorphisms w_{s} and w do not commute with the diamond and Hecke operators, and instead satisfy

w_{s}⟨a; b⟩ = ⟨a; b^{−1}⟩w_{s},  w_{s}T_{n} = ⟨1; n⟩T_{n}w_{s},  w_{s}^{2} = ⟨−p^{s}; 1⟩, (25)
w⟨a; b⟩ = ⟨a^{−1}; b⟩w,  wT_{n} = ⟨n; 1⟩T_{n}w,  w^{2} = ⟨1; −N⟩. (26)

Moreover, the operators w_{s} are not defined over ℚ, but only over ℚ(ζ_{s}) and ℚ(ζ_{N}) respectively. More precisely, for all σ ∈ G_{ℚ},

(27)  (w_{s})^{σ} = ⟨1; ϵ_{s}^{−1}(σ)⟩ w_{s},  (w)^{σ} = ⟨ϵ_{N}^{−1}(σ); 1⟩ w,

where ϵ_{N} denotes the mod N cyclotomic character. The actions of w and w_{s} on differentials, functions, and de Rham cohomology via pullback will commonly be denoted by w and w_{s} rather than by the more strictly correct (but more notationally cumbersome) w^{∗} and w_{s}^{∗}.

The U_{p} operator. The Hecke operator U_{p} on X_{s} is realised by a correspondence of bi-degree p, which is given on geometric points by the rule

(28)  U_{p}(A, i_{N}, i_{p}) = ∑_{φ:A⟶A′}(A′, φ ∘ i_{N}, φ ∘ i_{p}),

where the sum is taken over the p distinct isogenies φ : A ⟶ A′ of degree p whose kernel is not equal to p^{s−1} · Im(i_{p}). The Hecke operator U_{p} acts on modular forms of weight two and its action on differentials of the form f(τ)dτ = ∑_{n≥1} a_{n}q^{n} \frac{dq}{q} is given by the familiar rule

(29)  U_{p}(f(τ)dτ) = \frac{1}{p} ∑_{j=0}^{p−1} f (\frac{t + j}{p} ) dτ = ∑_{n≥1} a_{np}q^{n}.

Note that while U_{p} commutes with the diamond operators, it does not commute with its adjoint U_{p}^{*} relative to the Poincaré pairings ⟨ , ⟩_{s}, which is given by the formula

U_{p}^{*} = w_{s}U_{p}w_{s}^{−1}.

A modular form (or cohomology class) ϕ on X_{s} is said to be ordinary if there exists a polynomial P (T) ∈ ℚ̄_{p}[T] such that P (U_{p})(ϕ) = 0 and P (0) is a p-adic unit. If instead ϕ has a similar property relative to U_{p}^{∗}, it is designated as anti-ordinary.

The ordinary and anti-ordinary projectors, which play a key role in the constructions of this article, are defined as

(30)  e_{ord} := lim_{n→∞} U_{p}^{n!},  e_{ord}^{∗} := lim_{n→∞}(U_{p}^{∗})^{n!}.

Projective systems: The curves X_{s} form projective systems in two different ways, relative to the collections of degeneracy maps

(31)  ϖ_{1}, ϖ_{2} : X_{s+1} → X_{s},  \{ \begin{matrix}ϖ_{1}(A, i_{N}, i_{p}) = (A, i_{N}, p · i_{p}), \\ ϖ_{2}(A, i_{N}, i_{p}) = (A, i_{N}, i_{p})/C, C := i_{p}(μ_{p}), \end{matrix}

where the quotient of the triple (A, i_{N}, i_{p}) by the subgroup scheme C is to be understood in the obvious sense. The degeneracy maps ϖ_{1} and ϖ_{2} are of degree p^{2} if s ≥ 1, and of degree (p^{2} − 1) if s = 0. With respect to the analytic uniformisation by the upper-half plane, the map ϖ_{1} corresponds to the natural map Γ_{s+1} · τ ↦ Γ_{s} · τ, while ϖ_{2} corresponds to Γ_{s+1} · τ ↦ Γ_{s} · (pτ). Both maps commute with the good Hecke operators and the diamond operators, and factor through the natural projection μ : X_{s+1} ⟶ X_{s}^{♭}, where X_{s}^{♭} denotes the modular curve associated to Γ_{1}(Np^{s}) ∩ Γ_{0}(p^{s+1}). More precisely, they can be written as ϖ_{1} = π_{1} ∘ μ, ϖ_{2} = π_{2} ∘ μ, and fit into the diagram below in which the vertical map is a Galois cover (with Galois group ℤ/pℤ, generated by suitable diamond operators) and the two horizontal maps are non-Galois morphisms of degree p:

(32)  \xymatrix{X_{s+1} \ar[d]_{μ} \ar[dr]^{ϖ_{1},ϖ_{2}} \\ X_{s}^{♭} \ar[r]_{π_{1},π_{2}} & X_{s}.}

The Hecke operator U_{p} acting on Ω_{1}(X_{s}) or on H_{dR}^{1}(X_{s}) is described by the standard formula

(33)  U_{p} = (π_{2})_{∗}π_{1}^{∗},

as can be verified directly from the formula for U_{p} in terms of the coordinate τ given in (29) (cf. also, for example, [DS05, Ex. 7.9.3]). It follows from this that

(34)  (ϖ_{2})_{∗}ϖ_{1}^{∗} = (π_{2})_{∗}μ_{∗}μ^{∗}π_{1}^{∗} = pU_{p}.

The endomorphism in (34) is the diagonal arrow in the commutative diagram:

\xymatrix{H_{dR}^{1}(X_{s}) \ar[r]^{ϖ_{1}^{∗}} \ar[dr]_{pU_{p}} \ar[d]_{(ϖ_{2})_{∗}} & H_{dR}^{1}(X_{s+1}) \ar[d]^{(ϖ_{2})_{∗}} \\ H_{dR}^{1}(X_{s−1}) \ar[r]_{ϖ_{1}^{∗}} & H_{dR}^{1}(X_{s})} ,

and it is apparent from this description that π_{1}^{∗}, ϖ_{1}^{∗}, (π_{2})_{∗} and (ϖ_{2})_{∗} commute with U_{p}, while π_{2}^{∗}, ϖ_{2}^{∗}, (π_{1})_{∗} and (ϖ_{1})_{∗} commute with the adjoint U_{p}^{*}. In particular, the Hecke operator U_{p} acts naturally on the inverse limit of the (étale or de Rham) cohomology groups H^{1}(X_{s}) taken relative to the maps (ϖ_{2})_{∗}. Whenever inverse limits of H^{1}(X_{s}) are discussed, it should always be understood that the transition maps are given by the pushforward maps (ϖ_{2})_{∗}.

Group ring-valued pairings. The action of the diamond operators on X_{s} can be combined with Poincaré duality to define group-ring valued pairings

(35) ⟨ , ⟩_{Γ_{s}} : H_{et}^{1}(X̄_{s}, ℤ_{p}) × H_{et}^{1}(X̄_{s}, ℤ_{p}) ⟶ ℤ_{p}(−1)[Γ_{s}], H_{dR}^{1}(X_{s}/ℚ) × H_{dR}^{1}(X_{s}/ℚ) ⟶ ℚ[Γ_{s}]

by the formula

⟨a, b⟩_{Γ_{s}} = ∑_{σ∈Γ_{s}}⟨a^{σ}, b⟩_{s} · σ^{−1} = ∑_{σ∈Γ_{s}}⟨a, b^{σ}⟩_{s} · σ.

As before, the same symbols are reserved for the pairings arising from Poincaré duality on de Rham or étale cohomology, leaving it to the context to determine which is being used. The pairings ⟨ , ⟩_{Γ_{s}} are Λ_{s}-linear and anti-linear (with respect to the involution sending a group-like element to its inverse) in the first and second argument respectively.

The pairings ⟨ , ⟩_{Γ_{s}} and ⟨ , ⟩_{s} are related by the formula, valid for all characters ϵ of Γ_{s},

(36)  ϵ(⟨a, b⟩_{Γ_{s}}) = ⟨θ_{ϵ}a, b⟩_{s} = ⟨a, θ_{ϵ^{−1}}b⟩_{s} = d_{s}^{−1}⟨θ_{ϵ}a, θ_{ϵ^{−1}}b⟩_{s},

where

(37)  θ_{ϵ} := ∑_{σ∈Γ_{s}} ϵ(σ) · σ^{−1} ∈ ℂ_{p}[Γ_{s}].

The first identity in (36) is a direct consequence of the definitions, while the second follows from the identity θ_{ϵ}^{2} = d_{s}θ_{ϵ}. A convenient modification of the ⟨ , ⟩_{Γ_{s}} pairings (in both their étale and de Rham incarnations) is given by setting

(38)  [a, b]_{Γ_{s}} := ⟨a, ww_{s} · U_{p}^{s} · b⟩_{Γ_{s}}.

All the Hecke operators on X_{s}, including U_{p}, are self-adjoint relative to this modified pairing, which is also Λ_{s}-linear relative to both its arguments. In addition, the pairings [ , ]_{Γ_{s}} have the virtue of being compatible with the maps ϖ_{2∗} and the ring homomorphisms

p_{s+1} : ℂ_{p}[Γ_{s+1}] ⟶ ℂ_{p}[Γ_{s}]

induced from the natural homomorphism Γ_{s+1} ⟶ Γ_{s}, in the following sense:

Lemma 1.1. For either □ ∈ {et, dR}, the diagram below commutes:

\xymatrix{H_{□}^{1}(X_{s+1}) × H_{□}^{1}(X_{s+1}) \ar[r]^{[ , ]_{Γ_{s+1}}} \ar[d]_{ϖ_{2∗}×ϖ_{2∗}} & ℂ_{p}[Γ_{s+1}] \ar[d]_{p_{s+1}} \\ H_{□}^{1}(X_{s}) × H_{□}^{1}(X_{s}) \ar[r]^{[ , ]_{Γs}} & ℂ_{p}[Γ_{s}]}

Proof. This follows from a direct calculation (in which the symbol ⟨ , ⟩_{X_{s}^{♭}} refers to the Poincaré pairing on the curve X_{s}^{♭}):

p_{s+1}([η_{s+1}, ϕ_{s+1}]_{Γ_{s+1}}) = ∑_{σ∈Γ_{s}}⟨σμ_{∗}η_{s+1}, μ_{∗}ww_{s+1}U_{p}^{s+1}ϕ_{s+1}⟩_{X_{s}^{♭}} · σ^{−1}

= ∑_{σ∈Γ_{s}}⟨σμ_{∗}η_{s+1}, ww_{s+1}U_{p}^{s}π_{1}^{∗}π_{2∗}μ_{∗}ϕ_{s+1}⟩_{X_{s}^{♭}} · σ^{−1}

= ∑_{σ∈Γ_{s}}⟨σμ_{∗}η_{s+1}, π_{2}^{∗}ww_{s}U_{p}^{s}ϕ_{s}⟩_{X_{s}^{♭}} · σ^{−1} = ∑_{σ∈Γ_{s}}⟨σπ_{2∗}μ_{∗}η_{s+1}, ww_{s}U_{p}^{s}ϕ_{s}⟩_{X_{s}} · σ^{−1}

= ∑_{σ∈Γ_{s}}⟨ση_{s}, ww_{s}U_{p}^{s}ϕ_{s}⟩_{X_{s}} · σ^{−1} = [η_{s}, ϕ_{s}]_{Γ_{s}}.

□

Define

H_{□}^{1}(X_{∞}) := \underleftarrow{lim}_{s} H_{□}^{1}(X_{s}),

the inverse limit being taken relative to the maps (ϖ_{2})_{∗}. Lemma 1.1 shows that the pairings [ , ]_{Γ_{s}} can be packaged into Λ-adic pairings with values in the appropriate completed group rings

(39)  [ , ]_{Γ} :H_{dR}^{1}(X_{∞}) × H_{dR}^{1}(X_{∞}) ⟶ ℂ_{p}[[Γ]],
(40)  [ , ]_{Γ} :H_{et}^{1}(X_{∞}, ℤ_{p}) × H_{et}^{1}(X_{∞}, ℤ_{p}) ⟶ ℤ_{p}[[Γ]](−1).

The étale incarnation of [ , ]_{Γ} is even G_{ℚ}-equivariant, after endowing ℤ_{p}[[Γ]] with the tautological G_{ℚ}-action whereby σ ∈ G_{ℚ} acts as multiplication by the group-like element ϵ_{cyc}(σ) ∈ ℤ_{p}^{×}.

1.3. Twisted diagonal cycles. For any integer s ≥ 1 define

W_{s,s} := X_{0}(Np) × X_{1}(Np^{s}) × X_{1}(Np^{s}) = X_{0}(Np) × X_{s} × X_{s}.

This three-fold is equipped with a natural action of the group G_{s}^{(N)} × G_{s}^{(N)} via the diamond operators acting on the second and third factors. Let

δ : X_{s} ⟶ X_{s} × X_{s} × X_{s}, X_{s}^{♭} ⟶ X_{s}^{♭} × X_{s}^{♭} × X_{s}^{♭}

denote the natural diagonal embeddings.

Fix a system {ζ_{s}} of compatible p^{s}-th roots of unity and set

(41)  ^{♭}Δ_{s,s} := (j_{1} ∘ ϖ_{2}^{s−1} ∘ w_{s}, Id, w_{s})_{∗}δ_{∗}(X_{s}) ∈ C^{2}(W_{s,s})(ℚ(ζ_{s})),

where ϖ_{2}^{s−1} := ϖ_{2} ∘ ⋯ ∘ ϖ_{2} : X_{s} → X_{1} and j_{1} : X_{1} → X_{0}(Np) are the maps introduced in (31) and (19) respectively. This codimension two cycle is just a copy of the curve X_{s} embedded diagonally in the three-fold product X_{s}^{3}, but with a twist by w_{s} in the first and third factors, projected to W_{s,s} in the natural way. It is the twist by w_{s} which causes ^{♭}Δ_{s,s} to only be defined over ℚ(ζ_{s}) in general. The cycle ^{♭}Δ_{s,s} will be referred to as the twisted diagonal of level s. Let

(Id, U_{p}, Id) : C^{2}(W_{s,s}) ⟶ C^{2}(W_{s,s})

be the endomorphism induced by the correspondence U_{p} acting on the middle factor of the triple product W_{s,s} and whose action on cycles is spelled out in the proof of Proposition 1.2 below. The correspondence (Id, U_{p}, Id) is realised as a copy of X_{0}(Np) × X_{s}^{♭} × X_{s} embedded in W_{s,s}^{2} via the map

(Id_{1}, π_{1}, Id_{3}, Id_{1}, π_{2}, Id_{3}) : X_{0}(Np) × X_{s}^{♭} × X_{s} ⟶ (X_{0}(Np) × X_{s} × X_{s}) × (X_{0}(Np) × X_{s} × X_{s}),

where Id_{1} and Id_{3} refer to the identity maps on the first and third factors respectively in the direct product, and π_{1}, π_{2} : X_{s}^{♭} ⟶ X_{s} are defined on the second factor. A direct calculation using the definition of the action of correspondences shows that

(42)  (Id, U_{p}, Id)_{∗}δ_{∗}(X_{s}) = (π_{1}, π_{2}, π_{1})_{∗}δ_{∗}(X_{s}^{♭}),  (Id, U_{p}^{∗}, Id)_{∗}δ_{∗}(X_{s}) = (π_{2}, π_{1}, π_{2})_{∗}δ_{∗}(X_{s}^{♭}).

The next proposition studies the compatibilities of the cycles ^{♭}Δ_{s,s} for varying s under the maps

(43)  ϖ_{22} = (Id, ϖ_{2}, ϖ_{2}) : W_{s+1,s+1} ⟶ W_{s,s},  s ≥ 1

induced by the degeneracy maps of (31).

Proposition 1.2. For all s ≥ 1, ϖ_{22,∗}(^{♭}Δ_{s+1,s+1}) = p · (Id, U_{p}, Id)_{∗}(^{♭}Δ_{s,s}) in C^{2}(W_{s,s}).

Proof. The commutativity of the diagram

\xymatrix{X_{s+1} \ar[r]^{δ} \ar[d]_{μ} & X_{s+1} × X_{s+1} × X_{s+1} \ar[d]_{(μ,μ,μ)} \ar[r]^{(j_{1}ϖ_{2}^{s}w_{s+1}, Id, w_{s+1})} & X_{0}(Np) × X_{s+1} × X_{s+1} \ar[d]^{(Id,μ,μ)} \\ X_{s}^{♭} \ar[r]^{δ} & X_{s}^{♭} × X_{s}^{♭} × X_{s}^{♭} \ar[d]_{(π_{1},π_{2},π_{1})} \ar[r]^{(j_{1}ϖ_{2}^{s−1}π_{2}w_{s+1}, Id, w_{s+1})} & X_{0}(Np) × X_{s}^{♭} × X_{s}^{♭} \ar[d]^{(Id,π_{2},π_{2})} \\  & X_{s} × X_{s} × X_{s} \ar[r]^{(j_{1}ϖ_{2}^{s−1}w_{s}, Id, w_{s})} & X_{0}(Np) × X_{s} × X_{s},}

in which the composition of the rightmost vertical arrows is the map ϖ_{22}, implies that

ϖ_{22∗}(^{♭}Δ_{s+1,s+1}) = (Id, π_{2}, π_{2})_{∗} ∘ (Id, μ, μ)_{∗} ∘ (j_{1}ϖ_{2}^{s}w_{s+1}, Id, w_{s+1})_{∗} ∘ δ_{∗}(X_{s+1})
= (j_{1}ϖ_{2}^{s−1}w_{s}, Id, w_{s})_{∗} ∘ (π_{1}, π_{2}, π_{1})_{∗} ∘ δ_{∗} ∘ μ_{∗}(X_{s+1})
= p · (j_{1}ϖ_{2}^{s−1}w_{s}, Id, w_{s})_{∗} ∘ (π_{1}, π_{2}, π_{1})_{∗} ∘ δ_{∗}(X_{s}^{♭})
= p · (j_{1}ϖ_{2}^{s−1}w_{s}, Id, w_{s})_{∗} ∘ (Id, U_{p}, Id)_{∗}δ_{∗}(X_{s}),

where the last equality follows from (42). It follows that

ϖ_{22∗}(^{♭}Δ_{s+1,s+1}) = p · (Id, U_{p}, Id)_{∗}(j_{1}ϖ_{2}^{s−1}w_{s}, Id, w_{s})_{∗} ∘ δ_{∗}(X_{s})
= p · (Id, U_{p}, Id)_{∗}(^{♭}Δ_{s,s}),

as was to be shown. □

Proposition 1.2 shows that, after formally regularizing the cycles ^{♭}Δ_{s,s} by the rule

(44)  ^{♭}Δ_{s,s}^{reg} := \frac{1}{p^{s}} · (Id, U_{p}, Id)^{−s}(^{♭}Δ_{s,s}),

the family {^{♭}Δ_{s,s}^{reg}} becomes part of a projective system under the push-forward maps ϖ_{22,∗}. The factor \frac{1}{p^{s}} arising in (44) is problematic, introducing unwanted denominators which present a genuine obstruction to piecing the cycles ^{♭}Δ_{s,s} into a “two-variable p-adic family”. This is what motivates the passage to a suitable quotient of W_{s,s} by a group of automorphisms of (essentially) p-power order, which will now be explained.

Given s ≥ 1, let D_{s} denote the “diagonal at N” and “anti-diagonal at p” subgroup of G_{s}^{(N)} × G_{s}^{(N)} consisting of elements of the form (⟨a; b⟩, ⟨a; b^{−1}⟩) with (a, b) ∈ (ℤ/Nℤ)^{×} × (ℤ/p^{s}ℤ)^{×}, and let

pr_{s} : W_{s,s} ⟶ W_{s} := W_{s,s}/D_{s}

be the natural projection from W_{s,s} to its quotient by the action of this group.

By (25), the action of D_{s} on W_{s,s} preserves the cycle ^{♭}Δ_{s,s}. Identifying ^{♭}Δ_{s,s} with the curve X_{s}, the element (⟨a; b⟩, ⟨a; b^{−1}⟩) ∈ D_{s} acts as the diamond operator ⟨a; b⟩ on this curve. Let ^{♭}Δ_{s} ∈ C^{2}(W_{s})(ℚ(ζ_{s})) be the natural image of ^{♭}Δ_{s,s} in the quotient W_{s}. Since pr_{s} induces a morphism of degree d_{s} := \frac{1}{2}φ(N)(p − 1)p^{s−1} from ^{♭}Δ_{s,s} to ^{♭}Δ_{s}, the latter cycle is defined by the relation

(45)  (pr_{s})_{∗}(^{♭}Δ_{s,s}) =: d_{s} · (^{♭}Δ_{s}) in C^{2}(W_{s})(ℚ(ζ_{s})).

Note that the curve ^{♭}Δ_{s} is isomorphic to an embedded copy of X_{0}(Np^{s}) in the three-fold W_{s}, but that the corresponding closed immersion X_{0}(Np^{s}) ≃ ^{♭}Δ_{s} ↪ W_{s} is only defined over ℚ(ζ_{s}).

By a slight abuse of notation, let

(46)  ϖ_{11}, ϖ_{22} : W_{s+1} = W_{s+1,s+1}/D_{s+1} ⟶ W_{s} = W_{s,s}/D_{s}

denote the natural projection map “from level s + 1 to level s” induced from the maps ϖ_{11} and ϖ_{22} above, by passing to the quotient via pr_{s+1} and pr_{s}.

Proposition 1.3. For all s ≥ 1, ϖ_{22,∗}(^{♭}Δ_{s+1}) = (Id, U_{p}, Id)_{∗}(^{♭}Δ_{s}).

Proof. Proposition 1.2 asserts that ϖ_{22,∗}(^{♭}Δ_{s+1,s+1}) = p · (Id, U_{p}, Id)_{∗}(^{♭}Δ_{s,s}) in C^{2}(W_{s,s}). Applying the pushforward map induced by pr_{s} : W_{s,s} ⟶ W_{s} to this identity, using the fact that pr_{s} ∘ ϖ_{22} = ϖ_{22} ∘ pr_{s+1}, and that pr_{s} also commutes with (Id, U_{p}, Id), one obtains

ϖ_{22,∗} pr_{s+1,∗}(^{♭}Δ_{s+1,s+1}) = p · (Id, U_{p}, Id) pr_{s,∗}(^{♭}Δ_{s,s}).

The sought-for equality then follows from (45), because C^{2}(W_{s}) is torsion free. □

It would also have been possible to adapt the proof of Proposition 1.2 to the cycles ^{♭}Δ_{s} on the quotients W_{s} to show directly that Proposition 1.3 holds at the level of cycles. In reducing Proposition 1.3 to Proposition 1.2, it was expedient to work with the groups C^{2}(W_{s,s}) and C^{2}(W_{s}) of codimension 2 cycles in W_{s,s} and W_{s}, rather than the Chow groups CH^{2}(W_{s,s}) and CH^{2}(W_{s}) of rational equivalence classes, because the latter are not necessarily torsion free. This is no longer necessary, and from now on ^{♭}Δ_{s,s} and ^{♭}Δ_{s} will be identified with their classes in CH^{2}(W_{s,s}) and CH^{2}(W_{s}), respectively.

These cycle classes can be made null-homologous by applying to them a suitable correspondence, denoted

(47)  ε_{s,s} := (ε, ε_{s}, ε_{s}) : X_{0}(Np) × X_{s} × X_{s},

where

• ε is a correspondence on X_{0}(Np) with coefficients in ℤ_{p} which annihilates H^{0}(X_{0}(Np)) and H^{2}(X_{0}(Np)) but fixes the f-isotypic subspace H^{1}(X_{0}(Np))[f] of the introduction (defined as an eigenspace only for the good Hecke operators T_{ℓ} with ℓ ∤ Np). To fix ideas, recall that the prime p is assumed to be a non-Eisenstein prime for f, meaning that f is not congruent to an Eisenstein series modulo p. There is therefore an auxiliary prime ℓ ∤ Np for which ℓ + 1 − a_{ℓ}(f) lies in ℤ_{p}^{×}. Set

ε := (ℓ + 1 − T_{ℓ})/(ℓ + 1 − a_{ℓ}(f)).

• ε_{s} is the correspondence on X_{s} defined by setting ε_{s} := 2^{−1} · (1 − ⟨−1; 1⟩), where the standard convention has been adopted of identifying a morphism φ : V ⟶ W between varieties V and W with the correspondence on V × W associated to its graph.

The modified twisted diagonal cycle of level s is then defined by

(48)  Δ_{s,s} := ε_{s,s}(^{♭}Δ_{s,s}) ∈ CH^{2}(W_{s,s}).

Proposition 1.4. For all s ≥ 1, the class of Δ_{s,s} lies in CH^{2}(W_{s,s})_{0}.

Proof. In order to lighten the notations in the proof, write H^{i}(V) as a shorthand for the étale cohomology H_{et}^{i}(\overline{V} , ℤ_{p}) of V with coefficients in ℤ_{p}, and write W_{s,s} = C_{1} × C_{2} × C_{3}. This variety is a product of curves, whose integral cohomology is torsion-free. By the Künneth decomposition theorem (cf. [Mi, Thm. 22.4]) H^{4}(W_{s,s}) can therefore be expressed as a sum of three terms

H^{2}(C_{1}) ⊗ H^{2}(C_{2} × C_{3}) + H^{2}(C_{2}) ⊗ H^{2}(C_{1} × C_{3}) + H^{2}(C_{3}) ⊗ H^{2}(C_{1} × C_{2}).

By (48), cl(Δ_{s,s}) = ε_{s,s}(cl(^{♭}Δ_{s,s})). The correspondence ε annihilates the first term in the above sum, while ε_{s} annihilates the second and third term, since it annihilates H^{2}(X_{s}). It follows that ε_{s,s} annihilates the target of the cycle class map and hence a fortiori that Δ_{s,s} is null-homologous. □

Remark 1.5. The element ε_{s,s} has the virtue of annihilating H_{dR}^{4}(W_{s,s}) while fixing much of the interesting part of the middle cohomology H_{dR}^{3}(W_{s,s}). More precisely, if ηf̆ is any class in H_{dR}^{1}(X_{0}(Np))[f], and ηğ ∈ H_{dR}^{1}(X_{s})[g] and ηh̆ ∈ H_{dR}^{1}(X_{s})[h] are classes in the isotypic subspaces attached to eigenforms g and h with nebentypus characters whose prime-to-p parts are the odd Dirichlet characters χ and χ^{−1} respectively, then ε_{s,s}(ηf̆ ⊗ ηğ ⊗ ηh̆) = ηf̆ ⊗ ηğ ⊗ ηh̆.

Recall that the group G_{s}^{(N)} ×G_{s}^{(N)} acts freely on W_{s,s}, and that this action descends to an action on W_{s} factoring through the quotient (G_{s}^{(N)}×G_{s}^{(N)})/D_{s} ≃ G_{s}^{(N)}. More generally, the linear endomorphism of CH^{∗}(W_{s,s}) induced by any element ε ∈ ℤ[G_{s}^{(N)} × G_{s}^{(N)}] descends to a linear map on CH^{∗}(W_{s}) which shall be denoted by the same symbol. With these notations, letting Δ_{s} denote the natural image of Δ_{s,s} in the quotient W_{s}, the classes of the cycles Δ_{s} and ^{♭}Δ_{s} are related by

(49)  Δ_{s} = ε_{s,s}(^{♭}Δ_{s}) ∈ CH^{2}(W_{s})_{0}(ℚ(ζ_{s})).

Moreover, similarly as above it again holds that

(50)  (pr_{s})_{∗}(Δ_{s,s}) = d_{s} · Δ_{s} and ϖ_{22,∗}(Δ_{s+1}) = (Id, U_{p}, Id)_{∗}(Δ_{s}).

The first equality in (50) follows exactly as in (45), while the second is a consequence of Proposition 1.3 because the correspondences ε_{s+1,s+1} and ε_{s,s} commute with the diamond operators, with (Id, U_{p}, Id), and with the projections ϖ_{22,∗} in the sense that ε_{s,s} ∘ ϖ_{22,∗} = ϖ_{22,∗} ∘ ε_{s+1,s+1}.

The cycles Δ_{s,s} and Δ_{s} are only defined over ℚ(ζ_{s}); the following proposition describes the action of the Galois group Gal (ℚ(ζ_{s})/ℚ) = (ℤ/p^{s}ℤ)^{×} on these cycles.

Proposition 1.6. For all σ ∈ G_{ℚ}, σ(Δ_{s,s}) = (Id, Id, ⟨1; ϵ_{s}^{−1}(σ)⟩)(Δ_{s,s}).

Proof. Equation (27) implies that

(51)  σ(^{♭}Δ_{s,s}) = (Id, Id, ⟨1; ϵ_{s}^{−1}(σ)⟩)(^{♭}Δ_{s,s}).

Note the the correspondence ε_{s,s} is defined over ℚ and hence commutes with the action of G_{ℚ} on cycles. It also commutes with the diamond operators and hence with the morphism (Id, Id, ⟨1; ϵ_{s}^{−1}(σ)⟩). Applying it to (51) and invoking equation (48), one obtains that σ(Δ_{s,s}) = (Id, Id, ⟨1; ϵ_{s}^{−1}(σ)⟩)(Δ_{s,s}). Proposition 1.6 follows because pr_{s} is defined over ℚ. □

In light of Proposition 1.6, the varieties W_{s,s}^{†} and W_{s}^{†} are defined to be the twists of W_{s,s} and W_{s} respectively by the cocycle σ ∈ G_{ℚ} ↦ ⟨ϵ_{s}^{−1}⟩(σ) = (1, 1, ⟨1; ϵ_{s}^{−1}(σ)⟩). Note that W_{s,s}^{†} = X_{0}(Np) × X_{s} × X_{s}^{†} and W_{s}^{†} = W_{s,s}^{†}/D_{N,p}^{†}, where D_{N,p}^{†} is the subgroup of diamond operators acting “diagonally at N” and “anti-diagonally at p” on W_{s,s}^{†}, defined in a similar way as for D_{s}.

Thanks to Proposition 1.6, the cycle Δ_{s} can be viewed as a codimension two cycle in W_{s}^{†} with coefficients in ℤ_{p} which is defined over ℚ, and whose associated class in the Chow group gives rise to a canonical element

(52)  Δ_{s} ∈ CH^{2}(W_{s}^{†})_{0}(ℚ).

1.4. Global cohomology classes. Let

(53)  AJ_{et} : CH^{2}(W_{s}^{†})_{0}(ℚ) ⟶ H^{1}(ℚ, H_{et}^{3}(W̄_{s}^{†}, ℤ_{p})(2)),

denote the p-adic étale Abel-Jacobi map arising from (18). Of crucial importance for the results of this paper are the images of the cycles Δ_{s} under this map:

(54)  κ_{s}^{(1)} := AJ_{et}(Δ_{s}) ∈ H^{1}(ℚ, H_{et}^{3}(W̄_{s}^{†}, ℤ_{p})(2)) = H^{1}(ℚ, H_{et}^{3}(W̄_{s}, ℤ_{p})(2 ⟨ϵ_{s}^{−1}⟩)).

As was already observed, the ℤ_{p}-modules H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p}) and H_{et}^{3}(W̄_{s}^{†}, ℤ_{p}) are equipped with a natural structure of modules over the rings ℤ_{p}[G_{s}^{(N)} × G_{s}^{(N)}] and ℤ_{p}[(G_{s}^{(N)} × G_{s}^{(N)})/D_{s}] ≃ ℤ_{p}[G_{s}^{(N)}] via the action of the diamond operators.

