BEILINSON-FLACH ELEMENTS AND EULER SYSTEMS II: THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR HASSE-WEIL-ARTIN L-SERIES

MASSIMO BERTOLINI, HENRI DARMON AND VICTOR ROTGER

Abstract. Let E be an elliptic curve over ℚ and let ϱ be an odd, irreducible two-dimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-Weil-Artin L-series L(E, ϱ, s), namely, the implication

L(E, ϱ, 1) ≠ 0 ⇒ (E(H) ⊗ ϱ)^{Gal(H/ℚ)} = 0, where H is the finite extension of ℚ cut out by ϱ. The proof relies on p-adic families of global Galois cohomology classes arising from Beilinson-Flach elements in a tower of products of modular curves.

Contents

Introduction 1 1. Strategy of proof 3 2. A p-adic Beilinson formula 6 2.1. Siegel units and Eisenstein series 6 2.2. Beilinson-Flach elements 7 2.3. Etale and syntomic regulators 7 2.4. The rigid cohomology of modular curves 10 2.5. Analytic description of the syntomic regulator 11 2.6. Hida’s p-adic L-function 15 3. Λ-adic classes 17 3.1. Norm-compatible systems of elements 17 3.2. The Λ-adic Beilinson-Flach cohomology class 18 3.3. The explicit reciprocity law 19 References 21

Introduction

This paper completes the project undertaken in [BDR12] of proving the Birch and Swinnerton-Dyer conjecture in analytic rank 0 for the ϱ-isotypic components of Mordell-Weil groups of elliptic curves, when ϱ is an odd, irreducible two-dimensional Artin representation.

In the early 90’s, Kato unveiled a new approach to the Birch and Swinnerton-Dyer conjecture and to the cyclotomic Iwasawa main conjecture for (modular) elliptic curves, resting on Beilinson elements in the second K-groups of modular curves and on their p-adic deformations. This approach led to a proof of the implication

L(E, χ, 1) ≠ 0 ⇒ Hom_{G_{ℚ}}(χ, E(ℚ̄) ⊗ ℂ) = 0,

for all elliptic curves E over ℚ and all Dirichlet characters χ (viewed as Galois characters via class field theory). The article [BCDDPR] places Kato’s strategy in the broader framework of “Euler systems of Garrett-Rankin-Selberg type”. Roughly speaking, the global objects exploited by Kato are distinguished elements in the higher Chow group CH^{2}(X_{1}(Np^{s}), 2) attached to a pair of modular units, whose logarithmic derivatives are Eisenstein series of weight two. Replacing either one, or both, of these Eisenstein series by cuspidal eigenforms leads to the study of Beilinson-Flach elements in CH^{2}(X_{1}(Np^{s})^{2}, 1) and of Gross-Kudla-Schoen diagonal cycles in the Chow group CH^{2}(X_{1}(Np^{s})^{3}) of codimension two cycles on the triple product of X_{1}(Np^{s}). Chapters 2.2 and 2.3. of [BCDDPR] explain how the images of these elements under p-adic étale regulators and Abel-Jacobi maps, when made to vary in p-adic families and specialised at suitable classical weight one points, might be parlayed into a proof of the implications

(1)  L(E, ϱ, 1) ≠ 0 ⇒ Hom_{G_{ℚ}}(ϱ, E(ℚ̄) ⊗ ℂ) = 0,
(2)  L(E, ϱ_{1} ⊗ ϱ_{2}, 1) ≠ 0 ⇒ Hom_{G_{ℚ}}(ϱ_{1} ⊗ ϱ_{2}, E(ℚ̄) ⊗ ℂ) = 0,

where ϱ, ϱ_{1} and ϱ_{2} are odd, irreducible two-dimensional (complex) Artin representations for which ϱ_{1} ⊗ ϱ_{2} is self-dual. The details of this program are carried out in [DR2] to obtain the proof of (2). The goal of the present work is to deliver on the other promise made in [BCDDPR] by proving the following result:

Theorem A. Let E be an elliptic curve over ℚ and let ϱ be an odd, irreducible, two-dimensional Artin representation. Assume that the conductors of E and ϱ are prime to each other. Then (1) holds.

Prior to Theorem A, the only irreducible two-dimensional Artin representations for which (1) had been proved were those induced from ring class characters of

(i) imaginary quadratic fields, by a series of works building on the methods of Kolyvagin [Ko89]; (ii) real quadratic fields, under a mild analytic non-vanishing hypothesis, by Corollary A1 of [DR2].

In addition to Artin representations with projective image isomorphic to A_{4}, S_{4} and A_{5}, Theorem A also applies to a large class of two-dimensional representations induced from general ray class characters of quadratic fields:

Theorem B. Let E be an elliptic curve over ℚ of conductor N, let K be a quadratic field of discriminant D, and let ψ : Gal (H/K) ⟶ ℂ^{×} be a ray class character of conductor 𝔣 ⊲ O_{K}. Assume that gcd(N, D · norm(𝔣)) = 1, and that ψ is of mixed signature if K is real quadratic. Then

L(E/K, ψ, 1) ≠ 0 ⇒ E(H)^{ψ} = 0,

where

E(H)^{ψ} := {P ∈ E(H) ⊗ ℂ s.t. σP = ψ(σ)P, ∀σ ∈ Gal (H/K)} .

Even for imaginary quadratic fields, Theorem B goes well beyond what can be obtained by the methods of Kolyvagin which apply only to ring class characters. For real quadratic K, there is simply no overlap between Theorem B and [DR2, Cor. A1], since ring class characters are either totally even or totally odd.

It is worth pointing out that the representation ϱ arising in Theorem A is typically not self-dual. Because of this, the root number arising in the functional equation of L(E, ϱ, s) does not control the parity of its order of vanishing at s = 1, and it is expected that L(E, ϱ, 1) vanishes only in rare, sporadic instances, so that the non-vanishing hypothesis in Theorem A is almost always satisfied. Any quantitative non-vanishing statement along these lines would find immediate, unconditional applications to the arithmetic of elliptic curves via Theorems A and B. It should also be noted that L(E, ϱ, 1) ≠ 0 if and only if L(E, ϱ^{∨}, 1) ≠ 0, where ϱ^{∨} is the contragredient representation of ϱ. Likewise, the triviality of the complex vector space Hom_{G_{ℚ}}(ϱ, E(ℚ̄) ⊗ ℂ) = (E(ℚ̄) ⊗ ϱ^{∨})^{G_{ℚ}} is equivalent to that of Hom_{G_{ℚ}}(ϱ^{∨}, E(ℚ̄) ⊗ ℂ).

The proof of Theorem A is based on so-called Beilinson-Flach elements whose definition is introduced in §2 below, and on their variation in p-adic families. For the purpose of the introduction, suffice it to say that Beilinson-Flach elements are distinguished elements in the higher Chow group CH^{2}(X_{1}(N)^{2}, 1), which can also be interpreted as the motivic cohomology group H_{M}^{3}(X_{1}(N)^{2}, ℚ(2)), They were first introduced in the 1980’s by Beilinson [Bei, Ch. 2, §6], who related their image under his complex regulator to the value at s = 2 of the Rankin L-series L(f ⊗ g, s) attached to the convolution of weight two newforms f and g on Γ_{1}(N).

In the early 1990’s, Flach [Fl] exploited (a slight variant of) these elements to prove the finiteness of the Selmer group of the symmetric square of an elliptic curve. Shortly thereafter, these finiteness results found a spectacular application in Wiles’ epoch-making proof of the Shimura-Taniyama conjecture and Fermat’s Last Theorem. Flach’s approach to bounding the Selmer group of the symmetric square featured prominently in Wiles’ original strategy, but was later obviated by the simpler and more flexible approach based on “Taylor-Wiles systems” introduced in [TW].

The power and versatility of Taylor-Wiles systems for questions surrounding the deformation theory of Galois representations may explain why, with a few exceptions ([MWe], [We02], …), Beilinson-Flach elements garnered relatively less attention in the intervening decades. Yet their arithmetic relevance is not confined to the symmetric square motive of a modular form: guided by an analogy between Beilinson-Flach elements and the study of Gross-Kudla-Schoen diagonal cycles undertaken in [DR1] and [DR2], the article [BDR12] established a p-adic analogue of Beilinson’s formula for Beilinson-Flach elements, pointing out a number of arithmetic applications of this formula based on the “Euler system philosophy” propounded by Kato and Perrin-Riou, and lying ostensibly outside the scope of the Taylor-Wiles method. (Cf. the introduction of loc.cit., and [BCDDPR, Sec. 2.2.].)

The recent work of Kings, Lei, Loeffler and Zerbes makes significant strides in furthering the program initiated in [BDR12], applying the ideas of the present work to more general settings and introducing key technical improvements along the way. Thus, [LLZ1] constructs a full cyclotomic Euler system of Beilinson-Flach elements and applies it to the study of the Selmer group of the tensor product of two motives attached to cusp forms, while [LLZ2] applies similar techniques to study the Iwasawa theory of elliptic curves over the full two-variable ℤ_{p}-extension of an imaginary quadratic field. The recent preprint [KLZ] exploits an interpolation of the Beilinson-Flach elements in all weights at once based on the notion of “Rankin-Iwasawa classes” arising in [Ki13], to establish a general explicit reciprocity law relating these families to Hida’s three variable p-adic Rankin L-function, with important applications to the associated three variable Main conjecture. It is worth pointing out that the approach to Theorem A in the present work rests solely on p-adic families of Beilinson-Flach elements of minimal level and on their local behaviour at p; it can therefore be compared to the strategy followed by Coates-Wiles [CoWi] to prove the finiteness of Mordell-Weil groups of CM elliptic curves. By exploiting the full collection of Beilinson-Flach elements and their Euler system properties, notably their “tame deformations” at auxiliary primes ℓ ≠ p, [KLZ] also establishes (just as in Thaine and Rubin’s strengthening of the Coates-Wiles method) the finiteness of certain p-parts of the relevant Selmer and Shafarevich-Tate groups in the setting of Theorem A.

Shortly after [BDR12], Dasgupta [Da] had the idea of exploiting Beilinson-Flach elements to factor Hida’s p-adic Rankin L-function of the tensor square f ⊗ f into the Kubota-Leopoldt p-adic zetafunction and the Coates-Schmidt p-adic L-function attached to the symmetric square of f. The main result of [Da] is a deep manifestation of the Artin formalism for Hida’s p-adic L-series and rests on a comparison between circular units and a new construction of units in number fields arising from Beilinson-Flach elements. Dasgupta’s approach, which is inspired by a theorem of Gross [Gr] for the Katz p-adic L-function of an imaginary quadratic field and builds on the ideas of the present work, also makes essential use of the improvements described in [KLZ].

Acknowledgements: The authors are grateful to Samit Dasgupta, Guido Kings, Antonio Lei, David Loeffler and Sarah Zerbes for many stimulating exchanges related to the circle of ideas explored in this paper. The work of the second author on this paper was partially supported by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach. The third author was supported by MTM20121-34611 during the ellaboration of this work.

1. Strategy of proof

Not surprisingly, modularity plays a key part in the proof of Theorem A. The elliptic curve E is associated to a weight 2 newform f = f_{E} ∈ S_{2}(N_{f}) thanks to [BCDT], while the Artin representation ϱ is associated to a weight 1 newform g = g_{ϱ} ∈ S_{1}(N_{g}, χ) thanks to [KW09]. The Hasse-Weil-Artin L-series L(E, ϱ, s) can therefore be identified with the convolution L-series L(f ⊗ g, s), whose analytic continuation follows from Rankin’s method.

Let L ⊂ ℚ^{ab} denote a field of coefficients over which ϱ can be described, and let H be the finite extension of ℚ cut out by ϱ (i.e., the fixed field of the kernel of ϱ). Write

E(H)_{L}^{ϱ} := hom_{G_{ℚ}}(V_{ϱ}, E(H) ⊗ L)

for the ϱ-isotypic part of the Mordell-Weil group of E. Theorem A asserts the triviality of this vector space when L(E, ϱ, 1) ≠ 0.

