DIAGONAL CYCLES AND EULER SYSTEMS I: A p-ADIC GROSS-ZAGIER FORMULA

HENRI DARMON AND VICTOR ROTGER

Abstract. This article is the first in a series devoted to studying generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a p-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the p-adic Abel-Jacobi map to special values of certain p-adic Lfunctions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula.

Cet article est le premier d’une série consacrée aux cycles de Gross-Kudla-Schoen généralisés appartenant aux groupes de Chow de produits de trois variétés de Kuga-Sato, et aux systèmes d’Euler qui leur sont associés. La série au complet repose sur une variante p-adique de la formule de Gross-Zagier qui relie l’image des cycles de Gross-Kudla-Schoen par l’application d’Abel-Jacobi p-adique aux valeurs spéciales de certaines fonctions L p-adiques attachées à la convolution de Garrett-Rankin de trois familles de Hida de formes modulaires cuspidales. L’objectif principal de cet article est de décrire et de démontrer cette variante.

MSC: 11F12, 11G05, 11G35, 11G40.

Keywords: Gross-Kudla-Schoen cycle, Garrett-Rankin p-adic L-function, p-adic Abel-Jacobi map, Chow group, Coleman integration.

Contents

1. Introduction 2 2. Cohomology and modular forms 7 2.1. The de Rham cohomology of curves over p-adic rings 7 2.2. p-adic modular forms 10 2.3. Nearly holomorphic modular forms and the Shimura-Maass derivative 14 2.4. Nearly overconvergent modular forms and the d operator 16 2.5. Periods of modular forms 20 2.6. Hida families 22 3. Diagonal cycles and p-adic Abel-Jacobi maps 26 3.1. Generalised Gross-Kudla-Schoen cycles 26 3.2. A formula for AJ_{p}(Δ_{k,ℓ,m}) in terms of Coleman integration 28 3.3. Proof of the formula: Besser’s finite polynomial cohomology 31 3.4. A formula for AJ_{p}(Δ_{k,ℓ,m}) in terms of p-adic modular forms 34 4. Garrett-Rankin triple product L-functions 39 4.1. Classical L-functions 39 4.2. p-adic L-functions 40 5. The p-adic Gross-Zagier formula 45 References 46

1. Introduction

This article is the first in a series devoted to studying generalised diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a p-adic Gross-Zagier formula relating

• the image under the p-adic Abel-Jacobi map of certain generalised Gross-Kudla-Schoen cycles in the product of three Kuga-Sato varieties, to

• the special value of the p-adic L-function of [HaTi] attached to the Garrett-Rankin triple convolution of three Hida families of modular forms, at a point lying outside its region of interpolation.

In order to precisely state the main result, let

f = ∑ a_{n}(f)q^{n} ∈ S_{k}(N_{f}, χ_{f}),
g = ∑ a_{n}(g)q^{n} ∈ S_{ℓ}(N_{g}, χ_{g}),
h = ∑ a_{n}(h)q^{n} ∈ S_{m}(N_{h}, χ_{h})

be three normalized primitive cuspidal eigenforms of weights k, ℓ, m ≥ 2, levels N_{f}, N_{g}, N_{h} ≥ 1, and Nebentypus characters χ_{f}, χ_{g}, and χ_{h}, respectively. Let N := lcm(N_{f}, N_{g}, N_{h}) and assume that

χ_{f} · χ_{g} · χ_{h} = 1,

so that in particular k + ℓ + m is even.

The triple (k, ℓ, m) is said to be balanced if the largest weight is strictly smaller than the sum of the other two. A triple of weights which is not balanced will be called unbalanced, and the largest weight in an unbalanced triple will be referred to as the dominant weight.

Section 4.1 recalls the definition of the Garrett-Rankin L-function L(f, g, h; s) attached to the triple tensor product

V_{p}(f, g, h) := V_{p}(f) ⊗ V_{p}(g) ⊗ V_{p}(h)

of the (compatible systems of) p-adic Galois representations V_{p}(f), V_{p}(g) and V_{p}(h) attached to f, g and h respectively. This L-function satisfies a functional equation relating its values at s and k + ℓ + m − 2 − s. In particular, the parity of the order of vanishing of L(f, g, h; s) at the central critical point c := \frac{k+ℓ+m−2}{2} is controlled by the sign ε ∈ \{±1\} in this functional equation, a quantity that can be expressed as a product ε = ∏_{v|N∞} ε_{v}, ε_{v} ∈ {±1}, of local root numbers indexed by the places dividing N∞. The following hypothesis is assumed throughout:

H: The local root numbers ε_{v} at all the finite primes v|N are equal to +1.