Lemma 1.7. For every d ≥ 0, the group H_{et}^{d}(W̄_{s,s}^{†}, ℤ_{p}) is free over ℤ_{p} and admits a canonical direct sum decomposition (the Künneth formula):

(55)  H_{et}^{d}(W̄_{s,s}^{†}, ℤ_{p}) = ⨁ ; \begin{matrix}i + j + k = d \\ 0 ≤ i, j, k ≤ 2 \end{matrix} H_{et}^{i}(X̄_{0}(Np), ℤ_{p}) ⊗ H_{et}^{j}(X̄_{s}, ℤ_{p}) ⊗ H_{et}^{k}(X̄_{s}^{†}, ℤ_{p}).

Proof. This follows from e.g. Theorem 22.4 of [Mi], in light of the fact that the integral cohomology of smooth projective curves is torsion-free. □

The natural projection pr_{s} : W_{s,s}^{†} ⟶ W_{s}^{†} induces functorial maps between the corresponding étale cohomology groups in both directions

pr_{s}^{∗} : H_{et}^{3}(W̄_{s}^{†}, ℤ_{p}) ⟶ H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p})^{D_{s}}, pr_{s,∗} : H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p})_{D_{s}} ⟶ H_{et}^{3}(W̄_{s}^{†}, ℤ_{p}),

the second map being related to the first by taking duals and invoking Poincaré duality. These two maps become isomorphisms after tensoring with ℚ_{p}, but working integrally requires a bit more care.

Recall the correspondence ε_{s,s} on W_{s,s} and W_{s} introduced in (47). By an abuse of notation, continue to denote with the same symbol the correspondence on W_{s,s}^{†} and W_{s}^{†} defined exactly in the same way, taking into account that both X_{s} and X_{s}^{†} are equipped with a canonical action of diamond operators.

Lemma 1.8. The kernels and cokernels of the maps pr_{s}^{∗} and pr_{s,∗} are annihilated by the element ε_{s,s}.

Proof. The Hochschild-Serre spectral sequence H^{p}(D_{s}, H_{et}^{q}(W̄_{s,s}^{†}, ℤ_{p})) ⇒ H_{et}^{p+q}(W̄_{s}^{†}, ℤ_{p}) shows that pr_{s}^{∗} sits in the middle of an exact sequence

\xymatrix{K \ar[r] & H_{et}^{3}(W̄_{s}^{†}, ℤ_{p}) \ar[r]^{^{pr{∗ \atop s}}} & H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p})^{D_{s}} \ar[r] & H^{2}(D_{s}, H_{et}^{2}(W̄_{s,s}^{†}, ℤ_{p})),}

where the kernel K is canonically a subquotient of H^{2}(D_{s}, H_{et}^{1}(W̄_{s,s}^{†}, ℤ_{p})) ⊕ H^{1}(D_{s}, H_{et}^{2}(W̄_{s,s}^{†}, ℤ_{p})). Since ε_{s,s} annihilates each term in the Künneth decomposition (55) of H_{et}^{d}(W̄_{s,s}^{†}, ℤ_{p}) when 0 ≤ d ≤ 2, the assertion of the lemma for pr_{s}^{∗} follows. The result for pr_{s,∗} then follows by taking duals. □

Since the projector ε_{s,s} of (47) is integrally defined over ℤ_{p}, Lemma 1.8 makes it possible to define classes

(56) κ_{s}^{(2)} := ε_{s,s} pr_{s,∗}^{−1}(ε_{s,s}κ_{s}^{(1)}) ∈ H^{1}(ℚ, H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p})(2))_{D_{s}} = H^{1}(ℚ, H_{et}^{3}(W̄_{s,s}, ℤ_{p})(2 ⟨ϵ_{s}^{−1}⟩))_{D_{s}},

where the first application of ε_{s,s} ensures the existence of the inverse image, and the second guarantees its uniqueness. Let

(57)  V_{0}(Np) = H_{et}^{1}(X̄_{0}(Np), ℤ_{p})(1),  V_{1}(Np^{s}) = H_{et}^{1}(X̄_{s}, ℤ_{p})(1),

denote the Tate modules of the jacobians of X_{0}(Np) and X_{s}, and let

V_{1}(Np^{s})^{†} = H_{et}^{1}(X̄_{s}^{†}, ℤ_{p})(1) = V_{1}(Np^{s})(−1)(⟨ϵ_{s}^{−1}⟩).

Define also

(58)  V_{s,s} := V_{0}(Np) ⊗ (V_{1}(Np^{s}) ⊗ V_{1}(Np^{s})^{†}) ,  V_{s} := V_{0}(Np) ⊗ (V_{1}(Np^{s}) ⊗Λ̃_{s} V_{1}(Np^{s})^{†}) .

The projection from H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p}) onto the (1, 1, 1)-component in the Künneth decomposition (55) induces a G_{ℚ}-equivariant map on the D_{s}-coinvariants:

pr_{111} : H_{et}^{3}(W̄_{s,s}^{†}, ℤ_{p})(2)_{D_{s}} ⟶ (V_{s,s})_{D_{s}} = V_{s}.

The classes κ_{s}^{(3)} are now defined by setting

(59)  κ_{s}^{(3)} := pr_{111}(κ_{s}^{(2)}) ∈ H^{1}(ℚ, V_{s}).

By a slight abuse of notation, the symbol ϖ_{22,∗} is also used to denote the proper push-forward maps ϖ_{22,∗} : V_{s+1} ⟶ V_{s} induced on étale cohomology by the projections ϖ_{22} of (46). Denote also by (1 ⊗ U_{p} ⊗ 1) the endomorphism acting on V_{s} via the action of U_{p} on the second factor in (59).

Proposition 1.9. For all s ≥ 1, ϖ_{22,∗}(κ_{s+1}^{(3)}) = (1 ⊗ U_{p} ⊗ 1)(κ_{s}^{(3)}).

Proof. The classes κ_{s}^{(3)} arise from the cycles Δ_{s} by applying succesively the étale Abel-Jacobi map, the inverse of the push-forward map pr_{s∗}^{−1} induced by the projection pr_{s}, the correspondence ε_{s,s} constructed using diamond operators and the projection pr_{111} arising from the Künneth decomposition. It is immediate to verify that ϖ_{22∗} commutes with these four maps. Since in addition ϖ_{22∗} commutes with (Id, U_{p}, Id) as proved in the paragraph following (34), the result follows from (50). □

In order to make the classes κ_{s}^{(3)} fit into families that are compatible under the natural projections, and take values in manageable Λ̃ := lim Λ̃_{s}-modules (locally free of finite rank, say) it is desirable to cut down the modules V_{s} by applying Hida’s ordinary projector e_{ord} = lim U_{p}^{n!} to them. Write

V_{1}^{ord}(Np^{s}) := e_{ord}V_{1}(Np^{s}), V_{1}^{ord}(Np^{s})^{†} := e_{ord}V_{1}(Np^{s})^{†},
V_{s}^{ord} := V_{0}(Np) ⊗ (V_{1}^{ord}(Np^{s}) ⊗Λ̃_{s} V_{1}^{ord}(Np^{s})^{†})
(60)  and set  κ_{s} := (Id, U_{p}, Id)^{−s}(1 ⊗ e_{ord} ⊗ e_{ord})κ_{s}^{(3)} ∈ H^{1}(ℚ, V_{s}^{ord}).

Note that (60) is well-defined because U_{p} acts invertibly on V_{1}^{ord}(Np^{s}). Define the Λ̃^{(N)}[G_{ℚ}]-modules

(61)  𝕍_{1}^{ord}(Np^{∞}) := \underleftarrow{lim}_{s} V_{1}^{ord}(Np^{s}), 𝕍_{1}^{ord}(Np^{∞})^{†} := \underleftarrow{lim}_{s} V_{1}^{ord}(Np^{s})^{†} and

(62)  𝕍_{∞}^{ord} := \underleftarrow{lim}_{s} V_{s}^{ord} ≃ V_{0}(Np) ⊗ (𝕍_{1}^{ord}(Np^{∞}) ⊗Λ̃ 𝕍_{1}^{ord}(Np^{∞})^{†})

where the inverse limits are taken relative to the maps induced from ϖ_{2} and ϖ_{22} by covariant functoriality on the étale cohomology of the towers of curves X_{s} and of threefolds W_{s}.

Since U_{p} commutes with ϖ_{2∗}, it follows from Proposition 1.9 that the classes κ_{s} form a system of global classes satisfying the crucial compatibility ϖ_{22,∗}(κ_{s+1}) = κ_{s}. Hence the classes κ_{s} can be pieced together into the Λ-adic class

(63)  κ_{∞} := \underleftarrow{lim}_{s} κ_{s} ∈ H^{1}(ℚ, 𝕍_{∞}^{ord}).

The next two sections explain how the Λ̃^{(N)}[G_{ℚ}]- module 𝕍_{∞}^{ord} realises one-parameter families of p-adic Galois representations interpolating the Kummer self-dual twists of the triple tensor product of the Galois representations associated to f, g and h, as (g, h) range over certain pairs of classical specialisations of common weight ℓ of Hida families of tame level Np. The classes κ_{∞} likewise will give rise to p-adic families of Galois cohomology classes with values in these representations.

1.5. Modular forms and Galois representations in families.

1.5.1. Classical modular forms and Galois representations. Let S_{k}(M)_{R} = S_{k}(Γ_{1}(M))_{R} denote the space of classical modular forms of weight k ≥ 1 and level M ≥ 1 with coefficients in some ring R.

If χ is a Dirichlet character mod M, let S_{k}(M, χ)_{R} denote the subspace of S_{k}(M)_{R} on which diamond operators act through χ. Let

(64)  ϕ = q + ∑_{n≥2} a_{n}(ϕ)q^{n} ∈ S_{k}(N_{ϕ}p^{s}, χ)_{ℂ}

be a normalized cuspform of weight k ≥ 1, level N_{ϕ}p^{s}, s ≥ 1 and character χ : (ℤ/N_{ϕ}p^{s}ℤ)^{×} → ℂ^{×}. Here p is taken to be a prime number not dividing N_{ϕ}. Write χ = χ_{0}χ_{p} as the product of Dirichlet characters of conductor dividing N_{ϕ} and p^{s}, respectively.

Assume ϕ is an eigenform (with respect to all good Hecke operators T_{ℓ}, ℓ ∤ N_{ϕ}p) and let K_{ϕ} denote the finite extension of ℚ_{p} generated by the fourier coefficients a_{n}(ϕ) of ϕ. Write O_{ϕ} for its valuation ring. Eichler-Shimura, Deligne and Serre associated to ϕ a two-dimensional representation

ϱ_{ϕ} : G_{ℚ} ⟶ GL(V_{ϕ}) ≃ GL_{2}(O_{ϕ})

which is unramified at all primes not dividing N_{ϕ}p and satisfies Tr(ϱ_{ϕ}(Frob_{ℓ})) = a_{ℓ}(ϕ) for any such ℓ ∤ N_{ϕ}p, where Frob_{ℓ} denotes an arithmetic Frobenius element at ℓ.

If ϕ is ordinary at p, then the Hecke polynomial T^{2} − a_{p}(ϕ)T + χ(p)p^{k−1} has at least one root which is a p-adic unit. If k ≥ 2 or χ_{p} ≠ 1, this root is unique. Fix such a root, call it α_{ϕ}, and let ψ_{ϕ} : G_{ℚ_{p}} ⟶ O_{ϕ}^{×} be the unramified character such that ψ_{ϕ}(Frob_{p}) = α_{p}(ϕ) ∈ O_{ϕ}^{×}. By [W88, Theorem 2], the restriction of V_{ϕ} to a decomposition group at p takes the form

(65)  V_{ϕ}_{|G_{ℚp}} : ( \begin{matrix}ψ_{ϕ}^{−1}χϵ_{cyc}^{k−1} & ∗ \\ 0 & ψ_{ϕ} \end{matrix} )

on a suitable basis. Here ϵ_{cyc} = \underleftarrow{lim}_{s} ϵ_{s} is the cyclotomic character introduced in (15).

Definition 1.10. We let V_{ϕ}^{+} denote the one-dimensional G_{ℚ_{p}}-submodule of V_{ϕ} on which G_{ℚ_{p}} acts via ψ_{ϕ}^{−1}χϵ_{cyc}^{k−1}, and write V_{ϕ}^{−} for the unramified quotient on which G_{ℚ_{p}} acts via ψ_{ϕ}.

When k ≥ 2, the representation V_{ϕ} arises as the ϕ-isotypic component of H_{et}^{k−1}(Ē_{k}, O_{ϕ})(k−1), where E_{k} is the (k − 1)-dimensional Kuga-Sato variety fibered over X_{s}. In particular, when ϕ has weight 2, V_{ϕ} is a quotient of V_{1}(N_{ϕ}p^{s}) ⊗ O_{ϕ} and ϖ_{ϕ} : V_{1}(N_{ϕ}p^{s}) ⊗ O_{ϕ} → V_{ϕ} denotes the canonical projection.

If N is any multiple of N_{ϕ}, let V_{ϕ}(Np^{s}) denote the ϕ-isotypical component of V_{1}(Np^{s}) ⊗ O_{ϕ}, which is isomorphic to a direct sum of several copies of V_{ϕ}, and write ϖ_{ϕ} : V_{1}(Np^{s}) ⊗ O_{ϕ} → V_{ϕ}(Np^{s}) for the resulting projection. Define V_{ϕ}(Np^{s})^{+} and V_{ϕ}(Np^{s})^{−} analogously as above, and likewise for V_{1}^{ord}(Np^{s})^{+} and V_{1}^{ord}(Np^{s})^{−}. It will also be convenient to introduce the representation

(66)  V_{ϕ}^{∗} := V_{ϕ} ⊗ χ_{p}^{−1}ϵ_{cyc}^{1−k},  V_{ϕ}^{∗}_{|G_{ℚp}} : ( \begin{matrix}ψ_{ϕ}^{−1}χ_{0} & ∗ \\ 0 & ψ_{ϕ}ϵ_{cyc}^{1−k}χ_{p}^{−1} \end{matrix} ) ,

whose restriction to G_{ℚ_{p}} contains an étale submodule (instead of having an étale quotient like V_{ϕ}). If k = 2, note that V_{ϕ}^{∗} ≃ V_{ϕ}(−1)(⟨ϵ_{s}^{−1}⟩) ≃ V_{ϕ}(−1)(χ_{p}^{−1}), which is consistent with (57). Geometrically, the Galois representation V_{ϕ}^{∗} is naturally realised in the anti-ordinary quotient of H_{et}^{1}(X̄_{s}^{†}, ℤ_{p}).

1.5.2. Λ-adic modular forms. A Dirichlet character ϵ of conductor dividing p^{s}, regarded either as a character of Γ_{s} or as a locally constant character of Γ = \underleftarrow{lim} Γ_{s} ≃ 1 + pℤ_{p}, gives rise to a natural ring homomorphism, denoted ϵ : ℂ_{p}[[Γ]] ⟶ ℂ_{p}, by extending ϵ to a linear homomorphism on the group ring ℂ_{p}[Γ_{s}].

Let ω : ℤ_{p}^{×} → μ_{p−1} denote the Teichmüller character, so that zω^{−1}(z) ∈ 1 + pℤ_{p} for any z ∈ ℤ_{p}^{×}.

Definition 1.11. A Λ-adic modular cuspform of tame level N and character χ : (ℤ/Nℤ)^{×} → ℂ_{p}^{×} is a formal q-series

\underline{ϕ} := ∑_{n} a_{n}(\underline{ϕ})q^{n} ∈ Λ[[q]]

with the property that, for all finite order Dirichlet characters ϵ : Γ ⟶ Γ_{s} ⟶ ℂ_{p}^{×}, the specialisation

ϵ(\underline{ϕ}) := ∑_{n} ϵ(a_{n}(\underline{ϕ}))q^{n} ∈ ℂ_{p}[[q]]

is the q-expansion of a modular cuspform of weight 2, level Np^{s} and character χϵω^{−1}.

Write S_{Λ}(N, χ) for the space of such Λ-adic modular forms, and S_{Λ}(N) for the direct sum of the spaces S_{Λ}(N, χ) as χ ranges over all characters mod N.

The natural projection of \underline{ϕ} to ℤ_{p}[Γ_{s}][[q]], denoted \underline{ϕ}_{s}, can be written as

\underline{ϕ}_{s} = ∑_{σ∈Γ_{s}} ϕ_{s}^{σ} · σ^{−1} ∈ ℤ_{p}[Γ_{s}][[q]]

where ϕ_{s} ∈ S_{2}(Np^{s}) = Ω^{1}(X_{s}) is a classical modular form that will be referred to as the level s specialisation of \underline{ϕ}. With these notations, note that ϵ(\underline{ϕ}) = ϵ(\underline{ϕ}_{s}) = θ_{ϵ}ϕ_{s}. Of course, the system \{ϕ_{s}\}_{s≥1} of classical modular forms determines \underline{ϕ} completely, and vice-versa.

While the forms ϕ_{s} fail to be compatible under pushforward by either of the transition maps ϖ_{1} or ϖ_{2}, they do satisfy

μ_{∗}(ϕ_{s+1}) = π_{1}^{∗}(ϕ_{s}),  (ϖ_{1})_{∗}(ϕ_{s+1}) = pϕ_{s},  (ϖ_{2})_{∗}(ϕ_{s+1}) = U_{p}ϕ_{s},

where the first relation is merely a restatement of the fact that the forms \underline{ϕ}_{s} are compatible under the natural projections ℤ_{p}[Γ_{s+1}] ⟶ ℤ_{p}[Γ_{s}], and the second and third follow directly from the first in light of (33). In particular, the systems of modular forms {p^{−s}ϕ_{s}} and {U_{p}^{−s}ϕ_{s}} are compatible under the pushforward maps (ϖ_{1})_{∗} and (ϖ_{2})_{∗}, respectively. It is thus natural to define the maps

℧_{s} : S_{2}(Np^{s}) → H_{dR}^{1}(X_{s}) and ℧_{∞} : S_{Λ}(N) → H_{dR}^{1}(X_{∞})

given by ℧_{s}(ϕ) = U_{p}^{−s}ω_{ϕ} and ℧_{∞}(\underline{ϕ}) = \{U_{p}^{−s}ω_{ϕ_{s}}\}_{s}.

For any integrally closed subring O of O_{ℂ_{p}}, set

H_{dR}^{1}(X_{∞})_{O} := \{\underline{η} ∈ H_{dR}^{1}(X_{∞}) such that [η, ℧_{∞}(\underline{ϕ})]_{Γ} ∈ O[[Γ]]] for all \underline{ϕ} ∈ S_{Λ}(Np)\} .

Recall the group ring element θ_{ϵ} ∈ ℂ_{p}[Γ_{s}] introduced in (37). The next lemma records the behaviour of the Λ-adic pairing [ , ]_{Γ} and the inclusion map ℧_{∞} under specialisation.

Lemma 1.12. Let \underline{η} = \{η_{s}\} and \underline{ϕ} = \{ϕ_{s}\} be elements of H_{dR}^{1}(X_{∞})_{O} and S_{Λ_{O}}(N). For all Dirichlet characters ϵ of conductor p^{s′} with 1 ≤ s′ ≤ s,

ϵ([\underline{η}, ℧_{∞}(\underline{ϕ})]_{Γ}) = ⟨η_{s}, ww_{s}θ_{ϵ}(ϕ_{s})⟩_{s} = \frac{1}{d_{s}} ⟨θ_{ϵ}(η_{s}), ww_{s}θ_{ϵ}(ϕ_{s})⟩_{s}.

Proof. The first equality follows directly from the definitions, since

ϵ([\underline{η}, ℧_{∞}(\underline{ϕ})]_{Γ}) = ϵ ( ∑_{σ∈Γ_{s}}⟨ση_{s}, ww_{s}U_{p}^{s}U_{p}^{−s}ϕ_{s}⟩_{s} · σ^{−1}) = ϵ ( ∑_{σ∈Γ_{s}}⟨η_{s}, ww_{s}σϕ_{s}⟩_{s} · σ^{−1})
= ⟨η_{s}, ww_{s}θ_{ϵ}(ϕ_{s})⟩_{s} = ⟨w^{−1}w_{s}^{−1}η_{s}, θ_{ϵ}(ϕ_{s})⟩_{s}.

The second identity is a direct consequence of the first in light of the fact that θ_{ϵ} and θ_{ϵ^{−1}} are adjoint to each other relative to Poincaré duality, that θ_{ϵ^{−1}}ww_{s} = ww_{s}θ_{ϵ}, and that θ_{ϵ}^{2} = d_{s}θ_{ϵ}. □

Note that the formula of Lemma 1.12 does not depend on the value of s ≥ s′, since

⟨w_{s+1}η_{s+1}, ϵ(\underline{ϕ}_{s+1})⟩_{s+1} = ⟨w_{s+1}η_{s+1}, ϖ_{1}^{∗}ϵ(\underline{ϕ}_{s})⟩_{s+1} = ⟨ϖ_{1∗}w_{s+1}η_{s+1}, ϵ(\underline{ϕ}_{s})⟩_{s}
= ⟨w_{s}(ϖ_{2∗}η_{s+1}), ϵ(\underline{ϕ}_{s})⟩_{s} = ⟨w_{s}(η_{s}), ϵ(\underline{ϕ}_{s})⟩_{s}.

1.5.3. Hida families. Let O be a finite extension of ℤ_{p} containing the values of all Dirichlet characters of conductor Np. For any ℤ_{p}-module M, write M_{O} = M ⊗_{ℤ_{p}} O. Assuming that p does not divide φ(N) = |(ℤ/Nℤ)^{×}|, the semi-local ring Λ̃_{O}^{(N)} decomposes as a sum over the characters ψ : (ℤ/Npℤ)^{×} ⟶ O^{×}:

Λ̃_{O}^{(N)} = ⊕_{ψ}Λ_{O}^{ψ}, Λ_{O}^{ψ} := e_{ψ}(Λ̃_{O}) ≃ Λ_{O},

where e_{ψ} is the O-algebra homomorphism defined on group-like elements by

e_{ψ}(⟨a; b⟩) = ψ(⟨a; b⟩)⟨1; bω^{−1}(b)⟩ ∈ Λ_{O}.

The weight space Ω is defined to be the rigid analytic space underlying the formal spectrum of Λ, so that for any complete ℤ_{p}-algebra R, the set of R-rational points of Ω is

Ω(R) = Hom(Λ, R) = Hom_{cts}(Γ, R^{×}).

An element of Ω is said to be classical if it corresponds to a character of Γ, denoted ν_{k,ϵ}, of the form

ν_{k,ϵ}(n) = n^{k−1}ϵ(n), for all n ∈ 1 + pℤ_{p},

for some integer k and some continuous character ϵ of finite order (hence, conductor p^{s} for some s ≥ 1, factoring through a primitive character of Γ/Γ^{p^{s−1}} ≃ ℤ/p^{s−1}ℤ ⊂ (ℤ/p^{s}ℤ)^{×}).

More generally, if Λ̃ is a finite flat extension of Λ, let Ω̃ := Hom(Λ̃, ℂ_{p}) denote the space of continuous algebra homomorphisms and write w : Ω̃ → Ω for the natural “weight map” induced by pull-back from the structure map Λ ↪ Λ̃. A point in Ω̃ is said to be classical if its image under w is classical.

Generalising Definition 1.11 slightly, a Λ-adic ordinary modular form of tame level N and character χ : (ℤ/Nℤ)^{×} → O^{×} with coefficients in Λ̃ is a pair (Ω̃_{cl}, \underline{ϕ}), where

(i) Ω̃_{cl} is a dense subset of the classical points in Ω̃ for the rigid analytic topology; (ii) \underline{ϕ} := ∑ \underline{a}_{n}q^{n} ∈ Λ̃[[q]] is an element of SΛ̃(N, χ) := S_{Λ}(N, χ) ⊗_{Λ} Λ̃, such that for all x ∈ Ω̃_{cl} of

weight w(x) = ν_{k,ϵ}, with k ≥ 2 and character ϵ of conductor p^{s}, the power series

ϕ_{x} := ∑_{n=1}^{∞} \underline{a}_{n}(x)q^{n} ∈ ℂ_{p}[[q]], \underline{a}_{n}(x) := x(\underline{a}_{n}),

is the q-expansion of a classical ordinary cusp form in S_{k}(Np^{s}, χϵω^{1−k}; ℂ_{p}).

The Λ̃-module of Λ-adic ordinary modular forms of tame level N and character χ is denoted Sord Λ̃(N, χ).

Definition 1.13. An element \underline{ϕ} ∈ Sord Λ̃(N, χ) whose classical specializations ϕ_{x} are eigenforms (that is to say, simultaneous eigenvector for the good Hecke operators T_{ℓ} with ℓ ∤ Np) is called a Λ-adic eigenform or a Hida family.

If the subset Ω̃_{cl} ⊂ Ω̃ can be clearly inferred from the context, such a form will simply be denoted as \underline{ϕ}, suppressing the datum of Ω̃_{cl} from the notation.

If in addition the specialisation ϕ_{x} of a Λ-adic eigenform \underline{ϕ} does not share the same eigenvalues with any eigenform of level N_{0}p^{r} with N_{0} | N, N_{0} < N and r ≤ s, then \underline{ϕ} is called a Λ-adic newform. Note that a form \underline{ϕ} as in Definition 1.13 above is in particular a Λ-adic modular form in the weaker sense of Definition 1.11, and hence the material of §1.5.2 applies to it.

Recall the Hecke algebras 𝕋_{1}(N) and 𝕋_{1}(Np^{s}) that were introduced in Section 1.2, and let

𝕋_{1}(Np^{∞}) := e_{ord} (\underleftarrow{lim}_{s}(𝕋_{1}(Np^{s}) ⊗ ℤ_{p})) .

It follows from a theorem of Hida (cf. [Hi86], [Em99]), that the algebra 𝕋_{1}(Np^{∞}) is locally free of finite rank as a Λ̃^{(N)}-module, and the algebra

𝕋_{1}^{χ}(Np^{∞}) = e_{χω^{−1}}𝕋_{1}(Np^{∞})_{O} = 𝕋_{1}(Np^{∞}) ⊗_{e_{χω^{−1}}} Λ_{O}

is therefore free of finite rank over the local ring Λ_{O}. Let L denote the fraction field of Λ_{O}, and write

𝕋_{1}^{χ}(Np^{∞})_{L} := 𝕋_{1}^{χ}(Np^{∞}) ⊗_{Λ_{O}} L.

It is a finite-dimensional étale L-algebra, hence decomposes as a product of finite extensions L_{j} of L:

𝕋_{1}^{χ}(Np^{∞})_{L} = ⊕_{j=1}^{t}L_{j}.

Furthermore, for each 1 ≤ j ≤ t, the integral closure of the image of 𝕋_{1}^{χ}(Np^{∞}) in L_{j}, denoted Λ_{j}, is a finite flat extension of Λ_{O}. The Λ-adic newforms \underline{ϕ} of character χ (of conductor dividing N) and primitive tame level N_{ϕ} (with N_{ϕ} ranging over the divisors of N that are multiples of the conductor of χ) are in bijection with the resulting homomorphisms

η_{j} : 𝕋_{1}^{χ}(Np^{∞}) ⟶ Λ_{j}.

If \underline{ϕ} is one such Λ-adic eigenform, let Λ_{ϕ} denote the corresponding finite flat extension of Λ and corresponding homomorphism η_{ϕ} : 𝕋_{1}(N_{0}p^{∞})_{O} → 𝕋_{1}^{χ}(Np^{∞}) → Λ_{ϕ}. Let

(67)  \underline{ϵ}_{cyc} : G_{ℚ} ⟶ Λ^{×}

be the Λ-adic cyclotomic character characterised by ν_{k,ϵ} ∘ \underline{ϵ}_{cyc} = ϵ · ϵ_{cyc}^{k−1} · ω^{1−k}.

Theorem 1.14 (Hida). Let \underline{ϕ} be a Λ-adic newform of tame level N_{ϕ} and tame character χ. The modules

(68)  𝕍\underline{ϕ} := 𝕍_{1}^{ord}(N_{ϕ}p^{∞}) ⊗_{𝕋_{1}(N_{ϕ}p^{∞})_{O},η_{ϕ}} Λ_{ϕ},  𝕍∗ ; \underline{ϕ} := 𝕍\underline{ϕ} ⊗ \underline{ϵ}_{cyc}^{−1}

are both locally free of rank two over Λ_{ϕ}.

The representations 𝕍\underline{ϕ} and 𝕍∗ ; \underline{ϕ} will be referred to respectively as the ordinary and anti-ordinary Λ-adic representation of G_{ℚ} attached to the Hida family \underline{ϕ}. Their restrictions to a decomposition group at p are reducible and can be written in upper triangular form as

(69)  (𝕍\underline{ϕ})_{|G_{ℚp}} : ( \begin{matrix}Ψ−1 ; \underline{ϕ}χ\underline{ϵ}_{cyc} & ∗ \\ 0 & Ψ\underline{ϕ} \end{matrix} ) ,  (𝕍∗ ; \underline{ϕ})_{|G_{ℚp}} : ( \begin{matrix}Ψ−1 ; \underline{ϕ}χ & ∗ \\ 0 & Ψ\underline{ϕ}\underline{ϵ}_{cyc}^{−1} \end{matrix} ) ,

where Ψ\underline{ϕ} : G_{ℚ_{p}} ⟶ Λ_{ϕ}^{×} is the unramified character defined by Ψ\underline{ϕ}(Frob_{p}) = a_{p}(\underline{ϕ}) ∈ Λ_{ϕ}^{×}. As in Definition 1.10, let 𝕍+ ; \underline{ϕ} denote the one-dimensional G_{ℚ_{p}}-submodule of 𝕍\underline{ϕ} on which G_{ℚ_{p}} acts via

Ψ−1 ; \underline{ϕ}χ\underline{ϵ}_{cyc}, and write 𝕍− ; \underline{ϕ} for the unramified quotient on which G_{ℚ_{p}} acts via Ψ\underline{ϕ}.

More generally, if N is any multiple of N_{ϕ}, set

𝕍\underline{ϕ}(N) := 𝕍_{1}^{ord}(Np^{∞}) ⊗_{𝕋_{1}(Np^{∞})_{O},η_{ϕ}} Λ_{ϕ}, 𝕍∗ ; \underline{ϕ}(N) := 𝕍∗ ; \underline{ϕ}(N) ⊗ \underline{ϵ}_{cyc}^{−1},

and write ϖ\underline{ϕ} and ϖ∗ ; \underline{ϕ} for the canonical projections

ϖ\underline{ϕ} : 𝕍_{1}^{ord}(Np^{∞}) ⟶ 𝕍\underline{ϕ}(N), ϖ∗ ; \underline{ϕ} : 𝕍_{1}^{ord}(Np^{∞})^{†} ⟶ 𝕍∗ ; \underline{ϕ}(N).

The quotient 𝕍\underline{ϕ}(N) is (non-canonically) isomorphic to a finite direct sum of the Λ-adic representations 𝕍\underline{ϕ}, and likewise for 𝕍∗ ; \underline{ϕ}(N). If x ∈ Ω̃_{cl} is a classical point of weight w(x) = ν_{k,ϵ}, with k ≥ 2 and character ϵ of conductor p^{s}, it gives rise to G_{ℚ}-equivariant specialization homomorphisms (which are all denoted by x)

x : 𝕍\underline{ϕ} ⟶ V_{ϕ_{x}},  x : 𝕍∗ ; \underline{ϕ} ⟶ V_{ϕ_{x}}^{∗},  x : 𝕍\underline{ϕ}(N) ⟶ V_{ϕ_{x}}(Np^{s}),  x : 𝕍∗ ; \underline{ϕ}(N) ⟶ V_{ϕ_{x}}^{∗}(Np^{s}).

where V_{ϕ_{x}} and V_{ϕ_{x}}^{∗} are the Galois representations associated in (66) to the eigenform ϕ_{x}.