The standard approach to bounding E(H)_{L}^{ϱ} consists in embedding it in an appropriate Selmer group attached to the choice of a rational prime p, which throughout the paper is assumed to be prime to N_{f}N_{g}, and ordinary for f, i.e., p does not divide a_{p}(f). To define this Selmer group, let

V_{f} := V_{p}(E) := H^{1}(Eℚ̄, ℚ_{p})(1) = (lim_{←,n} E[p^{n}]) ⊗ ℚ_{p}

denote Deligne’s p-adic representation attached to f, which occurs as a direct factor of the étale cohomology H_{et}^{1}(X_{1}(N)ℚ̄, ℚ_{p})(1), and let V_{g} = V_{g,σ} denote the Artin representation attached to g, viewed as a p-adic representation by fixing some embedding σ of its coefficient field L into L_{p} ⊂ ℚ̄_{p}. The four dimensional L_{p}-vector space

V_{p}(E) ⊗ V_{g} = V_{f} ⊗ V_{g},

where the tensor product is taken over ℚ_{p}, is a p-adic representation of G_{ℚ}. The connecting homomorphism of Kummer theory arising from the p-power descent exact sequence on E(H), composed with the inverse of the restriction map from ℚ to H, gives a linear injection of L_{p}-vector spaces

E(H)_{L_{p}}^{ϱ} := E(H)_{L}^{ϱ} ⊗_{L} L_{p} ⟶ H^{1}(H, V_{p}(E) ⊗ V_{g}^{∨})^{Gal (H/ℚ)} ⟶^{res_{H}^{−1}} H^{1}(ℚ, V_{f} ⊗ V_{g}^{∨}),

denoted δ, where V_{g}^{∨} := V_{g} ⊗ det(V_{g})^{−1} is the L_{p}-linear dual of the representation V_{g}. For each place v of ℚ the corresponding map on local points yields an injection

δ_{v} : E(H_{v})_{L_{p}}^{ϱ} ⟶ H^{1}(ℚ_{v}, V_{f} ⊗ V_{g}^{∨}),

where Gal (H/ℚ) acts in the natural way on the local points E(H_{v}) := ⊕_{w|v}E(H_{w}). The image of δ_{v} in H^{1}(ℚ_{v}, V_{f} ⊗V_{g}^{∨}) is called the finite part of the local cohomology at v, and is denoted H_{fin}^{1}(ℚ_{v}, V_{f} ⊗V_{g}^{∨}). Likewise, H_{fin}^{1}(ℚ_{v}, V_{f} ⊗V_{g}) is defined to be the image of E(H_{v})^{ϱ^{∨}} by the appropriate local coboundary map.

When v = p, the finite part of the local cohomology admits an alternate description in terms of the general local conditions attached by Bloch and Kato [BK, §3] to any p-adic representation V of G_{ℚ_{p}}:

H_{exp}^{1}(ℚ_{p}, V) := ker (H^{1}(ℚ_{p}, V) → H^{1}(ℚ_{p}, V ⊗ B_{cris}^{ϕ=1})) ,
(3)  H_{fin}^{1}(ℚ_{p}, V) := ker (H^{1}(ℚ_{p}, V) → H^{1}(ℚ_{p}, V ⊗ B_{cris})) = Ext_{cris}^{1}(ℚ_{p}, V),
H_{geom}^{1}(ℚ_{p}, V) := ker (H^{1}(ℚ_{p}, V) → H^{1}(ℚ_{p}, V ⊗ B_{dR})) = Ext_{dR}^{1}(ℚ_{p}, V),

where B_{cris} ⊂ B_{dR} are Fontaine’s rings of periods for cristalline and de Rham representations, respectively, and ϕ denotes the cristalline frobenius. The subscripts exp, fin, and geom stand for “exponential”, “finite”, and “geometric” respectively, and one has the obvious inclusions

H_{exp}^{1}(ℚ_{p}, V) ⊂ H_{fin}^{1}(ℚ_{p}, V) ⊂ H_{geom}^{1}(ℚ_{p}, V).

When V = V_{f} ⊗ V_{g} or V_{f} ⊗ V_{g}^{∨}, the condition p ∤ N_{f}N_{g} shows, as in the proof of equation (18), that the Dieudonné module of V has no ϕ-invariant vectors (for reasons of weight), and thus

δ_{p}(E(H_{p})_{L_{p}}^{ϱ}) = H_{exp}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{∨}) = H_{fin}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{∨}) = H_{geom}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{∨}).

The cup product in Galois cohomology combined with the Weil pairing V_{f} × V_{f} → ℚ_{p}(1) gives rise to the perfect local Tate pairing

⟨ , ⟩_{v} : H^{1}(ℚ_{v}, V_{f} ⊗ V_{g}) × H^{1}(ℚ_{v}, V_{f} ⊗ V_{g}^{∨}) ⟶ H^{2}(ℚ_{v}, ℚ_{p}(1)) = ℚ_{p}.

For all v, the finite subspaces H_{fin}^{1}(ℚ_{v}, V_{f} ⊗ V_{g}) and H_{fin}^{1}(ℚ_{v}, V_{f} ⊗ V_{g}^{∨}) are orthogonal complements of each other under the local Tate pairing. Let

H_{sing}^{1}(ℚ_{v}, V_{f} ⊗ V_{g}) := \frac{H^{1}(ℚ_{v}, V_{f} ⊗ V_{g})}{H_{fin}^{1}(ℚ_{v}, V_{f} ⊗ V_{g})}

denote the singular quotient of the local cohomology at v, and write

∂_{v} : H^{1}(ℚ, V_{f} ⊗ V_{g}) ⟶ H_{sing}^{1}(ℚ_{v}, V_{f} ⊗ V_{g})

for the natural projection, which is referred to as the residue map at v. Both the finite part and the singular quotient of H^{1}(ℚ_{p}, V_{f} ⊗ V_{g}) are two-dimensional over L_{p}. By local Tate duality, similar statements hold for the finite part and the singular quotient of H^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{∨}).

The Selmer group attached to (E, V_{g}^{∨}) is the subgroup of H^{1}(ℚ, V_{f} ⊗V_{g}^{∨}), denoted H_{fin}^{1}(ℚ, V_{f} ⊗V_{g}^{∨}), which fits into the cartesian square

\xymatrix{H_{fin}^{1}(ℚ, V_{f} ⊗ V_{g}^{∨}) \ar[r] \ar[d] & H^{1}(ℚ, V_{f} ⊗ V_{g}^{∨}) \ar[d] \\ ∏_{v} H_{fin}^{1}(ℚ_{v}, V_{f} ⊗ V_{g}^{∨}) \ar[r] & ∏_{v} H^{1}(ℚ_{v}, V_{f} ⊗ V_{g}^{∨}).}

The Selmer group attached to (E, V_{g}) is defined likewise, by replacing V_{g}^{∨} by V_{g}.

The standard approach for bounding E(H)_{L}^{ϱ} rests on the following statement whose proof is explained in Prop. 6.3 of [DR2].

Lemma 1.1. Assume that the residue map ∂_{p} attached to the p-adic representation V_{f} ⊗ V_{g,σ} is a surjective map of L_{p}-vector spaces, for all embeddings σ : L ⟶ L_{p}. Then E(H)_{L}^{ϱ} = 0.

The proof of theorem A has thus been reduced to the problem of constructing two independent classes in the global cohomology H^{1}(ℚ, V_{f} ⊗V_{g}) whose images generate the singular quotient H_{sing}^{1}(ℚ_{p}, V_{f} ⊗V_{g}). Let α_{g} and β_{g} be the two (not necessarily distinct) roots of the Hecke poynomial

x^{2} − a_{p}(g)x + χ(p) = (x − α_{g})(x − β_{g})

attached to g at p, and let g_{α} and g_{β} denote the associated p-stabilisations of g satisfying

U_{p}g_{α} = α_{g}g_{α}, U_{p}g_{β} = β_{g}g_{β}.

When α_{g} ≠ β_{g}, these stabilisations are distinct, and the unramified representation V_{g} decomposes as a direct sum

V_{g} = V_{g}^{α} ⊕ V_{g}^{β}

of one-dimensional eigenspaces for the arithmetic frobenius element at p attached to these two eigenvalues. When α_{g} = β_{g}, set V_{g}^{α} := V_{g}. Theorem A will now follow from the following result.

Theorem 1.2. There exists a global cohomology class

κ(f, g_{α}) ∈ H^{1}(ℚ, V_{f} ⊗ V_{g})

attached to to the p-stabilised form g_{α}, satisfying:

(1) The residue ∂_{p}(κ(f, g_{α})) belongs to H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}^{α});

(2) it is non-zero if and only if L(f ⊗ g, 1) ≠ 0.

To see why Theorem 1.2 implies Theorem A, choose the descent prime p in such a way that the frobenius element σ_{p} acts on V_{g} with distinct eigenvalues α_{g} and β_{g}. Such a choice is always possible, in view of the Chebotarev density theorem and the fact that the two-dimensional V_{g} is irreducible and hence the image of ϱ contains a non-scalar matrix. When L(E, ϱ, 1) ≠ 0, Theorem 1.2 implies that the residues of the global classes κ(f, g_{α}) and κ(f, g_{β}) generate the two-dimensional L_{p}-vector space H_{sing}^{1}(ℚ_{p}, V_{f} ⊗ V_{g}). Since this reasoning applies to the representation V_{g,σ} for any embedding σ of L into L_{p}, Theorem A follows from Lemma 1.1.

The “singular” class κ(f, g_{α}) of Theorem 1.2 is not expected to arise from the images of algebraic cycles or elements in K-theory under an appropriate étale Abel-Jacobi or regulator map, since such images are typically cristalline, at least in scenarios of good reduction of the ambient variety. It is constructed instead as a p-adic limit of classes κ(f, g_{x}) attached to (sufficiently many) weight two specialisations g_{x} of a Hida family specialising to g_{α} in weight one. These “weight two classes” are obtained via a geometric construction, as the images of Beilinson-Flach elements under the p-adic étale regulator map to Galois cohomology.

Section 2 describes the construction of a geometric class κ(f, g) attached to a classical eigenform g of weight two and p-power nebentypus character, and summarises its key properties, including the relation between its p-adic logarithm and the values of Hida’s p-adic Rankin L-function attached to f and g. This relation extends the main result of [BDR12] to settings where the modular curve has bad reduction at the prime p, and relies crucially on the work of Nekovar and Niziol [NeNi] on syntomic cohomology and p-adic regulators for arbitrary semistable varieties, which in turn allowed Besser, Loeffler and Zerbes [BLZ] to generalize Besser’s finite polynomial cohomology to this setting.

Section 3 explains how the weight two classes κ(f, g), as g ranges over all the classical specialisations of weight two and p-power nebentypus character of a Hida family \underline{g}, can be interpolated into a Λ-adic cohomology class κ(f, \underline{g}). Since these ordinary weight two points are dense in the rigid analytic topology on the relevant components of the Coleman-Mazur eigencurve, it follows that κ(f, \underline{g}) is uniquely determined by its weight two specialisations.

The class κ(f, g_{α}) of Theorem 1.2 is obtained by letting \underline{g} be a Hida family specialising to g_{α} in weight one, and specialising the resulting κ(f, \underline{g}) to g_{α}. The p-adic Beilinson formula of Section 2 is then parlayed into the explicit reciprocity law relating the Bloch-Kato dual exponential of κ(f, g_{α}) to the special value of the same p-adic L-function at the point attached to g_{α}. Since this latter point lies in the range of classical interpolation defining the p-adic L-function, it is directly related to a classical L-value, and this is what leads to the proof of Theorem 1.2.

2. A p-adic Beilinson formula

2.1. Siegel units and Eisenstein series. Fix a modulus M = Np^{s} ≥ 3 with p ∤ N and s ≥ 0. Let

Y_{s} := Y_{1}(Np^{s}) ⊂ X_{s} := X_{1}(Np^{s})

be canonical models over ℚ of the (affine and projective, respectively) modular curves classifying triples (A, i_{N}, i_{p}) where A is a (generalised) elliptic curve and i_{N} : μ_{N} → A and i_{p} : μ_{p^{s}} → A are embeddings of finite group schemes. The standard notations Y_{0}(Np) and X_{0}(Np)/ℚ are adopted for the affine and projective modular curves classifying pairs (A, C_{Np}) where C_{Np} ⊂ A is a cyclic subgroup scheme of A of order Np. We will denote by w_{M} : X_{1}(M) ⟶ X_{1}(M) the standard Atkin-Lehner involution, and will simply denote it as w when the level M is clear from the context.

For a ∈ (ℤ/Mℤ)^{×}, let 𝔤_{a;M} be the Siegel unit denoted g_{0,a/M} in [Ka98], whose q-expansion is given by

(4)  𝔤_{a;M} := (1 − ζ_{M}^{a})q^{1/12} ∏_{n=1}^{∞}(1 − ζ_{M}^{a}q^{n})(1 − ζ_{M}^{−a}q^{n}),

where ζ_{M} is a fixed primitive M-th root of unity. The unit 𝔤_{a;M} can naturally be viewed as belonging to O_{Y_{1}(M)}^{×} ⊗ ℚ, and its q-expansion is defined over ℚ(μ_{M}), while the unit

𝔤_{a;M}^{w} := w𝔤_{a;M}

is defined over ℚ. If χ is a primitive even Dirichlet character of conductor M, taking values in a field denoted ℚ(χ), let

G(χ) = ∑_{a=1}^{M} χ(a)ζ_{M}^{a}

denote the Gauss sum attached to χ and the choice of ζ_{M}, and set

(5)  𝔤_{χ} = \frac{1}{φ(M)} · ∑_{a=1}^{M} χ^{−1}(a) ⊗ 𝔤_{a;M} ∈ O_{Y_{1}(M)}^{×} ⊗ ℚ(χ),  𝔤_{χ}^{w} := w𝔤_{χ}.