This assumption holds in a broad collection of settings of arithmetic interest. For instance, it is satisfied in either of the following two cases:

• gcd(N_{f}, N_{g}, N_{h}) = 1, or,

• N = N_{f} = N_{g} = N_{h} is square-free and a_{v}(f)a_{v}(g)a_{v}(h) = −1 for all primes v | N. Assumption H implies that ε = ε_{∞} depends only on the local sign at ∞, which in turn depends only on whether the weights of (f, g, h) are balanced or not:

ε = ε_{∞} = \{ \begin{matrix}−1 & if (k, ℓ, m) is balanced; \\ 1 & if (k, ℓ, m) is unbalanced. \end{matrix}

In particular, the L-function L(f, g, h, s) necessarily vanishes (to odd order) at its central point c when (k, ℓ, m) is balanced.

A p-ADIC GROSS-ZAGIER FORMULA FOR DIAGONAL CYCLES 3

Let E denote the universal generalised elliptic curve fibered over X = X_{1}(N). For any n ≥ 0, let E^{n} be the n-th Kuga-Sato variety over X_{1}(N). It is an n + 1-dimensional variety obtained by desingularising the n-fold fiber product of E over X_{1}(N). (Cf. [Sc] for a more detailed account of its construction.) The p-adic Galois representation V_{p}(f, g, h) occurs in the middle cohomology of the triple product

(1)  W := E^{k−2} × E^{ℓ−2} × E^{m−2}.

When (k, ℓ, m) is balanced and assumption H is satisfied, the conjectures of Bloch-Kato and Beilinson-Bloch predict (because of the vanishing of L(f, g, h, c)) that there should then exist a non-trivial cycle in the Chow group ℚ ⊗ CH^{c}(W)_{0} of rational equivalence classes of nullhomologous cycles of codimension c on the variety W of (1). Section 3.1 introduces cycles Δ_{f,g,h} ∈ ℚ ⊗ CH^{c}(W)_{0} which are natural candidates to fulfill these expectations, and whose construction we now briefly summarize.

Set r = \frac{k+ℓ+m−6}{2}. As explained in §3.1, there exists an essentially unique, natural way of embedding the Kuga-Sato variety E^{r} in the variety W. Its image gives rise to an element in the Chow group CH^{r+2}(W) which, suitably modified, becomes homologically trivial. In this way, we obtain a cycle

Δ_{k,ℓ,m} ∈ CH^{r+2}(W)_{0} := ker(CH^{r+2}(W) ⟶^{cl} H_{dR}^{2r+4}(W/ℂ)).

In the special case where k = ℓ = m = 2, the cycle Δ_{2,2,2} is just the modified diagonal considered by Gross–Kudla [GrKu] and Gross–Schoen [GrSc].

The cycles Δ_{f,g,h} alluded to above are defined as the (f, g, h)-isotypical component of the null-homologous cycle Δ_{k,ℓ,m} with respect to the action of the Hecke operators.

It is natural to conjecture that the heights of these cycles in the sense of Beilinson and Bloch are well-defined (cf. [GrKu] and [GrSc] for more details on the necessary definitions), and can be directly related to the first derivative of the triple product L-function L(f, g, h, s) at the central point:

(2)  h(Δ_{f,g,h}) =^{?} (Explicit non-zero factor) × L′(f, g, h, r + 2).

When (k, ℓ, m) = (2, 2, 2), this was predicted in [GrKu] and has recently been proved by X. Yuan, S. Zhang and W. Zhang in [YZZ].

Remark 1.1. It would be natural to relax assumption H to the weaker condition

(3)  H_{even}: The set of primes v|N for which ε_{v} = −1 is of even cardinality.

This is sufficient to guarantee that ε = ε_{∞}, and can be dealt with at the cost of replacing Kuga-Sato varieties with more general objects arising from the self-fold products of certain families of abelian surfaces (or genus two curves) fibered over Shimura curves rather than classical modular curves. Hypothesis H may thus be regarded as analogous to the classical Heegner or Gross-Zagier hypothesis imposed in the study of the Rankin-Selberg L-function L(f ⊗ θ_{K}, s) attached to a single eigenform f and the weight one theta series of an imaginary quadratic field K. Both are meant to avoid having to deal with Shimura curves associated with a quaternion division algebra, and make it possible to confine one’s attention to classical modular curves. Much of our study extends to the setting of H_{even} by appealing to the work of P. Kassaei [Kas99] and R. Brasca [Br]; in our exposition we have tried to present our results in a way that suggests the modifications necessary to deal with arbitrary Shimura curves.