1.6. The Λ-adic class attached to two Hida families. Let f ∈ S_{2}(N_{f}) be the newform associated to the elliptic curve E in the introduction and let \underline{g} and \underline{h} be two Λ-adic newforms of tame level N_{g} and N_{h} and tame character χ and χ^{−1}, respectively. The prime p arising as the residual characteristic of Λ is taken here not to divide neither N = lcm(N_{f}, N_{g}, N_{h}) nor φ(N). Moreover, as we assumed in (47), we also take p to be a non-Eisenstein prime for f.

Recall the Λ-adic representation 𝕍_{∞}^{ord} := V_{0}(Np) ⊗ (𝕍_{1}^{ord}(Np^{∞}) ⊗Λ̃ 𝕍_{1}^{ord}(Np^{∞})^{†}), and define

𝕍_{f\underline{gh}} := V_{f} ⊗ (𝕍\underline{g} ⊗_{Λ} 𝕍∗ ; \underline{h}), 𝕍_{f\underline{gh}}(N) := V_{f}(Np) ⊗ (𝕍\underline{g}(N) ⊗_{Λ} 𝕍∗ ; \underline{h}(N)),

which are modules over the ring

Λ_{f\underline{gh}} := O_{f} ⊗ (Λ\underline{g} ⊗_{Λ} Λ\underline{h}),

a finite flat extension of the Iwasawa algebra Λ. The rigid analytic space Ω_{f\underline{gh}} attached to the formal spectrum of Λ_{f\underline{gh}} is a finite covering of the weight space Ω. The “classical points” in Ω_{f\underline{gh}} correspond to pairs of classical points of the form (y, z) ∈ Ω\underline{g} × Ω\underline{h} satisfying w(y) = w(z), that is to say, having the same weight and the same nebentypus at p.

Let g_{y} and h_{z} denote as in the previous section the specialisations of \underline{g} and \underline{h} at the points y and z, respectively. Following similar notations as above, let K = K_{fg_{y}h_{z}} denote the finite extension of ℚ_{p} generated by K_{f}, K_{g_{y}} and K_{h_{z}} and let O = O_{fg_{y}h_{z}} denote the valuation ring of K. The specialisation V_{fg_{y}h_{z}} of 𝕍_{f\underline{gh}} at the classical point (y, z) of weight ℓ and nebentype character ϵ is a Galois representation of rank 8 over O, and there are natural identifications

(70)  V_{fg_{y}h_{z}} = V_{f} ⊗ V_{g_{y}} ⊗ V_{h_{z}}^{∗},  V_{fg_{y}h_{z}}(N) := V_{f}(Np) ⊗ V_{g_{y}}(Np^{s}) ⊗ V_{h_{z}}^{∗}(Np^{s}).

The reader will note that this Galois representation is pure of weight −1, since a geometric Frobenius element at ℓ acting on V_{f} has eigenvalues of complex absolute value ℓ^{−1/2}, while its eigenvalues on V_{g_{y}} ⊗ V_{h_{z}}^{∗} have complex absolute value 1. Note that V_{f\underline{gh}} interpolates only the Kummer self-dual Tate twists of the tensor product of the representations V_{f}, V_{g_{y}} and V_{h_{z}}.

The canonical projections ϖ_{f}, ϖ\underline{g}, and ϖ∗ ; \underline{h} associated to f, \underline{g} and \underline{h} give rise to a surjective Λ-module homomorphism

(71)  ϖ_{f,\underline{g},\underline{h}} : 𝕍_{∞}^{ord} ⟶ 𝕍_{f\underline{gh}}(N).

Definition 1.15. The one-variable Λ-adic cohomology class attached to the triple (f, \underline{g}, \underline{h}) is the class

κ(f, \underline{gh}) := ϖ_{f,\underline{g},\underline{h}}(κ_{∞}) ∈ H^{1}(ℚ, 𝕍_{f\underline{gh}}(N)).

This class takes values in the Galois representation 𝕍_{f\underline{gh}}(N), which is (non-canonically) isomorphic to a finite sum of copies of the representation 𝕍_{f\underline{gh}}.

2. Local Λ-adic cohomology classes

2.1. Local Galois representations. Let f ∈ S_{2}(N_{f}) be a newform and let p ∤ N_{f} be a prime at which f is ordinary. Let \underline{g} and \underline{h} be two Λ-adic newforms of tame level N_{g} and N_{h} and character χ and χ^{−1}, respectively. Set N = lcm(N_{f}, N_{g}, N_{h}) as usual. This chapter analyses the restrictions to G_{ℚ_{p}} of the Galois representation 𝕍_{f\underline{gh}} and of the Λ-adic cohomology class κ(f, \underline{gh}).

Lemma 2.1. The Galois representation 𝕍_{f\underline{gh}} is endowed with a four-step filtration

0 ⊂ 𝕍_{f\underline{gh}}^{++} ⊂ 𝕍_{f\underline{gh}}^{+} ⊂ 𝕍_{f\underline{gh}}^{−} ⊂ 𝕍_{f\underline{gh}}

by G_{ℚ_{p}}-stable Λ_{f\underline{gh}}-submodules of ranks 0, 1, 4, 7 and 8 respectively. The Galois group G_{ℚ_{p}} acts on the successive quotients for this filtration as a direct sum of Λ_{f\underline{gh}}-adic characters. More precisely:

𝕍_{f\underline{gh}}^{++}  ≃  Λ_{f\underline{gh}}(\underline{Υ}^{+}),
𝕍_{f\underline{gh}}^{+}/𝕍_{f\underline{gh}}^{++} = 𝕍_{f\underline{gh}}^{f} ⊕ 𝕍_{f\underline{gh}}^{g} ⊕ 𝕍_{f\underline{gh}}^{h} ≃ Λ_{f\underline{gh}}(\underline{Υ}_{f}^{+}) ⊕ Λ_{f\underline{gh}}(\underline{Υ}+ ; \underline{g}) ⊕ Λ_{f\underline{gh}}(\underline{Υ}+ ; \underline{h}),
𝕍_{f\underline{gh}}^{−}/𝕍_{f\underline{gh}}^{+} = 𝕍_{f\underline{gh}}^{f} ⊕ 𝕍_{f\underline{gh}}^{g} ⊕ 𝕍_{f\underline{gh}}^{h} ≃ Λ_{f\underline{gh}}(\underline{Υ}_{f}^{−}) ⊕ Λ_{f\underline{gh}}(\underline{Υ}− ; \underline{g}) ⊕ Λ_{f\underline{gh}}(\underline{Υ}− ; \underline{h}),
𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{−} ≃  Λ_{f\underline{gh}}(\underline{Υ}^{−}),

where the characters arising on the right are described in the following table:

Subquotient Rank  Galois action
𝕍_{f\underline{gh}}^{++}  1 \underline{Υ}^{+} = (ψ_{f}Ψ\underline{g}Ψ\underline{h})^{−1} × ϵ_{cyc}\underline{ϵ}_{cyc}
𝕍_{f\underline{gh}}^{+}/𝕍_{f\underline{gh}}^{++}  3 \underline{Υ}_{f}^{+} = ψ_{f}Ψ−1 ; \underline{g}Ψ−1 ; \underline{h} × \underline{ϵ}_{cyc}

\underline{Υ}+ ; \underline{g} = ψ_{f}^{−1}Ψ\underline{g}Ψ−1 ; \underline{h} × χ^{−1} × ϵ_{cyc}

\underline{Υ}+ ; \underline{h} := ψ_{f}^{−1}Ψ−1 ; \underline{g}Ψ\underline{h} × χ × ϵ_{cyc}

𝕍_{f\underline{gh}}^{−}/𝕍_{f\underline{gh}}^{+}  3 \underline{Υ}− ; \underline{h} = ψ_{f}Ψ\underline{g}Ψ−1 ; \underline{h} × χ^{−1}

\underline{Υ}− ; \underline{g} = ψ_{f}Ψ−1 ; \underline{g}Ψ\underline{h} × χ

\underline{Υ}_{f}^{−} = ψ_{f}^{−1}Ψ\underline{g}Ψ\underline{h} × ϵ_{cyc}\underline{ϵ}_{cyc}^{−1}

𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{−}  1 \underline{Υ}^{−} = ψ_{f}Ψ\underline{g}Ψ\underline{h} × \underline{ϵ}_{cyc}^{−1}.

Proof. The lemma follows directly from (65), (68) and (69). □

Since 𝕍_{f\underline{gh}}(N) is a finite sum of copies of the representation 𝕍_{f\underline{gh}}, in view of the above lemma it also admits a natural filtration

0 ⊂ 𝕍_{f\underline{gh}}(N)^{++} ⊂ 𝕍_{f\underline{gh}}(N)^{+} ⊂ 𝕍_{f\underline{gh}}(N)^{−} ⊂ 𝕍_{f\underline{gh}}(N)

of Λ_{f\underline{gh}}[G_{ℚ_{p}}]-modules, and

(72)  𝕍_{f\underline{gh}}^{+}(N)/𝕍_{f\underline{gh}}^{++}(N) =: 𝕍_{f\underline{gh}}^{f}(N) ⊕ 𝕍_{f\underline{gh}}^{g}(N) ⊕ 𝕍_{f\underline{gh}}^{h}(N).

For each ϕ = f, g, h, the subquotient 𝕍_{f\underline{gh}}^{ϕ}(N) is a direct sum of finitely many copies of 𝕍_{f\underline{gh}}^{ϕ}, on which

G_{ℚ_{p}} acts via the character \underline{Υ}+ ; \underline{ϕ}.

Let

(73)  κ_{p}(f, \underline{gh}) := res_{p}(κ(f, \underline{gh})) ∈ H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}})

denote the natural image of the global class κ(f, \underline{gh}) in the local cohomology at p, and set

ξ_{f\underline{gh}} = (1 − α_{f}a_{p}(\underline{g})a_{p}(\underline{h})^{−1}χ^{−1}(p))(1 − α_{f}a_{p}(\underline{g})^{−1}a_{p}(\underline{h})χ(p)).

Proposition 2.2. The class ξ_{f\underline{gh}} · κ_{p}(f, \underline{gh}) belongs to the natural image of H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}^{+}(N)) in

H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N)) under the map induced from the inclusion 𝕍_{f\underline{gh}}^{+}(N) ↪ 𝕍_{f\underline{gh}}(N).

Proof. Recall the natural projection Λ ⟶ Λ_{s} to the finite group ring of level p^{s}. The Galois representation 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s} is realised in the middle p-adic étale cohomology of the quotient W_{s} of X_{0}(Np) × X_{s} × X_{s}^{†}, and the natural image of κ_{p}(f, \underline{gh}) in H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s}) is proportional to the image of the cycle Δ_{s} under the p-adic étale Abel-Jacobi map. The three-fold W_{s} is smooth and proper over ℚ_{p}, and the purity conjecture for the monodromy filtration is known to hold by the work of Saito (cf. [Sa97], [Ne2, (3.2)]). By Theorem 3.1 of [Sa97], it follows that the natural image of κ_{p}(f, \underline{gh}) belongs to H_{fin}^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s}), where

(74)  H_{fin}^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s}) ⊂ H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s})

denotes the finite submodule in the sense of Bloch-Kato, as defined in [BK93, §3] or [Ne2, (2.2)]. Lemma 1.15 implies that 𝕍_{f\underline{gh}}(N)⊗_{Λ} Λ_{s} is an ordinary p-adic representation in the sense of Greenberg (cf. [Gr89, (4)], [Fl90]). Hence the finite subspace may be described concretely as

H_{fin}^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s}) = ker (H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s}) ⟶ H^{1}(I_{p}, (𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{+}(N)) ⊗_{Λ} Λ_{s})) , where I_{p} denotes the inertia group at p. (See [Fl90, Lemma 2, p. 125] for a proof of this useful fact. Note that H_{fin}^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N)) = H_{g}^{1}(ℚ_{p}, V_{f\underline{gh}}(N)) in this case, by [Ne2, (3.1) and (3.2)].) By passing to the inverse limit and using the fact that \underleftarrow{lim}_{s} 𝕍_{f\underline{gh}}(N) ⊗_{Λ} Λ_{s} = 𝕍_{f\underline{gh}}(N), one concludes that

κ_{p}(f, \underline{gh}) ∈ ker (H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N)) ⟶ H^{1}(I_{p}, 𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{+}(N))) .

The kernel of the restriction map H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{+}(N)) ⟶ H^{1}(I_{p}, 𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{+}(N)) is natu-

rally a quotient of H^{1}(ℚ_{p}^{ur}/ℚ_{p}, (𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{+}(N))^{I_{p}}).

The local Galois representation (𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{+}) lies in the middle of an exact sequence

0 ⟶ 𝕍_{f\underline{gh}}^{f−} ⊕ 𝕍_{f\underline{gh}}^{g−} ⊕ 𝕍_{f\underline{gh}}^{h−} ⟶ (𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{+}) ⟶ (𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{−}) ⟶ 0,

which induces an isomorphism 𝕍_{f\underline{gh}}^{g−} ⊕ 𝕍_{f\underline{gh}}^{h−} ⟶ (𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{+})^{I_{p}} by Lemma 2.1. The term

H^{1}(ℚ_{p}^{ur}/ℚ_{p}, 𝕍_{f\underline{gh}}^{g−} ⊕ 𝕍_{f\underline{gh}}^{h−}) = (Λ_{f\underline{gh}}(\underline{Υ}− ; \underline{g}) ⊕ Λ_{f\underline{gh}}(\underline{Υ}− ; \underline{h}))/(Frob_{p} − 1),

is annihilated by ξ_{f\underline{gh}}, as can be read off from the table in Lemma 2.1, and the same is therefore true of H^{1}(ℚ_{p}^{ur}/ℚ_{p}, (𝕍_{f\underline{gh}}/𝕍_{f\underline{gh}}^{−})^{I_{p}}) as well as of H^{1}(ℚ_{p}^{ur}/ℚ_{p}, (𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{−}(N))^{I_{p}}). It follows that

ξ_{f\underline{gh}} · κ_{p}(f, \underline{gh}) belongs to the kernel of the natural map to H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N)/𝕍_{f\underline{gh}}^{+}(N)). □

It will be convenient, for the discussion that follows and in the remainder of this article to replace the ring Λ_{f\underline{gh}} and all modules over it (such as 𝕍_{f\underline{gh}} and its various subquotients) with their localisations by the multiplicative set generated by ξ_{f\underline{gh}}. Since the element ξ_{f\underline{gh}} is non-zero at all classical points of weight ℓ ≥ 1, the passage to this localisation is harmless and still makes it permissible to specialise the Λ-adic class κ(f, \underline{gh}) at these points. With this convention in place, the following is immediate:

Corollary 2.3. The class κ_{p}(f, \underline{gh}) belongs to the image of H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}^{+}(N)) in H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}(N))

under the map induced from the inclusion 𝕍_{f\underline{gh}}^{+}(N) ↪ 𝕍_{f\underline{gh}}(N).

The decomposition (72) induces a corresponding decomposition of the local cohomology group H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}^{+}(N)/𝕍_{f\underline{gh}}^{++}(N)) into a direct sum of three contributions. Write

(75) κ_{p}^{f}(f, \underline{gh}) ∈ H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}^{f}(N)), κ_{p}^{g}(f, \underline{gh}) ∈ H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}^{g}(N)), κ_{p}^{h}(f, \underline{gh}) ∈ H^{1}(ℚ_{p}, 𝕍_{f\underline{gh}}^{h}(N))

for the projections of the local class κ_{p}(f, \underline{gh}) in each direct summand.

The reader will note that the Λ_{f\underline{gh}}-adic characters \underline{Υ}_{g}^{+} and \underline{Υ}_{h}^{+} agree with the cyclotomic character when restricted to the inertia group at p, and thus are not “truly varying”, while the character \underline{Υ}_{f}^{+} is somewhat more interesting, since it involves the Λ-adic cyclotomic character \underline{ϵ}_{cyc}.

2.2. Classical specializations. Let (y, z) ∈ Ω_{g} × Ω_{h} be a classical point of weight ℓ and character ϵ. The goal of this section is to analyze the specialization

κ(f, g_{y}, h_{z}) := κ(f, \underline{gh})_{(y,z)} ∈ H^{1}(ℚ, V_{fg_{y}h_{z}}(N))

of the global Λ-adic class κ(f, \underline{gh}) at (y, z), with special emphasis on its local behaviour at p.

A one-dimensional character Υ : G_{ℚ_{p}} ⟶ ℂ_{p}^{×} is said to be of Hodge-Tate weight −j if it is equal to a finite order character times the j-th power of the cyclotomic character. The following is an immediate corollary of Lemma 2.1 of the previous chapter.

Corollary 2.4. The Galois representation V_{fg_{y}h_{z}} is endowed with a four-step G_{ℚ_{p}}-stable filtration

0 ⊂ V_{fg_{y}h_{z}}^{++} ⊂ V_{fg_{y}h_{z}}^{+} ⊂ V_{fg_{y}h_{z}}^{−} ⊂ V_{fg_{y}h_{z}},

and the action of G_{ℚ_{p}} on its successive quotients is via the one-dimensional representations described in the following table:

Subquotient Galois action Hodge-Tate

ℓ = 2 ℓ = 1

weights

V_{fg_{y}h_{z}}^{++}  Υ^{+}(y, z) := (ψ_{f}ψ_{g_{y}}ψ_{h_{z}})^{−1} × ϵ × ω^{1−ℓ} × ϵ_{cyc}^{ℓ}  −ℓ  −2 −1
V_{fg_{y}h_{z}}^{+}/V_{fg_{y}h_{z}}^{++} Υ_{f}^{+}(y, z) := ψ_{f}ψ_{g_{y}}^{−1}ψ_{h_{z}}^{−1} × ϵ × ω^{1−ℓ} × ϵ_{cyc}^{ℓ−1}  1 − ℓ  −1  0
Υ_{g_{y}}^{+}(y, z) := ψ_{f}^{−1}ψ_{g_{y}}ψ_{h_{z}}^{−1} × χ^{−1} × ϵ_{cyc}  −1  −1 −1
Υ_{h_{z}}^{+}(y, z) := ψ_{f}^{−1}ψ_{g_{y}}^{−1}ψ_{h_{z}} × χ × ϵ_{cyc}  −1  −1 −1
V_{fg_{y}h_{z}}^{−}/V_{fg_{y}h_{z}}^{+} Υ_{h_{z}}^{−}(y, z) := ψ_{f}ψ_{g_{y}}ψ_{h_{z}}^{−1} × χ^{−1}  0  0  0
Υ_{g_{y}}^{−}(y, z) := ψ_{f}ψ_{g_{y}}^{−1}ψ_{h_{z}} × χ  0  0  0
Υ_{f}^{−}(y, z) := ψ_{f}^{−1}ψ_{g_{y}}ψ_{h_{z}} × ϵ^{−1} × ω^{ℓ−1} × ϵ_{cyc}^{2−ℓ}  ℓ − 2  0 −1
V_{fg_{y}h_{z}}/V_{fg_{y}h_{z}}^{−} Υ^{−}(y, z) := ψ_{f}ψ_{g_{y}}ψ_{h_{z}} × ϵ^{−1} × ω^{ℓ−1} × ϵ_{cyc}^{1−ℓ}  ℓ − 1  1  0

2.2.1. Weight two specializations. When ℓ = 2, the classes κ(f, g_{y}, h_{z}) are then, by their very construction, directly related to the étale Abel-Jacobi images of twisted diagonal cycles. Recall that α_{g_{y}} denotes the eigenvalue of U_{p} acting on the ordinary modular form g_{y}.

Proposition 2.5. Assume that the classical points (y, z) have weight ℓ = 2 and character ϵ of conductor p^{s}. Then

κ(f, g_{y}, h_{z}) = α_{g_{y}}^{−s}ϖ_{fg_{y}h_{z}}(AJ_{et}(Δ_{s})) ∈ H^{1}(ℚ, V_{fg_{y}h_{z}}(N)).

Proof. By Definition 1.15 and equation (63), κ(f, g_{y}, h_{z}) = ϖ_{fg_{y}h_{z}} ∘ϖ_{f\underline{g}\underline{h}}(κ_{∞}) = ϖ_{fg_{y}h_{z}}(κ_{s}). But since ϖ_{fg_{y}h_{z}}(1 ⊗ U_{p} ⊗ 1) = α_{g_{y}}ϖ_{fg_{y}h_{z}} and the projection ϖ_{fg_{y}h_{z}} factors through the ordinary projection e_{ord} ⊗ e_{ord}, equation (60) implies that κ(f, g_{y}, h_{z}) = α_{g_{y}}^{−s}ϖ_{fg_{y}h_{z}}(κ_{s}^{(3)}). The projection ϖ_{fg_{y}h_{z}} also

factors through the Künneth projection pr_{111} of equation (59), and thus κ(f, g_{y}, h_{z}) = α_{g_{y}}^{−s}ϖ_{fg_{y}h_{z}}(κ_{s}^{(2)}). Now, using equation (56) and the fact that ϖ_{fg_{y}h_{z}}ε_{s,s} = ϖ_{fg_{y}h_{z}}, it can be seen that κ(f, g_{y}, h_{z}) = α_{g_{y}}^{−s}ϖ_{fg_{y}h_{z}}(κ_{s}^{(1)}). Proposition 2.5 now follows from the definition of κ_{s}^{(1)} given in (54). □

Corollary 2.6. For all classical (y, z) ∈ Ω_{g} × Ω_{h} of weight two and character ϵ of conductor p^{s}, the class κ_{p}(f, g_{y}, h_{z}) belongs to the image of H^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}^{+}(N)) in H^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) under the map induced from the inclusion V_{fg_{y}h_{z}}^{+}(N) ↪ V_{fg_{y}h_{z}}(N).

Proof. This follows by specialising Corollary 2.3 to the point (y, z). (It could also be deduced from Proposition 2.5 in light of the known properties of étale Abel-Jacobi images of algebraic cycles.) □

2.2.2. Weight one specializations. The richest arithmetic phenomena arise when studying the specialisation of κ(f, \underline{gh}) at a point (y, z) of weight ℓ = 1 and trivial nebentype character at p, which will be assumed in this section. The specializations of \underline{g} and \underline{h} at y and z need not be classical in general; let us assume however that they are: the departure point of this section is the assumption that there exist newforms g ∈ S_{1}(N_{g}, χ) and h ∈ S_{1}(N_{h}, χ^{−1}) such that

g_{y} = g_{α} ∈ S_{1}(N_{g}p, χ), h_{z} = h_{α} ∈ S_{1}(N_{h}p, χ^{−1}).

Recall from the introduction that associated to g and h there exist Artin representations ϱ_{g} and ϱ_{h} with coefficients in a finite extension L/ℚ. Setting ϱ = ϱ_{g} ⊗ ϱ_{h}, we have V_{gh} := V_{g} ⊗ V_{h} ≃ V_{ϱ} ⊗_{L} L_{p}.

After labelling the two (distinct, by the hypothesis imposed in the introduction) roots of the p-th Hecke polynomial x^{2} − a_{p}(g) + χ(p) as α_{g} and β_{g}, one may define the global cohomology class

κ(f, g_{α}, h_{α}) := κ(f, \underline{gh})_{y,z} ∈ H^{1}(ℚ, V_{fgh}(N))

with values in the Galois representation V_{fgh}(N) = V_{f}(Np) ⊗ V_{g}(Np) ⊗ V_{h}(Np) ≃ (V_{f} ⊗ V_{gh})^{a} for some a ≥ 1. Let also κ_{p}^{f}(f, g_{α}, h_{α}) ∈ H^{1}(ℚ_{p}, V_{fgh}^{f}(N)) denote the associated local cohomology class arising likewise from (75).

The weight one setting distinguishes itself in (at least) two further respects. Firstly, recall the running assumption that α_{g} ≠ β_{g}, α_{h} ≠ β_{h}. Let us denote \underline{g}_{α}, \underline{g}_{β}, \underline{h}_{α}, \underline{h}_{β} the Hida families passing through g_{α}, g_{β}, h_{α}, h_{β}, respectively. One thus obtains four (a priori distinct) global classes

(76)  κ(f, g_{α}, h_{α}), κ(f, g_{α}, h_{β}), κ(f, g_{β}, h_{α}), κ(f, g_{β}, h_{β}) ∈ H^{1}(ℚ, V_{fgh}(N)).

Note that for primes ℓ ≠ p, it follows from [Ne2, (2.5) and (3.2)] that H^{1}(ℚ_{ℓ}, V_{fgh}(N)) = 0 and hence a fortiori the restriction to G_{ℚ_{ℓ}} of the above cohomology classes are all trivial. This is in general not true for ℓ = p, as we shall see later in Theorem 6.4.

Since V_{gh} is an Artin representation which is unramified at p, and for which both V_{g} and V_{h} are regular, the local cohomology group H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}(N)) decomposes into a direct sum of four vector spaces over L_{p} = K_{fgh}:

H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}(N)) = H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}^{αα}(N)) ⊕ H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}^{αβ}(N))
⊕H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}^{βα}(N)) ⊕ H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}^{ββ}(N)),

where V_{gh}^{αα}(N) := V_{g}(N)^{α_{g}} ⊗ V_{h}(N)^{α_{h}} denotes the G_{ℚ_{p}}-stable subspace in V_{gh}(N) formed from the tensor product of the two eigenspaces in V_{g}(N) and V_{h}(N) on which Frob_{p} acts with eigenvalue α_{g} and α_{h}, respectively, and likewise for the other spaces.

Proposition 2.7. The local class κ_{p}(f, g_{α}, h_{α}) belongs to the kernel of the natural homomorphism

H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}(N)) ⟶ H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{gh}^{αα}(N)).

Proof. This follows from Corollary 2.3 after observing that the specialisation of 𝕍_{f\underline{gh}}^{+} at (y, z) is contained in

V_{f}(Np) ⊗ (V_{gh}^{αβ}(N) ⊕ V_{gh}^{βα}(N) ⊕ V_{gh}^{ββ}(N)) .
□

Although the Galois representation V_{fgh}, which is independent of the choice of stabilisations of g and h, is crystalline under our running assumptions, the same need not be true of the extension classes in (76). This can be seen by observing that the Hodge-Tate weights attached to the specializations of the characters \underline{Υ}_{f}^{+} and \underline{Υ}_{f}^{−}, which in weight ℓ ≥ 2 are < 0 and ≥ 0 respectively, acquire Hodge-Tate weights that are ≥ 0 and < 0 respectively when ℓ = 1, as summarized in the table of Corollary 2.4. The local condition H_{fin}^{1}(ℚ_{p}, V_{fgh}(N)) at p defining the Selmer group of V_{fgh}(N) is given by

(77)  H_{fin}^{1}(ℚ_{p}, V_{fgh}(N)) = ker (H^{1}(ℚ_{p}, V_{fgh}(N)) ⟶ H^{1}(ℚ_{p}, V_{f}(Np)^{−} ⊗ V_{gh}(N))) ,

where V_{f}(Np)^{−} is the unramified quotient of V_{f}(Np) of Definition 1.10 and V_{gh}(N) = V_{g}(N) ⊗ V_{h}(N). Let

∂_{p} : H^{1}(ℚ, V_{fgh}(N)) ⟶ H^{1}(ℚ_{p}, V_{f}(Np)^{−} ⊗ V_{gh}(N))

be the natural projection to the “singular quotient” of the cohomology group at p, whose kernel is H_{fin}^{1}(ℚ_{p}, V_{fgh}(N)).

Proposition 2.8. The class ∂_{p}(κ(f, g_{α}, h_{α})) belongs to H^{1}(ℚ_{p}, V_{f}(Np)^{−} ⊗ V_{gh}^{ββ}(N)), and is non-zero if and only κ_{p}^{f}(f, g_{α}, h_{α}) ≠ 0.

Proof. The second row in the table in Corollary 2.4 shows that the specialisation of \underline{Υ}_{f}^{+} at (y, z) is the unramified character that sends Frob_{p} to α_{f}(α_{g}α_{h})^{−1} = α_{f}β_{g}β_{h} and hence the local class κ_{p}^{f}(f, g_{α}, h_{α}) belongs to H^{1}(ℚ_{p}, V_{f}(N)^{−} ⊗ V_{gh}^{ββ}(N)). Since

𝕍_{f\underline{gh}}^{+}/(𝕍_{f\underline{gh}}^{+} ∩ (V_{f}^{+} ⊗ (𝕍\underline{g} ⊗_{Λ} 𝕍∗ ; \underline{h}))) = O(\underline{Υ}_{f}^{+}),

it follows from (72) that ∂_{p}(κ(f, g_{α}, h_{α})) = κ_{p}^{f}(f, g_{α}, h_{α}) ∈ H^{1}(ℚ_{p}, V_{f}(N)^{−} ⊗ V_{gh}^{ββ}(N)). □

Note that analogous statements hold for the three other classes attached in (76) to the other pstabilisations of g and h. In particular, if the four associated singular parts are non-zero, they are necessarily linearly independent over L_{p}, and therefore generate the target of ∂_{p}.

2.3. Dieudonné modules and the Bloch-Kato logarithm. Let B_{dR} denote Fontaine’s ring of de Rham p-adic periods. Given a representation V of G_{ℚ_{p}} with coefficients in a finite extension K/ℚ_{p}, the de Rham Dieudonné module of V is defined as

D_{dR}(V) := (V ⊗ B_{dR})^{G_{ℚp}} .

It is a finite-dimensional K-vector space equipped with a descending exhaustive filtration Fil^{j}D_{dR}(V) by K-vector subspaces. Falting’s comparison theorem (cf. e.g.[Fa97]) asserts that there is an isomorphism of filtered modules

(78)  D_{dR}(V_{0}(Np)) ≃ H_{dR}^{1}(X_{0}(Np)/ℚ_{p})(1),  D_{dR}(V_{1}(Np^{s})) ≃ H_{dR}^{1}(X_{s}/ℚ_{p})(1).

For V = V_{0}(Np) and V_{1}(Np^{s}), the filtration on D_{dR}(V) is given by

Fil^{−1}D_{dR}(V) = D_{dR}(V) ⊋ Fil^{0}D_{dR}(V) ⊋ Fil^{1}D_{dR}(V) = 0

and (78) identifies Fil^{0}D_{dR}(V) with Ω^{1}(X_{0}(Np)_{/ℚ_{p}}) and Ω^{1}(X_{s/ℚ_{p}}) respectively.

Let B_{cris} ⊂ B_{dR} denote Fontaine’s ring of crystalline p-adic periods. The crystalline Dieudonné modules associated to the Galois representations V_{f} and V_{f}(Np) are

D_{f} := (B_{cris} ⊗ V_{f})^{G_{ℚp}} , D_{f}(Np) := (B_{cris} ⊗ V_{f}(Np))^{G_{ℚp}} .

These modules are vector spaces over ℚ_{p} equipped with a Hodge filtration and a ℚ_{p}-linear action of a Frobenius automorphism Φ, with eigenvalues α_{f} and β_{f}. As filtered modules, forgetting the operator Φ, there is canonical isomorphism D_{f}(Np) ≃ D_{dR}(V_{f}(Np)) and (78) identifies D_{f}(Np) with H_{dR}^{1}(X_{0}(Np)/ℚ_{p})(1)[f]. Via this identification, the regular differential form ωf̆ associated to a test vector f̆ ∈ S_{2}(Np)[f] gives rise to an element of Fil^{0}D_{f}(Np). Likewise, given a dual test vector f̆^{∗} ∈ S_{2}(Np)^{∨}[f], let ηf̆^{∗} ∈ D_{f}(Np) denote the unique class such that Φ(ηf̆^{∗}) = α_{f} · ηf̆^{∗} and

(79)  ⟨ηf̆^{∗}, w_{1}ω⟩_{X_{0}(Np)} = f̆^{∗}(ω)  for all ω ∈ e_{ord}Ω^{1}(X_{0}(Np)).