Given a pair (χ_{1}, χ_{2}) of Dirichlet characters and an integer k ≥ 2 such that χ_{1}χ_{2}(−1) = (−1)^{k}, define also the Eisenstein series

(6)  E_{k}(χ_{1}, χ_{2}) = \{ \begin{matrix}\frac{1}{2} · L(χ_{2}, 1 − k) & if χ_{1} = 1 \\ 0 & otherwise \end{matrix} \} + ∑_{n=1}^{∞} (∑_{d|n} χ_{1}(n/d)χ_{2}(d)d^{k−1}) q^{n}.

A direct calculation using (4), (5) and (6) shows that the logarithmic derivative of 𝔤_{χ} is

(7)  d log 𝔤_{χ} = \frac{2G(χ^{−1})}{φ(M)} · E_{2}(χ, 1)\frac{dq}{q} ∈ Ω_{Y_{1}(M)}^{1}.

2.2. Beilinson-Flach elements. The Beilinson-Flach elements of this section are elements in the higher Chow group over ℚ of the surface

S_{s} := X_{0}(Np) × X_{s}.

For any field F, let C^{2}(S_{s}, 1)_{F} denote the group of finite formal linear combinations D = ∑_{i} a_{i}(Z_{i}, u_{i}) of pairs (Z_{i}, u_{i}) with coefficients a_{i} ∈ F, where Z_{i} are irreducible curves embedded in S_{s} and u_{i} are rational functions on Z_{i}. Write Z^{2}(S_{s}, 1)_{F} ⊂ C^{2}(S_{s}, 1)_{F} for the subspace of elements D satisfying

div(D) := ∑ a_{i}div(u_{i}) = 0.

Any pair {φ, ψ} of non-zero rational functions on S_{s} gives rise to the element ∑_{i}(Z_{i}, u_{i}) in Z^{2}(S_{s}, 1)_{F}, where {Z_{i}} is the set of irreducible components in S_{s} of the divisors of φ or ψ, and, after setting a_{i} = ord_{Z_{i}}(φ) and b_{i} = ord_{Z_{i}}(ψ), the rational function u_{i} on Z_{i} is given by the tame symbol u_{i} = ∏(−1)^{a_{i}b_{i}}(\frac{φ^{b_{i}}}{ψ^{a_{i}}})_{|Z_{i}}. Writing B^{2}(S_{s}, 1)_{F} for the subspace of Z^{2}(S_{s}, 1)_{F} formed by such elements, Bloch’s higher Chow group of the surface S_{s} with coefficients in F is defined as

CH^{2}(S_{s}, 1)_{F} = Z^{2}(S_{s}, 1)_{F} /B^{2}(S_{s}, 1)_{F} .

Letting

(8)  π̃_{s} : X_{s} ⟶ X_{0}(Np)

denote the natural forgetful projection of modular curves which is compatible with the U_{p} correspondence acting on both curves, define

(9)  ι_{s} = (π̃_{s}, Id) : X_{s} ↪ S_{s},

to be the closed embedding given by ι_{s}(x) = (π̃_{s}(x), x). Let

Δ_{s} := ι_{s}(X_{s}) ⊂ S_{s}

denote the resulting embedded curve in S_{s}.

Lemma 2.1. For any u ∈ O_{Y_{s}}^{×}, there exists a negligible element θ_{s} ∈ C^{2}(S_{s}, 1)_{F} satisfying

(10)  div θ_{s} = div(Δ_{s}, u).

Proof. The proof is given in Lemma 3.1. of [BDR12], and relies on the fact the the group of degree zero divisors supported at the cusps is torsion in the jacobian of X_{s} by the Manin-Drinfeld theorem. □

Thanks to Lemma 2.1, the class (Δ_{s}, u) − θ_{s} lies in Z^{2}(S_{s}, 1)_{ℚ}, and its image

BF(u) := [(Δ_{s}, u) − θ_{s}] in CH_{neg}^{2}(S_{s}, 1)_{ℚ}(ℚ)

is independent of the particular choice of θ_{s} satisfying (10). It is called the Beilinson-Flach element attached to the modular unit u ∈ O_{Y_{s}}^{×}.

Definition 2.2. The elements

BF(a; Np^{s}) := BF(𝔤_{a;Np^{s}}^{w}), BF_{s} := BF(1; Np^{s}) in CH_{neg}^{2}(S_{s}, 1)_{ℚ}(ℚ)

are called the Beilinson-Flach elements of level Np^{s}.

2.3. Etale and syntomic regulators. Let φ = ∑ a_{n}(φ)q^{n} ∈ S_{k}(N_{φ}, χ) be a normalized newform of weight k, level N_{φ} and character χ, and let ω_{φ} := φ(q)\frac{dq}{q} be the associated regular differential form on X_{1}(N_{φ}). Let K_{φ} be the finite extension of ℚ_{p} generated by the fourier coefficients of φ, let O_{φ} be its valuation ring, and let V_{φ} denote the two-dimensional Galois representation over O_{φ} associated by Deligne to φ. This Galois representation is pure of weight −1 and is characterized up to isomorphism by the property that for any prime ℓ ∤ pN_{φ}, the characteristic polynomial of an arithmetic Frobenius element Fr_{ℓ} is equal to the Hecke polynomial x^{2} − a_{ℓ}(φ)x + χ(ℓ)ℓ^{k−1}. Write α_{φ} and β_{φ} be the roots of this Hecke polynomial at ℓ = p, ordered so that ord_{p}(α_{φ}) ≤ ord_{p}(β_{φ}).

If N is a multiple of N_{φ} and 𝕋_{N} is the Hecke algebra generated by the good Hecke operators T_{ℓ} with ℓ ∤ N, the φ-isotypic component of a 𝕋_{N}-module S is defined to be

(11)  S[φ] := {v ∈ S ⊗ O_{φ} : T_{ℓ}(v) = a_{ℓ}(φ)v for all ℓ ∤ N}.

Generalising the setting of §1, and modifying slightly the notations therein, let f ∈ S_{2}(Np)[f_{0}] now denote a normalised eigenform with trivial Nebentypus character arising from a newform f_{0} on Γ_{0}(N_{f}) for some N_{f} | N. In the context of §1, f_{0} is the eigenform associated to the elliptic curve E. More specifically, we take

(12)  f(q) = ∑_{d| \frac{Np}{N_{f}}} λ_{d}f_{0}(q^{d}) ∈ S_{2}(Np)[f_{0}]

to be an eigenvector for all Hecke operators U_{ℓ}, ℓ | Np, and assume f is ordinary, i.e., that α_{f} = α_{f_{0}} is a p-adic unit.

In addition, for the rest of this section let g ∈ S_{2}(Np^{s}, χ · χ_{p}) be an eigenform of weight 2, level Np^{s} and nebentype character χχ_{p}, that we have factored in such a way that

cond(χ) | N, cond(χ_{p}) | p^{s}.

Assume g arises from a newform g_{0} of level N_{g}p^{s} for some N_{g} | N and that the conductor of χ_{p} is equal to p^{s}. In the terminology introduced by Mazur-Wiles [MWi84], g is said to be primitive at p. Assume also that g is ordinary at p, i.e., that α_{g} = a_{p}(g) is a p-adic unit.

For any variety X over ℚ, the notation X̄ := X × ℚ̄ will be used to designate the base change of X to ℚ̄. Let

(13)  V_{0}(Np) = H_{et}^{1}(X̄_{0}(Np), ℤ_{p})(1)  and V_{s} = H_{et}^{1}(X̄_{s}, ℤ_{p})(1) for s ≥ 1

denote the Galois representation afforded by the Tate module of X_{0}(Np) and X_{s}, respectively.

Let

pr_{1,1} : H_{et}^{2}(S̄_{s}, ℤ_{p})(2) ⟶ V_{0}(Np) ⊗ V_{s}

denote the natural projection of G_{ℚ}-modules induced by the Künneth decomposition, and let

reg_{et} : CH^{2}(S_{s}, 1)(ℚ) ⟶ H^{1}(ℚ, H_{et}^{2}(S̄_{s}, ℤ_{p})(2)) ⟶^{pr_{1,1}} H^{1}(ℚ, V_{0}(Np) ⊗ V_{s})

denote the composition of the p-adic étale regulator map (as defined for instance in chapter 2 of [Fl]) with the map induced by pr_{1,1} in Galois cohomology.

The image under reg_{et} of the subgroup generated by negligible element vanishes, for the cohomology class of such elements take values in the Künneth component

H_{et}^{2}(X̄_{0}(Np), ℤ_{p}) ⊗ H_{et}^{0}(X̄_{s}, ℤ_{p})(2) ⊕ H_{et}^{0}(X̄_{0}(Np), ℤ_{p}) ⊗ H_{et}^{2}(X̄_{s}, ℤ_{p})(2).

Hence the étale regulator descends to a map which we continue to denote with the same symbol

(14)  reg_{et} : CH_{neg}^{2}(S_{s}, 1)(ℚ) ⟶ H^{1}(ℚ, V_{0}(Np) ⊗ V_{s}).

Definition 2.3. The Beilinson-Flach cohomology classes of level Np^{s} are defined to be

κ(a; Np^{s}) := reg_{et}(BF(a; Np^{s})) ∈ H^{1}(ℚ, V_{0}(Np) ⊗ V_{s}),
κ_{s} := κ(1; Np^{s}).

We now turn to providing an analytic description of the restriction to G_{ℚ_{p}} of the classes κ_{s}, which will pave the way for establishing its relation to the values of the Hida-Rankin p-adic L-function associated to Hida families passing through f and g. A first step towards this task is the description of H_{fin}^{1}(ℚ_{p}, V_{fg}) in terms of the de Rham cohomology of the relevant modular curves.

Recall Fontaine’s ring B_{cris} of cristalline periods, and let

(15) D_{f} := (B_{cris} ⊗_{ℚ_{p}} V_{f})^{G_{ℚp}} , D_{g} := (B_{cris} ⊗_{ℚ_{p}} V_{g})^{G_{ℚp(μ_{p^{s}})}}, D_{fg} := (B_{cris} ⊗_{ℚ_{p}} V_{fg})^{G_{ℚp(μ_{p^{s}})}}

be the (potentially crystalline) Dieudonné modules associated to V_{f}, V_{g} and V_{fg}, respectively. Note that D_{fg} ≃ D_{f} ⊗ D_{g} in the category of filtered frobenius vector spaces over the field K. Let

(16)  g^{∗} = ḡ = ∑ ā_{n}(g)q^{n} ∈ S_{2}(Np^{s}, χ^{−1}χ_{p}^{−1}),  g^{w} := wg

be the modular forms which both belong to the automorphic representation attached to the contragredient of g; these modular forms are eigenvectors for the good Hecke operators, and

V_{g^{∗}} = V_{g} ⊗ χ^{−1}χ_{p}^{−1} = V_{g}^{∨}(1).

The Dieudonné module of the Kummer dual V_{fg}^{∨}(1) = V_{fg^{∗}}(−1) of V_{fg}, denoted D_{fg^{∗}}(−1), is canonically identified with D_{fg^{∗}} but with a shift in the Hodge filtration. The dimensions of the relevant pieces of the Hodge filtration on D_{fg} and D_{fg^{∗}}(−1) are summarised in the table below:

Fil^{−2} Fil^{−1} Fil^{0} Fil^{1} Fil^{2}
D_{fg}  4  3  1  0  0
D_{fg^{∗}}(−1) 4  4  3  1  0.

The Bloch-Kato logarithm log_{p} of [BK] associated to the Galois representation V_{fg} gives an isomorphism

log_{p} : H_{exp}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}) ⟶ \frac{D_{fg}}{Fil^{0}D_{fg} + D_{fg}^{ϕ=1}} ,

where H_{exp}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}) is as defined in (3), with ℚ_{p} replaced by ℚ_{p}(μ_{p^{s}}). The cristalline frobenius ϕ acts on D_{fg} with eigenvalues

(17)  α_{f}α_{g},  α_{f}β_{g},  β_{f}α_{g},  β_{f}β_{g}.

These eigenvalues are algebraic numbers of absolute value p (relative to any complex embedding of ℚ̄) and therefore D_{fg}^{ϕ=1} = 0. It follows from [BK, Cor. 3.8.4] that

(18)  H_{exp}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}) = H_{fin}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}).