In this work we do not focus on (2), but rather on a p-adic analogue. Our main result relates the image of Δ_{f,g,h} under the p-adic Abel-Jacobi map

(4)  AJ_{p} : CH^{r+2}(W)_{0}(ℚ_{p}) ⟶ Fil^{r+2}H_{dR}^{2r+3}(W/ℚ_{p})^{∨}

to the special value of a triple product p-adic L-function attached to three Hida families of modular forms, which we now describe in more detail.

Fix an odd prime number p ∤ N at which f, g and h are ordinary. Let

f : Ω_{f} ⟶ ℂ_{p}[[q]], g : Ω_{g} ⟶ ℂ_{p}[[q]], h : Ω_{h} ⟶ ℂ_{p}[[q]]

denote the Hida families of overconvergent p-adic modular forms passing though f, g and h, respectively, as constructed in [Hi86a] and [Hi86b], and briefly reviewed in §2.6 below. The spaces Ω_{f}, Ω_{g} and Ω_{h} are finite rigid analytic coverings of suitable subsets of the weight space

Ω := Hom_{cts}(ℤ_{p}^{×}, ℂ_{p}^{×}),

which contains the integers ℤ as a dense subset via the natural inclusion k ↦ (x ↦ x^{k}). A point x ∈ Ω_{f} is said to be classical if its image in Ω, denoted κ(x), belongs to ℤ^{≥2}, and the set of classical points in Ω_{f} is denoted by Ω_{f,cl}. Part of the requirement that f be a Hida family is that the formal q-series f_{x}^{(p)} := f(x) should correspond to a normalised eigenform of weight κ(x) on Γ_{1}(N) ∩ Γ_{0}(p), for almost all x ∈ Ω_{f,cl}. For all but finitely many such x, the form f_{x}^{(p)} is the ordinary p-stabilisation of a normalised eigenform on Γ_{1}(N), denoted f_{x}.

The natural domain of definition of the triple product p-adic L-functions is the p-adic analytic space

Σ := Ω_{f} × Ω_{g} × Ω_{h}.

Let Σ_{cl} := Ω_{f,cl} × Ω_{g,cl} × Ω_{h,cl} ⊂ Σ denote its subset of “classical points”. This set is naturally partitioned into four disjoint subsets:

Σ_{f} = {(x, y, z) ∈ Σ_{cl}, such that κ(x) ≥ κ(y) + κ(z)};
Σ_{g} = {(x, y, z) ∈ Σ_{cl}, such that κ(y) ≥ κ(x) + κ(z)};
Σ_{h} = {(x, y, z) ∈ Σ_{cl}, such that κ(z) ≥ κ(x) + κ(y)};
Σ_{bal} = {(x, y, z) ∈ Σ_{cl}, such that (κ(x), κ(y), κ(z)) is balanced.}.

Section 4 exploits the strategy pioneered by Hida [Hi88b] and subsequently extended by Harris and Tilouine [HaTi] to construct three a priori distinct p-adic L-functions of three variables, denoted

ℒ_{p} ^{f}(f, g, h), ℒ_{p} ^{g}(f, g, h), ℒ_{p} ^{h}(f, g, h) : Σ ⟶ ℂ_{p},

which interpolate the square-roots of the central critical values of the classical L-function L(f_{x}, g_{y}, h_{z}, s), as (x, y, z) ranges over Σ_{f}, Σ_{g}, and Σ_{h} respectively. The precise interpolation property defining the three p-adic L-functions is spelled out in Theorem 4.7 of Section 4.2.

Given (x, y, z) ∈ Σ_{bal}, the Heegner assumption H can be used to show that the classical L-function L(f_{x}, g_{y}, h_{z}, s) vanishes at its central point for reasons of sign. The central critical derivative L′(f_{x}, g_{y}, h_{z}, \frac{κ(x)+κ(y)+κ(z)−2}{2}) is then a natural object of arithmetic interest. In the p-adic realm, the three distinct p-adic avatars of the classical L-function, namely, ℒ_{p} ^{f}(f, g, h), ℒ_{p} ^{g}(f, g, h), and ℒ_{p} ^{h}(f, g, h), need not vanish at the balanced point (x, y, z), since this point lies outside the region of classical interpolation. The corresponding p-adic special values can be viewed as different p-adic avatars of the complex leading term, and one might expect them to encode similar information related to the motive of V_{f_{x}} ⊗ V_{g_{y}} ⊗ V_{h_{z}}.