As in (1.5.1), let ϕ ∈ S_{2}(Np^{s}, χ_{ϕ}) be an eigenform of weight two, level Np^{s} and nebentype χ_{ϕ} = χ_{0}χ_{p}. Assume that ϕ is primitive at p in the sense of [MW84, §3], i.e., the p-part of the primitive level of ϕ and the primitive conductor of χ_{p} are both p^{s}. The Galois representation V_{ϕ} (as well as V_{ϕ}(Np^{s}), of course) is potentially cristalline; more precisely, it becomes cristalline when restricted to G_{ℚ_{p}(ζ_{s})}. The crystalline Dieudonné modules

D_{ϕ} := (B_{cris} ⊗ V_{ϕ})^{G_{ℚp(ζs)}}, D_{ϕ}(Np^{s}) := (B_{cris} ⊗ V_{ϕ}(Np^{s}))^{G_{ℚp(ζs)}}

are vector spaces over K = K_{ϕ} equipped with

• a Hodge filtration by K ⊗_{ℚ_{p}} ℚ_{p}(ζ_{ϕ})-submodules of D_{ϕ} ⊗_{ℚ_{p}} ℚ_{p}(ζ_{ϕ}), such that (78) identifies

D_{ϕ}(Np^{s}) with H_{dR}^{1}(X_{s}/ℚ_{p}(ζ_{s})) ⊗ K(1)[ϕ];

• a K-linear action of a Frobenius operator Φ, with eigenvalues α_{ϕ} = a_{p}(ϕ) and β_{ϕ} = χ_{0}(p)\overline{a_{p}(ϕ)}. Given a dual test vector ϕ^{∗} ∈ S_{2}^{ord}(Np^{s})^{∨}[ϕ], let us denote by η_{ϕ^{∗}} ∈ D_{ϕ} the unique class satisfying Φ(η_{ϕ^{∗}}) = α_{ϕ} · η_{ϕ^{∗}} and

(80)  ⟨η_{ϕ^{∗}}, w_{s}ω⟩_{s} = ϕ^{∗}(ω)  for all ω ∈ Ω^{1}(X_{s})^{ord}.

If \underline{ϕ}^{∗} := \{ϕ_{s}^{∗}\}_{s≥1} is a collection of elements ϕ_{s}^{∗} ∈ S_{2}^{ord}(Np^{s})^{∨} which are compatible under the maps (ϖ_{1}^{∗})^{∨} dual to the pullback by ϖ_{1}, then the corresponding elements η_{ϕ_{s}^{∗}} in H_{dR}^{1}(X_{s/ℚ_{p}(ζ_{s})}) are compatible under the transition maps ϖ_{2∗}, and hence describe an element of H_{dR}^{1}(X_{∞}).

The weight two specialisations of the Λ-adic test vectors ˘ \underline{g} ∈ S_{Λ_{g}}^{ord}(N, χ)[\underline{g}] and ˘ \underline{h} ∈ S_{Λ_{h}}^{ord}(N, χ^{−1})[\underline{h}] in the introduction provide natural examples of eigenforms ϕ that are primitive at p. Namely, let ϵ be a Dirichlet character of conductor p^{s} and let (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} be a classical point of weight (2, ϵ). Assuming that ϵ ≠ ω, the eigenforms g_{y}, h_{z} and ğ_{y} ∈ S_{2}(Np^{s}, χϵω^{−1}) and h̆_{z} ∈ S_{2}(Np^{s}, χ^{−1}ϵω^{−1}) are primitive at p. Let

D_{fg_{y}h_{z}}(N) := (B_{cris} ⊗_{ℚ_{p}} V_{fg_{y}h_{z}}(N))^{G_{ℚp(ζs)}} ≃ D_{f}(Np) ⊗ D_{g_{y}}(Np^{s}) ⊗ D_{h_{z}}(Np^{s})(−1)

denote the Dieudonné module associated to the Galois representation V_{fg_{y}h_{z}}(N) introduced in (70).

Let H_{exp}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) ⊆ H_{fin}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) ⊆ H^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) be the subspaces introduced in [BK93] by Bloch and Kato and let

log_{p} : H_{exp}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) ⟶ D_{fg_{y}h_{z}}(N)/(Fil^{0}(D_{fg_{y}h_{z}}(N)) + D_{fg_{y}h_{z}}(N)^{Φ=1})

denote the Bloch-Kato logarithm map of loc. cit.

Since the eigenvalues of the Frobenius operator Φ acting on D_{fg_{y}h_{z}}(N) are α_{f}α_{g_{y}}α_{h_{z}}, …, β_{f}β_{g_{y}}β_{h_{z}}, they all are algebraic numbers of complex absolute value p^{3/2}, and it follows from [BK93, Corollary 3.8.4] that D_{fg_{y}h_{z}}(N)^{Φ=1} = 0 and H_{exp}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) = H_{fin}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)). Bloch-Kato’s logarithm map may therefore be recast as a map

(81) log_{p} : H_{fin}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) ⟶ D_{fg_{y}h_{z}}(N)/Fil^{0}(D_{fg_{y}h_{z}}(N)) ≃ Fil^{0}(D_{fg_{y}h_{z}}(N))^{∨},

where the latter isomorphism is induced by Poincaré duality.

The p-adic syntomic Abel-Jacobi map is then defined as the composition

(82) \xymatrix{AJ_{p} : CH^{2}(W_{s})_{0}(ℚ_{p}(ζ_{s})) \ar[r]^{ϖ_{fgy hz} ∘AJ_{et}} & H_{fin}^{1}(ℚ_{p}(ζ_{s}), V_{fg_{y}h_{z}}(N)) \ar[r]^{^{log^{p}}} & Fil^{0}(D_{fg_{y}h_{z}}(N))^{∨}.}

2.4. Ohta’s periods and Λ-adic test vectors. Let ℤ̂_{p}^{ur} denote the completion of the ring of integers in the maximal unramified extension ℚ̂_{p}^{ur} of ℚ_{p}. Recall the convention whereby the symbol ˆ⊗ denotes the completed tensor product, so that, for example, Λ ˆ⊗ ℤ̂_{p}^{ur} is isomorphic to the power series ring ℤ̂_{p}^{ur}[[T]].

Define D(V_{1}^{ord}(Np^{s})^{−}) := (V_{1}^{ord}(Np^{s})^{−} ⊗ ℤ̂_{p}^{ur})^{G_{ℚp}}. Since the modules V_{1}^{ord}(Np^{s})^{−} are unramified, they are admissible for the period ring ℤ̂_{p}^{ur} and hence D_{dR}(V_{1}^{ord}(Np^{s})^{−}) = D(V_{1}^{ord}(Np^{s})^{−}). Define also

𝔻(𝕍_{1}^{ord}(Np^{∞})^{−}) := (𝕍_{1}^{ord}(Np^{∞})^{−} ˆ⊗ℤ̂_{p}^{ur})^{G_{ℚp}} = \underleftarrow{lim}_{s} D(V_{1}^{ord}(Np^{s})^{−}).

Recall that the comparison theorem of (78) leads to a natural inclusion

S_{2}^{ord}(Np^{s}) = e_{ord}Ω^{1}(X_{s}) ⟶ e_{ord}H_{dR}^{1}(X_{s}) = D_{dR}(V_{1}^{ord}(Np^{s})).

Write

cmp_{s} : S_{2}^{ord}(Np^{s}) ⟶ D(V_{1}^{ord}(Np^{s})^{−})

for the natural map obtained by composing this inclusion with the natural projection

D_{dR}(V_{1}^{ord}(Np^{s})) ⟶ D(V_{1}^{ord}(Np^{s})^{−})

arising from the G_{ℚ_{p}}-invariant filtration on V_{1}^{ord}(Np^{s}).

We begin by recalling a theorem of Ohta [Oh95], as refined subsequently by Wake [Wa] which provides a p-adic interpolation of the maps cmp_{s}, relating the space S_{Λ}^{ord}(N) of ordinary Λ-adic modular forms to the Galois representation 𝕍_{1}^{ord}(Np^{∞}).

Proposition 2.9 (Ohta). There is a canonical isomorphism

cmp_{∞} : S_{Λ}^{ord}(N) ⟶ 𝔻(𝕍_{1}^{ord}(Np^{∞})^{−})

characterised by the commutativity of the diagram

\xymatrix{S_{Λ}^{ord}(N) \ar[r]^{^{cmp^{∞}}} \ar[d]_{℧_{s}} & 𝔻(𝕍_{1}^{ord}(Np^{∞})^{−}) \ar[d] \\ S_{2}^{ord}(Np^{s}) \ar[r]^{_{cmp_{s}}} & D(V_{1}^{ord}(Np^{s})^{−})}

where the left vertical arrow sends the ordinary Λ-adic form \underline{g} = \{g_{s}\} to ℧_{s}(g_{s}) = U_{p}^{−s}g_{s}, and the right vertical arrow denotes the natural specialisation map of level p^{s}.

The reader will recall that the system {℧_{s}(g_{s})} of ordinary modular forms on X_{s} are compatible under the pushforward maps ϖ_{2∗}, in harmony with the fact that the inverse limit defining 𝔻(𝕍_{1}^{ord}(Np^{∞})^{−})) is also taken relative to these maps. The statement differs slightly from the one in Ohta’s work, which focusses on the anti-ordinary part of the étale cohomology of the tower X_{s}, in which the inverse limits are taken relative to the maps ϖ_{1∗}. Working with the ordinary part obviates the application of the automorphism w_{s} to the specialisation at level s which occurs in Ohta’s result.

Since the U_{p} operator (and hence, the ordinary projector, as well) are self-adjoint relative to the twisted Λ-adic Weil pairing [ , ]_{Γ} of (40), this pairing induces a pairing on the ordinary parts, which shall be denoted by the same symbol:

(83)  [ ,]_{Γ} : 𝕍_{1}^{ord}(Np^{∞}) × 𝕍_{1}^{ord}(Np^{∞}) ⟶ Λ(\underline{ϵ}_{cyc}).

The submodule 𝕍_{1}^{ord}(Np^{∞})^{+} is isotropic for this pairing, which therefore induces a natural Λ(\underline{ϵ}_{cyc})- valued duality between 𝕍_{1}^{ord}(Np^{∞})^{+} and the quotient 𝕍_{1}^{ord}(Np^{∞})^{−}. (This duality is the one described, for instance, in Theorem B of [Oh95].) Combining [ , ]_{Γ} with Ohta’s map cmp_{∞} yields canonical G_{ℚ_{p}}- equivariant Λ-linear maps, as described in [Wa, Theorem 1.3]:

(84)  \underline{c}^{+} :S_{Λ}^{ord}(N) ⟶ Hom_{Λ}(𝕍_{1}^{ord}(Np^{∞})^{+}, Λ(\underline{ϵ}_{cyc}) ˆ⊗ ℤ̂_{p}^{ur}),
\underline{c}^{−} :S_{Λ}^{ord}(N)^{∨} ⟶ Hom_{Λ}(𝕍_{1}^{ord}(Np^{∞})^{−}, Λ ˆ⊗ ℤ̂_{p}^{ur}),

which are given by the explicit formulae

(85)  \underline{c}^{+}(\underline{ϕ})(\underline{ξ}) = [℧_{∞}(\underline{ϕ}), \underline{ξ}]_{Γ} = \{⟨U_{p}^{−s}ϕ_{s}, ww_{s}U_{p}^{s}ξ_{s}⟩_{Γ_{s}}\}_{s≥1}
= {⟨ϕ_{s}, ww_{s}ξ_{s}⟩_{Γ_{s}}}_{s≥1} = {−⟨ξ_{s}, (ww_{s})^{−1}ϕ_{s}⟩_{Γ_{s}}}_{s≥1} .
(86)  \underline{c}^{−}(\underline{ϕ}^{∗})(\underline{ξ}) = [\underline{ϕ}^{∗}, \underline{ξ}]_{Γ} = \{−⟨ξ_{s}, (ww_{s})^{−1}U_{p}^{s}η_{ϕ_{s}^{∗}}⟩_{Γ_{s}}\}_{s≥1} .

After tensoring over Λ with Λ_{g} and taking \underline{g}-isotypic parts in (84), one deduces Λ_{g}-linear G_{ℚ_{p}}- equivariant homomorphisms

(87)  \underline{c}_{g}^{+} :S_{Λ_{g}}(N)[\underline{g}] ⟶ Hom_{Λ_{g}}(𝕍_{g}^{+}(N), Λ_{g}(\underline{ϵ}_{cyc}) ˆ⊗ ℤ̂_{p}^{ur}),
(88)  \underline{c}_{g}^{−} :S_{Λ_{g}}(N)^{∨}[\underline{g}] ⟶ Hom_{Λ_{g}}(𝕍_{g}^{−}(N), Λ_{g} ˆ⊗ ℤ̂_{p}^{ur}),

which are described by the rules, deduced from (85) and (86),

(89)  \underline{c}_{g}^{+}(˘\underline{g})(\underline{ξ}) = \{−⟨ξ_{s}, (ww_{s})^{−1}ğ_{s}⟩_{Γ_{s}}\}_{s≥1} ,
(90)  \underline{c}_{g}^{−}(˘\underline{g}^{∗})(\underline{ξ}) = \{α_{\underline{g}_{s}}^{s}⟨ξ_{s}, (ww_{s})^{−1}η_{}_{ğ_{s}^{∗}}⟩_{Γ_{s}}\}_{s≥1} .

If Λ̃ is any complete Noetherian ℤ_{p}-algebra and Ψ : G_{ℚ_{p}} ⟶ Λ̃^{×} is an unramified character, it is convenient to define

Λ̃{Ψ} := (Λ̃ ˆ⊗ ℤ̂_{p}^{ur})^{Ψ}

to be the Λ̃-submodule of Λ̃ ˆ⊗ ℤ̂_{p}^{ur} on which G_{ℚ_{p}} acts via the character Ψ. As explained in [LZ12, Prop. 3.3], Λ̃{Ψ} is non-canonically isomorphic to Λ̃(Ψ) as a Λ̃[G_{ℚ_{p}}]-module. The advantage of working with Λ̃{Ψ} is that one has a canonical identification

(91)  𝔻(Λ̃{Ψ}) = (((Λ̃ ˆ⊗ ℤ̂_{p}^{ur})^{Ψ}) ˆ⊗ ℤ̂_{p}^{ur})^{G_{ℚp}} = Λ̃,

the last equality being simply induced from the multiplication on the coefficients.

The G_{ℚ_{p}}- equivariance of the maps (87) and (88) above shows that \underline{c}_{g}^{+}(˘\underline{g}) and \underline{c}_{g}^{−}(˘\underline{g}^{∗}) give rise to G_{ℚ_{p}}-equivariant Λ_{g}-linear maps

\underline{c}_{g}^{+}(˘\underline{g}) : 𝕍_{g}(N)^{+} ⟶ Λ\{Ψ−1 ; \underline{g}\}(\underline{ϵ}_{cyc}), \underline{c}_{g}^{−}(˘\underline{g}^{∗}) : 𝕍_{g}(N)^{−} ⟶ Λ\{Ψ\underline{g}\},

with similar remarks applying, of course, to the elements \underline{c}_{h}^{+}(˘\underline{h}) and \underline{c}_{h}^{−}(˘\underline{h}^{∗}). The maps \underline{c}_{g}^{+}(˘\underline{g}) and \underline{c}_{g}^{−}(˘\underline{g}^{∗}) also admit simpler variants associated to the fixed forms f̆ and f̆^{∗},

c_{f}^{+}(f̆) : V_{f}(Np)^{+} ⟶ ℚ_{p}\{ψ_{f}^{−1}\}(ϵ_{cyc}), \underline{c}_{f}^{−}(f̆^{∗}) : V_{f}(Np)^{−} ⟶ ℚ_{p}\{ψ_{f}\}.

Recall from Lemma 2.1 that there are canonical isomorphisms of Λ_{f\underline{gh}}[G_{ℚ_{p}}]-modules

𝕍_{f\underline{gh}}^{f}(N) ≃ V_{f}(Np)^{−} ⊗ 𝕍\underline{g}(N)^{+} ⊗_{Λ} 𝕍\underline{h}(N)^{+}(\underline{ϵ}_{cyc}^{−1}),
𝕍_{f\underline{gh}}^{g}(N) ≃ V_{f}(Np)^{+} ⊗ 𝕍\underline{g}(N)^{−} ⊗_{Λ} 𝕍\underline{h}(N)^{+}(\underline{ϵ}_{cyc}^{−1}),
𝕍_{f\underline{gh}}^{h}(N) ≃ V_{f}(Np)^{+} ⊗ 𝕍\underline{g}(N)^{+} ⊗_{Λ} 𝕍\underline{h}(N)^{−}(\underline{ϵ}_{cyc}^{−1}).

Choose a triple of Λ-adic test vectors

(92)  f̆ ∈ S_{2}(Np)[f],  ˘  \underline{g} ∈ S_{Λ_{g}}^{ord}(N, χ)[\underline{g}],  ˘  \underline{h} ∈ S_{Λ_{h}}^{ord}(N, χ^{−1})[\underline{h}],

and a triple of dual test vectors

(93)  f̆^{∗} ∈ S_{2}(Np)^{∨}[f],  ˘  \underline{g}^{∗} ∈ S_{Λ_{g}}^{ord}(N, χ^{−1})^{∨}[\underline{g}],  ˘  \underline{h}^{∗} ∈ S_{Λ_{h}}^{ord}(N, χ)^{∨}[\underline{h}]

as in the introduction.

We can then define the local Λ-adic cohomology classes

κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) := c_{f}^{−}(f̆^{∗}) ⊗ \underline{c}_{g}^{+}(˘\underline{g}) ⊗ \underline{c}_{h}^{+}(˘\underline{h})(κ_{p}^{f}(f, \underline{gh})) ∈ H^{1}(ℚ_{p}, Λ_{f\underline{gh}}\{ψ_{f}Ψ−1 ; \underline{g}Ψ−1 ; \underline{h}\}(\underline{ϵ}_{cyc})),
(94) κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) := c_{f}^{+}(f̆) ⊗ \underline{c}_{g}^{−}(˘\underline{g}^{∗}) ⊗ \underline{c}_{h}^{+}(˘\underline{h})(κ_{p}^{g}(f, \underline{gh})) ∈ H^{1}(ℚ_{p}, Λ_{f\underline{gh}}\{ψ_{f}^{−1}Ψ\underline{g}Ψ−1 ; \underline{h}χ^{−1}\}(ϵ_{cyc})),
κ_{p}^{h}(f̆, \underline{ğh̆^{∗}}) := c_{f}^{+}(f̆) ⊗ \underline{c}_{g}^{+}(˘\underline{g}) ⊗ \underline{c}_{h}^{−}(˘\underline{h}^{∗})(κ_{p}^{h}(f, \underline{gh})) ∈ H^{1}(ℚ_{p}, Λ_{f\underline{gh}}\{ψ_{f}^{−1}Ψ−1 ; \underline{g}Ψ\underline{h}χ\}(ϵ_{cyc})).

The modules in which the above classes take values are non-canonically isomorphic to Λ_{f\underline{gh}}(\underline{Υ}+ ; \underline{f}),

Λ_{f\underline{gh}}(\underline{Υ}+ ; \underline{g}) and Λ_{f\underline{gh}}(\underline{Υ}+ ; \underline{h}), respectively, but elements of the former incorporate a choice of period (in the style of [LZ12]) for the relevant unramified Galois characters.

As a piece of notation, we let

(95)  κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) ∈ H^{1}(ℚ_{p}, K_{x}{ψ_{f}Ψ_{g_{y}}^{−1}Ψ_{h_{z}}^{−1}}(ϵ_{cyc}^{ℓ−1}ϵω^{1−ℓ})),
κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z}) ∈ H^{1}(ℚ_{p}, K_{x}{ψ_{f}^{−1}Ψ_{g_{y}}Ψ_{h_{z}}^{−1} × χ^{−1}}(ϵ_{cyc})),

denote the local cohomology classes obtained by specialising the Λ-adic classes of (94) at a classical point x := (y, z) ∈ Ω_{g} ×_{Ω}Ω_{h} of weight (ℓ, ϵ) for some positive integer ℓ ≥ 1 and character ϵ of conductor p^{s}, corresponding to a ring homomorphism x : Λ_{f\underline{gh}} ⟶ K_{x} to a finite extension K_{x} of ℚ_{p}.

Our most immediate goal is now to relate the images of the classes κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) and κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) under Perrin-Riou’s Λ-adic logarithm map to the restriction to Ω_{f\underline{gh}} of the Garrett-Hida p-adic L-functions

ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h}) and ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h}), respectively. (A similar relation exists between ℒ_{p} ^{h_{α}}(˘\underline{f}, ˘ \underline{g}, ˘ \underline{h}^{∗}) and κ_{p}^{h}(f̆, \underline{ğh̆^{∗}}), since g and h play symmetric roles in all the constructions; the details have been omitted because the latter relation plays no role in the proof of the main theorems of the article.)

The family κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) and the p-adic L-function ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h}) play a crucial role in the proofs of Theorems A and C, while κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) and ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h}) are the main ingredient in the proofs of Theorems B and D. It is worth remarking that the family of cohomology classes κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}), whose underlying representation space involves the Λ-adic cyclotomic character, is of greater arithmetic complexity than the family κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}), whose restriction to the inertia group at p is equal to ϵ_{cyc}, and whose specialisations are therefore independent of (y, z) when restricted to inertia.

2.5. The logarithm of weight two specialisations. Let ϵ ≠ ω be a Dirichlet character of conductor p^{s} and let x := (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} be a classical point of weight (2, ϵ). Put K = K_{x} = K_{fg_{y}h_{z}} as above and let κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) and κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z}) be the local cohomology classes introduced in (95). Since the Hodge-Tate weights of the target Galois representations are all −1, the Bloch-Kato p-adic logarithm maps induce isomorphisms

(96)  log_{p} : H^{1}(ℚ_{p}, K_{x}{ψ_{f}Ψ_{g_{y}}^{−1}Ψ_{h_{z}}^{−1}}(ϵ_{cyc}ϵω^{−1})) ⟶ K
log_{p} : H^{1}(ℚ_{p}, K_{x}{ψ_{f}^{−1}Ψ_{g_{y}}Ψ_{h_{z}}^{−1} × χ^{−1}}(ϵ_{cyc})) ⟶ K.

We refer to (142) in §5.1 below for a more detailed discussion of this fact in a more general setting and a discussion of the periods involved in the above isomorphisms.

The goal of this section is to give an explicit formula for the Bloch-Kato p-adic logarithms of the classes κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) and κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z}) in terms of the image of the twisted diagonal cycles introduced in the previous chapter under the p-adic syntomic Abel-Jacobi map (82).

Recall the de Rham cohomology classes of (79) and (80), and let

ω∘ ; f̆:= (ww_{s})^{−1}ωf̆,  ω∘ ; ğ_{y} := (ww_{s})^{−1}ωğ_{y},  ω∘ ; h̆_{z} = (ww_{s})^{−1}ηh̆_{z},
η∘ ; f̆^{∗} := (ww_{s})^{−1}ηf̆^{∗},  η_{ğ_{y}^{∗}}^{∘} := (ww_{s})^{−1}η_{}_{ğ_{y}^{∗}},  η∘ ; h̆_{z}^{∗} = (ww_{s})^{−1}ηh̆_{z}^{∗},

viewed as elements of the anti-ordinary part of H_{dR}^{1}(X_{0}(Np)_{/ℚ_{p}}) and H_{dR}^{1}(X_{s/ℚ_{p}(ζ_{s})}), respectively. The triple tensor classes

η∘ ; f̆^{∗} ⊗ ω∘ ; ğ_{y} ⊗ ω∘ ; h̆_{z},  ω∘ ; f̆ ⊗ η_{ğ_{y}^{∗}}^{∘} ⊗ ω∘ ; h̆_{z},  ω∘ ; f̆ ⊗ ω∘ ; ğ_{y} ⊗ η∘ ; h̆_{z}^{∗},

are thus elements in the anti-ordinary part of H_{dR}^{3}(W_{s,s/ℚ_{p}(ζ_{s})}^{†}) = H_{dR}^{3}(W_{s,s/ℚ_{p}(ζ_{s})}). Since the N-part (resp. the p-part) of the nebentype characters of g_{y} and h_{z} are inverse one of another (resp. equal), these classes are invariant under the action of the group D_{s} of diamond operators acting diagonally at N and anti-diagonally at p on W_{s,s}, hence they descend to classes in H_{dR}^{3}(W_{s/ℚ_{p}(ζ_{s})}), denoted

η∘ ; f̆^{∗} ⊗ ω∘ ; ğ_{y}ω∘ ; h̆_{z}, ω∘ ; f̆ ⊗ η_{ğ_{y}^{∗}}^{∘}ω∘ ; h̆_{z}, ω∘ ; f̆ ⊗ ω∘ ; ğ_{y}η∘ ; h̆_{z}^{∗} ∈ H_{dR}^{3}(W_{s/ℚ_{p}(ζ_{s})}),

which are characterized by the relation

(97)  pr_{s}^{∗}(η∘ ; f̆^{∗} ⊗ ω∘ ; ğ_{y}ω∘ ; h̆_{z}) = η∘ ; f̆^{∗} ⊗ ω∘ ; ğ_{y} ⊗ ω∘ ; h̆_{z},

and likewise for the other two. Faltings’ comparison isomorphism (78) together with Künneth’s formula (55) allows to regard them as elements of Fil^{0}(D_{fg_{y}h_{z}}(N)) and it thus makes sense to evaluate the functional AJ_{p}(Δ_{s}) at these classes.

Proposition 2.10. For all arithmetic points x = (y, z) of weight (2, ϵ) as above, the specialisations

κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) := κ_{p}^{f}(f̆^{∗}, \underline{ğh̆})(y, z), κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z}) := κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆})(y, z)

satisfy the formulae

log_{p}(κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z})) = α_{g}^{−s}AJ_{p}(Δ_{s})(η∘ ; f̆^{∗} ⊗ ω∘ ; ğ_{y}ω∘ ; h̆_{z}),
log_{p}(κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z})) = AJ_{p}(Δ_{s})(ω∘ ; f̆ ⊗ η_{ğ_{y}^{∗}}^{∘}ω∘ ; h̆_{z}).

Proof. Let ⟨ , ⟩_{W_{s}} denote the Poincaré pairing (in both its étale and de Rham incarnations) on the (1, 1, 1)-Künneth component of the middle cohomology of W_{s}, which is related to the Poincaré duality on the curves X_{0}(Np) and X_{s} by the rules

⟨η_{1} ⊗ η_{2}η_{3}, ω_{1} ⊗ ω_{2}ω_{3}⟩_{W_{s}} = ⟨η_{1}, ω_{1}⟩_{X_{0}(Np)}⟨η_{2}, ω_{2}⟩_{s}⟨η_{3}, ω_{3}⟩_{s}.

Let ⟨ , ⟩_{W_{s},Γ_{s}} denote the group-ring valued pairing constructed from the above by setting

⟨η_{1} ⊗ η_{2}η_{3}, ω_{1} ⊗ ω_{2}ω_{3}⟩_{W_{s},Γ_{s}} = ⟨η_{1}, ω_{1}⟩_{X_{0}(Np)}⟨η_{2}, ω_{2}⟩_{Γ_{s}}⟨η_{3}, ω_{3}⟩_{Γ_{s}}.

The formulae for \underline{c}_{g}^{+}(˘\underline{g}) and \underline{c}_{h}^{+}(˘\underline{h}) described in equation (89) lead to the following equality in the local cohomology group H^{1}(ℚ_{p}, K_{x}(Υ_{f}^{+}(y, z))), where κ_{p}(f, \underline{gh})_{s} denotes the natural projection of the ordinary Λ-adic class κ_{p}(f, \underline{gh}) to the s-th component in the inverse limit:

(98)  κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) := ν_{(y,z)}(κ_{p}^{f}(f̆^{∗}, ˘ \underline{g}, ˘ \underline{h})) = ν_{(y,z)}(⟨κ_{p}(f, \underline{gh})_{s}, η_{f^{∗}}^{∘} ⊗ ω∘ ; ğ_{s}ω∘ ; h̆_{s}⟩_{W_{s},Γ_{s}}).

It then follows from Lemma 1.12 that

(99)  κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) = ⟨κ_{p}(f, g_{y}, h_{z})_{s}, η_{f^{∗}}^{∘} ⊗ ω∘ ; ğ_{y}ω∘ ; h̆_{z}⟩_{W_{s}}.

Proposition 2.5 thus yields κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) = α_{g_{y}}^{−s}⟨AJ_{et}(Δ_{s}), η∘ ; f̆^{∗} ⊗ ω∘ ; ğ_{y}ω∘ ; h̆_{z}⟩_{W_{s}} and the first assertion of the proposition follows by appying the Bloch-Kato logarithm to both sides. The second is proved by the same calculation, but replacing (98) by the analogous formula arising from both (89) and (90),

namely κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z}) = α_{g_{y}}^{s}ν_{(y,z)}(⟨κ(f, \underline{gh}), η_{f}^{∘} ⊗ η_{ğ_{s}^{∗}}^{∘}ω∘ ; h̆_{s}⟩_{W_{s},Γ_{s}}). □

For the later calculations, it will be convenient to dispose of a simpler formula for log_{p}(κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z})) and log_{p}(κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z})) involving the product variety W_{s,s} rather than its quotient W_{s}, and the classes ηf̆^{∗} ⊗ ωğ_{y}ωh̆_{z} and ωf̆ ⊗ η_{}_{ğ_{y}^{∗}}ωh̆_{z} instead of their anti-ordinary counterparts. To do this, define the cycle

Δ_{s,s}^{∘} := ((ww_{1})^{−1}, (ww_{s})^{−1}, (ww_{s})^{−1})_{∗}Δ_{s,s} ∈ CH^{2}(W_{s,s})_{0}(ℚ_{p}(ζ_{N}ζ_{s})).

Equations (41) and (49) show that Δ_{s,s}^{∘} is given by the simple equation

(100) Δ_{s,s}^{∘} = ((ww_{1})^{−1}, (ww_{s})^{−1}, (ww_{s})^{−1})_{∗}ε_{s,s∗}(j_{1} ∘ ϖ_{2}^{s−1} ∘ w_{s}, Id, w_{s})_{∗}δ_{∗}(X_{s})

= ε_{s,s∗}(j_{1} ∘ ϖ_{1}^{s−1}, w_{s}^{−1}, Id)_{∗}(w^{−1}, w^{−1}, w^{−1})_{∗}δ_{∗}(X_{s}) = ε_{s,s∗}(j_{1} ∘ ϖ_{1}^{s−1}, w_{s}^{−1}, Id)_{∗}δ_{∗}(X_{s}).

Corollary 2.11. For all arithmetic points x = (y, z) of weight (2, ϵ),

log_{p}(κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z})) = α_{g}^{−s}d_{s}^{−1}AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ_{y}ωh̆_{z}),
log_{p}(κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z})) = d_{s}^{−1}AJ_{p}(Δ_{s,s}^{∘})(ωf̆ ⊗ η_{}_{ğ_{y}^{∗}}ωh̆_{z}).