The Bloch-Kato logarithm can therefore be recast in the current context as a map

(19)  log_{p} : H_{fin}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}) ⟶ \frac{D_{fg}}{Fil^{0}(D_{fg})} = Fil^{0}(D_{fg^{∗}}(−1))^{∨},

where the last identification arises from the natural duality between D_{fg} and D_{fg^{∗}}(−1), relative to which Fil^{0}(D_{fg}) and Fil^{0}(D_{fg^{∗}}(−1)) are orthogonal complements of each other.

Assume furthermore that none of the eigenvalues in (17) are equal to p. This assumption is satisfied with at most finitely many exceptions when f is fixed and g runs over the weight two specialisations of a Hida family, hence will suffice for the applications in this paper. It implies, by another application of [BK, cor. 3.8.4], that H_{exp}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg^{∗}}(−1)) = H_{fin}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg^{∗}}(−1)), and hence, by duality, that

(20)  H_{fin}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}) = H_{geom}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}).

The argument used in the proof of Lemma 2.8 of [Fl] shows that the image of the étale regulator reg_{et}^{fg} consists of extensions of Galois representations occurring in the étale cohomology of varieties with potentially semistable reduction at p, which hence are de Rham at p. This image is therefore contained in H_{geom}^{1}(ℚ_{p}(μ_{p^{s}}), V_{fg}), and thus by (18) in the finite subspace of the local cohomology.

Let

(21)  D_{0}(Np) := (B_{dR} ⊗ V_{0}(Np))^{G_{ℚp}} ,  D_{s} := (B_{dR} ⊗ V_{s})^{G_{ℚp(μ_{p^{s}})}},

be the de Rham Dieudonné filtered modules attached to the Galois representations V_{0}(Np) and V_{s}. Note that these p-adic Galois representations are B_{dR} but not B_{cris}-admissible, but that their f_{0} and g_{0}-isotypic parts, denoted V_{0}(Np)[f] ≃ V_{f}^{i} and V_{s}[g] ≃ V_{g}^{j}, are cristalline over ℚ_{p} and ℚ_{p}(μ_{p^{s}}) respectively.

The étale regulator map (14) on CH^{2}(S_{s}, 1)_{neg} and the Bloch-Kato logarithm associated to the Galois representation V_{0}(Np) ⊗ V_{s} give rise to a diagram of maps

(22) \xymatrix{CH^{2}(S_{s}, 1)_{neg}(ℚ_{p}(μ_{p^{s}})) \ar[r]^{^{reg^{et}}} & H_{geom}^{1}(ℚ_{p}(μ_{p^{s}}), V_{0}(Np) ⊗ V_{s}) \\  & H_{exp}^{1}(ℚ_{p}(μ_{p^{s}}), V_{0}(Np) ⊗ V_{s}) \ar[r]^{_{log_{p}}} \ar@{^{(}->}[u] & Fil^{0}(D_{0}(Np) ⊗ D_{s}(−1))^{∨}.}

Let

(23)  \xymatrix{reg_{p}^{fg} : CH^{2}(S_{s}, 1)(ℚ_{p}(μ_{p^{s}}))[fg] \ar[r] & Fil^{0}(D_{0}(Np)[f] ⊗ D_{s}(−1)[g^{∗}])^{∨}}

be the map obtained by applying the (f, g)-isotypic projection to (22), invoking (18) and (20) to show that the vertical inclusion becomes an isomorphism after restricting to these (f, g)-isotypic subspaces.

The main goal of the next sections is proving an analytic formula for the value of reg_{p}^{fg}(BF_{s}) at a suitable vector in Fil^{0}(D_{0}(Np)[f] ⊗ D_{s}(−1)[g^{∗}]), which we now proceed to describe.

By the comparison theorem between étale and de Rham cohomology, there are isomorphisms of two-dimensional filtered frobenius modules

(24)  D_{f} ≃ H_{dR}^{1}(X_{0}(N_{f})_{/ℚ_{p}})(1)[f_{0}], D_{g} ≃ H_{dR}^{1}(X_{1}(N_{g}p^{s})_{/ℚ_{p}(μ_{p^{s}})})(1)[g_{0}]

such that

Fil^{0}D_{f} ≃ Ω^{1}(X_{0}(N_{f}))[f_{0}] = ⟨ω_{f_{0}}⟩ and Fil^{0}D_{g} ≃ Ω^{1}(X_{1}(N_{g}p^{s}))[g_{0}] = ⟨ω_{g_{0}}⟩.

For any projective curve X/F, write

⟨ , ⟩_{X} : H_{dR}^{1}(X_{/F}) × H_{dR}^{1}(X_{/F}) ⟶ F

for the canonical Poincaré pairing on X. Let η_{f_{0}} be an element in the one-dimensional unit-root subspace of D_{f}, normalised so that

⟨ω_{f_{0}}, η_{f_{0}}⟩_{X_{0}(N_{f})} = 1.

Fix a finite extension K/ℚ_{p} containing K_{f} and K_{g}, and let O denote its valuation ring. For each positive divisor d of Np/N_{f}, write π_{d} : X_{1}(Np) ⟶ X_{1}(N_{f}) for the standard degeneracy map induced by multiplication by d on the upper half plane. Let

π_{d∗} : V_{0}(Np)[f_{0}] ⟶ V_{f}, V_{f} := V_{0}(N_{f})[f_{0}] = H_{et}^{1}(X̄_{0}(N_{f}), O)(1)[f_{0}]

be the map in étale cohomology induced by π_{d} by covariant functoriality. Our choice of f in S_{2}(Np)[f_{0}] stated in equation (12) determines the projection

(25)  ϖ_{f} := ∑ λ_{d}π_{d∗} : V_{0}(Np) ⊗ O ⟶ V_{f}.

Committing a slight abuse of notation, denote by ϖ_{f} the map induced by (25) by functoriality on Dieudonné modules and the first isomorphism in (24), and write ϖ_{f}^{∗} for the dual map arising from Poincaré duality:

(26)  ϖ_{f} : D_{0}(Np) ⟶ D_{f},  ϖ_{f}^{∗} : D_{f} ⟶ D_{0}(Np).

The maps ϖ_{g} : D_{s} ⟶ D_{g} and ϖ_{g}^{∗} : D_{g} ⟶ D_{s} are defined similarly, and we let

ϖ_{f}^{∗} ⊗ ϖ_{g}^{∗} : D_{fg} ⟶ D_{0}(Np) ⊗ D_{s}

denote the map induced by ϖ_{f}^{∗} and ϖ_{g}^{∗} and the Künneth decomposition.

Define

η_{f} := ϖ_{f}^{∗}(η_{f_{0}}) ∈ D_{0}(Np)[f],  ω_{g} := ϖ_{g}^{∗}(ω_{g_{0}}) ∈ D_{s}[g],  ω_{g^{w}} := wω_{g} ∈ D_{s}[g^{∗}].

Note that η_{f} is ordinary, by the compatibility of ϖ_{f}^{∗} with the U_{p}-operator. Since η_{f} belongs to Fil^{−1}(D_{f}) = D_{f} and ω_{g^{w}} belongs to Fil^{0}(D_{g^{∗}}) ⊂ Ω^{1}(X_{s}), the tensor product η_{f} ⊗ ω_{g^{w}} belongs to Fil^{−1}(D_{0}(Np)[f] ⊗ D_{s}[g^{∗}]). Because of the assumption on g leading to (20), the expression

reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}})

is defined. The main result of §2.5 is a formula for its value. As a preparation for that, we first recall in §2.4 a few needed facts on the rigid geometry of the modular curve X_{1}(Np^{s}).

2.4. The rigid cohomology of modular curves. This section briefly summarises the description of the de Rham cohomology of X_{s} viewed as a curve over a p-adic field, following the treatment in [DR2, §3.1], with suitable modifications aimed at giving a similar description of the cohomology of the affine curve Y_{s}. We adopt the same notations as in loc.cit., which the reader is encouraged to consult for a more detailed description of the objects that are used here.

More specifically, let F be any extension of ℚ_{p}(μ_{p^{s}}) over which X_{s} admits a semistable model, and over which the closed points of Σ := X_{s} − Y_{s} are defined.

The étale cohomology groups of X_{s} and Y_{s} are related by the Gysin exact sequence of semistable representations of G_{F} = Gal (F̄/F):

(27)  0 ⟶ H_{et}^{1}(X̄_{s}, ℚ_{p}) ⟶ H_{et}^{1}(Ȳ_{s}, ℚ_{p}) ⟶ ℚ_{p}(−1)^{t} ⟶ 0,

where t = |Σ| − 1.

Let D_{st} be Fontaine’s comparison functor attached to the ring of semistable periods B_{st}. By applying D_{st} to the sequence (27), and invoking the comparison theorem between étale and de Rham cohomology, one obtains a corresponding sequence of filtered Frobenius monodromy (FFM) modules

(28)  0 ⟶ H_{dR}^{1}(X_{s}_{/F}) ⟶^{ι} H_{dR}^{1}(Y_{s}_{/F}) ⟶ F (−1)^{t} ⟶ 0.

(See for example [CI10], Proposition 7.6 for details.) Note that the above FFM modules are isomorphic as filtered Frobenius modules to those obtained by applying the construction of the potentially crystalline Dieudonné module of Section 2.3, in which B_{st} is replaced by B_{cris}. The use of B_{st} equips these modules with the additional structure of a monodromy operator.

In particular, it follows from (28) that the Frobenius operator Φ associated to Y_{s} acts on the subspace of H_{dR}^{1}(Y_{s}_{/F}) generated by classes of Eisenstein series with eigenvalues of absolute value p.

In order to make (28) more explicit, we recall, following [CI10], the Maier-Vietoris exact sequences describing the de Rham cohomology groups of X_{s} and Y_{s}. Fix a proper semistable model X_{s} of X_{s} over the ring of integers O_{F} of F, whose special fiber C is a union of smooth geometrically connected components C_{1}, …, C_{n} intersecting at ordinary double points defined over the residue field 𝔽 of F. Since s ≥ 1, there are at least two components, i.e., n ≥ 2. Write G for the dual graph of C, and V, resp. E for its set of vertices, resp. oriented edges. Recall that

V = {C_{1}, …, C_{n}}, E = {(x, C_{i}, C_{j}) : x ∈ C_{i} ∩ C_{j}}.

Given e = (x, C_{i}, C_{j}) in E, we say that s(e) := C_{i} is the source of e and t(e) := C_{j} is the target of e. Write

red : X_{s}(ℂ_{p}) ⟶ C(𝔽̄)

for the reduction map, and identify X_{s}(ℂ_{p}) with the rigid-analytic space over F attached to X_{s}. The wide-open space associated to v ∈ V is defined as W_{v} := red^{−1}(v). Its underlying connected affinoid space A_{v} is obtained as the inverse image by reduction of the component v with the singular points removed. The wide-open annulus associated to e = (x, v_{i}, v_{j}) ∈ E is defined as W_{e} := red^{−1}(x) (so that W_{e} = W_{v_{i}} ∩ W_{v_{j}}). The collection {W_{v}}_{v∈V} forms an admissible covering of X_{s}(ℂ_{p}) by wide open spaces. Likewise, an admissible covering of Y_{s}(ℂ_{p}) is given by {W′_{v}}_{v∈V}, where W′_{v} := W_{v} − Σ. Note that

W′_{v_{i}} ∩ W′_{v_{j}} = W_{v_{i}} ∩ W_{v_{j}} = W_{e},

with e = (x, v_{i}, v_{j}) ∈ E. Since H_{dR}^{1}(X_{s}/F) and H_{dR}^{1}(Y_{s}_{/F}) are identified with the Cech hypercohomology groups of their associated admissible coverings, they arise as the middle terms in the following commutative diagram originating from the Maier-Vietoris sequences attached to these coverings:

(29)

[figure: 0; \frac{⊕_{e∈E}H_{dR}^{0}(W_{e})}{δ(⊕_{v∈V} H_{dR}^{0}(W_{v}))}; i; =; H_{dR}^{1}(X_{s}/F); r; ι; ker (⊕_{v∈V}H_{dR}^{1}(W_{v}) ⟶^{δ} ⊕_{e∈E}H_{dR}^{1}(W_{e})); ῑ; 0; 0; \frac{⊕_{e∈E}H_{dR}^{0}(W_{e})}{δ(⊕_{v∈V} H_{dR}^{0}(W_{v}))}; i′; H_{dR}^{1}(Y_{s}_{/F}); r′; _{ker (⊕_{v∈V}H_{dR}^{1}(W′_{v}) ⟶^{δ} ⊕_{e∈E}H_{dR}^{1}(W_{e}))}; 0]

In (29), the various coboundary maps δ are defined by the rule δ{α_{v}}(e) := α_{t(e)}|_{W_{e}} − α_{s(e)}|_{W_{e}}, and we have used the fact that H_{dR}^{0}(W_{v}) = H_{dR}^{0}(W′_{v}), since Σ is finite. Furthermore, the map i sends the class of a collection {λ_{e}} of scalars with λ_{e} ∈ ℂ_{p} to the class of the hypercocycle ({0_{v}}, {λ_{e}}), while the map r sends the class of the hypercocycle ({ω_{v}}, {f_{e}}) to the class of {ω_{v}}. Likewise for i′ and r′.