Remark 1.2. One can also envisage a fourth p-adic L-function

L_{p}^{bal}(f, g, h)(x, y, z, s) : Ω_{f} × Ω_{g} × Ω_{h} × Ω ⟶ ℂ_{p}

involving a further “cyclotomic” variable s and interpolating the classical special values of L(f_{x}, g_{y}, h_{z}, s)–but not their square roots–in the critical range

(x, y, z) ∈ Σ_{bal}, 1 ≤ s ≤ κ(x) + κ(y) + κ(z) − 3.

The construction of such an L-function is described in [BoPa] (see also the references therein). Under hypothesis H, the function L_{p}^{bal}(f, g, h) should vanish identically on the “central critical hyperplane” 2s = κ(x) + κ(y) + κ(z) − 2, and the presence of the cyclotomic variable is therefore key to ensuring its non-triviality. This “balanced” p-adic L-function plays no role in this article.

In stating our main result, it will be convenient to assume for simplicity that N_{f} = N_{g} = N_{h}, i.e., that the three eigenforms f, g and h are new of the same level N. (This assumption, which is too restrictive for most interesting arithmetic applications, will be relaxed in the body of the text.) To any classical newform ϕ of weight r + 2 on Γ_{1}(N) which is ordinary at p, there corresponds a cohomology class

ω_{ϕ} ∈ Fil^{r+1}H_{dR}^{r+1}(E^{r}/ℚ̄) ⊂ Fil^{r+1}H_{dR}^{r+1}(E^{r}/ℂ_{p}),

where the inclusion is induced from our fixed embedding ℚ̄ ⊂ ℂ_{p}. The ϕ-isotypic component of H_{dR}^{r+1}(E^{r}/ℂ_{p}) is two-dimensional over ℂ_{p} and (because ϕ is ordinary) it admits a onedimensional unit root subspace, denoted H_{dR}^{r+1}(E^{r}/ℂ_{p})^{u-r}, on which the Frobenius endomorphism acts as multiplication by a p-adic unit. This unit root subspace is complementary to the middle step in the Hodge filtration, and hence, there is a unique element η_{ϕ}^{u-r} ∈ H_{dR}^{r+1}(E^{r}/ℂ_{p})^{u-r} satisfying

⟨ω_{ϕ}, η_{ϕ}^{u-r}⟩ = 1,

where ⟨ , ⟩ denotes the non-degenerate Poincaré pairing on H_{dR}^{r+1}(E^{r}/ℂ_{p}). One thus obtains a natural basis (ω_{ϕ}, η_{ϕ}^{u-r}) of the ϕ-isotypic component of H_{dR}^{r+1}(E^{r}/ℂ_{p}), for any ordinary classical newform ϕ ∈ S_{r+2}(Γ_{1}(N)).

If (f, g, h) is a triple of newforms of level N and balanced weights (k, ℓ, m) = (r_{1} + 2, r_{2} + 2, r_{3} + 2), then

(5)  η_{f}^{u-r} ⊗ ω_{g} ⊗ ω_{h} ∈ H_{dR}^{r_{1}+1}(E^{r_{1}}) ⊗ Fil^{r_{2}+1}H_{dR}^{r_{2}+1}(E^{r_{2}}) ⊗ Fil^{r_{3}+1}H_{dR}^{r_{3}+1}(E^{r_{3}})
⊂ Fil^{r+2} (H_{dR}^{r_{1}+1}(E^{r_{1}}) ⊗ H_{dR}^{r_{2}+1}(E^{r_{2}}) ⊗ H_{dR}^{r_{3}+1}(E^{r_{3}}))
⊂ Fil^{r+2}H_{dR}^{2r+3}(W/ℚ̄_{p}),

where the last inclusion arises from the Künneth decomposition. In particular, the class η_{f}^{u-r} ⊗ ω_{g} ⊗ ω_{h} lies in the domain of AJ_{p}(Δ) when (k, ℓ, m) is balanced.

For any f ∈ S_{k}(N, χ), we shall always write

(x^{2} − a_{p}(f)x + χ_{f}(p)p^{k−1}) = (x − α_{f})(x − β_{f}), with ord_{p}(α_{f}) ≤ ord_{p}(β_{f}),

so that in particular α_{f} is a p-adic unit when f is ordinary.