Proof. This follows directly from Proposition 2.10 in light of the functorial properties of Abel-Jacobi maps with respect to automorphisms. □

Corollary 2.11 motivates the study of the image of the twisted diagonal cycle Δ_{s,s}^{∘} under the p-adic syntomic Abel-Jacobi map, which is taken up in the next two chapters.

3. The syntomic Abel-Jacobi map on products of semistable curves

3.1. The cohomology of semistable curves. This section recalls the description (following [CI99], and [CI10]) of the de Rham cohomology of a semistable curve and the attendant structures with which it is equipped. Let F be a finite extension of ℚ_{p}, let O_{F} denote its ring of integers, fix an uniformizer π_{F} and let k_{F} = O_{F} /π_{F} denote the residue field. Let F_{0} denote the maximal unramified subfield of F, which is isomorphic to the fraction field of the ring of Witt vectors of k_{F}. Write σ_{0} ∈ Gal (F_{0}/ℚ_{p}) for the Frobenius element generating the Galois group of F_{0}/ℚ_{p}. Let X be a smooth, connected, proper curve over F and assume it possesses a regular semistable model X over O_{F}, meaning that locally at every point, X is either smooth over Spec(O_{F}) or étale over Spec(O_{F} [X, Y]/(XY − π_{F})).

Let X̃ denote the special fiber of X. It is assumed that the irreducible components of X̃ are smooth, geometrically irreducible, and defined over k_{F}, that there are at least two of them, and that distinct components intersect only in ordinary double points defined over k_{F}. Let G denote the dual graph of the special fiber, whose set V(G) of vertices is in bijection with the irreducible components of X̃ and whose set E(G) of oriented edges is in bijection with the singular points of the special fiber, together with an ordering of the two components which intersect at that point. Given v ∈ V(G), let X̃_{v} denote the associated component, and let X̃_{v}^{sm} denote the smooth locus of X̃_{v}, that is to say, the complement in X̃_{v} of the singular points of X̃. If e ∈ E(G) is an edge of G, write s(e) and t(e) for its source and target respectively, and x_{e} for the associated ordinary double point, satisfying X̃_{s(e)} ∩ X̃_{t(e)} = {x_{e}}. Write also ē for the reversed edge, satisfying s(ē) = t(e) and t(ē) = s(e); note that xē = x_{e}.

Let

red : X(ℂ_{p}) ⟶ X̃(𝔽̄_{p})

denote the reduction map and, for each v ∈ V(G), let

W_{v} := red^{−1}(X̃_{v}(𝔽̄_{p})), A_{v} := red^{−1}(X̃_{v}^{sm}(𝔽̄_{p})) ⊂ W_{v}

denote the wide open subspace and underlying affinoid associated to the component X̃_{v}. If e is an edge of G, the wide opens W_{s(e)} and W_{t(e)} intersect in the wide open annulus

W_{e} := W_{s(e)} ∩ W_{t(e)} = red^{−1}({x_{e}}).

The collection {W_{v}}_{v∈V(G)} gives rise to an admissible cover of X(ℂ_{p}) by wide open subsets defined over F, whose nerve is encoded in the graph G. The de Rham cohomology group H_{dR}^{1}(X) can be described rigid analytically as the set of hypercocycles

{({ω_{v}}_{v}, {f_{e}}_{e}) ∈ ∏_{v∈V(G)}Ω^{1}(W_{v}) × ∏_{e∈E(G)}O_{W_{e}} such that df_{e} = ω_{t(e)} − ω_{s(e)}, fē = −f_{e} for all e ∈ E(G)} ,

taken modulo the hyper-coboundaries, which are hypercocycles of the form (df_{v}, f_{t(e)} − f_{s(e)}) attached to a collection {f_{v}}_{v} of elements of O_{W_{v}} indexed by v ∈ V(G).

The non-trivial rigid cohomology groups of W = W_{v} or W_{e} are

H_{rig}^{0}(W) = F and H_{rig}^{1}(W) = Ω^{1}(W)/dO_{W}.

Following [CI99, (1.1)], the Mayer-Vietoris sequence attached to this admissible covering places H_{dR}^{1}(X) in the middle of a short exact sequence

(101)  0 ⟶ \frac{⊕_{e}H_{rig}^{0}(W_{e})^{−}}{δ(⊕_{v}H_{rig}^{0}(W_{v}))} ⟶^{i} H_{dR}^{1}(X) ⟶^{r} ker (⊕_{v}H_{rig}^{1}(W_{v}) →^{δ} ⊕_{e}H_{rig}^{1}(W_{e})^{−}) ⟶ 0,

where • ⊕_{e}H_{rig}^{i}(W_{e})^{−} is the (−1)-eigen-subspace of ⊕_{e}H_{rig}^{i}(W_{e}) which respect to the involution e ↦ ē;

• the indices e and v run over E(G) and V(G) respectively;

• the map δ sends {κ_{v}}_{v} to the class of {κ_{t(e)} − κ_{s(e)}}_{e};

• the map i sends a collection {λ_{e}}_{e} to the class of the hypercocycle ({0}_{v}, {λ_{e}}_{e});

• the map r is the natural restriction map sending the class of ({ω_{v}}_{v}, {f_{e}}_{e}) to {ω_{v}}_{v}. The cohomology group H_{dR}^{1}(X) is an F -vector space equipped with the following further structures:

(i) A decreasing Hodge filtration of F -vector spaces given by Fil^{i} = H_{dR}^{1}(X) for i ≤ 0, Fil^{1} = Ω^{1}(X) and Fil^{i} = {0} for i > 1.

(ii) A canonical F_{0}-structure H_{log−cris}^{1}(X̃) afforded by the log-crystalline cohomology of X̃ with coefficients in F_{0}. This is a F_{0}-vector space for which there is a canonical comparison isomorphism

(102)  H_{log−cris}^{1}(X̃) ⊗_{F_{0}} F ≃ H_{dR}^{1}(X).

(iii) A σ_{0}-semilinear Frobenius operator φ on H_{log−cris}^{1}(X̃) arising from the Frobenius on k_{F}.

(iv) A F_{0}-linear monodromy map N : H_{log−cris}^{1}(X̃) ⟶ H_{log−cris}^{1}(X̃) which via (102) extends to a F -linear map

(103)  N : H_{dR}^{1}(X) ⟶ H_{dR}^{1}(X),

denoted with the same letter. It sends the class of ({ω_{v}}_{v}, {f_{e}}_{e}) to the class of ({0}_{v}, {λ_{e}}_{e}) with λ_{e} = ∂_{W_{e}}ω_{s(e)}. Here, ∂_{W_{e}} : H_{rig}^{1}(W_{e}) ⟶ H_{rig}^{0}(W_{e}) denotes the p-adic annular residue on W_{e}, taken relative to an orientation on W_{e} which is determined by e.

(v) A section

(104)  s : H_{dR}^{1}(X) ⟶ ⊕_{e}H_{rig}^{0}(W_{e})^{−}/δ (⊕_{v}H_{rig}^{0}(W_{v}))

of the map i in the exact sequence (101), defined by the rule

s({ω_{v}}_{v}, {f_{e}}_{e}) := {λ_{e}}_{e}, λ_{e} := f_{e} + F_{ω_{s(e)}} − F_{ω_{t(e)}},

where F_{ω_{v}} denotes the Coleman primitive of the differential ω_{v}. This primitive is a canonically defined up to addition of a single constant of integration. It is analytic on each residue disk in A_{v} and its restriction to the annulus W_{e} (for e any edge having v as either source or target) is generated by a single logarithmic function over the ring of rigid analytic functions on W_{e}.

Set d = [F_{0} : ℚ_{p}]. Since Φ := φ^{d} is F_{0}-linear, it extends via the comparison isomorphism (102) to a F -linear operator on H_{dR}^{1}(X) that is still written as Φ.

The action of Φ on H_{dR}^{1}(X) admits a direct description in terms of the exact sequence (101), in which the action of Φ on the left and right-hand terms is realized concretely by choosing a system Φ_{v} of characteristic 0 lifts of Frobenius to a system A_{v} ⊂ W_{v}[ϵ) ⊂ W_{v} of Frobenius neighbourhoods of the underlying affinoids A_{v} in W_{v}, indexed by a real parameter 0 < ϵ < 1. While Φ_{v} is only defined on W_{v}[ϵ) for suitable ϵ > 0 and does not preserve this domain, the map ω_{v} ↦ Φ^{∗}ω_{v} still induces a well-defined transformation

(105)  [figure: Φ : \frac{Ω^{1}(W_{v})}{dO_{Wv}}; ^{res^{Wv [ϵ)}}; \frac{Ω^{1}(W_{v}[ϵ))}{dO_{Wv [ϵ)}}; Φ^{∗}; \frac{Ω^{1}(W_{v}[ϵ′))}{dO_{Wv [ϵ′)}}; ^{res{−1 \atop Wv [ϵ′)}}; \frac{Ω^{1}(W_{v})}{dO_{Wv}}]

where res_{W_{v}[ϵ)} is the isomorphism induced by restriction to the wide open W_{v}[ϵ). This action of Frobenius on the cohomology does not depend on the choice of characteristic zero liftings {Φ_{v}}. Following the definitions in [CI99], the action of Φ then extends to the full H_{dR}^{1}(X) via the requirement that Φ be compatible with the section s of (104).

Definition 3.1. A class in H_{dR}^{1}(X) is said to be pure if it is in the kernel of both N and s.

The set of pure classes, denoted H_{dR}^{1}(X)_{pure}, is a subgroup of H_{dR}^{1}(X) which is preserved by the action of Frobenius and is identified via the map r of (101) with

(106)  H_{dR}^{1}(X)_{pure} = ⨁_{v} \frac{Ω_{sk}^{1}(W_{v})}{dO_{W_{v}}},

where Ω_{sk}^{1}(W_{v}) ⊂ Ω^{1}(W_{v}) denotes the space of rigid differentials on W_{v} of the second kind, i.e., have vanishing annular residues. This identification is compatible with the action of Φ on both groups.

Proposition 3.2. The eigenvalues of Φ on H_{dR}^{1}(X)_{pure} have complex absolute value \sqrt{p}.

Proof. Equation (106) shows that H_{dR}^{1}(X)_{pure} is isomorphic as a Frobenius module to the crystalline cohomology of the disjoint union of the components X̃_{v} of the special fiber X̃. More precisely, there is an exact sequence

(107)  0 → H_{cris}^{1}(X̃_{v}) → H_{cris}^{1}(X̃_{v}^{sm}) → ⊕_{e}F (−1) →^{Σ} F (−1) → 0

where the comparison theorems allow us to identify

H_{cris}^{1}(X̃_{v}) ⊗_{F_{0}} F ≃ \frac{Ω_{sk}^{1}(W_{v})}{dO_{W_{v}}} and H_{cris}^{1}(X̃_{v}^{sm}) ⊗_{F_{0}} F ≃ \frac{Ω^{1}(W_{v})}{dO_{W_{v}}} = H_{rig}^{1}(W_{v}).

Since the components X̃_{v} are smooth and projective, the theorem follows from the Riemann hypothesis for curves. □

In light of Proposition 3.2, it would have been more appropriate to use the expression pure of weight 1 to designate the classes in H_{dR}^{1}(X)_{pure}. The vaguer but more concise terminology adopted in Definition 3.1 was preferred because the qualifier weight 1 is also used in the setting of modular forms with a different meaning. It follows from Proposition 3.2 that the exact sequence (107) has a natural Frobenius-equivariant splitting and hence the injective map r : H_{dR}^{1}(X)_{pure} ⟶ ⊕_{v}H_{rig}^{1}(W_{v}) admits a natural Frobenius-equivariant section

(108)  r^{ι} : ⊕_{v}H_{rig}^{1}(W_{v}) ⟶ H_{dR}^{1}(X)_{pure}.

The Poincaré duality on H_{dR}^{1}(X) descends to a perfect pairing on H_{dR}^{1}(X)_{pure}, which is described by the formula

(109)  ⟨{ω_{v}}, {η_{v}}⟩_{X} = ∑_{e∈E(G)} res_{W_{e}}(F_{e}η_{s(e)}),

where the sum is taken over all the oriented edges of G and F_{e} is an analytic primitive of the restriction to the annulus W_{e} of ω_{s(e)}, which exists because ω_{v} has vanishing annular residues for all v ∈ V(G), and is well-defined up to a constant. (The choice of constant in the local primitive F_{e} does not affect the term on the right in (109), since η_{v} also has vanishing residues for all v.)

3.2. An analytic recipe for the syntomic Abel-Jacobi map. The goal of this section is to give an analytic expression for the syntomic Abel-Jacobi map on the Chow group of codimension two cycles on the triple product of semistable curves, in terms of Coleman’s theory of p-adic integration of rigid differential forms. It will be convenient to place oneself in the abstract setting given by conditions (a)- (d) below. While ostensibly somewhat special, this setting will nonetheless suffice for the applications in this paper.

(a) The ambient three-fold is a product W = X_{1} × X_{2} × X_{3} of three semistable curves over F. For i = 1, 2, 3, write G_{i} for the dual graph of the special fiber of X_{i}, and let Φ_{i} = {Φ_{i,v}}_{v∈V(G_{i})} denote a system of liftings of Frobenius to each of the basic wide open spaces in the admissible covering of X_{i}(ℂ_{p}) described in the previous section. Denote by Φ_{23} := Φ_{2} ⊗ Φ_{3} the corresponding Frobenius endomorphism on H_{dR}^{1}(X_{2})_{pure} ⊗ H_{dR}^{1}(X_{3})_{pure} ⊂ H_{dR}^{2}(X_{2} × X_{3}). Our second assumption, concerning the cycle Δ itself, is:

(b) The cycle Δ ∈ CH^{2}(W) is obtained from the data of a semistable curve X, of three semistable morphisms π_{i} : X ⟶ X_{i} and of three correspondences ε_{i} ∈ Corr(X_{i}), by setting

Δ := (ε_{1}, ε_{2}, ε_{3})_{∗}(π_{1}, π_{2}, π_{3})_{∗}(X).

(c) The correspondences ε_{i} annihilate the H^{0} and H^{2} terms in the cohomology of X_{i}. By the same reasoning as in the proof of Proposition 1.4, this implies that Δ is null-homologous, i.e., that it belongs to CH^{2}(W)_{0}.

Let

(110)  AJ_{p} : CH^{2}(W)_{0} ⟶ (Fil^{2}H_{dR}^{3}(W))^{∨}

denote the p-adic syntomic Abel-Jacobi map, as discussed e.g. in [Be00, Introduction and §5], [BLZ, §3] and [Ne1, (3.7)]. It follows from [NeNi, Theorem B] that (110) is canonically identified with the composition of the étale Abel-Jacobi map with Bloch-Kato’s logarithm, as in (82).

Even under assumptions (a), (b), and (c), we fall short of giving a formula for the full image AJ_{p}(Δ) but only for its values on cohomology classes of the form

η_{1} ⊗ ω_{2} ⊗ ω_{3} ∈ H_{dR}^{1}(X_{1}) ⊗ Ω^{1}(X_{2}) ⊗ Ω^{1}(X_{3}) ⊂ Fil^{2}(H_{dR}^{3}(W)),

assuming that:

(d) η_{1}, ω_{2} and ω_{3} belong to the pure subspaces of H_{dR}^{1}(X_{1}), H_{dR}^{1}(X_{2}), and H_{dR}^{1}(X_{3}) respectively. Our recipe for describing AJ_{p}(Δ)(η_{1} ⊗ ω_{2} ⊗ ω_{3}) rests on the following lemma.

Lemma 3.3. There exists a polynomial P ∈ ℚ[X] satisfying:

(1) P (Φ_{23}) annihilates the class of ω_{2} ⊗ ω_{3} in H_{dR}^{2}(X_{2} × X_{3});

(2) P (Φ_{X}) acts invertibly on ker N ⊂ H_{dR}^{1}(X).

Proof. By Proposition 3.2, there is a polynomial P satisfying the first condition, all of whose roots have complex absolute value p. But any such P automatically satisfies the second condition, since Frobenius acts on the kernel of N with eigenvalues of complex absolute value either 1 or \sqrt{p}. □

Fix a polynomial P as in the above lemma. Since P (Φ_{23}) annihilates the class of ω_{2} ⊗ ω_{3} in H_{dR}^{2}(X_{2} × X_{3}), there is a system of rigid 1-forms

(111)  ρ := ρ_{P}^{v,w}(ω_{2}, ω_{3}) ∈ Ω^{1}(W_{v} × W_{w}),  v ∈ V(G_{2}), w ∈ V(G_{3})

satisfying

dρ_{P}^{v,w}(ω_{2}, ω_{3}) = P (Φ_{23}) ((ω_{2} ∧ ω_{3})_{|W_{v}×W_{w}}) .

The system ρ_{P}^{v,w}(ω_{2}, ω_{3}) of rigid 1-forms is well-defined, up to a system of closed 1-forms on each product of basic wide opens in X_{2} × X_{3}. One next defines a system {ξ̃_{P}^{v} (ω_{2}, ω_{3})}_{v∈V(G_{X})} of rigid 1-forms on each basic wide open of X by pulling back the differentials ρ_{P}^{v,w}(ω_{2}, ω_{3}) via the maps (π_{2}, π_{3}) : X ⟶ X_{2} × X_{3} and (ε_{2}, ε_{3})X_{2} × X_{3} ⟶ X_{2} × X_{3}:

{ξ̃_{P}^{v} (ω_{2}, ω_{3})}_{v∈V(G_{X})} = (π_{2}, π_{3})^{∗}(ε_{2}, ε_{3})^{∗} ({ρ_{P}^{v,w}(ω_{2}, ω_{3})}) .

Condition (c) combined with the Künneth formula for H_{rig}^{1}(X_{2} × X_{3}) implies that for each v ∈ V(G_{X}), the class of the differential ξ̃_{P}^{v} (ω_{2}, ω_{3}) in H_{rig}^{1}(W_{v}) does not depend on the choice of system ρ_{P}^{v,w}(ω_{2}, ω_{3}) satisfying (111). Hence the class of ξ̃_{P} (ω_{2}, ω_{3}) ∈ ⊕_{v∈V(G_{X})}H_{rig}^{1}(W_{v}) is independent of this choice. Let

ξ_{P} (ω_{2}, ω_{3}) := r^{ι}(ξ̃_{P} (ω_{2}, ω_{3})) ∈ H_{dR}^{1}(X)_{pure},

where r^{ι} is the Frobenius-equivariant section introduced in (108).

Applying the second condition in Lemma 3.3 shows that there is a unique class

ξ(ω_{2}, ω_{3}) ∈ H_{dR}^{1}(X)_{pure} satisfying P (Φ_{C})ξ(ω_{2}, ω_{3}) = ξ_{P} (ω_{2}, ω_{3}).

Theorem 3.4. For all Δ ∈ CH^{2}(W)_{0} and classes η_{1}, ω_{2} and ω_{3} satisfying conditions (a) − (d),

AJ_{p}(Δ)(η_{1} ⊗ ω_{2} ⊗ ω_{3}) = ⟨π_{1}^{∗}ε_{1}^{∗}(η_{1}), ξ(ω_{2}, ω_{3})⟩_{X}.

Proof. The proof of this formula closely follows the one given in [DR13, §3.3] for the Gross-Kudla-Schoen diagonal cycle on the cube of a modular curve with good reduction at p, which exploits Besser’s finite polynomial cohomology for smooth varietes over ℤ_{p} introduced in [Be00]. This theory has been extended recently to the setting of semistable varieties by Besser, Loeffler and Zerbes in [BLZ], building crucially on prior work of Nekovar and Niziol [NeNi]. Namely, [BLZ, Proposition 3.3] asserts in this case that

AJ_{p}(Δ)(η_{1} ⊗ ω_{2} ⊗ ω_{3}) = tr_{X} ((π_{1}, π_{2}, π_{3})^{∗}(ε_{1}, ε_{2}, ε_{3})^{∗}(η̃_{1} ⊗ ω̃_{2} ⊗ ω̃_{3})) where

• η̃_{1} ∈ H_{fp}^{1}(X_{1}, 0), ω̃_{2} ∈ H_{fp}^{1}(X_{2}, 1), ω̃_{3} ∈ H_{fp}^{1}(X_{3}, 1) are respectively lifts of η_{1}, ω_{2}, ω_{3} to P -syntomic cohomology in the sense of [BLZ, §2.2]. In order to lighten the notations, the polynomials implicit in the construction shall be omitted.

• (π_{1}, π_{2}, π_{3})^{∗}(ε_{1}, ε_{2}, ε_{3})^{∗}(η̃_{1} ⊗ ω̃_{2} ⊗ ω̃_{3}) ∈ H_{fp}^{3}(X , 2) stands for the pull-back to X of the tensor product of these classes in syntomic cohomology.

• tr_{X} : H_{fp}^{3}(X , 2) ⟶^{∼} F is the trace isomorphism of [BLZ, Definition 3.1].

Let ⟨ , ⟩_{fp} : H_{fp}^{1}(X , 0) × H_{fp}^{2}(X , 2) ⟶ F denote the cup-product described in [BLZ, §1.3, §2.4]. By the compatibility between the trace and cup-product in syntomic cohomology afforded by [BLZ, Theorem 2.20], the above quantity may be recast as

tr_{X} ((π_{1}, π_{2}, π_{3})^{∗}(ε_{1}, ε_{2}, ε_{3})^{∗}(η̃_{1} ⊗ ω̃_{2} ⊗ ω̃_{3})) = ⟨π_{1}^{∗}ε_{1}^{∗}(η̃_{1}), (π_{2}, π_{3})^{∗}(ε_{2}, ε_{3})^{∗}(ω̃_{2} ⊗ ω̃_{3})⟩_{fp}.

According to the explicit description of finite polynomial cohomology provided in [BLZ, Definition 2.4], the class ω̃_{2} ⊗ ω̃_{3} in H_{fp}^{2}(X^{2}, 2) may be represented by the pair (ρ, ω_{2} ⊗ ω_{3}) where ρ is as in (111), and hence the class (π_{2}, π_{3})^{∗}(ε_{2}, ε_{3})^{∗}(ω̃_{2} ⊗ω̃_{3}) in H_{fp}^{2}(X , 2) is represented by the image of ξ = ξ(ω_{2}, ω_{3}) under the natural morphism i : H_{dR}^{1}(X) ⟶ H_{fp}^{2}(X , 2), i(ξ) = P (Φ)(ξ). It follows that

⟨π_{1}^{∗}ε_{1}^{∗}(η̃_{1}), (π_{2}, π_{3})^{∗}(ε_{2}, ε_{3})^{∗}(ω̃_{2} ⊗ ω̃_{3})⟩_{fp} = ⟨π_{1}^{∗}ε_{1}^{∗}(η̃_{1}), i(ξ(ω_{2}, ω_{3}))⟩_{fp} = ⟨π_{1}^{∗}ε_{1}^{∗}(η_{1}), ξ(ω_{2}, ω_{3})⟩_{X},

where the second equality follows as in [Be00, (14)]. □

4. Bloch-Kato logarithms of weight two specializations

As was explained at the end of §2.4, the aim of this chapter is to show that the one-variable p-adic L-functions

(112)  ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆}) := ℒ_{p} ^{f_{α}}(˘\underline{f}^{∗}, ˘ \underline{g}, ˘ \underline{h})_{|Ω_{f\underline{gh}}}, ℒ_{p} ^{g_{α}}(f̆, \underline{ğ^{∗}h̆}) := ℒ_{p} ^{g_{α}}(˘\underline{f}, ˘ \underline{g}^{∗}, ˘ \underline{h})_{|Ω_{f\underline{gh}}}

arising as the restriction to Ω_{f\underline{gh}} of the Garrett-Hida p-adic L-functions discussed in the introduction can be recovered as the image of the Λ-adic classes κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) and κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) and under Perrin-Riou’s Λ-adic logarithm map. We shall do so by comparing their values at classical points (y, z) of weight 2. Corollary 2.11 asserts that the Bloch-Kato logarithm of the specialisation of these classes at such points may be computed as the value of the functional AJ_{p}(Δ_{s,s}^{∘}) at the classes ηf̆^{∗} ⊗ ωğ_{y} ⊗ ωh̆_{z} and ωf̆ ⊗ η_{}_{ğ_{y}^{∗}} ⊗ ωh̆_{z}, respectively.

Armed with Theorem 3.4, we can now proceed with this program, starting with an analysis of the above forms in the rigid analytic cohomology of the modular curves X_{0}(Np) and X_{s}.

4.1. The rigid geometry of modular curves. Let X_{1}(Np^{s}) be the proper, flat, regular model of X_{s} over ℤ_{p}[ζ_{s}] considered in [KM85, §12]. Its special fiber is the union of a finite number of reduced Igusa curves Ig_{j} over 𝔽_{p}, meeting (not always transversally) at their supersingular points. Among these irreducible components there are two which are isomorphic to the Igusa curve Ig(Np^{s}) representing the moduli problem (Γ_{1}(N), Ig(p^{s})) over 𝔽_{p} specified in [KM85, §12]. These are called the good components, and are labelled Ig_{∞} and Ig_{0}. More precisely, Ig_{∞} classifies ordinary elliptic curves with Γ_{1}(N)-level structure and a chosen section of order p^{s} in its canonical subgroup of order p^{s}.

Let F/ℚ_{p}(ζ_{s}) be a finite extension over which X_{s} acquires semistable reduction, with valuation ring O_{F} and residue field k_{F}. Let π : X_{s} → X_{1}(Np^{s}) × O_{F} be a semistable model of X_{s} over O_{F}, obtained by applying sufficiently many blow-ups to the base change of X_{1}(Np^{s}) to O_{F}. The map π is birational and induces an isomorphism between the generic fibers and also between two of the components of the closed fibre of X_{s} with Ig_{∞} × k_{F} and Ig_{0} × k_{F}, respectively. By an abuse of notation, continue to denote Ig_{∞} and Ig_{0} the two components of the closed fibre of X_{s} corresponding to the two latter. Let

red : X_{s}(F) ⟶ X̃_{s}(k_{F})

denote the “mod p” reduction map. The inverse images of the irreducible components of the special fiber of X_{s} give a finite collection of wide open subsets of X_{s}(F) which form an admissible cover of this rigid analytic space, as in the discussion in §3.1. Let W O(X_{s}) denote the collection of wide open spaces in this admissible cover.

Denote by W_{∞}(p^{s}) and W_{0}(p^{s}) the wide open spaces in W O(X_{s}) arising as the inverse image of Ig_{∞} and Ig_{0}, respectively, under the reduction map. The inverse image of the smooth locus of Ig_{∞}, denoted A_{∞}(p^{s}), is the underlying affinoid of W_{∞}(p^{s}) and an analogous definition is made for A_{0}(p^{s}).

The F -valued points of the rigid analytic space A_{∞}(p^{s}) are in bijection with triples (A, i_{N}, i_{p}) where A is an ordinary (generalised) elliptic curve over O_{F}, i_{N} : μ_{N} ⟶ A[N] is an injective map of group schemes over O_{F}, and i_{p} : μ_{p^{s}} ⟶ A{p^{s}} is an isomorphism between μ_{p^{s}} and the canonical subgroup A{p^{s}} of A of order p^{s}. As explained in [Co97] and [BrEm10, Lemme 4.4.1], the F -vector spaces

H_{rig}^{1}(W_{∞}(p^{s})) = \frac{Ω_{rig}^{1}(W_{∞}(p^{s}))}{dO_{W_{∞}(p^{s})}}, H_{rig}^{1}(W_{0}(p^{s})) = \frac{Ω_{rig}^{1}(W_{0}(p^{s}))}{dO_{W_{0}(p^{s})}}

are equipped with a natural action of Hecke operators T_{ℓ} for ℓ ∤ Np. As already explained in (105), they are also equipped with natural F -linear Frobenius endomorphisms, which are defined by choosing characteristic zero lifts Φ_{∞} and Φ_{0} of the Frobenius endomorphism in characteristic p to a system of wide open neighbourhoods of the affinoids A_{∞}(p^{s}) and A_{0}(p^{s}) in W_{∞}(p^{s}) and W_{0}(p^{s}), respectively. In the setting at hand, a natural choice of Φ_{∞} is given by the Deligne-Tate mapping defined on triples (A, i_{N}, i_{p}) as above by the rule

Φ_{∞}(A, i_{N}, i_{p}) = (A/A{p}, π_{{p}} ∘ i_{N}, ĩ_{p}),

where (1) A{p} is the canonical subgroup of A of order p;

(2) π_{{p}} is the natural isogeny A ⟶ A/A{p};

(3) ĩ_{p} : μ_{p^{s}} ⟶ A sends ζ_{s} to π_{{p}}(P_{s+1}) where P_{s+1} ∈ A{p^{s+1}} satisfies pP_{s+1} = i_{p}(ζ_{s}).

Let (𝔾_{m}/q^{ℤ}, i_{N}, i_{p}) denote the Tate curve, where i_{N} and i_{p} are induced by the natural embeddings μ_{N}, μ_{p^{s}} ↪ 𝔾_{m}. Then Φ_{∞} acts on it by the rule Φ_{∞}(𝔾_{m}/q^{ℤ}, i_{N}, i_{p}) = (𝔾_{m}/q^{pℤ}, i_{N}^{p}, i_{p}), and therefore

(113)  Φ_{∞} = p⟨p; 1⟩V

where V is the operator on p-adic modular forms whose action is given by V (∑_{n=1}^{∞} a_{n}q^{n}) = ∑_{n} a_{n}q^{np} on q-expansions. To define Φ_{0}, recall the Atkin-Lehner automorphism w_{s} attached to the choice of primitive p^{s}-th root of unity ζ_{s}. It interchanges the wide opens W_{∞}(p^{s}) and W_{0}(p^{s}), and one defines

(114)  Φ_{0} := w_{s}^{−1}Φ_{∞}w_{s} = ⟨p^{−1}; 1⟩w_{s}Φ_{∞}w_{s}.

For each W ∈ W O(X_{s}), recall that

res_{W} : H_{dR}^{1}(X_{1}(Np^{s})) ⟶ H_{rig}^{1}(W)

is the the map induced by restriction to W, and write res_{0} and res_{∞} for res_{W_{0}(p^{s})} and res_{W_{∞}(p^{s})}.

Let H_{dR}^{1}(X_{s}/F)^{prim} denote the subspace of the de Rham cohomology of X_{s} associated to the primitive subspace S_{2}(Np^{s})_{F}^{prim} in the sense of [MW84, §3].

Theorem 4.1. [Co97], [BrEm10] The restriction maps res_{∞} and res_{0} induce an isomorphism

H_{dR}^{1}(X_{s/F})^{prim} ≃ H_{rig}^{1}(W_{∞}(p^{s}))^{pure} ⊕ H_{rig}^{1}(W_{0}(p^{s}))^{pure},

where H_{rig}^{1}(W_{∞}(p^{s}))^{pure} denotes the subspace of the de Rham cohomology which is pure of weight one, i.e., the space generated by classes of rigid differentials with vanishing annular residues. The restriction morphism is equivariant with respect to

• the Hecke operators T_{ℓ} for ℓ ∤ Np acting on both sides;

• the Frobenius endomorphism Φ acting on H_{dR}^{1}(X_{s/F});

• the Frobenius endomorphism (Φ_{∞}, Φ_{0}) acting on H_{rig}^{1}(W_{∞}(p^{s})) ⊕ H_{rig}^{1}(W_{0}(p^{s})).