2.5. Analytic description of the syntomic regulator. We now give an analytic description of the image under the p-adic syntomic regulator of the Beilinson-Flach element BF_{s} attached to the Siegel unit 𝔤_{1,Np^{s}}. Recall the modular forms f ∈ S_{2}(Np) and the primitive form g ∈ S_{2}(Np^{s}, χχ_{p}) considered in §2.3, arising from newforms f_{0} and g_{0} of level N_{f} and N_{g}p^{s}, respectively.

Lemma 2.4. We have the equality

reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = reg_{p}^{fg}(BF(𝔤_{χ^{−1}χ_{p}^{−1}}^{w}))(η_{f} ⊗ ω_{g^{w}}).

Proof. This follows by noting that

BF_{s} = ∑ BF(𝔤_{ψ}^{w}),

where the sum is being taken over the even characters ψ of modulus Np^{s}. It follows from the equivariance of reg_{p} under the group of diamond operators acting functorially on both source and target that reg_{p}^{fg}(BF(𝔤_{ψ}^{w}))(η_{f} ⊗ ω_{g^{w}}) is zero for all ψ ≠ χ^{−1}χ_{p}^{−1}. □

In light of this lemma, let

(30) ω_{E} := E(q)\frac{dq}{q} = \frac{φ(Np^{s})}{2} · dlog𝔤_{χ^{−1}χ_{p}^{−1}} = G(χχ_{p})E_{2}(χ^{−1}χ_{p}^{−1}, 1)\frac{dq}{q} ,  E ∈ M_{2}(Np^{s}, χ^{−1}χ_{p}^{−1})

be the differential of the third kind on X_{1}(Np^{s}) and the associated Eisenstein series introduced in equation (7) of §2.1.

Choose a polynomial P (x) ∈ F [x] such that the following conditions hold:

(1) P (Φ × Φ) annihilates the class of ω_{g^{w}} ⊗ ω_{E^{w}} in H_{dR}^{2}(Y_{s}^{2}_{/F}),

(2) P (Φ) acts invertibly on H_{dR}^{1}(Y_{s}_{/F}).

The polynomial P exists because Φ × Φ acts with eigenvalues of complex absolute value p^{3/2} on the subspace of H_{dR}^{2}(Y_{s}^{2}_{/F}) generated by the frobenius translates of classes of the form ω_{g^{w}} ⊗ ω_{E^{w}}, while Φ acts on H_{dR}^{1}(Y_{s}_{/F}) with eigenvalues of absolute value 1, \sqrt{p} and p (as is deduced for example by analysing the Mayer-Vietoris sequence (29)).

Just as in [DR2, §3.2], condition (1) in the choice of P means that P (Φ × Φ) (ω_{g^{w}} ⊗ ω_{E^{w}}) is exact and hence there exists a system of rigid one-forms

ρ_{P} (g^{w}, E^{w}) = {ρv_{1},v_{2}_{P}(g^{w}, E^{w}) ∈ Ω_{rig}^{1}(W′_{v_{1}} × W′_{v_{2}})}

indexed by pairs (v_{1}, v_{2}) ∈ V × V, satisfying

(31)  dρv_{1},v_{2}_{P}(g^{w}, E^{w}) = P (Φ × Φ) (ω_{g^{w}} ⊗ ω_{E^{w}} |_{W′_{v_{1}} ×W′_{v_{2}}} ) .

Note that the choice of the system of one-forms ρv_{1},v_{2}_{P}(g^{w}, E^{w}) is not unique, as it is well-defined only up to systems of closed rigid one-forms on W′_{v_{1}} × W′_{v_{2}}. Nevertheless, we shall use this system to construct a canonical class in H_{dR}^{1}(X_{s}/F) associated to g^{w} and E^{w}, independent of all the choices made.

For any a ∈ (ℤ/Np^{s}ℤ)^{×}, let ⟨a⟩ ∈ Aut(Y_{s}) denote the diamond operator associated to a on Y_{s}, and for any pair (a, b) of such elements let ⟨a, b⟩ denote the automorphism of Y_{s} × Y_{s} given by the product of ⟨a⟩ and ⟨b⟩ acting on the first and second variable, respectively.

Fix an a ∈ (ℤ/Np^{s}ℤ)^{×} such that χχ_{p}(a) ≠ 1 for the rest of the section, and define the linear homomorphism

(32)  δ_{a}^{∗} : ⊕_{v_{1},v_{2}}Ω_{rig}^{1}(W′_{v_{1}} × W′_{v_{2}}) ⟶ ⊕_{v}Ω_{rig}^{1}(W′_{v}), δ_{a}^{∗} = diag^{∗} ∘ (Id − ⟨a, 1⟩^{∗}) ∘ (Id − ⟨1, a⟩^{∗})

where diag^{∗} denotes the map induced by pull-back under the diagonal embedding W′_{v} ⟶ W′_{v} × W′_{v}.

Lemma 2.5. The map δ_{a}^{∗} sends collections of closed 1-forms to collections of exact 1-forms.

Proof. By the Künneth formula, there is a decomposition

H_{dR}^{1}(W′_{v} × W′_{v}) ≃ (H^{0} ⊗ H^{1}) ⊕ (H^{1} ⊗ H^{0}),

where we use the shorthands H^{0} = H_{dR}^{0}(W′_{v}) and H^{1} = H_{dR}^{1}(W′_{v}). Since the endomorphism Id−⟨a, 1⟩^{∗} vanishes on the first factor, and Id − ⟨1, a⟩^{∗} vanishes on the second, the claim follows. □

Corollary 2.6. The image Ξ_{P,a}(g^{w}, E^{w}) of the system of closed rigid 1-forms δ_{a}^{∗}(ρ_{P} (g^{w}, E^{w})) in ⊕_{v}H_{dR}^{1}(W′_{v}) does not depend on the choice of solution of the differential equation (31).

As the notation indicates, the class Ξ_{P,a}(g^{w}, E^{w}) does depend on both our choices of P and a. In order to eliminate this ambiguity, define the class

(33)  Ξ_{P} (g^{w}, E^{w}) := (1 − χχ_{p}(a))^{−1}(1 − χ^{−1}χ_{p}^{−1}(a))^{−1}Ξ_{P,a}(g^{w}, E^{w}) ∈ ⊕_{v∈V}H_{dR}^{1}(W′_{v}).

This class still depends on P, but is independent of the choice of a as above. Indeed, this follows from the very definition of Ξ_{P} (g^{w}, E^{w}) and the fact that ⟨a⟩ acts on the g^{w}-isotypic (resp. E^{w}-isotypic) component of H_{dR}^{1}(Y_{s}_{/F}) as multiplication by χ^{−1}χ_{p}^{−1}(a) (resp. χχ_{p}(a)).

Following [DR2, Def. 3.1], define the subspace H_{dR}^{1}(X_{s}_{/F})^{(1)} of pure classes of weight one of H_{dR}^{1}(X_{s}_{/F}) by the condition that Φ acts via eigenvalues of complex absolute value \sqrt{p} on its elements. Likewise, define the subspace ⊕_{v}H_{dR}^{1}(W′_{v})^{(1)} of ⊕_{v}H_{dR}^{1}(W′_{v}) by the same condition. Weight considerations show that ⊕_{v}H_{dR}^{1}(W′_{v})^{(1)} is contained in the kernel of the map δ appearing in the commutative diagram (29), and that the map r′ ∘ ι defined in the same diagram induces an isomorphism between the above spaces of pure weight one classes. Denote by

spl_{X} : ⊕_{v}H_{dR}^{1}(W′_{v})^{(1)} ⟶ H_{dR}^{1}(X_{s}_{/F})^{(1)}

the (Frobenius-equivariant) inverse isomorphism. In light of condition (2) in our choice of P, define the class

(34)  Ξ(g^{w}, E^{w}) := P (Φ)^{−1}spl_{X} (Ξ_{P} (g^{w}, E^{w})) ∈ H_{dR}^{1}(X_{s}/F)^{(1)}.

It is immediate to check that Ξ(g^{w}, E^{w}) not depend on our choice of P.

Let π_{s} and ϖ_{s} be the natural degeneracy maps arising in the diagram

\xymatrix{π̃_{s} : X_{s} \ar[r]^{π_{s}} & X_{0}(Np^{s}) \ar[r]^{ϖ_{s}} & X_{0}(Np),}

where π̃_{s} is the morphism appearing in equation (8). The degeneracy map ϖ_{s} that we choose is the unique one which is compatible with the U_{p} operators in level Np^{s} and Np. Let

η_{f,s} := ϖ_{s}^{∗}η_{f},  η̃_{f,s} := π_{s}^{∗}η_{f,s}.

The class η_{f,s} is ordinary, by the ordinariness of η_{f} combined with the compatibility of ϖ_{s} with U_{p}.

Proposition 2.7. The equality

(35)  reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = \frac{2}{φ(Np^{s})}⟨η̃_{f,s}, Ξ(g^{w}, E^{w})⟩_{X_{s}}

holds.

Proof. This formula follows by applying the same argument as in the proof of [BDR12, Proposition 3.3], with the following difference. While in loc. cit. we work with the modular curve X_{1}(N) which admits a smooth model over ℤ_{p}, here the modular curve X_{s} has bad reduction at p and we are bound to work with a semistable model of this curve. Besser’s theory of finite polynomial cohomology has been extended recently to the setting of semistable varieties by Besser, Loeffler and Zerbes in [BLZ], building on previous work of Nekovar and Niziol [NeNi]. The proof of [BDR12, Proposition 3.3] applies verbatim after replacing Besser’s [Bes, Prop. 6.3] with [BLZ, Proposition 3.12], taking into account that

ω_{E^{w}} = \frac{φ(Np^{s})}{2} · d log(𝔤_{χ^{−1}χ_{p}^{−1}}^{w}).

□

The fact that the nebentype characters of g^{w} and E^{w} are inverse one of another implies that the class Ξ(g^{w}, E^{w}) arises as the pull-back of a class – denoted with the same symbol by abuse of notation – in the cohomology of X_{0}(Np^{s}) under π_{s}.

Proposition 2.8. For all s ≥ 1,

(36)  reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = ⟨η_{f,s}, Ξ(g^{w}, E^{w})⟩_{X_{0}(Np^{s})}.

Proof. Because the degree of π_{s} is \frac{1}{2}φ(Np^{s}), and the pullback and pushforward maps are adjoint to each other relative to the Poincaré pairing,

(37)  ⟨η̃_{f,s}, Ξ(g^{w}, E^{w})⟩_{X_{s}} = \frac{1}{2}φ(Np^{s})⟨η_{f,s}, Ξ(g^{w}, E^{w})⟩_{X_{0}(Np^{s})}.

The result follows. □

It will be convenient to consider the anti-ordinary counterparts of the classes η_{f} and η_{f,s}:

η_{f}^{w} := wη_{f}, η_{f,s}^{w} := wη_{f,s},

where we recall that w refers in the above equations to the Atkin-Lehner involutions w_{Np} and w_{Np^{s}} of levels Np and Np^{s} respectively.

Let {W_{∞}, W_{0}} denote the standard admissible covering of X_{0}(Np) by wide-open neighbourhoods of the Hasse domain of X_{0}(N), as described by Coleman in [Co94]. By construction, arguing as in [DR2, Lemma 4.6], the ordinary unit root class η_{f} ∈ H_{dR}^{1}(X_{0}(Np)) is supported on W_{0}, and has vanishing residues at the supersingular annuli, while the anti-ordinary unit root class η_{f}^{w} is supported on W_{∞}.

Let U_{p}^{w} = wU_{p}w be the adjoint operator of U_{p} with respect to the Poincaré pairing on X_{0}(Np^{s}), and let

e_{ord} = lim U_{p}^{n!} and e_{ord}^{w} = lim(U_{p}^{w})^{n!} = we_{ord}w

denote Hida’s ordinary and anti-ordinary projectors, respectively. Pull-back under ϖ_{s} gives rise to an isomorphism between the ordinary subspaces of H_{dR}^{1}(X_{0}(Np^{s})) and H_{dR}^{1}(X_{0}(Np)). Hence e_{ord}Ξ(g, E) arises as the pull-back under ϖ_{s} of some ordinary class in H_{dR}^{1}(X_{0}(Np)), that we again denote with the same symbol. In the context of the next theorem, e_{ord}Ξ(g, E) is to be understood in the latter sense. Since e_{ord}Ξ(g, E) belongs to the subspace of the cohomology which is pure of weight one, its restriction to the wide open W_{∞} has vanishing residues along the supersingular annuli. Therefore, it also makes sense to pair it with the (restriction of) the anti-ordinary class η_{f}^{w}.

Theorem 2.9. For all s ≥ 1,

reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = α_{f}^{s−1} · ⟨η_{f}^{w}, e_{ord}Ξ(g, E)⟩_{W_{∞}}.

Proof. Since the class η_{f,s} is ordinary, it is fixed by e_{ord} and thus

(38)  ⟨η_{f,s}, Ξ(g^{w}, E^{w})⟩_{X_{0}(Np^{s})} = ⟨e_{ord}η_{f,s}, Ξ(g^{w}, E^{w})⟩_{X_{0}(Np^{s})}
= ⟨η_{f,s}, e_{ord}^{w}Ξ(g^{w}, E^{w})⟩_{X_{0}(Np^{s})}
= ⟨η_{f,s}^{w}, e_{ord} Ξ(g, E)⟩_{X_{0}(Np^{s})},

where the self-adjointness of the involution w has been exploited to derive the last equation. But since e_{ord}Ξ(g, E) arises as a pullback under ϖ_{s}, we can write

(39)  ⟨η_{f,s}^{w}, e_{ord} Ξ(g, E)⟩_{X_{0}(Np^{s})} = ⟨ϖ_{s}(η_{f,s}^{w}), e_{ord} Ξ(g, E)⟩_{X_{0}(Np)}
= ⟨ϖ_{s}wϖ_{s}^{∗}(η_{f}), e_{ord} Ξ(g, E)⟩_{X_{0}(Np)}
= ⟨w ∘ U_{p}^{s−1}(η_{f}), e_{ord} Ξ(g, E)⟩_{X_{0}(Np)}
= α_{f}^{s−1}⟨η_{f}^{w}, e_{ord} Ξ(g, E)⟩_{W_{∞}},

where the last identity follows from the fact that the anti-ordinary unit root class η_{f}^{w} is supported on W_{∞}. The theorem now follows from Proposition 2.8 combined with (38) and (39). □

We are now in a position to state the main result of this section, namely, the evaluation of the p-adic syntomic regulator attached to BF_{s} in terms of the q-expansions of concrete (overconvergent) p-adic modular forms. For a p-adic modular form φ = ∑ a_{n}(φ)q^{n}, let

φ^{[p]} = ∑_{p∤n} a_{n}(φ)q^{n}

denote its “p-deprived” counterpart.

The Atkin-Serre operator d = q \frac{d}{dq} sends p-adic overconvergent modular forms of weight 0 to overconvergent modular forms of weight two. The image of the p-depleted classical weight two Eisenstein series E_{2}^{[p]}(χ^{−1}χ_{p}^{−1}, 1) under d^{−1} is the p-adic overconvergent Eisenstein series of weight 0:

d^{−1}E_{2}^{[p]}(χ^{−1}χ_{p}^{−1}, 1) = E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1}).

The product g^{[p]} × E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1}) is therefore an overconvergent cusp form of weight two with trivial nebentypus character, and hence its image under the ordinary projection e_{ord} corresponds to a classical weight two cusp form on Γ_{0}(Np), which can be viewed as an element of the first de Rham cohomology of X_{0}(Np). The inner product ⟨ , ⟩_{X_{0}(N)} in the theorem below denotes the usual Poincaré duality on this de Rham cohomology group.

Theorem 2.10. For all s ≥ 1,

reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = G(χχ_{p}) · \frac{α_{f}^{s−1}}{1 − χ^{−1}(p)α_{f}^{−1}α_{g}} · ⟨η_{f}^{w}, e_{ord}(g^{[p]} · E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1}))⟩_{X_{0}(Np)}.

Proof. The restriction to W_{∞} of e_{ord}Ξ(g, E) is the cohomology class of a specific overconvergent weight two p-adic modular form representing a rigid analytic differential form on W_{∞} with vanishing annular residues. The class e_{ord}Ξ(g, E) is defined by exactly the same analytic recipe as the class denoted e_{ord}ξ(ωğ, ωh̆) in equation (85) of [DR2], after replacing ğ by g and setting

h̆ = w_{p^{s}}E (= G(χ)E_{2}(χ^{−1}, χ_{p})).

In [DR2], the form h̆ was assumed to be cuspidal while w_{p^{s}}E is an Eisenstein series, but the calculations of loc.cit. remain valid in the latter setting. In particular, the expression

w_{p^{s}} h̆ = G(χ_{p}^{−1})α_{h}^{−s}ωh̆^{ι}

that appears in the statement of Prop. 4.13 of [DR2] can be replaced by ω_{E}, yielding

(40)  e_{ord}Ξ_{P} (g, E) = e_{ord}ξ_{P} (ω_{g}, ω_{w_{p^{s}} E}) = −2e_{ord}(g × d^{−1}E^{[p]})
(41)  = −2G(χχ_{p})e_{ord}(g × E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1})),

where P (x) is the linear polynomial 2(1−p^{−1}χ^{−1}(p)α_{g}x), and Ξ_{P} (g, E) is defined as in (33). Combining equation (40) with Theorem 2.9, we obtain

(42)  reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = −2G(χχ_{p})α_{f}^{s−1} · ⟨η_{f}^{w}, P (Φ)^{−1}e_{ord}(g · E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1}))⟩_{W_{∞}}.

Note that g can be replaced by g^{[p]} in equation (42), as a direct calculation shows that

(g^{[p]} − g) · E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1})

belongs to the kernel of U_{p}, and a fortiori of e_{ord}. (See for instance Lemma 2.17 of [DR1].) Finally, for any class Ξ in H_{dR}^{1}(W_{∞}), we have

⟨η_{f}^{w}, ΦΞ⟩_{W_{∞}} = α_{f}^{−1}⟨Φη_{f}^{w}, ΦΞ⟩_{W_{∞}} = pα_{f}^{−1}⟨η_{f}^{w}, Ξ⟩_{W_{∞}} = β_{f}⟨η_{f}^{w}, Ξ⟩_{W_{∞}},

which in turn implies

(43)  ⟨η_{f}^{w}, P (Φ)^{−1}Ξ⟩_{W_{∞}} = P (β_{f})^{−1}⟨η_{f}^{w}, Ξ⟩_{W_{∞}}.

In light of the explicit description of P (x) given above, combined with equations (42) and (43),

reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = G(χχ_{p}) · \frac{α_{f}^{s−1}}{1 − χ^{−1}(p)α_{f}^{−1}α_{g}} · ⟨η_{f}^{w}, e_{ord}(g^{[p]} · E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1}))⟩_{W_{∞}}.

Proceeding just as in the proof of Theorem 2.9, Theorem 2.10 follows from the compatibility between the de Rham pairings on W_{∞} ⊂ X_{0}(Np) and on X_{0}(Np), using the fact that η_{f}^{w} is anti-ordinary and supported on W_{∞}. □

2.6. Hida’s p-adic L-function. A point of the rigid analytic space

Ω := Hom_{cts}(ℤ_{p}^{×}, ℂ_{p}^{×})

is said to be arithmetic if it is of the form

ν_{k,ϵ}(x) := x^{k−1}ϵ(x), (for all x ∈ ℤ_{p}^{×}),

where k ≥ 2 is an integer, and ϵ is a finite order character factoring through (ℤ/p^{s}ℤ)^{×}. The space Ω is referred to as the weight space and plays a basic role in the theory of p-adic familes of modular forms.

A Hida family of tame character χ : (ℤ/Nℤ)^{×} ⟶ ℂ_{p}^{×} is a pair (Ω_{φ}, \underline{φ}), where

(1) Ω_{φ} is a rigid analytic space equipped with a finite morphism

wt : Ω_{φ} ⟶ Ω,

(2) \underline{φ} = ∑_{n=1}^{∞} a_{n}(\underline{φ})q^{n} is a formal q-series with coefficients in the ring Λ_{φ} := A(Ω_{φ}) of rigid analytic functions on Ω_{φ}, such that, for all points x ∈ Ω_{φ} of arithmetic weight wt(x) = ν_{k,ϵ}, the specialisation

φ_{x} := ∑_{n=1}^{∞} a_{n}(\underline{φ})(x)q^{n}

is the q-series of a normalised eigenform of weight k, level Np^{s} and character χ_{φ_{x}} = χϵω^{1−k}, where ω is the Teichmüller character of conductor p.

Let g be the weight one modular form considered in Section 1, and let g_{α} be one of its ordinary p-stabilisations. A theorem of Hida ensures the existence of a Hida family, denoted \underline{g}, of tame character χ which specialises to g_{α} at a suitable point y_{0} ∈ Ω\underline{g} lying above ν_{1,1}. For any arithmetic point y ∈ Ω\underline{g} above ν_{ℓ,ϵ}, for some ℓ ≥ 1 and some character ϵ of p-power conductor, denote by g_{y} ∈ S_{ℓ}(Np^{s}, χϵω^{1−ℓ}) the specialisation of \underline{g} at y.

Write \underline{E}(1, χ^{−1}) for the ordinary Hida family of Eisenstein series, whose specialisation at the point ν_{ℓ,ϵ} in weight space is equal to the classical Eisenstein series

\underline{E}(1, χ^{−1})_{ν_{ℓ,ϵ}} = E_{ℓ}(1, χ^{−1}ϵω^{1−ℓ})

of weight ℓ and character χ^{−1}ϵω^{1−ℓ}. The Iwasawa algebra admits an involution inducing the map ν_{ℓ,ϵ} ↦ ν_{2−ℓ,ϵ^{−1}} on weight space. Applying the change of scalars to \underline{E}(1, χ^{−1}) via this involution yields a Λ-adic modular form , denoted \underline{E}^{∗}(1, χ^{−1}), whose specialisation at the arithmetic point ν_{ℓ,ϵ} is the overconvergent modular form

\underline{E}^{∗}(1, χ^{−1})_{ν_{ℓ,ϵ}} = E_{2−ℓ}(1, χ^{−1}ϵ^{−1}ω^{ℓ−1}).

The expression g_{y}^{[p]} · E_{0}^{[p]}(1, χ^{−1}χ_{p}^{−1}) appearing in Theorem 2.10 is the specialisation at the point y of weight ν_{2,χ_{p}ω} of the Λ-adic modular form

\underline{g}\underline{E}^{∗[p]}(1, χ^{−1}),

whose specialisation at a point y of weight ν_{ℓ,ϵ} is given by

(44)  \{\underline{g}^{[p]} · \underline{E}^{∗[p]}(1, χ^{−1})\}_{x} = g_{y}^{[p]} · E_{2−ℓ}^{[p]}(1, χ^{−1}ϵ^{−1}ω^{ℓ−1}).

Note that for all arithmetic y, this specialisation is an overconvergent modular form of weight 2 and trivial character, whose ordinary projection is therefore classical of level Np; hence,

e_{ord}(\underline{g}^{[p]} · \underline{E}^{∗[p]}(1, χ^{−1}) belongs to S_{2}(Γ_{0}(Np), ℚ_{p}) ⊗ Λ\underline{g}.

The reader is referred to Section 4.5 of [DR2] for details on dual families of modular forms, which motivate the definition of the Λ-adic modular form of equation (44).

Let L_{p}(\underline{f}, \underline{g})(x, y, j) be the 3-variable Rankin p-adic L-function on Ω\underline{f} ×Ω\underline{g}×Ω introduced in [BDR12, §2.2], where \underline{f} is the Hida family passing through f. It interpolates the algebraic part of the critical values L(f_{x} ⊗ g_{y}, j), where f_{x} and g_{y} are classical specialisations of \underline{f} and \underline{g} whose weights k and ℓ satisfy

(45)  k > ℓ,  \frac{k + ℓ − 1}{2} ≤ j ≤ k − 1.