A similar but simpler analysis also holds for the modular curve X_{0}(Np), since it admits a proper regular model over ℤ_{p}, whose special fiber is the union of two irreducible components, each isomorphic to the special fiber of the smooth integral model of X_{0}(N). Write {W_{∞}, W_{0}} for the standard admissible covering of X_{0}(Np) by wide open neighbourhoods of the Hasse domain of X_{0}(N). As above, the group H_{dR}^{1}(X_{0}(Np)) is again endowed with the action of a Frobenius map Φ and the Hecke operator U_{p}, which mutually commute (by functoriality of log-crystalline cohomology with respect to correspondences).

The ordinary unit root subspace

H_{dR}^{1}(X_{0}(Np)_{/ℚ_{p}})^{ord,ur} ⊂ e_{ord}H_{dR}^{1}(X_{0}(Np)_{/ℚ_{p}})

is defined to be the subspace spanned by the eigenvectors of Φ whose eigenvalue is a p-adic unit.

Lemma 4.2. The map H_{dR}^{1}(X_{0}(Np)_{/F})^{ord,ur} ⟶ H_{rig}^{1}(W_{∞}) induced by restriction is the zero map, and for any eigenform ϕ of weight 2 and level dividing N the map

(115)  H_{dR}^{1}(X_{0}(Np)_{/F})^{ord,ur}[ϕ] ⟶ H_{rig}^{1}(W_{0})^{ur}[f] ≃ H_{dR}^{1}(X_{0}(N)_{/F})^{ur}[ϕ]

is an isomorphism.

Proof. This follows from noting that the action of Φ = Φ_{∞} and U on H_{rig}^{1}(W_{∞}) are related by the rule Φ_{∞} = ⟨p⟩pV = ⟨p⟩pU^{−1}. □

4.2. Test vectors and their associated de Rham cohomology classes. As was explained in the introduction and recalled above, the Λ-adic cohomology classes and the Garrett-Hida p-adic Lfunctions considered in this work depend on a choice of test vectors f̆, ˘ \underline{g}, ˘ \underline{h} and dual test vectors f̆^{∗}, ˘ \underline{g}^{∗}, ˘\underline{h}^{∗} that were fixed in (92) and (93). We focus first on the former, and analyze the regular differential forms associated to f̆ and the weight two specialisations of ˘ \underline{g} and ˘ \underline{h}.

Assume that f̆ ∈ S_{2}(Np)[f] arises as the ordinary p-stabilisation of some test vector in S_{2}(N)[f], and recall that ωf̆ ∈ Ω^{1}(X_{0}(Np)) denotes the regular differential form asociated to f̆. Recall also

the maps \xymatrix{X_{s} \ar[r]^{ϖ_{1}^{s−1}} & X_{1} \ar[r]^{j_{1}} & X_{0}(Np)} that arise in the equation (100) describing the cycle Δ_{s,s}^{∘}, and define

(116)  ω_{f̆,s} := (ϖ_{1}^{s−1})^{∗} ∘ j_{1}^{∗}(ωf̆),

to be the pull-back of ωf̆ to X_{s} via these projections.

Let (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} be a classical point of weight 2 and character ϵ ≠ ω of primitive conductor p^{s}. In order to lighten notations, set g = g_{y}, h = h_{z} and

ğ = ğ_{y} ∈ S_{2}(Np^{s}, χϵω^{−1}),  h̆ = h̆_{z} ∈ S_{2}(Np^{s}, χ^{−1}ϵω^{−1}).

These notations are in force only in Chapter 4, and g and h will revert to denoting modular forms of weight 1 in subsequent chapters, as was done in the introduction. Let h^{ι} ∈ S_{2}(N_{h}p^{s}, χ^{−1}ωϵ^{−1}) be the newform whose q-expansion coefficients are determined by

a_{m}(h^{ι}) = \{ \begin{matrix}ωϵ^{−1}(m) · a_{m}(h) & if p ∤ m; \\ χ^{−1}(p)\overline{a}_{p}(h)^{r} & if m = p^{r}. \end{matrix}

If h̆ = ∑_{d|N/N_{h}} λ_{d} · h(q^{d}), set also

h̆^{ι} := ∑_{d|N/N_{h}} λ_{d}h^{ι}(q^{d}) ∈ S_{2}(Np^{s}, χ^{−1}ωϵ^{−1}),

so that for all n with p ∤ n, the fourier coefficient a_{n}(h̆^{ι}) is given by a_{n}(h̆^{ι}) := (ωϵ^{−1})(n) · a_{n}(˘\underline{h}).

For any Dirichlet character α, let 𝔤(α) ∈ ℚ̄_{p} denote the Gauss sum attached to α, on which G_{ℚ_{p}} acts through this character.

Proposition 4.3. The identity w_{s}(h̆) = 𝔤(ω^{−1}ϵ)α_{h}^{−s}h̆^{ι} holds in S_{2}(N_{h}p^{s}, χ^{−1}ωϵ^{−1}).

Proof. This follows from [AL78, Theorem 2.1] (with k = 2 and Q = p^{s}), taking into account that w_{s} commutes with the level-raising operators at the divisors of N/N_{h}, which are prime to p. □

For any regular differential form ω ∈ Ω^{1}(X_{s/F}), let ω_{∞} and ω_{0} denote its restrictions to W_{∞}(p^{s}) and W_{0}(p^{s}) respectively, and write [ω_{∞}], [ω_{0}] for their classes in cohomology.

Proposition 4.4. For ω = ω_{f̆,s}, ωğ and ωh̆,

(117)  Φ_{∞}[ω_{∞}] = β · [ω_{∞}],  Φ_{0}[ω_{0}] = α · [ω_{0}],

where (α, β) = (α_{f}, β_{f}), (a_{p}(g), χ(p)\overline{a_{p}(g)}) and (a_{p}(h), χ^{−1}(p)\overline{a_{p}(h)}) respectively.

Proof. Let us prove the claim for ω = ωğ, as the other cases follow similarly. Together with V, let U = U_{p} be the usual operator on p-adic modular forms defined on q-expansions as in (29). A direct calculation shows that

U(ğ) = a_{p}(g)ğ, UV = V U = 1 on H_{rig}^{1}(W_{∞}(p^{s})),

hence V ([ω_{ğ,∞}]) = a_{p}(g)^{−1} · [ω_{ğ,∞}]. The first equality in (117) now follows from the fact that the canonical lift Φ_{∞} of Frobenius agrees with p⟨p; 1⟩V, as stated in (113), and that a_{p}(g) is a Weil number of weight one, i.e., a complex number of absolute value \sqrt{p}. To study the action of Φ_{0} on ω_{ğ,0}, note that

Φ_{0}(ω_{ğ,0}) = w_{s}^{−1}Φ_{∞}w_{s}(ω_{ğ,0}) = w_{s}^{−1}β_{g^{ι}}w_{s}ω_{ğ,0},

where the first equality follows from (114) and the second from Proposition 4.3. The second equality in (117) now follows from the fact that β_{g^{ι}} = a_{p}(g) = α_{g}. □

If ϕ(q) is any overconvergent p-adic modular form, its p-depletion is defined to be the overconvergent modular form whose q-expansion is given by

ϕ^{[p]}(q) := (1 − V U)ϕ(q) = ∑_{p∤n} a_{n}(ϕ)q^{n},

and ω_{ϕ}^{[p]} denotes the associated rigid differential on a wide open neighbourhood of A_{∞}(p^{s}) in W_{∞}(p^{s}).

Corollary 4.5. (1) The class of the rigid differential ω_{f̆,∞}^{[p]} = (1 − β_{f}^{−1}Φ_{∞})(ω_{f̆,∞}) is trivial in

H_{rig}^{1}(W_{∞}), i.e., ω_{f̆,∞}^{[p]} = dF̆ for some F̆ ∈ O_{W_{∞}}. The q-expansion of F̆ is

F̆(q) = ∑_{p∤n} \frac{a_{n}(f̆)}{n}q^{n}.

(2) The class of the rigid differential ω_{ğ,∞}^{[p]} = (1 − β_{g}^{−1}Φ_{∞})(ω_{ğ,∞}) is trivial in H_{rig}^{1}(W_{∞}(p^{s})), i.e.,

ω_{ğ,∞}^{[p]} = dĞ for some Ğ_{∞} ∈ O_{W_{∞}(p^{s})}. The q-expansion of G is given by

Ğ(q) = ∑_{p∤n} \frac{a_{n}(ğ)}{n}q^{n}.

(3) The class of the rigid differential ω_{h̆,0}^{[p]} = (1 − α_{h}^{−1}Φ_{0})(ω_{h̆,0}) is trivial in H_{rig}^{1}(W_{0}(p^{s})), i.e.,

ω_{h̆,0}^{[p]} = dH̆^{∘} for some H̆∘ ∈ O_{W_{0}(p^{s})}. The q-expansion of w_{s}H̆^{∘} is given by
(w_{s}H̆^{∘})(q) = 𝔤(ω^{−1}ϵ)α_{h}^{−s}H^{ι}(q),  H^{ι}(q) = ∑_{p∤n} \frac{a_{n}(h̆^{ι})}{n}q^{n},  a_{n}(h̆^{ι}) = (ωϵ^{−1})(n)a_{n}(h̆).

Proof. The triviality statements follow from Prop. 4.4 and the q-expansion of the primitives in (1) and (2) are obtained by a direct calculation. A similar reasoning, using Prop. 4.3, implies (3). □

We turn now to analyze the dual test vectors and their associated de Rham cohomology classes. Given a dual test vector f̆^{∗} ∈ S_{2}^{ord}(Γ_{0}(Np))^{∨}[f], recall that ηf̆^{∗} ∈ H_{dR}^{1}(X_{0}(Np))^{ord,ur}[f] denotes the unique class satisfying

(118)  Φ(ηf̆^{∗}) = α_{f} · ηf̆^{∗}, and ⟨ηf̆^{∗}, w_{1}ω⟩_{X_{0}(Np)} = f̆^{∗}(ω) ∀ω ∈ S_{2}^{ord}(Γ_{0}(Np)).

As in equation (116), define

(119)  η_{f̆^{∗},s} := (ϖ_{1}^{s−1})^{∗} ∘ j_{1}^{∗}(ηf̆^{∗}).

Likewise, set

ğ^{∗} = ğ_{y}^{∗} ∈ S_{2}(Np^{s}, χ^{−1}ϵω^{−1})^{∨}, h̆^{∗} = h̆_{z}^{∗} ∈ S_{2}(Np^{s}, χϵω^{−1})^{∨},

and let η_{}_{ğ^{∗}} and ηh̆^{∗} ∈ H_{dR}^{1}(X_{s/F}) be the de Rham cohomology classes associated to them as in (80). Note that

(120)  Φ(η) = α · η and U_{p}(η) = α · η

for η = η_{f̆^{∗},s}, η_{}_{ğ^{∗}} and ηh̆^{∗}, where α = α_{f}, α_{g} and α_{h}, respectively.

Lemma 4.6. The classes η_{}_{ğ^{∗}} and ηh̆^{∗} are supported on W_{0}(p^{s}).

Proof. Arguing as in Lemma 4.2 and taking (120) into account, it follows that the restriction of η_{}_{ğ^{∗}} and ηh̆^{∗} to W_{∞}(p^{s}) vanish, and hence they are supported on W_{0}(p^{s}) by Theorem 4.1. □

Remark 4.7. Lemma 4.6 is consistent with the fact that the value of ⟨η_{}_{ğ^{∗}}, w_{s}ω⟩_{s} depends only on the restriction of the ordinary class ω to W_{∞}(p^{s}), i.e., on the overconvergent modular form attached to ω.

4.3. Twisted diagonal cycles under the syntomic Abel-Jacobi map. We are now in a position to evaluate the image of the twisted diagonal cycles under the syntomic Abel-Jacobi map.

4.3.1. Evaluation of AJ_{p}(Δ_{s,s}^{∘}) at ηf̆^{∗} ⊗ ωğ ⊗ ωh̆.

Lemma 4.8. The data (Δ, η_{1}, ω_{2}, ω_{3}) = (Δ_{s,s}^{∘}, ηf̆^{∗}, ωğ, ωh̆) satisfies conditions (a)-(d) of §3.2.

Proof. Condition (a) is clear, since W_{s,s} is a triple product of curves with semistable reduction over F. The cycle Δ_{s,s}^{∘} is described as in condition (b) of Section 3.2 by setting

X = X_{s},  (π_{1}, π_{2}, π_{3}) = (j_{1} ∘ ϖ_{1}^{s−1}, w_{s}, Id),  (ε_{1}, ε_{2}, ε_{3}) = (ε, ε_{s}, ε_{s}).

Condition (c) is built into the assumptions, while Condition (d) follows from the fact that the form ηf̆^{∗} belongs to the f-isotypic subspace attached to an eigenform of prime-to-p level, and ğ and h̆ are primitive at p. □

Let ξ(ωğ, ωh̆) ∈ H_{dR}^{1}(X_{s/F}) be the cohomology class attached to the data (Δ_{s,s}^{∘}, ωğ, ωh̆) following the recipe of §3.2. Then Theorem 3.4 applies and asserts that

(121)  AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = ⟨η_{f̆^{∗},s}, ξ(ωğ, ωh̆)⟩_{s}.

Lemma 4.9. The class ξ(ωğ, ωh̆) arises as the pull-back of a class in H_{dR}^{1}(X_{0}(Np^{s})_{/F}) via j_{s}.

Proof. Let

ρ_{P}^{v,w}(ωğ, ωh̆) ∈ Ω^{1}(W_{v} × W_{w}), v, w ∈ V(G(X̃_{s})),

be a system of rigid 1-forms satisfying (111). The operators (⟨a; b⟩, ⟨a; b^{−1}⟩) in D_{s} leave the system {ρ_{P}^{v,w}} invariant up to systems of closed 1-forms in Ω^{1}(W_{v} × W_{w}); this follows because D_{s} fixes the class ωğ ⊗ ωh̆ ∈ H_{dR}^{2}(X_{s} × X_{s}). It follows that the class

(122)  ξ_{P} (ωğ, ωh̆) ∈ H_{dR}^{1}(X_{s/F})

introduced in §3.2 is well-defined, and fixed by the diamond operators. Since these operators commute with Frobenius, the same holds for ξ(ωğ, ωh̆), and the lemma follows. □

The degeneracy maps π_{1}, π_{2} : X_{s}^{♭} ⟶ X_{s} that were introduced in (32) induce similar maps from X_{0}(Np^{s+1}) ⟶ X_{0}(Np^{s}) by passing to the quotient by the action of the group (ℤ/Np^{s})^{×} of diamond operators of level Np^{s} on both sides. Let us continue to denote these maps as π_{1} and π_{2}, by a slight abuse of notation. The class η_{f̆^{∗},s} likewise arises as the pull-back via j_{s} of a class on H_{dR}^{1}(X_{0}(Np^{s})_{/F}), which shall be denoted by the same symbol. (Any ambiguity arising from this double use of notation shall be avoided by consistently specifying the modular curve on which the Poincaré pairing being computed. Furthermore, the object η_{f̆^{∗},s} will play only a provisional role, disappearing after the proof of Lemma 4.12.) With these new notations,

(123)  η_{f̆^{∗},s} = (π_{1}^{s−1})^{∗}(ηf̆^{∗}),

as follows directly from (119). Since d_{s} = deg(j_{s}), the functoriality of Poincaré duality relative to pull-backs now allows us to rewrite (121) in terms of the Poincaré pairing on X_{0}(Np^{s}), namely as

(124)  AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s} · ⟨η_{f̆^{∗},s}, ξ(ωğ, ωh̆)⟩_{X_{0}(Np^{s})}.

Recall the ordinary projector e_{ord} of (30) that was already exploited in the discussion of Hida theory in §1.5 as well as its anti-ordinary counterpart e_{ord}^{∗}. Both will now be considered as endomorphisms of H_{dR}^{1}(X_{0}(Np^{s})_{/F}). Just like U_{p} and U_{p}^{∗}, these endomorphisms are adjoint to each other relative to Poincaré duality.

Lemma 4.10. For all s ≥ 1,

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s}⟨η_{f̆^{∗},s}, e_{ord}^{∗}ξ(ωğ, ωh̆)⟩_{X_{0}(Np^{s})}.

Proof. This follows from (124) in light of the fact that the class η_{f̆^{∗},s} is ordinary and hence equal to its image under the ordinary projector, so that

⟨η_{f̆^{∗},s}, ξ(ωğ, ωh̆)⟩_{X_{0}(Np^{s})} = ⟨e_{ord}η_{f̆^{∗},s}, ξ(ωğ, ωh̆)⟩_{X_{0}(Np^{s})} = ⟨η_{f̆^{∗},s}, e_{ord}^{∗}ξ(ωğ, ωh̆)⟩_{X_{0}(Np^{s})}.
□

Lemma 4.11. The pulback (π_{1}^{s−1})^{∗} induces an isomorphism

(π_{1}^{s−1})^{∗} : e_{ord}H_{dR}^{1}(X_{0}(Np)) ⟶ e_{ord}H_{dR}^{1}(X_{0}(Np^{s})).

Proof. The second displayed equation in the proof of Thm. 4.1. of [Em99] (after replacing s by 1 and r by s) asserts that the homomorphism

e_{ord}((Γ_{1}(Np) ∩ Γ_{0}(p^{s}))^{ab} ⊗ ℤ_{p}) ⟶ e_{ord}(Γ_{1}(Np)^{ab} ⊗ ℤ_{p})

induced from the natural inclusion Γ_{1}(Np)∩Γ_{0}(p^{s}) ⊂ Γ_{1}(Np) is an isomorphism. The abelianisation of a congruence subgroup (tensored with ℤ_{p}) is naturally identified with the homology with ℤ_{p}-coefficients of the associated open modular curve. Taking a further quotient by the subgroup generated by the parabolic elements and passing to ℤ_{p}-duals, one concludes that the natural map

e_{ord}H^{1}(X_{1}(Np), ℤ_{p}) ⟶ e_{ord}H^{1}(X_{1}(Np) ×_{X_{0}(Np)} X_{0}(Np^{s}), ℤ_{p})

is an isomorphism. This map agrees with (π_{1}^{s−1})^{∗} (viewed as a map on singular cohomology). Hence Lemma 4.11 follows after taking invariants under the action of the group (ℤ/Npℤ)^{×} of diamond operators acting on both sides, and invoking the comparison isomorphism between singular and de Rham cohomology. □

Applying the ordinary projector to ξ^{ι}(ωğ, ωh̆) := w_{s}ξ(ωğ, ωh̆), it follows from Lemma 4.11 that

(125)  e_{ord}ξ^{ι}(ωğ, ωh̆) = (π_{1}^{s−1})^{∗}(ξ^{ι}(ωğ, ωh̆)_{1}),

for a suitable class ξ^{ι}(ωğ, ωh̆)_{1} ∈ e_{ord}H_{dR}^{1}(X_{0}(Np)_{/F}). Let ξ^{ι}(ωğ, ωh̆)_{W_{∞}} denote the restriction of the class ξ^{ι}(ωğ, ωh̆)_{1} to the Hasse domain W_{∞} viewed as a p-adic modular form of weight two.

Lemma 4.12. For all s ≥ 1,

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s}α_{f}^{s−1}f̆^{∗}(ξ^{ι}(ωğ, ωh̆)_{W_{∞}}).

Proof. Since w_{s} = w_{s}^{−1} on X_{0}(Np^{s}), one has e_{ord}^{∗} = w_{s}e_{ord}w_{s}. Lemma 4.10 implies that

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s}⟨η_{f̆^{∗},s}, w_{s}e_{ord}ξ^{ι}(ωğ, ωh̆)⟩_{X_{0}(Np^{s})}.

It folllows from (125) that

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s}⟨η_{f̆^{∗},s}, w_{s}(π_{1}^{s−1})^{∗}ξ^{ι}(ωğ, ωh̆)_{1}⟩_{X_{0}(Np^{s})}
= d_{s}⟨π_{2∗}^{s−1}η_{f̆^{∗},s}, w_{1}^{∗}ξ^{ι}(ωğ, ωh̆)_{1}⟩_{X_{0}(Np)},

where the facts that w_{s}^{∗}(π_{1}^{s−1})^{∗} = (π_{2}^{s−1})^{∗}w_{1}^{∗} and that the adjoint of π_{2}^{∗} relative to Poincaré duality is π_{2∗} have been used to derive the last equation. By (123) and (33),

π_{2∗}^{s−1}η_{f̆^{∗},s} = π_{2∗}^{s−1}(π_{1}^{s−1})^{∗}ηf̆^{∗} = U_{p}^{s−1}ηf̆^{∗} = α_{f}^{s−1}ηf̆^{∗},

and therefore

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s}α_{f}^{s−1}⟨ηf̆^{∗}, w_{1}^{∗}ξ^{ι}(ωğ, ωh̆)_{1}⟩_{X_{0}(Np)}
= d_{s}α_{f}^{s−1}f̆^{∗}(ξ^{ι}(ωğ, ωh̆)_{W_{∞}}),

where the last equality follows from equation (118) defining ηf̆^{∗}. □

Since the pull-back π_{1}^{∗} induces the identity on q-expansions, Lemma 4.12 implies that

(126)  AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = d_{s}α_{f}^{s−1}f̆^{∗}(e_{ord}ξ^{ι}(ωğ, ωh̆)),

in light of (125). We now turn to the calculation of the ordinary p-adic modular form e_{ord}ξ^{ι}(ωğ, ωh̆). Let P ∈ ℂ_{p}[t] be a polynomial satisfying the conditions of Lemma 3.3, i.e. such that

(i) P (Φ × Φ) annihilates the class of ω_{ğ,∞} ⊗ ω_{h̆,0} in H_{rig}^{2}(W_{∞}(p^{s}) × W_{0}(p^{s})).

(ii) None of the roots of P (t) are of complex absolute value \sqrt{p}.

In light of Proposition 4.4, one may take P (x) := 2(1 − β_{g}^{−1}α_{h}^{−1}x) = 2(1 − p^{−1}χ^{−1}(p)α_{g}α_{h}^{−1}x). The polynomial identity

P (xy) = (1 − β_{g}^{−1}x)(1 + α_{h}^{−1}y) + (1 + β_{g}^{−1}x)(1 − α_{h}^{−1}y)

combined with Corollary 4.5 implies that

P (Φ_{2} × Φ_{3})(ω_{ğ,∞} ⊗ ω_{h̆,0}) = (dĞ) ⊗ (1 + α_{h}^{−1}Φ_{0})ω_{h̆,0} + (1 + β_{g}^{−1}Φ_{∞})ω_{ğ,∞} ⊗ (dH̆^{∘}).

It follows that the rigid analytic differential 1-form ϱ_{P} (ωğ, ωh̆) ∈ Ω^{1}(W_{∞}(p^{s}) × W_{0}(p^{s})) satisfying

dϱ_{P} (ωğ, ωh̆) = P (Φ_{2} × Φ_{3})(ω_{ğ,∞} ⊗ ω_{h̆,0})

can be chosen to be

(127)  ϱ_{P} (ωğ, ωh̆) = Ğ ⊗ (1 + α_{h}^{−1}Φ_{0})ω_{h̆,0} − (1 + β_{g}^{−1}Φ_{∞})ω_{ğ,∞} ⊗ H̆^{∘},

where Ğ ∈ O_{W_{∞}(p^{s})} and H̆^{∘} ∈ O_{W_{0}(p^{s})} are the primitives described in Corollary 4.5.

After setting ξ_{P}^{ι} (ωğ, ωh̆) := w_{s}ξ_{P} (ωğ, ωh̆) and observing that the correspondences ε_{s} fix the differentials ωğ, ωh̆ as well as their primitives Ğ and H̆^{∘}, it follows that

(128)  ξ_{P} (ωğ, ωh̆) = ((w_{s}^{−1}Ğ) × (1 + χ^{−1}(p)α_{h}^{−1}Φ_{∞})ω_{h̆,∞} − (1 + β_{g}^{−1}Φ_{∞})(w_{s}^{−1}ω_{ğ,∞}) × H̆^{∘}) ,
(129)  ξ_{P}^{ι} (ωğ, ωh̆) = (Ğ × (1 + χ^{−1}(p)α_{h}^{−1}Φ_{∞})(w_{s}ω_{h̆,∞}) − (1 + β_{g}^{−1}Φ_{∞})ω_{ğ,∞} × (w_{s}H̆^{∘})) .

By Corollary 4.5, the q-expansion of this p-adic modular form is given by

(130) ξ_{P}^{ι} (ωğ, ωh̆)(q) = 𝔤(ω^{−1}ϵ)α_{h}^{−s} (Ğ × (1 + χ^{−1}(p)α_{h}^{−1}Φ_{∞})h̆^{ι} − (1 + β_{g}^{−1}Φ_{∞})ğ_{∞} × H̆^{ι}) (q)\frac{dq}{q} .

This is a rigid differential on a wide open neighbourhood of A_{∞}(p^{s}) in W_{∞}(p^{s}), and hence corresponds to an overconvergent modular form of weight 2 on W_{∞}(p^{s}). Since the nebentypus character of ğ and Ğ is χω^{−1}ϵ, and that of h̆^{ι} and H̆^{ι} is χ^{−1}ωϵ^{−1}, it follows that ξ_{P}^{ι} (ωğ, ωh̆) arises as a pullback under the projection j_{s} of a rigid differential on W_{∞}.

Proposition 4.13. The class of e_{ord}(ξ_{P}^{ι} (ωğ, ωh̆)) in H_{rig}^{1}(W_{∞}) is equal to the class of the weight two p-adic modular form

−2𝔤(ω^{−1}ϵ) × α_{h}^{−s} × ğH̆^{ι} ∈ Ω^{1}(W_{∞}).

Proof. If f_{1} and f_{2} are overconvergent modular forms, then f_{1}^{[p]} × (V f_{2}) lies in the kernel of U and hence a fortiori in the kernel of e_{ord}. Since Φ_{∞} = ⟨p⟩pV, after applying e_{ord} to (130) one obtains

e_{ord}(ξ_{P}^{ι} (ωğ, ωh̆)) = 𝔤(ω^{−1}ϵ) × α_{h}^{−s} × e_{ord}(Ğ × h̆_{∞}^{ι} − ğ_{∞} × H̆^{ι})
(131)  = 𝔤(ω^{−1}ϵ) × α_{h}^{−s} × e_{ord}(Ğ × h̆_{∞}^{ι[p]} − ğ_{∞}^{[p]} × H̆^{ι}).

Applying e_{ord} to the identity d(ĞH̆^{ι}) = Ğ × h̆_{∞}^{ι[p]} + ğ_{∞}^{[p]} × H̆^{ι}, and using the fact that the image of d on overconvergent forms is in the kernel of e_{ord}, it follows that

(132)  e_{ord}(Ğ × h̆_{∞}^{ι[p]}) = −e_{ord}(ğ_{∞}^{[p]} × H̆^{ι}),

and the result follows from (132) combined with (131). □

We can now state the main formula concerning the value of AJ_{p}(Δ_{s,s}^{∘}) at the vector ηf̆^{∗} ⊗ ωğ ⊗ ωh̆:

Theorem 4.14. Setting E_{f}(f, g, h) := −2(1 − χ^{−1}(p)α_{f}^{−1}α_{g}α_{h}^{−1})^{−1}, one has

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = E_{f}(f, g, h) × 𝔤(ω^{−1}ϵ) × α_{f}^{s−1}α_{h}^{−s} × d_{s} × f̆^{∗}(ğH̆^{ι}).

Proof. By combining (126) with Proposition 4.13,

AJ_{p}(Δ_{s,s}^{∘})(ηf̆^{∗} ⊗ ωğ ⊗ ωh̆) = −2 × 𝔤(ω^{−1}ϵ) × α_{f}^{s−1}α_{h}^{−s} × d_{s} × f̆^{∗}(P (Φ_{∞})^{−1}(ğH̆^{ι})).

But for all classes ξ ∈ H_{rig}^{1}(W_{∞}), f̆^{∗}(Φξ) = pα_{f}^{−1}f̆^{∗}(ξ) = β_{f} f̆^{∗}(ξ). More generally, f̆^{∗}(Q(Φ)ξ) = Q(β_{f})f̆^{∗}(ξ), for all rational functions Q with no pole at β_{f}. The result follows. □

4.3.2. Evaluation of AJ_{p}(Δ_{s,s}^{∘}) at ωf̆ ⊗ η_{}_{ğ^{∗}} ⊗ ωh̆. Thanks to Lemma 4.8, the data (Δ_{s,s}, ωf̆, η_{}_{ğ^{∗}}, ωh̆) satisfies conditions (a)-(d) of §3.2. The class w_{s}^{−1}η_{}_{ğ^{∗}} is anti-ordinary and supported on W_{∞}(p^{s}) by Lemma 4.6. Theorem 3.4 therefore applies and asserts that

(133)  AJ_{p}(Δ_{s,s})(ωf̆ ⊗ η_{}_{ğ^{∗}} ⊗ ωh̆) = ⟨w_{s}^{−1}η_{}_{ğ^{∗}}, ξ(ωf̆, ωh̆)⟩_{s} = ⟨η_{}_{ğ^{∗}}, w_{s}e_{ord}(ξ(ωf̆, ωh̆))⟩_{s}

where ξ(ωf̆, ωh̆) ∈ H_{dR}^{1}(X_{s/F}) is the class associated to the triple (Δ_{s,s}, ωf̆, ωh̆) as in §3.2 and the second equality follows as in Lemma 4.10.

In order to compute the class ξ(ωf̆, ωh̆) explicitly, consider the polynomial P (x) := 2(1 − β_{f}^{−1}β_{h}^{−1}x). It satisfies the conditions of Lemma 3.3, since the endomorphism P (Φ_{1} × Φ_{3}) annihilates the class of ω_{f̆,∞} ⊗ ω_{h̆,∞} in H_{rig}^{2}(W_{∞} × W_{∞}(p^{s})) by Proposition 4.4 and (120). The polynomial identity

P (xy) = (1 − β_{f}^{−1}x)(1 + β_{h}^{−1}y) + (1 + β_{f}^{−1}x)(1 − β_{h}^{−1}y)

combined with Corollary 4.5 implies that

P (Φ_{1} × Φ_{3})(ω_{f̆,∞} ⊗ ω_{h̆,∞}) = ω_{f̆,∞}^{[p]} ⊗ (1 + β_{h}^{−1}Φ_{∞})ω_{h̆,∞} + (1 + β_{f}^{−1}Φ_{∞})ω_{f̆,∞} ⊗ ω_{h̆,∞}^{[p]}.

It follows that the form ϱ_{P} (ωf̆, ωh̆) ∈ Ω^{1}(W_{∞} × W_{∞}(p^{s})) satisfying (111) can be chosen to be

(134)  ϱ_{P} (ωf̆, ωh̆) = F̆ ⊗ (1 + β_{h}^{−1}Φ_{∞})ω_{h̆,∞} − (1 + β_{f}^{−1}Φ_{∞})ω_{f̆,∞} ⊗ H̆,

where H̆ ∈ O_{W_{∞}(p^{s})} is the rigid analytic primitive of ω_{h̆,∞}^{[p]} following the definitions of Corollary 4.5 (with g and G replaced by h and H). By the analogous calculation that led to (128), the pull-back ξ_{P} (ωf̆, ωh̆) of ϱ_{P} (ωf̆, ωh̆) under the correspondence

(ε, ε_{s})(j_{1} ∘ ϖ_{1}^{s−1}, Id) : X_{0}(Np) × X_{s} ⟶ X_{0}(Np) × X_{s}

arising in the definition of the cycle Δ_{s,s}^{∘} is given by

(135)  ξ_{P} (ωf̆, ωh̆) = F̆ × (1 + χ(p)α_{h}Φ_{∞})ω_{h̆,∞} − (1 + β_{f}^{−1}Φ_{∞})ω_{f̆,s,∞} × H̆,

where the fact that ϖ_{1}^{∗} and j_{1}^{∗} induce the identity on q-expansions has been used to show that F̆ pulls back to the same overconvergent modular form (viewed this time as a rigid differential on O_{W_{∞}(p^{s})}). In this calculation, copious use has been made of the fact that the correspondence ε was chosen so as to leave ωf̆ and F̆ invariant, and likewise that the correspondence ε_{s} fixes ωh̆ and H̆.