Consider the restriction of L_{p}(\underline{f}, \underline{g})(x, y, j) to the set of points (x_{0}, y, ν_{ℓ,ϵ}), where x_{0} is such that \underline{f}_{x_{0}} = f, and y ∈ Ω\underline{g} is a point lying above ν_{ℓ,ϵ}. In the notations of [BDR12, §2.2.1], this amounts to considering only the values of L_{p}(\underline{f}, \underline{g})(k, ℓ, j) when k = 2, j = ℓ, t = 1 − ℓ and m = ℓ. In light of equations (27) and (24) in loc.cit., combined with the equality

d^{1−ℓ}E_{ℓ}^{[p]}(χ^{−1}ϵ^{−1}ω^{ℓ−1}, 1) = E_{2−ℓ}^{[p]}(1, χ^{−1}ϵ^{−1}ω^{ℓ−1}),

this restriction agrees, up to a non-zero fudge factor which depends only on the fixed form f, with the (p-adic analytic) function described for all such y by

(46)  L_{p}(f, \underline{g})(y) := ⟨η_{f}^{w}, e_{ord}(g_{y}^{[p]} · E_{2−ℓ}^{[p]}(1, χ^{−1}ϵ^{−1}ω^{ℓ−1}))⟩_{X_{0}(Np)}.

Note that the points of the form (x_{0}, y, ν_{ℓ,ϵ}) lie outside the range of classical interpolation for L_{p}(\underline{f}, \underline{g}), unless g_{y} is classical and of weight one. The next result relates the values of L_{p}(f, \underline{g}) at arithmetic points y of weight ν_{2,ϵ} to the p-adic regulators of Beilinson-Flach elements:

Theorem 2.11. Let ϵ be a character of conductor p^{s}, let y ∈ Ω\underline{g} be a point over ν_{2,ϵ} and let g = g_{y} ∈ S_{2}(Np^{s}, χϵω^{−1}) denote the specialization of \underline{g} at y. Then

reg_{p}^{fg}(BF_{s})(η_{f} ⊗ ω_{g^{w}}) = G(χϵω^{−1}) · \frac{α_{f}^{s−1}}{1 − χ(p)^{−1}α_{f}^{−1}α_{g}} · L_{p}(f, \underline{g})(y).

Proof. This follows directly by combining equation (46) with Theorem 2.10. □

Another important property of L_{p}(f, \underline{g}) pertains to its specialisations at classical weight one points attached to the Hida family \underline{g}.

Theorem 2.12. Let y ∈ Ω\underline{g} be a point over ν_{1,1}, corresponding to a classical p-stabilised weight one form g = g_{y} ∈ S_{1}(Np, χ). Then

L_{p}(f, \underline{g})(y) ≠ 0 ⇔ L(f ⊗ g, 1) ≠ 0.

Proof. By equation (46),

L_{p}(f, \underline{g})(y) = ⟨η_{f}^{w}, e_{ord} (g_{y}^{[p]} · E_{1}^{[p]}(1, χ^{−1}))⟩_{X_{0}(Np)}
(47)  = ⟨η_{f}^{w}, g_{y}^{[p]} · E_{1}^{[p]}(1, χ^{−1})⟩_{X_{0}(Np^{2})} (mod ℚ̄^{×}),

where the fact that g_{y}^{[p]}E_{1}^{[p]}(1, χ^{−1}) is a classical modular form of level Np^{2} has been used to deduce the second equality. The expression in (47) is an explicit non-zero multiple of the value of the classical complex Rankin-Selberg L-function L(f ⊗ g, s) of the convolution of f and g at s = 1. (See equation (27) and the one immediately following it in [BDR12, §2.2.2].) The theorem follows. □

3. Λ-adic classes

3.1. Norm-compatible systems of elements. For all s ≥ 0, the natural U_{p}-compatible projections

(48)  π_{s+1,s} : X_{s+1} → X_{s}

of modular curves give rise to maps

(49)  π_{s+1,s} : S_{s+1} ⟶ S_{s}

on the associated surfaces, that we continue to denote with the same symbol by a slight abuse of notation. Write also π_{s+1,s} : O_{X_{s+1}}^{×} ⟶ O_{X_{s}}^{×} for the norm maps on units induced by π_{s+1,s}.

Proposition 3.1. For all s ≥ 1,

π_{s+1,s}(𝔤_{a;Np^{s+1}}^{w}) = 𝔤_{a;Np^{s}}^{w}.

Proof. Lemma 2.12 of [Ka98] implies that the units g_{a;Np^{s+1}} are compatible relative to the pushforward via the maps π′_{s+1,s} : X_{s+1} ⟶ X_{s} that are compatible with the operator U′_{p}. This compatibility of the units 𝔤_{a;Np^{s}} relative to the π′_{s+1,s} := w_{p^{s}}π_{s+1,s}w_{p^{s+1}} implies the π_{s+1,s}-compatiblity of the units 𝔤_{a;Np^{s}}^{w}. See also §2.11 and §2.13 of [Ka98], or Thm. 2.24 of [LLZ1], for a convenient summary of these and further calculations in the more general setting treated in [Ka98]. (Recall that the unit 𝔤_{a;Np^{s}} is denoted g_{0, \frac{a}{Np^{s}}} in [Ka98] and [LLZ1].) □

Write

π_{s+1,s} : CH^{2}(S_{s+1}, 1) → CH^{2}(S_{s}, 1)

for the norm maps on higher Chow groups induced by push-forward under the maps π_{s+1,s} of (49). Note that π_{s+1,s} preserves the subspaces of negligible classes, and hence descends to a well-defined map

(50)  π_{s+1,s} : CH_{neg}^{2}(S_{s+1}, 1) → CH_{neg}^{2}(S_{s}, 1).

The norm compatibilities of the units 𝔤_{a;Np^{s}} are inherited by the associated Beilinson-Flach elements:

Proposition 3.2. For all s ≥ 1,

(51) π_{s+1,s}(BF(a; Np^{s+1})) = BF(a; Np^{s}),  π_{s+1,s}(BF_{s+1}) = BF_{s} in CH_{neg}^{2}(S_{s}, 1)(ℚ).

Proof. This follows directly from Proposition 3.1 in light of the fact that the map π_{s+1,s} of (49) maps the diagonally embedded X_{s+1} ≃ Δ_{s+1} ⊂ S_{s+1} to Δ_{s}, and is identified with the projection π_{s+1,s} of (48). □

Corollary 3.3. The Beilinson-Flach cohomology classes κ_{s} of Definition 2.3 are compatible under the norm maps, i.e.,

π_{s+1,s}(κ_{s+1}) = κ_{s}.

In particular, the classes κ_{s} can be packaged together into the inverse limit class

κ_{∞} := (κ_{s})_{s≥1} ∈ H^{1}(ℚ, V_{0}(Np) ⊗ 𝕍_{∞}),

where

𝕍_{∞} := lim_{←,s} V_{s}.

3.2. The Λ-adic Beilinson-Flach cohomology class. The module 𝕍_{∞} of the previous section becomes more manageable when replaced by its image under the ordinary projection. Let

V_{s}^{ord} := e_{ord}V_{s}, 𝕍_{∞}^{ord} := e_{ord}𝕍_{∞} = lim_{←,s} e_{ord}V_{s}.

Note that the action of the group D_{s} of diamond operators on X_{s} endows the G_{ℚ}-module 𝕍_{∞}^{ord} with a natural structure of module over the Iwasawa algebra Λ̃ = ℤ_{p}[[D_{∞}]] := \underleftarrow{lim} ℤ_{p}[D_{s}]. A theorem of Hida asserts that this module is finitely generated and locally free over this algebra. Its Hecke eigenspaces realise the Λ-adic representations attached to ordinary families of eigenforms.

The Λ-adic Beilinson-Flach cohomology class is defined by setting

κ_{s}^{ord} := e_{ord}κ_{s} ∈ H^{1}(ℚ, V_{0}(Np) ⊗ V_{s}^{ord}),
κ_{∞}^{ord} := e_{ord}κ_{∞} = (κ_{s}^{ord})_{s≥1} ∈ H^{1}(ℚ, V_{0}(Np) ⊗ 𝕍_{∞}^{ord}).

Recall the cusp form f = q + ∑_{n≥2} a_{n}(f)q^{n} ∈ S_{2}(Np) introduced in §2.3, on which U_{p} acts with eigenvalue α_{f}. Define

ψ_{f} : G_{ℚ_{p}} ⟶ O^{×}

to be the unramified character of G_{ℚ_{p}} such that ψ_{f}(Fr_{p}) = α_{f} and let ϵ_{cyc} : G_{ℚ_{p}} ⟶ ℤ_{p}^{×} denote the cyclotomic character. There is an exact sequence of G_{ℚ_{p}}-modules

(52)  0 → V_{f}^{+} → V_{f} → V_{f}^{−} → 0  with V_{f}^{+} ≃ O(ψ_{f}^{−1}ϵ_{cyc}) and V_{f}^{−} ≃ O(ψ_{f})

as O[G_{ℚ_{p}}]-modules.

Let

\underline{ϵ}_{cyc} : G_{ℚ_{p}} ⟶ Λ^{×}

denote the Λ-adic cyclotomic character satisfying the interpolation property

ν_{ℓ,ϵ} ∘ \underline{ϵ}_{cyc} = ϵϵ_{cyc}^{ℓ−1}ω^{1−ℓ}

for any ℓ ≥ 1 and for any Dirichlet character ϵ of p-power conductor.

Let \underline{g} = ∑_{n≥1} a_{n}(\underline{g})q^{n} be a Λ-adic cuspidal eigenform of tame level N and tame character χ, arising from a Λ-adic newform of level N\underline{g}, and let Λ\underline{g} denote the finite flat extension of the Iwasawa algebra Λ := ℤ_{p}[[1 + pℤ_{p}]] generated by the coefficients a_{n}(\underline{g}). Hida and Wiles associated to \underline{g} a twodimensional Galois representation 𝕍\underline{g} over Λ\underline{g}, again characterized up to isomorphism by the property that the characteristic polynomial of Fr_{ℓ} is T^{2} − a_{ℓ}(\underline{g})T + \underline{ϵ}_{cyc}(ℓ) for primes ℓ ∤ Np.

Wiles [Wi88] proved a Λ-adic analogue of (52), which asserts that (cf. also [Oh00])

(53)  0 → 𝕍+ ; \underline{g} → 𝕍\underline{g} → 𝕍− ; \underline{g} → 0,  with 𝕍+ ; \underline{g} ≃ Λ\underline{g}(ψ−1 ; \underline{g}χ\underline{ϵ}_{cyc}) and 𝕍− ; \underline{g} ≃ Λ\underline{g}(ψ\underline{g})

as Λ\underline{g}[G_{ℚ_{p}}]-modules. Here

ψ\underline{g} : G_{ℚ_{p}} ⟶ Λ\underline{g}^{×}

is the unramified character sending Fr_{p} to a_{p}(\underline{g}).

Fix once and for all a choice of the isomorphisms in (53). Similarly to the previous sections, the Λ-adic form \underline{g} gives rise to an epimorphism

ϖ\underline{g} : 𝕍_{∞}^{ord} ⟶ 𝕍\underline{g}

of Λ̃[G_{ℚ}]-modules. Set

𝕍_{f,\underline{g}} := V_{f} ⊗ 𝕍\underline{g}, ϖ_{f,\underline{g}} := ϖ_{f} ⊗ ϖ\underline{g} : V_{0}(Np) ⊗ 𝕍_{∞}^{ord} ⟶ 𝕍_{f,\underline{g}}.

Definition 3.4. The Λ-adic Beilinson-Flach cohomology class associated to the pair (f, \underline{g}) is

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

Let ℓ ≥ 1 be an integer and ϵ be a Dirichlet character of conductor p^{s} for some s ≥ 1. Let y be a point in Ω\underline{g} = Spf(Λ\underline{g}) such that wt(y) = ν_{ℓ,ϵ}, and assume that the specialization g_{y} of \underline{g} at y is a classical eigenform (which is always the case if ℓ ≥ 2). Then g_{y} belongs to M_{ℓ}(Np^{s}, χϵω^{1−ℓ}), and is actually a cuspform if ℓ ≥ 2. Define

(54)  κ(f, g_{y}) ∈ H^{1}(ℚ, V_{fg_{y}})

to be the specialization at y of the Λ-adic class κ(f, \underline{g}). Let

κ_{p}(f, \underline{g}) ∈ H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}), κ_{p}(f, g_{y}) ∈ H^{1}(ℚ_{p}, V_{fg_{y}})

denote the restriction to G_{ℚ_{p}} of the Λ-adic class κ(f, \underline{g}) and of its specialization κ(f, g_{y}).

Since the p-adic syntomic regulator map is the composition of the étale regulator with Bloch-Kato’s logarithm map, Theorem 2.11 translates into the following statement:

Theorem 3.5. Let y ∈ Ω\underline{g} be an arithmetic point of weight-character ν_{2,ϵ} for some character ϵ of conductor p^{s}. Then

log_{p} κ_{p}(f, g_{y})(η_{f} ⊗ ω_{g_{y}^{w}}) = G(χϵω^{−1}) · \frac{α_{f}^{s−1}}{1 − χ(p)^{−1}α_{f}^{−1}α_{g_{y}}^{−1}} · L_{p}(f, \underline{g})(y).