Arguing exactly as in the proof of Proposition 4.13, it follows that

(136)  e_{ord}ξ_{P} (ωf̆, ωh̆)_{W_{∞}} = 2F̆ · ω_{h̆,∞} and hence e_{ord}ξ(ωf̆, ωh̆)_{W_{∞}} = 2P (Φ_{∞})^{−1}(F̆ · ω_{h̆,∞}).

Theorem 4.15. Let E_{g}(f, g, h) := 2(1 − χ^{−1}(p)β_{f}^{−1}β_{g}β_{h}^{−1})^{−1} = 2(1 − χ(p)p^{−1}α_{f}α_{g}^{−1}α_{h})^{−1}. Then

AJ_{p}(Δ_{s,s}^{∘})(ωf̆ ⊗ η_{}_{ğ^{∗}}ωh̆) = E_{g}(f, g, h) × ğ^{∗}(F̆ × h̆).

Proof. Combining (133) and (136), and taking into account that the adjoint of w_{s}^{−1} with respect to ⟨ , ⟩_{s} is w_{s}, it follows that

AJ_{p}(Δ_{s,s}^{∘})(ωf̆ ⊗ η_{}_{ğ^{∗}}ωh̆) = 2⟨η_{}_{ğ^{∗}}, w_{s}P (Φ_{∞})^{−1}(F̆ · ω_{h̆,∞})⟩_{s} = E_{g}(f, g, h)⟨η_{}_{ğ^{∗}}, w_{s}(F̆ · ω_{h̆,∞})⟩_{s}.

The theorem is then a consequence of the definition of η_{}_{ğ^{∗}}. □

4.4. Garrett-Hida p-adic L-functions. We now come to the main results of this chapter, expressing the Bloch-Kato logarithms of the specialisation of the classes κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) and κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) at weight 2 points in terms of the values of the one-variable Garret-Hida p-adic L-functions introduced in (112).

Theorem 4.16. For all arithmetic points x = (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} of weight (2, ϵ),

log_{p}(κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z})) = E_{f}(f, g_{y}, h_{z}) × 𝔤(ω^{−1}ϵ) × α_{f}^{s−1}α_{g_{y}}^{−s}α_{h_{z}}^{−s} × ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆})(y, z).

Proof. Combining Corollary 2.11 with Theorem 4.14 leads to the equality

(137)  log_{p}(κ_{p}^{f_{α}}(f̆^{∗}, ğ_{y}, h̆_{z})) = E_{f}(f, g_{y}, h_{z}) × 𝔤(ω^{−1}ϵ) × α_{f}^{s−1}α_{g_{y}}^{−s}α_{h_{z}}^{−s} × f̆^{∗}(ğ_{y} × d^{−1}h̆_{z}^{ι}).

The formal q-expansion with coefficients in Λ_{h} which is given by ˘ \underline{H}^{ι}(q) := ∑_{p∤n}⟨n^{−1}⟩a_{n}(h̆)q^{n}, where ⟨n⟩ ∈ Λ denotes the group-like element associted to n ∈ ℤ_{p}^{×}, satisfies the following properties:

(i) For any arithmetic point z ∈ Ω_{h} of weight-character (2, ϵ) as above, ˘ \underline{H}_{z}^{ι} = d^{−1}h̆_{z}^{ι}.

(ii) For any point z ∈ Ω\underline{h} of weight k ≥ 2 with k ≡ 1 (mod p − 1) and trivial nebentypus character at p, ˘ \underline{H}_{z}^{ι} = d^{1−k}h̆_{z}. In particular, it is an overconvergent modular form of weight 2 − k.

If ˘ \underline{g} is any Λ-adic modular form of level N and character χ, then the product ˘ \underline{g} × ˘ \underline{H}^{ι} therefore specialises, at all points (y, z) with common weights (k, ϵ), to a p-adic overconvergent modular form of weight 2 on Γ_{0}(Np). Since ordinary overconvergent modular forms of weight 2 are classical, it follows that e_{ord}(˘\underline{g} × ˘ \underline{H}^{ι}) belongs to the space S_{2}(Γ_{0}(Np)) ⊗ Λ_{gh} of classical modular forms tensored with the ring Λ_{gh} = Λ_{g} ⊗_{Λ} Λ_{h}. Property (ii) together with the interpolation property (6) implies that f̆^{∗}(˘\underline{g} ˘\underline{H}^{ι}) = ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆}), since these two elements of Λ_{f\underline{gh}} coincide on a dense set of points in Ω_{g} ×_{Ω} Ω_{h}. In particular, f̆^{∗}(ğ_{y} × d^{−1}h̆_{z}^{ι}) = ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆})(y, z) and the theorem follows from (137). □

Theorem 4.17. For all arithmetic points x = (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} of weight (2, ϵ),

log_{p}(κ_{p}^{g}(f̆^{∗}, ğ_{y}, h̆_{z})) = E_{g_{y}}(f, g_{y}, h_{z}) × ℒ_{p} ^{g_{α}}(f̆, \underline{ğ^{∗}h̆})(y, z).

Proof. Combining the second statement in Corollary 2.11 with Theorem 4.15,

(138)  log_{p}(κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z})) = d_{s}^{−1} × E_{g_{y}}(f, g_{y}, h_{z}) × ğ_{y}^{∗}(F̆ × h̆_{z}).

By definition,

(139) ğ_{y}^{∗}(F̆ × h̆_{z}) = ⟨η_{}_{ğ_{y}^{∗}}, w_{s}(F̆ × h̆_{z})⟩_{s} = d_{s} × (y, z)(⟨η_{}_{ğ_{s}^{∗}}, w_{s}(F̆ × h̆_{s})⟩_{Γ_{s}}) = d_{s} × (y, z)(˘\underline{g}^{∗}(F̆ × ˘ \underline{h})_{s}),

where the penultimate equality follows from Lemma 1.12, and the expression ˘ \underline{g}^{∗}(F̆ × ˘ \underline{h})_{s} ∈ Λ_{f\underline{gh}} ⊗_{Λ} Λ_{s} occuring in the last term denotes the natural projection “to level s” of an element of Λ_{f\underline{gh}}. The interpolation property (6) of the p-adic L-function ℒ_{p} ^{g_{α}}(f̆, ˘ \underline{g}^{∗}, ˘ \underline{h}) at points of weight (2, ℓ, ℓ) and trivial nebentypus character shows that

˘\underline{g}^{∗}(F̆ × ˘ \underline{h}) = ℒ_{p} ^{g_{α}}(f̆, ˘ \underline{g}^{∗}, ˘ \underline{h}),

and hence, since ℒ_{p} ^{g_{α}}(f̆, \underline{ğ^{∗}h̆}) is just the restriction of ℒ_{p} ^{g_{α}}(f̆, ˘ \underline{g}^{∗}, ˘ \underline{h}) to the one-dimensional rigid analytic space Ω_{f\underline{gh}}, that

(140)  (y, z)(˘\underline{g}^{∗}(F̆ × ˘ \underline{h})_{s}) = ℒ_{p} ^{g_{α}}(f̆, \underline{ğ^{∗}h̆})(y, z).

The theorem follows by combining (138), (139) and (140). □

5. The explicit reciprocity law

5.1. Perrin-Riou’s Λ-adic regulator. Let Λ̃ be a finite flat extension of Λ and let Ψ : G_{ℚ_{p}} ⟶ Λ̃^{×} be an unramified Λ̃-valued character. Let

(141)  \underline{κ}^{g} ∈ H^{1}(ℚ_{p}, Λ̃\{Ψ\}(1)),  \underline{κ}^{f} ∈ H^{1}(ℚ_{p}, Λ̃\{Ψ\}(\underline{ϵ}_{cyc}))

be Λ̃-adic classes with values in the Λ̃[G_{ℚ_{p}}]-modules Λ̃\{Ψ\}(1) = Λ̃\{Ψ\}(ϵ_{cyc}) and Λ̃\{Ψ\}(\underline{ϵ}_{cyc})), respectively. The notations used to designate these classes are motivated by (94), which asserts that \underline{κ}^{g} = κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) and \underline{κ}^{f} = κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) yield particular instances of (141), after taking

Ψ = ψ_{f}^{−1}Ψ\underline{g}Ψ−1 ; \underline{h} × χ^{−1} and Ψ = ψ_{f}Ψ−1 ; \underline{g}Ψ−1 ; \underline{h}

for \underline{κ}^{g} and \underline{κ}^{f} respectively.

In general, the Λ-adic classes \underline{κ}^{g} and \underline{κ}^{f} gives rise to a collection of classical specialisations

κ_{x}^{g} ∈ H^{1}(ℚ_{p}, K_{x}{Ψ_{x}}(1)), κ_{x}^{f} ∈ H^{1}(ℚ_{p}, K_{x}{Ψ_{x}}(ϵ_{x})) where ϵ_{x} = ϵ_{cyc}^{ℓ−1}ω^{1−ℓ}ϵ,

as x ∈ Ω̃_{cl} ranges over the set of classical points of Ω̃ = Spf(Λ̃) satisfying w(x) = ν_{ℓ,ϵ} for some ℓ ≥ 1 and a Dirichlet character ϵ of conductor p^{s}, s ≥ 1.

One has canonical isomorphisms of Dieudonné modules for any x ∈ Ω̃_{cl},

(142)  D_{dR}(K_{x}{Ψ_{x}}(1)) := D_{dR}(K_{x}{Ψ_{x}}) ⊗ D_{dR}(K_{x}(1)) = D_{dR}(K_{x}(1)) ≃^{·t} K_{x}
D_{dR}(K_{x}{Ψ_{x}}(ϵ_{x})) := D_{dR}(K_{x}{Ψ_{x}}) ⊗ D_{dR}(K_{x}(ϵ_{x})) = D_{dR}(K_{x}(ϵ_{x})) ≃^{·t^{ℓ−1}} K_{x},

where (91) has been invoked in deriving the penultimate equalities, and the last isomorphism is given by multiplication by the corresponding power of Fontaine’s period t ∈ B_{dR}, the p-adic analogue of 2πi on which G_{ℚ_{p}} acts as multiplication by ϵ_{cyc}. (Note that the field K_{x} is assumed to contain ζ_{s}, and hence, that it contains the periods of the finite order character ω^{1−ℓ}ϵ.)

Assume that for any x ∈ Ω̃_{cl} the character Ψ_{x} is not trivial. Arguing as in [BDR2, Lemma 3.8], it then follows that the Bloch-Kato logarithm and dual exponential map induce isomorphisms

(143)  log_{p} : H^{1}(ℚ_{p}, K_{x}{Ψ_{x}}(1)) ⟶ K_{x} for any x ∈ Ω̃_{cl},
log_{p} : H^{1}(ℚ_{p}, K_{x}{Ψ_{x}}(ϵ_{x})) ⟶ K_{x} for any x ∈ Ω̃_{cl} such that ℓ ≥ 2,
exp_{p}^{∗} : H^{1}(ℚ_{p}, K_{x}{Ψ_{x}}(ϵ_{x})) ⟶ K_{x} for any x ∈ Ω̃_{cl} such that ℓ = 1.

Note that the logarithm maps quoted in (96) are particular cases of the above. The two propositions below describe how the images of the classes κ_{x}^{g} and κ_{x}^{f} under the maps in (143) can be interpolated p-adically. The p-adic interpolation of the logarithms of the classes κ_{x}^{g} is treated first in Proposition 5.1 below, inspired by the elegant treatment of the p-adic interpolation of unramified periods given in [LZ12]. The corresponding interpolation problem for the classes κ_{x}^{f}, which is taken up in Proposition 5.2, lies considerably deeper and relies crucially on the “explicit reciprocity law” of Perrin-Riou (as refined in [LZ12]).

Proposition 5.1. There exists an element L(\underline{κ}^{g}) ∈ Λ̃ such that L(\underline{κ}^{g})(x) = log_{p}(κ_{x}^{g}) for all x ∈ Ω̃_{cl}.

Proof. Let

G^{ur} = Gal (ℚ_{p}^{ur}/ℚ_{p}) = ℤ̂

be the Galois group of the maximal unramified extension of ℚ_{p}, which is canonically isomorphic to ℤ̂ as a profinite group via the canonical topological generator given by the Frobenius automorphism σ_{p} ∈ G^{ur}. Let ℤ_{p}[[G^{ur}]] be the completed group ring for G^{ur}, and write ℤ_{p}^{∘}[[G^{ur}]] for the same completed group ring viewed as a rank one module over itself and equipped with the “tautological” unramified action of G_{ℚ_{p}} in which the Frobenius automorphism acts as multiplication by the group-like element σ_{p}. If R is a p-adic ring, and α ∈ R^{×} is any invertible element of R, the assignment σ_{p} ↦ α extends uniquely to a continuous group homomorphism Ψ : G^{ur} ⟶ R^{×} since R^{×} is a profinite group, and this homomorphism extends by ℤ_{p}-linearity to a continuous ring homomorphism

Ψ : ℤ_{p}[[G^{ur}]] ⟶ R.

Let R{Ψ} denote the free R-module of rank one equipped with an action of G_{ℚ_{p}} in which G_{ℚ_{p}} acts on R via the unramified character Ψ. With these notations, the representation R(Ψ)(1) of G_{ℚ_{p}} can be written as

R(Ψ)(1) = ℤ_{p}^{∘}[[G^{ur}]](1) ˆ⊗_{Ψ}R.

It follows, after invoking Shapiro’s Lemma for the second equality, that

H^{1}(ℚ_{p}, R(Ψ)(1)) = H^{1}(ℚ_{p}, ℤ_{p}^{∘}[[G^{ur}]](1)) ˆ⊗_{Ψ}R = (\underleftarrow{lim}_{n} H^{1}(ℚ_{p^{n}}, ℤ_{p}(1))) ˆ⊗_{Ψ}R,

where the inverse limit is taken with respect the corestriction maps and ℚ_{p^{n}} denotes the unramified extension of ℚ_{p} of degree n for each n ≥ 1. Hilbert’s theorem 90 implies that

H^{1}(ℚ_{p}, R(Ψ)(1)) = (\underleftarrow{lim}_{n}(ℚ_{p^{n}}^{×} ⊗ ℤ_{p})) ˆ⊗_{Ψ}R = (\underleftarrow{lim}_{n}(O_{n}^{×} ⊗ ℤ_{p})) ˆ⊗_{Ψ}R,

where the inverse limit is taken with respect to the norm maps, and O_{n} denotes the ring of integers of ℚ_{p^{n}}. The standard p-adic logarithm log_{p} : O_{n}^{×} → O_{n} therefore gives rise to a natural homomorphism

(144)  log_{p} : H^{1}(ℚ_{p}, R(Ψ)(1)) ⟶ (\underleftarrow{lim}_{n} O_{n}) ˆ⊗_{Ψ}R,

where the inverse limit is taken relative to the trace maps. The elements of \underleftarrow{lim}_{n} O_{n} can be interpreted

as ℤ̂_{p}^{ur}-valued measures on G^{ur} by associating to an element a = {a_{n}} of this inverse limit the measure μ_{a} defined by

μ_{a}(σ · Gal (ℚ_{p}^{ur}/ℚ_{p^{n}})) := σa_{n}, ∀n ≥ 1, σ ∈ Gal (ℚ_{p^{n}}/ℚ_{p}).

The assignment

a ⊗ 1 ↦ ∫_{G^{ur}} Ψ(t)dμ_{a}(t)

gives rise to a natural homomorphism

(145)  (\underleftarrow{lim}_{n} O_{n}) ˆ⊗_{Ψ}R ⟶ (R ˆ⊗ℤ̂_{p}^{ur})^{σ_{p}=α^{−1}} = (R(Ψ) ˆ⊗ℤ̂_{p}^{ur})^{G_{ℚp}} .

Combining (144) and (145) allows one to parlay the logarithm map into a canonical homomorphism

(146)  H^{1}(ℚ_{p}, R(Ψ)(1)) ⟶ (R(Ψ) ˆ⊗ℤ̂_{p}^{ur})^{G_{ℚp}} .

Equivalently, after invoking (91), one obtains a canonical logarithm map

(147)  L : H^{1}(ℚ_{p}, R{Ψ}(1)) ⟶ R.

The homomorphism L is functorial in the sense that for all G^{ur}-equivariant homomorphisms φ : R{Ψ} ⟶ R′{Ψ′}, the diagram

\xymatrix{H^{1}(ℚ_{p}, R\{Ψ\}(1)) \ar[r]^{L} \ar[d]_{φ} & R \ar[d]_{φ} \\ H^{1}(ℚ_{p}, R′\{Ψ′\}(1)) \ar[r]^{L} & R′}

commutes. In particular, after setting R := Λ̃{Ψ} and letting φ : Λ̃{Ψ} ⟶ K_{x}{Ψ_{x}} be the homomorphism induced by the evaluation map at x, the element L(\underline{κ}^{g}) ∈ Λ̃ satisfies the interpolation property claimed in Proposition 5.1, in light of the fact that the map L : H^{1}(ℚ_{p}, K_{x}{Ψ_{x}}(1)) ⟶ K_{x} agrees with the p-adic logarithm map log_{p}. □

In order to treat the analogous question for the Λ-adic cohomology class \underline{κ}^{f}, define

(148)  ℧_{x} := (−1)^{1−ℓ} · Γ^{∗}(2 − ℓ) · 𝔤(ϵ^{−1}ω^{ℓ−1}) · Ψ_{x}(Frob_{p})^{−s} · E_{x} · p^{s(2−ℓ)} ∈ K_{x},

where

Γ^{∗}(2 − ℓ) = \frac{1}{(ℓ − 2)!} if ℓ ≥ 2; Γ^{∗}(2 − ℓ) = (1 − ℓ)! if ℓ ≤ 1,

and

E_{x} = (1 − p^{1−ℓ}Ψ_{x}(Frob_{p})^{−1})(1 − p^{ℓ−2}Ψ_{x}(Frob_{p}))^{−1} if ϵ = ω^{ℓ−1}; E_{x} = 1 otherwise.

Proposition 5.2. There exists an element L(\underline{κ}^{f}) ∈ Λ̃ such that for all x ∈ Ω̃_{cl} with w(x) = ν_{ℓ,ϵ},

L(\underline{κ}^{f})(x) = \{ \begin{matrix}℧_{x} · log_{p}(κ_{x}^{f}), & if ℓ ≥ 2, \\ ℧_{x} · exp_{p}^{∗}(κ_{x}^{f}), & if ℓ = 1. \end{matrix}

Proof. Let K_{∞} denote the abelian extension of ℚ_{p} through which the Λ̃-adic character Ψ\underline{ϵ}_{cyc} factors, let G = Gal (K_{∞}/ℚ_{p}), and let Λ_{G} := ℤ_{p}[[G]] be the completed group ring attached to G, equipped with its tautological action of G_{ℚ_{p}}. The decomposition K_{∞} = K_{∞}^{ur}ℚ_{p}(μ_{p^{∞}}), where K_{∞}^{ur} is the maximal unramified subfield of K_{∞}, determines a canonical decomposition G = U × ℤ_{p}^{×}. The unramified character Ψ : U ⟶ Λ̃^{×} extends by linearity to a natural G_{ℚ_{p}}-equivariant projection

(149)  θ_{Ψ} : Λ_{G} ⟶ Λ̃\{Ψ\}(\underline{ϵ}_{cyc}).

The p-adic regulator of [LZ12], specialised to the case where V is the trivial one-dimensional representation of G_{ℚ_{p}}, yields a “two-variable regulator map”

L^{G} : H^{1}(ℚ_{p}, Λ_{G}) ⟶ Λ_{G} ˆ⊗ ℤ̂_{p}^{ur}.

Let θ̃_{Ψ} : H^{1}(ℚ_{p}, Λ_{G}) ⟶ H^{1}(ℚ_{p}, Λ̃\{Ψ\}(\underline{ϵ}_{cyc})) denote the map induced by (149) in cohomology. Note that θ̃_{Ψ} is surjective because the cohomological dimension of ℚ_{p} is 1. Let ˜ \underline{κ}^{f} ∈ H^{1}(ℚ_{p}, Λ_{G}) be any lift of the class \underline{κ}^{f} under θ̃_{Ψ}, and set

(150)  L(\underline{κ}^{f}) := θ_{Ψ}(L^{G}(˜\underline{κ}^{f})) ∈ Λ̃\{Ψ\} ˆ⊗ ℤ̂_{p}^{ur}.

It is not hard to see (and follows, for instance, from Prop. 4.9. of [LZ12]) that L(\underline{κ}^{f}) is fixed under the action of Frob_{p} and hence L(\underline{κ}^{f}) belongs to 𝔻(Λ̃\{Ψ\}), which is canonically isomorphic to Λ̃ by (91).

Now [LZ12, Theorem 4.15], with j = 1 − ℓ and Φ^{n} = Ψ_{x}(Frob_{p})^{−s} applied to the character ω = ϵ_{cyc}^{ℓ−1} in the notation of loc.cit. implies that the element L(\underline{κ}^{f}) has the required interpolation property. □

5.2. The triple product p-adic L-functions via Λ-adic cohomology classes. We are now in position to prove the following result, which lends support to Perrin-Riou’s vision according to which p-adic L-functions ought to arise as the images of p-adic families of distinguished global elements (referred to loosely as the Iwasawa theoretic incarnations of “Euler systems”) under suitable p-adic regulator maps. Realising a p-adic L-function in this way has strong arithmetic consequences, some of which shall be explored in the remainder of this article.

Theorem 5.3. The following equalities hold:

ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆}) = α_{f}/2 · (1 − χ^{−1}(p)α_{f}^{−1}a_{p}(\underline{g})a_{p}(\underline{h})^{−1}) × L(κ_{p}^{f}(f̆^{∗}, \underline{ğh̆})),
ℒ_{p} ^{g_{α}}(f̆, \underline{ğ^{∗}h̆}) = 1/2 · (1 − χ(p)p^{−1}α_{f}a_{p}^{−1}(\underline{g})a_{p}(\underline{h})) × L(κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆})).

Proof. By specialising Propositions 5.1 and 5.2 to \underline{κ}^{g} := κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}) and \underline{κ}^{f} := κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}) respectively, one obtains from (95) that

log_{p}(κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z})) = α_{f}^{s}a_{p}(g_{y})^{−s}a_{p}(h_{z})^{−s} × L(κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}))(y, z),
log_{p}(κ_{p}^{g}(f̆, ğ_{y}^{∗}, h̆_{z})) = L(κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}))(y, z),

at all classical points x = (y, z) ∈ Ω_{g} ×_{Ω} Ω_{h} of weight-character (2, ϵ) with ϵω^{−1} of conductor p^{s} with s ≥ 1. Comparing this identity with Theorems 4.16 and 4.17 respectively shows that

ℒ_{p} ^{f_{α}}(f̆^{∗}, ˘ \underline{g}, ˘ \underline{h})(y, z) = α_{f}/2 · (1 − χ^{−1}(p)α_{f}^{−1}α_{g_{y}}α_{h_{z}}^{−1}) × L(κ_{p}^{f}(f̆^{∗}, \underline{ğh̆}))(y, z),
ℒ_{p} ^{g_{α}}(f̆, ˘ \underline{g}^{∗}, ˘ \underline{h})(y, z) = 1/2 · (1 − χ(p)p^{−1}α_{f}α_{g_{y}}^{−1}α_{h_{z}}) × L(κ_{p}^{g}(f̆, \underline{ğ^{∗}h̆}))(y, z),

for infinitely many points (y, z) of weight-character of the form (2, ϵ). But a non-zero element of Λ_{f\underline{gh}} can only vanish at finitely many points of Ω_{g} ×_{Ω} Ω_{h}. The theorem follows, since a_{p}(g_{y}) = α_{g_{y}} and a_{p}(h_{z}) = α_{h_{z}}. □

6. Application to the Birch and Swinnerton-Dyer conjecture

As in the introduction, let

f ∈ S_{2}(N_{f}),  g ∈ S_{1}(N_{g}, χ),  h ∈ S_{1}(N_{h}, χ^{−1})

be three newforms and assume that the level N_{f} of f is relatively prime to N_{g}N_{h}. Assume also that the weight two modular form f has rational fourier coefficients, and hence is associated to an elliptic curve E. (This last assumption is only made for notational simplicity and the arguments of this chapter would extend to the setting where E is replaced by any simple abelian variety quotient of J_{0}(N), at the cost of slight technical complications.) The goal of this chapter is to prove Theorems A, B, C and D and their corollaries stated in the introduction, concerning the arithmetic of the twist of E by the four-dimensional self-dual Artin representation attached to

ϱ_{gh} := ϱ_{g} ⊗ ϱ_{h} : G_{ℚ} ⟶ GL_{4}(L),

where ϱ_{g} and ϱ_{h} are the odd, irreducible, two-dimensional Artin representations attached to g and h.

6.1. Bounding Mordell-Weil groups. For this section, let W be a general d-dimensional self-dual Artin representation, with coefficients in a finite extension L of ℚ, and factoring through the Galois group of a finite extension H of ℚ. Our goal is to present general results, based on local Tate duality and the global Poitou-Tate exact sequence, for bounding the W -isotypic part of E(H), defined to be the L-vector space

E(H)_{L}^{W} := Hom_{Gal (H/ℚ)}(W, E(H) ⊗ L).

As in the introduction, fix a rational prime p and an embedding L ⊂ L_{p} of L into a finite extension of ℚ_{p}. Associated to E and W are the continuous p-adic representations

V_{p}(E) := H_{et}^{1}(Eℚ̄, ℚ_{p})(1) ⊗_{ℚ_{p}} L_{p},  W_{p} := W ⊗_{L} L_{p},  V_{p}(E) ⊗_{L_{p}} W_{p}

of G_{ℚ}, which are L_{p}-vector spaces of dimensions 2, d and 2d respectively. Restriction to the absolute Galois group G_{H} induces an isomorphism

(151)  H^{1}(ℚ, V_{p}(E) ⊗ W_{p}) ≃ (H^{1}(H, V_{p}(E)) ⊗ W_{p})^{Gal (H/ℚ)}
= Hom_{Gal (H/ℚ)}(W_{p}, H^{1}(H, V_{p}(E))),

where the self-duality of W_{p} is used to obtain the second equality. Thanks to this identification, the connecting homomorphism

δ : E(H) ⊗ L ⟶ H^{1}(H, V_{p}(E))

of Kummer theory gives rise to a homomorphism

(152)  δ : E(H)_{L}^{W_{p}} ⟶ H^{1}(ℚ, V_{p}(E) ⊗ W_{p}).

For each rational prime ℓ, the maps (151) and (152) admit local counterparts

H^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p}) ≃ Hom_{Gal (H/ℚ)}(W_{p}, ⊕_{λ|ℓ}H^{1}(H_{λ}, V_{p}(E))),
δ_{ℓ} : (⊕_{λ|ℓ}E(H_{λ}))_{L_{p}}^{W_{p}} ⟶ H^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p}),

for which the following diagram commutes:

(153)  \xymatrix{E(H)_{L}^{W} \ar[r]^{δ} \ar[d]^{_{res_{ℓ}}} & H^{1}(ℚ, V_{p}(E) ⊗ W_{p}) \ar[d]^{_{res_{ℓ}}} \\ _{(⊕_{λ|ℓ}E(H_{λ})){W_{p} \atop L_{p}}} \ar[r]^{δ_{ℓ}} & H^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p}).}

The Bloch-Kato submodule H_{fin}^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p}) of the local cohomology group H^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p}) coincides with the image of the local connecting homomorphism δ_{ℓ}, and the singular quotient is defined to be

H_{sing}^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p}) := \frac{H^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p})}{H_{fin}^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p})}.

When ℓ = p is a prime of good reduction for E at which W is unramified, the subspace H_{fin}^{1}(ℚ_{p}, V_{p}(E)⊗ W_{p}) consists of classes of cristalline extensions of Galois representations. For each rational prime ℓ, let

res_{ℓ} : H^{1}(ℚ, V_{p}(E) ⊗ W_{p}) ⟶ H^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p})

denote the restriction map from the global to the local cohomology at ℓ. The composition

∂_{ℓ} : H^{1}(ℚ, V_{p}(E) ⊗ W_{p}) ⟶ H_{sing}^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p})

of res_{ℓ} with the natural projection to the singular quotient is called the residue map at ℓ. The Selmer group Sel_{p}(E, W) is defined in terms of these maps as

(154) Sel_{p}(E, W) = H_{fin}^{1}(ℚ, V_{p}(E) ⊗ W_{p}) := ⋂_{ℓ} ker (H^{1}(ℚ, V_{p}(E) ⊗ W_{p}) ⟶^{∂_{ℓ}} H_{sing}^{1}(ℚ_{ℓ}, V_{p}(E) ⊗ W_{p})).

Lemma 6.1. The local cohomology group H^{1}(ℚ_{p}, V_{p}(E) ⊗ W_{p}) is a 2d-dimensional L_{p}-vector space. The finite subspace H_{fin}^{1}(ℚ_{p}, V_{p}(E) ⊗ W_{p}) and the singular quotient H_{sing}^{1}(ℚ_{p}, V_{p}(E) ⊗ W_{p}) are each d-dimensional and in perfect duality under the local Tate pairing.

Proof. This follows by choosing a G_{ℚ_{p}}-stable lattice in V_{p}(E) ⊗ W_{p}, applying (for instance) Theorem 2.17 of [DDT] to its finite G_{ℚ_{p}}-stable quotients, and passing to the limit. □

Proposition 6.2. If the map ∂_{p} is surjective, then the map

(155)  res_{p} : Sel_{p}(E, W) ⟶ H_{fin}^{1}(ℚ_{p}, V_{p}(E) ⊗ W_{p})

is the zero map.

Proof. The standard calculations arising from the global Poitou-Tate exact sequence imply that the image of the restriction map

res_{p} : Sel_{p}(E, W) ⟶ H^{1}(ℚ_{p}, V_{p}(E) ⊗ W_{p})

is d-dimensional. (This can be shown, for example, by choosing a G_{ℚ}-stable lattice in V_{p}(E) ⊗ W_{p}, applying Theorem 2.18 of [DDT] to its finite quotients, and taking the inverse limit.) It follows that the restriction of res_{p} to H_{fin}^{1}(ℚ_{p}, V_{p}(E) ⊗ W_{p}) is the zero map. □

Proposition 6.3. Assume that the residue map ∂_{p} attached to the representation V_{p}(E) ⊗ W_{p}^{σ} is a surjective map of L_{p}-vector spaces, for all σ ∈ Gal (L/ℚ). Then E(H)_{L}^{W} = 0.

Proof. By Proposition 6.2, the map res_{p} of (155) is the zero map, and therefore the commutativity of the diagram (153) implies that the natural map

res_{p} : E(H)_{L}^{W^{σ}} ⟶ ((⊕_{𝔭|p}E(H_{𝔭})) ⊗ L_{p})^{W^{σ}}

is the zero map, for each σ ∈ Gal (L/ℚ). Since the local connecting homomorphism δ_{ℓ} is injective, this implies that the vector space

⊕_{σ∈Gal (L/ℚ)}E(H)_{L}^{W^{σ}} = E(H)W̃ ; L , where W̃ = ⊕_{σ∈Gal (L/ℚ)}W^{σ},

has trivial image in the group ⊕_{𝔭|p}E(H_{𝔭}) ⊗ L of local points. But W̃ is a rational representation and hence admits an L-basis consisting of elements of E(H). Since the natural map E(H) → ⊕_{𝔭|p}E(H_{𝔭}) is injective modulo torsion, it follows that dim_{ℚ} E(H)W̃ ; ℚ = 0, and therefore the same is true of dim_{L} E(H)_{L}^{W}. The proposition follows. □

6.2. Proof of Theorem C. The most interesting application of Theorem 5.3 arises without a doubt when ℓ = 1. A simplified version of the following statement was stated in Theorem C of the introduction. As in loc. cit. let g ∈ S_{1}(N_{g}, χ) and h ∈ S_{1}(N_{h}, χ^{−1}) be classical newforms of weight 1 and let g_{α} ∈ S_{1}(N_{g}p, χ) and h_{α} ∈ S_{1}(N_{h}p, χ^{−1}) be ordinary p-stabilisations.

Let \underline{g} and \underline{h} be Hida families specialising to g_{y} = g_{α} and h_{z} = h_{α} at suitable weight one points y and z of the parameter spaces Ω\underline{g} and Ω\underline{h} respectively, and fix test vectors f̆^{∗}, ˘ \underline{g} and ˘ \underline{h} as in (92) and (93). Let

κ(f, g_{α}, h_{α}) ∈ H^{1}(ℚ, V_{fgh}(N))

be the global cohomology class introduced in (76).

Theorem 6.4. The central critical value L(f, g, h, 1) is non-zero if and only if the global cohomology class κ(f, g_{α}, h_{α}) is not crystalline at p.

Proof. Since the triplet (2, 1, 1) is unbalanced, with 2 as its dominant weight, the point (y, z) lies in the region of interpolation defining the p-adic L-function ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆}). By [DR13, Theorems 4.2 and 4.7], L(f, g, h, 1) ≠ 0 if and only if there exists a choice (f̆^{∗}, ğ_{α}, h̆_{α}) of test vectors such that ℒ_{p} ^{f_{α}}(f̆^{∗}, \underline{ğh̆})(y, z) ≠ 0. By Theorem 5.3, this in turn is equivalent to the non-vanishing of the value of L(κ_{p}^{f}(f̆^{∗}, \underline{ğh̆})) at the point (y, z), which Proposition 5.2 recasts as a non-zero multiple of exp_{p}^{∗}(κ_{p}^{f}(f̆^{∗}, ğ_{α}, h̆_{α})). Because the dual exponential map appearing in (143) is an isomorphism, one concludes that L(f, g, h, 1) ≠ 0 if and only if there is a triple (f̆^{∗}, ğ_{α}, h̆_{α}) of test vectors for which the local class κ_{p}^{f}(f̆^{∗}, ğ_{α}, h̆_{α}) is non-trivial.

As G_{ℚ_{p}}-modules, V_{f}(Np)^{−} ⊗ V_{gh}^{ββ}(N) is isomorphic to the direct sum of a finite number of copies of K_{x}{ψ_{f}Ψ_{g_{α}}^{−1}Ψ_{h_{α}}^{−1}}, and the local class κ_{p}^{f}(f, g_{α}, h_{α}) is determined by the collection of its projections to the cohomology groups H^{1}(ℚ_{p}, K_{x}{ψ_{f}Ψ_{g_{α}}^{−1}Ψ_{h_{α}}^{−1}}) indexed by the set of all possible triples (f̆^{∗}, ğ_{α}, h̆_{α}) of test vectors. The non-vanishing of κ_{p}^{f}(f, g_{α}, h_{α}) is therefore equivalent to the non-vanishing of the local class κ_{p}^{f}(f̆^{∗}, ğ_{y}, h̆_{z}) ∈ H^{1}(ℚ_{p}, K_{x}{ψ_{f}Ψ_{g_{α}}^{−1}Ψ_{h_{α}}^{−1}}) introduced in (95) for some such triple. Theorem 6.4 now follows from Proposition 2.8. □

6.3. Proof of Theorem A and its corollaries. We now specialise the results of the previous section to the setting where W = V_{gh} in order to prove Theorem A of the introduction and its corollaries. Henceforth, the Artin representation V_{gh} is viewed as having coefficients in a finite extension L of ℚ, just as the representation W of the previous section. In the notation of §1.6 we thus have L_{p} = K_{fgh}.

Proposition 6.5. Assume L(f, g, h, 1) ≠ 0. For all pairs of eigenvalues

λ ∈ {(α_{g}, α_{h}), (α_{g}, β_{h}), (β_{g}, α_{h}), (β_{g}, β_{h})},

there exists a global class κ_{λ} ∈ H^{1}(ℚ, V_{fgh}) whose natural image in H_{sing}^{1}(ℚ_{p}, V_{fgh}) is non-zero and belongs to H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{gh}^{λ}).

Proof. Up to re-ordering the eigenvalues of Frob_{p} acting on V_{g} and V_{h}, it may be assumed without loss of generality that λ = (α_{g}, α_{h}). By Theorem 6.4, the associated cohomology class κ(f, g_{α}, h_{α}) is non-cristalline, and therefore there exists a Galois-equivariant surjection j : V_{fgh}(N) ⟶ V_{fgh} for which κ_{λ} := j(κ(f, g_{α}, h_{α})) is also non-cristalline. Since the natural image of κ_{λ} in H_{sing}^{1}(ℚ_{p}, V_{fgh}) belongs to H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{gh}^{λ}) by Proposition 2.8, the proposition follows. □

Theorem 6.6. If L(E, V_{gh}, 1) ≠ 0, there exists a rational prime p for which the natural map

∂_{p} : H^{1}(ℚ, V_{f} ⊗ V_{gh}) ⟶ H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{gh})

is surjective.

Proof. The Chebotarev density theorem shows the existence of a prime p ≥ 5 whose associated Frobenius element acts on both V_{g} and V_{h} with distinct eigenvalues, and such that a_{p}(f) ≠ 0 (so that f is ordinary at p). By Proposition 6.5, the non-vanishing of L(E, V_{gh}, 1) implies the existence of global classes

(156)  κ_{α_{g},α_{h}}, κ_{α_{g},β_{h}}, κ_{β_{g},α_{h}}, κ_{β_{g},β_{h}} ∈ H^{1}(ℚ, V_{fgh})

whose image under ∂_{p} is non-zero in H_{sing}^{1}(ℚ_{p}, V_{fgh}). Since ∂_{p}(κ_{λ}) belongs to H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{gh}^{λ}) for each pair λ ∈ {(α_{g}, α_{h}), (α_{g}, β_{h}), (β_{g}, α_{h}), (β_{g}, β_{h})}, the four classes are linearly independent and generate this singular quotient. □

We can now prove Theorem A of the introduction:

Theorem 6.7. Assume that ϱ_{gh} is regular. If L(E, ϱ_{gh}, 1) ≠ 0, then E(H)_{L}^{ϱ_{gh}} = 0.

Proof. For all σ ∈ Gal (L/ℚ), the non-vanishing of L(E, ϱ_{gh}, 1) at the central point implies that the same is true for L(E, V_{gh}^{σ}, 1), for all σ ∈ Gal (L/ℚ). Furthermore, there exists a prime p for which V_{gh}^{σ} is regular, for all such σ. Theorem 6.6 implies that the map

∂_{p} : H^{1}(ℚ, V_{p}(E) ⊗ V_{gh}^{σ}) ⟶ H_{sing}^{1}(ℚ_{p}, V_{p}(E) ⊗ V_{gh}^{σ})

is surjective for each σ ∈ Gal (L/ℚ). The theorem now follows from Proposition 6.3. □

Turning to Corollary A1 of the introduction, let K be a quadratic field of discriminant prime to the conductor N_{E} of E. If ψ is a finite order ray class character of K, let ψ′ denote the character obtained from it by composing with the automorphism of K/ℚ. A ring class character is a character of K with trivial central character, which therefore satisfies ψ′ = ψ^{−1}. Let Π_{K} denote the set of all ring class characters of conductor prime to N_{E}. When K is a real quadratic field, the set Π_{K} can be partitioned into two subsets Π_{K}^{+} and Π_{K}^{−}, consisting of totally even and totally odd ring class characters, respectively. Let Π_{E/K}, Π_{E/K}^{+} and Π_{E/K}^{−} denote the subsets of Π_{K}, Π_{K}^{+} and Π_{K}^{−} respectively consisting of the characters for which L(E/K, ψ, 1) ≠ 0.

To the pair (E, K) one can associate the sign sgn(E, K) ∈ {±1} of the functional equation for the L-function L(E/K, s). It turns out that this is the very same sign that occurs in the functional equation of L(E/K, ψ, s) for any ψ ∈ Π_{K}. In particular, Π_{E/K} is empty when sgn(E, K) = −1.

When sgn(E, K) = 1 and K is imaginary quadratic, a non-vanishing theorem of Cornut and Vatsal shows that Π_{E/K} is infinite. If K is real quadratic, a simple argument involving congruences for L-values shows that Π_{E/K}^{+} is either empty or infinite, and likewise for Π_{E/K}^{−}. The scenario where Π_{E/K}^{±} = ∅ is highly unlikely, since it would provide us with a systematic supply of ring class characters for which the (primitive) L-function L(E/K, ψ, s) admits at least a double zero at the center.

Definition 6.8. The pair (E, K) is said to satisfy the non-vanishing hypothesis if Π_{E/K}^{+} and Π_{E/K}^{−} are both non-empty (and hence, infinite).

When K is real, a proof of this non-vanishing hypothesis does not seem out of reach of current techniques in analytic number theory, although the unavailability of an “anti-cyclotomic ℤ_{p}-extension” of K prevents a straightforward application of the methods of Cornut and Vatsal.

Lemma 6.9. For all ψ ∈ Π_{K}, there exists a ray class character ψ_{0} of K of conductor prime to N_{E} such that ψ = ψ_{0}/ψ′_{0}.

Proof. Let M/K be the cyclic extension of K (of degree n, say) which is cut out by the character ψ. This field is Galois over ℚ, and Gal (M/ℚ) is isomorphic to the dihedral group D_{n} of order 2n, furnished with a natural embedding into the semi-direct product (ℂ^{×}⋊ℤ/2ℤ) via the character ψ. Embed D_{n} into PGL_{2}(ℂ) by sending λ ∈ ℂ^{×} to the class of any diagonal matrix of the form ( \begin{matrix}λ_{1} & 0 \\ 0 & λ_{2} \end{matrix} ), where λ_{1}/λ_{2} = λ modulo the center, and sending some reflection in D_{n} to the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). This yields a projective representation r̄: G_{ℚ} → PGL_{2}(ℂ). By a classical result of Tate on the vanishing of H^{2}(ℚ, ℚ/ℤ(1)), the projective representation r̄ may be lifted to a linear representation r : G_{ℚ} → GL_{2}(ℂ). By construction, the representation r has dihedral projective image, and hence is induced from some character ψ_{0} of K. The restriction of r to G_{K} is then equal to the direct sum ψ_{0} ⊕ ψ′_{0}. It follows that the restriction of r̄ to G_{K} is identified with ψ_{0}/ψ′_{0}, and therefore that ψ_{0}/ψ′_{0} = ψ. To show that ψ_{0} can be chosen to be unramified at the primes of K dividing N_{E}, observe that for each prime ℓ|N_{E}, the linear representation r maps the inertia group at ℓ to the subgroup ℂ^{×} of scalar matrices, since r̄ is unramified at ℓ. Now choose a Dirichlet character χ : G_{ℚ} ⟶ ℂ^{×} whose restriction to I_{ℓ} agrees with the restriction of r, for all ℓ|N_{E}, and replace r by r ⊗ χ^{−1}. This substitution does not affect the projective representation attached to r, but leads to a representation r which is unramified at the primes dividing N_{E}; hence the same is true for the resulting character ψ_{0}. □

Note that, if K is a real quadratic field, the character ψ_{0} is of mixed signature (resp. totally even or odd) when ψ is totally odd (resp. totally even).

By considering the case where α is further restricted to be a quadratic character of K, it is clear that the non-vanishing hypothesis is satisfied if there are two quadratic Dirichlet characters χ (of the two possible signatures) for which

(157)  L(E, χ, 1) ≠ 0, and L(E, χχ_{K}, 1) ≠ 0,

where χ_{K} is the (fixed) Dirichlet character attached to K. Such simultaneous non-vanishing results for pairs of quadratic Dirichlet characters with fixed product seem to lie just beyond the reach of currently available techniques. However, a character χ satisfying (157) is widely expected to exist, and a quick inspection is usually enough to produce such a χ, for any given E and K. For example, if E is the elliptic curve of smallest conductor 11 and K = ℚ(\sqrt{5}) is the real quadratic field of smallest discriminant 5, a cursory inspection of the tables of Cremona reveals that L(E, χ, 1) ≠ 0 and L(E, χχ_{K}, 1) ≠ 0 when

(1) χ is the trivial character;

(2) χ is the odd quadratic character of conductor 4. Hence the pair (E, K) satisfies the non-vanishing hypothesis.

Theorem 6.10. Let E be an elliptic curve over ℚ and let K be a quadratic field for which sign(E, K) = 1. When K is real, assume that (E, K) satisfies the non-vanishing hypothesis. Then for all ring class characters ψ ∈ Π_{K},

L(E/K, ψ, 1) ≠ 0 ⇒ E(H)^{ψ} = 0.

Proof. If ψ is a quadratic (genus) character, the statement already follows from the approach of Gross-Zagier and Kolyvagin (or Kato), since the induced representation V_{ψ} decomposes as a sum of two quadratic Dirichlet characters.

Assume henceforth that ψ^{2} ≠ 1, which amounts to saying that V_{ψ} is irreducible. Choose a character α ∈ Π_{E/K} which differs from ψ and ψ^{−1}, and for which V_{ψ} ⊕ V_{α} is regular. If K is real, assume α to be of opposite signature to ψ. This choice is possible thanks to the non-vanishing hypothesis, and implies that complex conjugation acts on V_{ψ} ⊕ V_{α} with eigenvalues (1, 1, −1, −1).

Invoking Lemma 6.9, let ψ_{0} and α_{0} be ray class characters of K of conductors prime to N_{E} and satisfying ψ_{0}/ψ′_{0} = ψ and α_{0}/α′_{0} = α, and set

ψ_{g} := ψ_{0}α_{0}, ψ_{h} := (ψ′_{0}α_{0})^{−1}.

The two-dimensional representations

ϱ_{g} := Ind_{K}^{ℚ}ψ_{g}, ϱ_{h} := Ind_{K}^{ℚ}ψ_{h}

satisfy the following properties:

• det(ϱ_{g}) = det(ϱ_{h})^{−1}, because the central characters of ψ_{g} and ψ_{h} are inverses of each other,

• ϱ_{g} and ϱ_{h} are odd: this is automatic when K is imaginary; when K is real this follows because ψ and α have opposite signature, hence exactly one of ψ_{0} and α_{0} is of mixed signature, implying that both ψ_{g} and ψ_{h} are of mixed signature.

• ϱ_{g} and ϱ_{h} are irreducible: this amounts to saying that ψ_{g} ≠ ψ′_{g} and ψ_{h} ≠ ψ′_{h}; tracing the definitions, this holds because α has been chosen to be different from ψ and ψ^{−1};

• the conductors of ϱ_{g} and ϱ_{h} are prime to N_{E}, and hence the same is true for the levels of the

associated weight one modular forms. The tensor product ϱ_{gh} decomposes as

(158)  V_{gh} = V_{ψ_{g}} ⊗ V_{ψ_{h}} = V_{ψ_{g}ψ_{h}} ⊕ V_{ψ_{g}ψ′_{h}} = V_{ψ} ⊕ V_{α}.

Thorem 6.10 now follows from the regularity of V_{g} and V_{h} and from Theorem 6.7, since

L(E, V_{gh}, 1) = L(E, V_{ψ}, 1)L(E, V_{α}, 1) = L(E/K, ψ, 1)L(E/K, α, 1) ≠ 0.
□

We now turn to the proof of Corollary A2 of the introduction:

Theorem 6.11. Let E be an elliptic curve over ℚ and let K be a non-real quintic extension of ℚ with Galois group A_{5}, of discriminant prime to the conductor of E. Then

ord_{s=1}L(E/K, s) = ord_{s=1}L(E/ℚ, s) ⇒ rank(E(K)) = rank(E(ℚ)).

Proof. Let K̃ denote the Galois closure of K and fix an embedding of the group A_{5} into PGL_{2}(ℂ). There are exactly two conjugacy classes of such embeddings, and the further choice of an isomorphism Gal (K̃/ℚ) = A_{5} gives rise to a projective representation

ϱ̄_{g} : G_{ℚ} ↠ Gal (K̃/ℚ) ≃ A_{5} ⊂ PGL_{2}(ℂ).

Tate’s lifting theorem produces a linear lift ϱ_{g} : G_{ℚ} ⟶ GL_{2}(ℂ) of ϱ̄_{g}, which is odd since K is not totally real. An argument similar to the one concluding the proof of Lemma 6.9 also shows that ϱ_{g} can be chosen to unramified at the primes dividing N_{E}.

Assume that the field H cut out by ϱ_{g} is a cyclic extension of K̃ of degree a power of 2. The image of Gal (H/K̃) under ϱ_{g} consists of scalar matrices whose diagonal entries are 2-power roots of unity. Let σ be an automorphism of ℂ which agrees with complex conjugation on the 2-power roots of unity and sends \sqrt{5} to −\sqrt{5}, and let ϱ_{h} be the Artin representation obtained by applying σ to the matrix entries of ϱ_{g}. The tensor product ϱ_{gh} is a four-dimensional representation which factors through Gal (K̃/ℚ) = A_{5} since it is trivial on Gal (H/K̃). To identify this representation, observe that there are precisely two distinct irreducible two-dimensional representations, denoted ϱ_{1} and ϱ_{2}, of the non-trivial central extension Ã_{5} of A_{5}, whose traces lie in ℚ(\sqrt{5}) and which are conjugate to each other over this field. Viewing ϱ_{1} ⊗ ϱ_{2} as a representation of Gal (K̃/ℚ), one has

ϱ_{gh} = ϱ_{1} ⊗ ϱ_{2} = (Ind_{K}^{ℚ}1) − 1,

where the last equality of virtual representations can be seen, for instance, by consulting the Atlas of finite simple groups. The Artin formalism implies that

r_{an}(E, ϱ_{gh}) = r_{an}(E/K) − r_{an}(E/ℚ), r(E, ϱ_{gh}) = r(E/K) − r(E/ℚ).

By hypothesis, r_{an}(E, ϱ_{gh}) = 0. Theorem 6.7 can now be invoked to conclude that r(E, ϱ_{gh}) = 0, i.e., r(E, K) = r(E, ℚ). Theorem 6.11 follows. □

6.4. Proof of Theorems B and D. We conclude with the proof of Theorem D of the introduction, which in turn implies Theorem B. Assume for this section that

(i) L(E, ϱ_{gh}, 1) = 0; (ii) the Artin representations ϱ_{g} and ϱ_{h} are both regular at p.

Let V_{fgh} and V_{fgh}(N) = V_{f}(Np) ⊗ V_{g}(Np) ⊗ V_{h}(Np) denote the K_{fgh}-vector spaces obtained by specialising of 𝕍_{f\underline{gh}} and V_{f\underline{gh}}(N) at the weight one point (y, z) attached to (f, g, h). As discussed right after (76), the global cohomology classes

(159)  κ(f, g_{α}, h_{α}), κ(f, g_{α}, h_{β}) ∈ H^{1}(ℚ, V_{fgh}(N))

are both locally trivial at primes ℓ ≠ p. Hence the vanishing of L(E, ϱ_{gh}, 1) implies, by Theorem 6.4, that κ(f, g_{α}, h_{α}) and κ(f, g_{α}, h_{β}) belong to the Selmer group Sel_{p}(E, V_{gh}(N)), which can be identified with a finite sum of copies of the Selmer group Sel_{p}(E, ϱ_{gh}).

Lemma 6.12. The images of κ(f, g_{α}, h_{α}) and κ(f, g_{α}, h_{β}) in H_{fin}^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{g}^{α_{g}}(Np) ⊗ V_{h}(Np)) belong to H_{fin}^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{g}^{α_{g}}(Np) ⊗ V_{h}^{β_{h}}(Np)) and to H_{fin}^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{g}^{α_{g}}(Np) ⊗ V_{h}^{α_{h}}(Np)) respectively.

Proof. This follows directly from Proposition 2.7. □

Theorem 6.13. Assume that ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) ≠ 0 for some choice

f̆ ∈ S_{2}(Np)[f],  ğ^{∗} ∈ S_{1}(N, χ^{−1})^{∨}[g],  h̆ ∈ S_{1}(N, χ^{−1})[h]

of test vectors. Then there exist G_{ℚ}-equivariant projections j_{1}, j_{2} : V_{fgh}(N) → V_{fgh} for which the global classes

(160)  κ_{αα} := j_{1}(κ(f, g_{α}, h_{α})),  κ_{αβ} := j_{2}(κ(f, g_{α}, h_{β}))

are linearly independent in Sel_{p}(E, ϱ_{gh}).

Proof. Let \underline{g} and \underline{h} be Hida families specialising to g_{α} and to h_{α} respectively in weight one, i.e., for which g_{y} = g_{α} and h_{z} = h_{α} for suitable weight one points (y, z) ∈ Ω\underline{g} × Ω\underline{h}. By Proposition 5.1 combined with Theorem 5.3, the non-vanishing of

ℒ_{p} ^{g_{α}}(f̆, ğ^{∗}, h̆) := ℒ_{p} ^{g_{α}}(f̆, ˘ \underline{g}^{∗}, ˘ \underline{h})(y, z)

implies that the local class κ_{p}^{g}(f̆, ğ_{α}^{∗}, h̆_{α}) has non-zero p-adic logarithm. It follows a fortiori from (94) and (95) that the local class κ_{p}^{g}(f, \underline{gh})(y, z) is non-zero, and therefore, that the natural image of the global class κ(f, g_{α}, h_{α}) in H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{g}^{α_{g}}(Np) ⊗ V_{h}(Np)) is non-zero as well. The same argument in which h_{α} is replaced by h_{β} implies (in light of equation (9) of the introduction) the same conclusion for the global class κ(f, g_{α}, h_{β}).

After choosing a basis of Hom_{G_{ℚ}}(V_{fgh}(N), V_{fgh}) (of cardinality t, say), the resulting isomorphism V_{fgh}(N) ⟶ ⊕_{i=1}^{t}V_{fgh} induces an isomorphism

H^{1}(ℚ_{p}, V_{f}(Np) ⊗ V_{g}^{α_{g}}(Np) ⊗ V_{h}(Np)) ⟶ ⊕_{i=1}^{t}H^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{α_{g}} ⊗ V_{h})

on the quotients of the local cohomology groups at p. Hence there are G_{ℚ}-equivariant homomorphisms j_{1}, j_{2} : V_{fgh}(N) → V_{fgh} for which the global classes κ_{αα} and κ_{αβ} defined in (160) have non-trivial image in the quotient H^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{α_{g}} ⊗ V_{h}).

As argued above right after (159), the global classes κ(f, g_{α}, h_{α}) and κ(f, g_{α}, h_{β}) lie in the Selmer group Sel_{p}(E, V_{gh}(N)) and hence κ_{αα} and κ_{αβ} likewise belong to Sel_{p}(E, ϱ_{gh}). Lemma 6.12 further implies that their natural images in H^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{α_{g}} ⊗ V_{h}) belong to the complementary subspaces H_{fin}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{α_{g}} ⊗ V_{h}^{β_{h}}) and H_{fin}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{α_{g}} ⊗ V_{h}^{α_{h}}) respectively. These images are therefore linearly independent, and the theorem follows. □

References

[AL78] A.O.L. Atkin, W. Li, Twists of newforms and pseudo-eigenvalues of W -operators, Invent. Math. 48 (1978), no. 3, 221–243.

[BeDi] J. Bellaïche, M. Dimitrov, On the eigencurve at classical weight one points, to appear in Duke Math. J.

[BCDDPR] M. Bertolini, F. Castella, H. Darmon, S. Dasgupta, K. Prasanna, and V. Rotger, p-adic L-functions and Euler systems: a tale in two trilogies, in Automorphic forms and Galois representations, vol. 1, LMS Lecture Notes 414, CUP (2014) 52–102.

[BD97] M. Bertolini, H. Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications (with an appendix by B. Edixhoven), Annals Math. (2) 146 (1997), no. 1, 111–147.

[BD99] M. Bertolini, H. Darmon, Euler systems and Jochnowitz congruences, Amer. J. Math. 121 (1999), 259–281.

[BD05] M. Bertolini, H. Darmon, Iwasawa’s main conjecture for elliptic curves over anticyclotomic ℤ_{p}-extensions, Annals Math. (2) 162 (2005), no. 1, 1–64.

[BDD07] M. Bertolini, H. Darmon, S. Dasgupta, Stark-Heegner points and special values of L-series, in L-Functions and Galois Representations, London Math. Soc. 320, 1–23, D. Burns, K. Buzzard, J. Nekovar, eds., (2007).

[BDR1] M. Bertolini, H. Darmon and V. Rotger, Beilinson-Flach elements and Euler Systems I: syntomic regulators and p-adic Rankin L-series, J. Algebraic Geometry 24 (2015), 355–378.

[BDR2] M. Bertolini, H. Darmon, V. Rotger, Beilinson-Flach elements and Euler systems II: p-adic families and the Birch and Swinnerton-Dyer conjecture, J. Algebraic Geometry 24 (2015), 569–604.

[Be00] A. Besser, A generalization of Coleman’s p-adic integration theory, Invent. Math. 142 (2000), 397–434.

[BK93] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives in The Grothendieck Festschrift I, Progr. Math. 108, 333-400 (1993), Birkhauser.

[BLZ] A. Besser, D. Loefffler, S. Zerbes, Finite polynomial cohomology for general varieties, submitted.

[BrEm10] C. Breuil, M. Emerton, Representations p-adique ordinaires de GL_{2}(ℚ_{p}) et compatibilite local-global, Astérisque 331 (2010), 235–295.

[Co97] R. F. Coleman, Classical and overconvergent modular forms of higher level, J. Th. Nombres Bordeaux 9 (1997), 395–403.

[CI99] R. Coleman, A. Iovita, Frobenius and Monodromy Operators for Curves and Abelian Varieties, Duke J. Math. 97 (1999), 171–217.

[CI10] R. Coleman, A. Iovita, Hidden structures on semistable curves. Astérisque 331 (2010), 179–254.

[Da] H. Darmon, Integration on H_{p} × H and arithmetic applications, Annals Math. 154 (2001), 589–639.

[DDT] H. Darmon, F. Diamond, R. Taylor, Fermat’s last theorem, in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), 2–140, Int. Press, Cambridge, MA, 1997.

[DLR] H. Darmon, A. Lauder, V. Rotger, Stark points and p-adic iterated integrals attached to modular forms of weight one, to appear in Forum of Mathematics, Pi.

[DR13] H. Darmon, V. Rotger, Diagonal cycles and Euler systems I: a p-adic Gross-Zagier formula, Annales Scientifiques de l’Ecole Normale Supérieure 47, n. 4, 779–832.

[DR15] H. Darmon, V. Rotger, Elliptic curves of rank two and generalised Kato classes, submitted.

[DS05] F. Diamond, J. Shurman, A first course in Modular forms, Grad. Texts Math. 228, 2005 Springer-Verlag.

[Em99] M. Emerton, A new proof of a theorem of Hida, Internat. Math. Res. Notices 9 (1999), 453–472.

[Fa89] G. Faltings, Crystalline cohomology and p-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 25–80, Johns Hopkins Univ. Press, Baltimore, MD, 1989.

[Fa97] G. Faltings, Crystalline cohomology of semistable curves –the ℚ_{p}-theory, J. Alg. Geom. 6 (1997), 1–18.

[Fl90] M. Flach, A generalisation of the Cassels-Tate pairing, J. reine angew. Math. 412 (1990), 113–127.

[Gr89] R. Greenberg, Iwasawa theory for p-adic representations, Adv. Studies Pure Math. 17 (1989), 97–137.

[GZ86] B. Gross, D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320.

[Hi85] H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms I, Invent. Math. 79 (1985), 159-195.

[Hi86] H. Hida, Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cuspforms, Invent. Math. 85 (1986), 545–613.

[Hi93] H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993.

[Ja] U. Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), 207–245.

[Ka98] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques III, Astérisque 295 (2004), ix, 117–290.

[KM85] N. Katz, B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, Princeton, NJ, 1985.

[Ko89] V. Kolyvagin, Finiteness of E(Q) and LLI(E, Q) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 670–671; translation in Math. USSR-Izv. 32 (1989), no. 3, 523–541.

[Liu1] Y. Liu. Gross-Kudla-Schoen cycles and twisted triple product Selmer groups. Preprint.

[Liu2] Y. Liu, Bounding cubic-triple product Selmer groups of elliptic curves. Preprint.

[LLZ1] A. Lei, D. Loeffler, S. L. Zerbes, Euler systems for Rankin-Selberg convolutions of modular forms, Annals of Math. 180 (2014), no. 2, 653–771.

[LZ12] D. Loeffler, S. L. Zerbes, Iwasawa theory and p-adic L-functions over ℤ_{p}^{2}-extensions, Int. J. Number Theory 10 (2014), no. 8, 2045–2095.

[LRV13] M. Longo, V. Rotger, S. Vigni. Special values of L-functions and the arithmetic of Darmon points, J. reine angew. Math. 684 (2013), 199–244.

[MW84] B. Mazur and A. Wiles, Class fields of abelian extensions of ℚ, Invent. Math. 76 (1984), no. 2, 179–330.

[Mi] J.S. Milne, Lectures on Étale cohomology (v2.21), available at www.jmilne.org/mat/, 202 pp.

[Ne1] J. Nekovář, Syntomic cohomology and p-adic regulators, preprint 1998.

[Ne2] J. Nekovář, p-adic Abel-Jacobi maps and p-adic heights, in The Arithmetic and Geometry of Algebraic Cycles (Banff, Canada, 1998), CRM Proc. Lect. Notes 24 (2000), 367–379.

[Ne3] J. Nekovář, Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two, Canad. J. Math. 64 (2012), no. 3, 588–668.

[NeNi] J. Nekovář, W. Niziol, Syntomic cohomology and p-adic regulators for varieties over p-adic fields, preprint.

[Oh95] M. Ohta, On the p-adic Eichler-Shimura isomorphism for Λ-adic cusp forms, Crelle 463 (1995), 49–98.

[Pr] D.Prasad, Trilinear forms for representations of GL_{2} and local epsilon factors, Comp. Math. 75 1990,1-46.

[Sa97] T. Saito, Modular forms and p-adic Hodge theory, Invent. Math. 129 (1997), 607–620.

[SU] C. Skinner, E. Urban, Vanishing of L-functions and ranks of Selmer groups, Proceedings of the International Congress of Mathematicians, Vol. 2., (2006), 473–500.

[Wa] P. Wake, The Λ-adic Eichler-Shimura isomorphism and p-adic étale cohomology, submitted.

[W88] A. Wiles, On ordinary Λ-adic representations associated to modular forms, Inv. Math. 94 (1988), 529-573.

H. D.: Department of Mathematics, McGill University, Montréal, Canada

   E-mail address: darmon@math.mcgill.ca

V. R.: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

   E-mail address: victor.rotger@upc.edu