3.3. The explicit reciprocity law. The Λ\underline{g}-module 𝕍_{f\underline{g}} admits a natural G_{ℚ_{p}}-stable filtration, given by

𝕍_{f\underline{g}}^{++} := V_{f}^{+} ⊗ 𝕍+ ; \underline{g} ⊂ 𝕍_{f\underline{g}}^{+} := V_{f} ⊗ 𝕍+ ; \underline{g} + V_{f}^{+} ⊗ 𝕍\underline{g} ⊂ 𝕍_{f\underline{g}}.

Note that 𝕍_{f\underline{g}}^{++} and 𝕍_{f\underline{g}}^{+} have rank 1 and 3 over Λ\underline{g}, respectively.

Lemma 3.6. (i) There is an isomorphism 𝕍_{f\underline{g}}/𝕍_{f\underline{g}}^{+} ≃ Λ_{g}(ψ_{f}ψ\underline{g}) of Λ\underline{g}[G_{ℚ_{p}}]-modules.

(ii) The quotient 𝕍_{f\underline{g}}^{+}/𝕍_{f\underline{g}}^{++} decomposes as a Λ\underline{g}[G_{ℚ_{p}}]-module as

𝕍_{f\underline{g}}^{+}/𝕍_{f\underline{g}}^{++} ≃ 𝕍_{f\underline{g}}^{f} ⊕ 𝕍_{f\underline{g}}^{g} where 𝕍_{f\underline{g}}^{f} = Λ\underline{g}(ψ_{f}ψ−1 ; \underline{g}χ · \underline{ϵ}_{cyc}) and 𝕍_{f\underline{g}}^{g} = Λ\underline{g}(ψ_{f}^{−1}ψ\underline{g}ϵ_{cyc}).

Proof. The lemma follows directly from the definitions of 𝕍_{f\underline{g}}^{++} and 𝕍_{f\underline{g}}^{+}, and the formulae described in (52) and (53). □

The natural inclusion 𝕍_{f\underline{g}}^{+} ↪ 𝕍_{f\underline{g}} induces a homomorphism H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}^{+}) ↪ H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}), which is

injective because H^{0}(ℚ_{p}, 𝕍_{f\underline{g}}/𝕍_{f\underline{g}}^{+}) = 0. Thus H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}^{+}) is identified with a subgroup of H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}).

Define ξ_{fg} := 1 − α_{f}a_{p}(\underline{g}) ∈ Λ\underline{g}. For the remainder of the article, we replace the ring Λ\underline{g} and all modules over it by their localization at the multiplicative set generated by the powers of ξ_{fg}. Since this element vanishes at no classical point of weight ℓ ≥ 1, this modification does not affect any of the subsequent arguments concerning the specialization at these points.

Lemma 3.7. The local class κ_{p}(f, \underline{g}) belongs to H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}^{+}) ⊂ H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}).

Proof. It follows from known properties of the étale regulator map, as explained in [Ne98] and [Fl, §2], that for all points y of weight ν_{2,ϵ}, the specializations κ_{p}(f, g_{y}) belong to the image of H^{1}(ℚ_{p}, V_{fg_{y}}^{+}) in H^{1}(ℚ_{p}, V_{fg_{y}}). Since these points form a dense subset of Ω\underline{g} for the rigid analytic topology, we conclude that κ_{p}(f, \underline{g}) belongs to the kernel of the map

H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}) ⟶ H^{1}(I_{p}, 𝕍_{f\underline{g}}/𝕍_{f\underline{g}}^{+}).

In order to prove the lemma, it thus suffices to show that the kernel of the restriction map

H^{1}(ℚ_{p}, 𝕍_{f\underline{g}}/𝕍_{f\underline{g}}^{+}) ⟶^{res} H^{1}(I_{p}, 𝕍_{f\underline{g}}/𝕍_{f\underline{g}}^{+})

is trivial. Note that such kernel is a quotient of H^{1}(ℚ_{p}^{ur}/ℚ_{p}, (𝕍_{f\underline{g}}/𝕍_{f\underline{g}}^{+})^{I_{p}}), which by Lemma 3.6 (i) is isomorphic to H^{1}(ℚ_{p}^{ur}/ℚ_{p}, Λ\underline{g}(ψ_{f}ψ\underline{g})). This module is ξ_{gh}-torsion. In view of the redefinition of Λ\underline{g}, this implies the sought-after claim. □

Set \underline{Ψ} := ψ_{f}ψ−1 ; \underline{g}χ. The two previous lemmas allow us to define

κ_{p}^{f}(f, \underline{g}) ∈ H^{1}(ℚ_{p}, Λ_{g}(\underline{Ψ} · \underline{ϵ}_{cyc}))

as the projection of κ_{p}(f, \underline{g}) to the first factor 𝕍_{f\underline{g}}^{f}.

Let y ∈ Ω\underline{g} be a classical point of weight wt(y) = ν_{ℓ,ϵ} with ℓ ≥ 2 and ϵ ≠ ω^{ℓ−1}, and let K_{y}/ℚ_{p} denote the residue field of y. Write Ψ_{y} : G_{ℚ_{p}} ⟶ K_{y}^{×} for the specialisation at y of the unramified Λ-adic character \underline{Ψ}. The classical eigenform g_{y} ∈ S_{ℓ}(Np^{s}, χϵω^{1−ℓ}) is new at p^{s}, and the specialization of κ_{p}^{f}(f, \underline{g}) at y yields a local class

κ_{p}^{f}(f, g_{y}) ∈ H^{1}(ℚ_{p}, K_{y}(Ψ_{y} · ϵ · ϵ_{cyc}^{ℓ−1} · ω^{1−ℓ})).

Lemma 3.8. For all arithmetic points y of weight ν_{ℓ,ϵ} with ℓ ≥ 2, the local cohomology group H^{1}(ℚ_{p}, K_{y}(Ψ_{y} ·ϵ·ϵ_{cyc}^{ℓ−1}·ω^{1−ℓ})) is one-dimensional over K_{y} and equal to H_{exp}^{1}(ℚ_{p}, K_{y}(Ψ_{y} ·ϵ·ϵ_{cyc}^{ℓ−1}·ω^{1−ℓ})).

Proof. Write simply V = V_{fg_{y}}^{f} for the specialization at y of 𝕍_{f\underline{g}}^{f} and D = D_{dR}(V_{fg_{y}}^{f}) for its de Rham Dieudonné module. By [BK, Def. 3.10], [Bel, Ex. 2.19], the Bloch-Kato logarithm map is an isomorphism

(55)  log_{p} : H_{exp}^{1}(ℚ_{p}, V) → D/(Fil^{0}D + D^{ϕ=1}).

By [Bel, §2.1.3], dim_{K_{y}} H^{1}(ℚ_{p}, V) = e_{0} + e_{1} + 1 where e_{0} = dim H^{0}(ℚ_{p}, V) and e_{1} is the multiplicity of K_{y}(1) as a quotient of V. Since ℓ ≥ 2, e_{0} = 0. Since ϵ ≠ ω^{ℓ−1}, e_{1} = 0 and thus H^{1}(ℚ_{p}, V) is one-dimensional over K_{y}. Moreover, since the Hodge-Tate weight of V is 1 − ℓ ≤ −1, the subspace of D on which frobenius acts as the identity is D^{ϕ=1} = 0, and Fil^{1−ℓ}(D) ⊋ Fil^{2−ℓ}(D). Since (55) is an isomorphism and dim H_{exp}^{1}(ℚ_{p}, V) ≤ dim H^{1}(ℚ_{p}, V) = 1, it follows that in fact Fil^{1−ℓ}(D) = D ⊃ Fil^{0}(D) = 0, and thus (55) gives rise to an isomorphism

(56)  log_{p} : H_{exp}^{1}(ℚ_{p}, V) = H^{1}(ℚ_{p}, V) ⟶ D

of one-dimensional K_{y}-vector spaces. The result follows. □

For each arithmetic point y ∈ Ω\underline{g} of weight ν_{ℓ,ϵ} with ℓ ≥ 2, the choice of a period ℧_{y} ∈ D_{dR}(V_{fg_{y}^{∗}}^{g_{y}^{∗}})

attached to the dual V_{fg_{y}^{∗}}^{g_{y}^{∗}} of V_{fg_{y}}^{f} determines an isomorphism

(57)  log_{℧_{y}} : H^{1}(ℚ_{p}, K_{y}(Ψ_{y} · ϵ · ϵ_{cyc}^{ℓ−1} · ω^{1−ℓ})) ⟶ K_{y},  log_{℧_{y}} (κ) := log_{p}(κ)(℧_{y}).

Fix a choice of ℧_{y} by defining, for all arithmetic points y of weight ν_{2,ϵ} with ϵ ≠ 1 of conductor p^{s},

(58)  ℧_{y} := α_{f}^{1−s}G(χϵω^{−1})^{−1} · (η_{f} ⊗ ω_{g_{y}^{w}}).

Ohta’s Λ-adic Eichler-Shimura isomorphism can be used to show that these periods agree with those arising in Perrin-Riou’s generalised Coleman maps attached to families of cyclotomic twists of a fixed Galois representations and their generalisations to twists by unramified Λ-adic characters, as described in [LZ, Thm. 4.15]. (Cf. the discussions in [DR2, §5.2, §5.3], or in [KLZ, §7.4, §7.6, Thm. 7.6.1].) It follows that, for all κ ∈ H^{1}(ℚ, 𝕍_{f\underline{g}}^{f}), the function y ↦ log_{℧_{y}} (κ) interpolates to an element of the fraction field of Λ\underline{g}, denoted L(κ).

Recall the point y ∈ Ω_{g} of weight-character ν_{1,1} satisfying \underline{g}_{y} = g_{α}, and let Λ\underline{g}′ denote the localisation of Λ\underline{g} at the prime ideal ker(y).

Theorem 3.9. For the map L as above,

(59)  L(κ_{p}^{f}(f, \underline{g})) = L_{p}(f, \underline{g}) (mod (Λ\underline{g}′)^{×}).

Proof. Theorem 3.5 implies that the elements

L(κ_{p}^{f}(f, \underline{g})), (1 − χ(p)^{−1}α_{f}^{−1}α−1 ; \underline{g})^{−1}L_{p}(f, \underline{g})

agree on all points y ∈ Ω_{g} of weight ν_{2,ϵ}, as ϵ ranges over all the characters of p-power conductor. Since both elements belong to the fraction field of Λ_{g}, it follows that they must be equal. Theorem 3.9 follows after noting that the factor (1 − χ(p)^{−1}α_{f}^{−1}α−1 ; \underline{g}) does not lie in the kernel of the weight one point y attached to g_{α} (for reasons of weight, since α_{f} has complex absolute \sqrt{p} while α_{g_{y}} is a root of unity), and hence belongs to (Λ\underline{g}′)^{×}. □

At the weight one point y attached to g = g_{α}, we have H_{fin}^{1}(ℚ_{p}, V_{fg}) = 0, and hence the class κ_{p}^{f}(f, g_{α}) can only be cristalline if it is 0. Let

exp_{p}^{∗} : H^{1}(ℚ_{p}, V_{fg}) ⟶ Fil^{0}D(V_{fg})

denote the dual exponental map of Bloch and Kato.

Theorem 3.10. Let y =∈ Ω\underline{g} be the point over ν_{1,1}, corresponding to the classical p-stabilised weight one form g_{y} = g_{α} ∈ S_{1}(Np, χ). Then

exp_{p}^{∗}(κ_{p}^{f}(f, g_{α})) ≠ 0 ⇔ L(f ⊗ g, 1) ≠ 0.

Proof. Perrin-Riou’s local reciprocity law and its extension in [LZ, Thm. 4.15] asserts that, at the weight one point y attached to g_{α},

L(κ_{p}(f, \underline{g}))(y_{0}) ∼ exp^{∗} κ_{p}^{f}(f, g_{α}),

where the symbol ∼ denotes agreement up to an (explicit) non-zero element of Fil^{0}D_{dR}(V_{fg_{α}}). (The Euler factor that arises in the p-adic interpolation process is non-vanishing at y, since the associated Galois representation V_{fg} is pure of weight −1.) The result now follows by combining Theorems 3.9 and 2.12. □

Theorem 3.10 implies that the class κ(f, g_{y}) = κ(f, g_{α}) of equation (54) satisfies the two properties listed in Theorem 1.2. This concludes the proof of Theorems A and B of the Introduction.

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M.B.: Fakultaet fuer Mathematik, Universitaet Duisburg-Essen, Essen, Germany

   E-mail address: massimo.bertolini@uni-due.de

H. D.: Department of Mathematics and Statistics, McGill University, Montreal, 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