The main result of this article is

Theorem 1.3. Given (x, y, z) ∈ Σ_{bal}, let

(f, g, h) := (f_{x}, g_{y}, h_{z}), (k, ℓ, m) := (κ(x), κ(y), κ(z)),
c := (k + ℓ + m − 2)/2, k = ℓ + m − 2 − 2t (with t ≥ 0),

and let Δ := Δ_{k,ℓ,m} be the generalised diagonal cycle in E^{k−2} × E^{ℓ−2} × E^{m−2}. Then

ℒ_{p} ^{f}(f, g, h)(x, y, z) = (−1)^{t} \frac{E(f, g, h)}{t! · E_{0}(f)E_{1}(f)} × AJ_{p}(Δ)(η_{f}^{u-r} ⊗ ω_{g} ⊗ ω_{h}),

where

(6)  E(f, g, h) := (1 − β_{f}α_{g}α_{h}p^{−c}) × (1 − β_{f}α_{g}β_{h}p^{−c})
× (1 − β_{f}β_{g}α_{h}p^{−c}) × (1 − β_{f}β_{g}β_{h}p^{−c}) ,
(7)  E_{0}(f) := (1 − β_{f}^{2}χ_{f}^{−1}(p)p^{1−k}),
(8)  E_{1}(f) := (1 − β_{f}^{2}χ_{f}^{−1}(p)p^{−k}).

See Theorem 5.1 for a more general statement involving newforms of possibly different levels. Theorem 1.3 can be viewed as a p-adic analogue of the archimedean formula suggested in (2), and in fact the Euler factor E(f, g, h) that arises in the formula fits within the general conjectural description of the p-adic L-function of an arbitrary motive given by Panciskin in [Pan, p. 285].

One could also envisage a more direct p-adic analogue, relating the cyclotomic p-adic height of Δ_{f_{x},g_{y},h_{z}} to the derivative in the cyclotomic direction of the p-adic L-function L_{p}^{bal}(f, g, h)(x, y, z, s) alluded to in Remark 1.2. Such a formula, which is currently unavailable in the literature, would be closer in spirit to the archimedean formulae conjectured in [GrKu] and proved in [YZZ], and to the p-adic analogues of the Gross-Zager formula proved in [PR] and [Ne1], while Theorem 1.3 is better adapted to the arithmetic applications that the authors wish to pursue by exploiting the Euler system properties of p-adic families of diagonal cycles and associated explicit reciprocity laws in the spirit of Coates–Wiles and Kato–Perrin-Riou. Let us close this introduction by describing some of these applications.

I. The Euler system of diagonal cycles. Theorem 1.3 is available when f is replaced by g or h if the latter two forms are also assumed to be ordinary. The Abel-Jacobi image AJ_{p}(Δ) therefore encodes the values of the three distinct p-adic L-functions ℒ_{p} ^{f}(f, g, h), ℒ_{p} ^{g}(f, g, h) and ℒ_{p} ^{h}(f, g, h) at (x, y, z) ∈ Σ_{bal}. This suggests that these L-functions should be viewed as the different projections of a common Euler system obtained by p-adically interpolating the diagonal cycles themselves as (x, y, z) ranges over Σ_{bal}. (More precisely: their images κ(f_{x}, g_{y}, h_{z}) ∈ H^{1}(ℚ, V_{f_{x},g_{y},h_{z}}) under the p-adic étale Abel-Jacobi map, where V_{f_{x},g_{y},h_{z}} denotes the self-dual Tate twist of the triple tensor product V_{f_{x}} ⊗ V_{g_{y}} ⊗ V_{h_{z}} of the Deligne representations attached to the eigenforms f_{x}, g_{y} and h_{z}.) The sequel [DR] to this paper pieces these global classes together into an element

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

where V_{f,g,h} is a twist of the tensor product V_{f} ⊗ V_{g} ⊗ V_{h} of Hida’s Λ-adic representations attached to f, g, and h respectively, interpolating the representations V_{f_{x},g_{y},h_{z}}. The three Garrett-Rankin p-adic L-functions are then obtained from the image of \underline{κ}(f, g, h) under a homomorphism interpolating the Bloch-Kato logarithms attached to the specialisations V_{f_{x}} ⊗ V_{g_{y}} ⊗V_{h_{z}} in the range (x, y, z) ∈ Σ_{bal}. The resulting construction of the Garrett-Rankin p-adic L-functions in terms of diagonal cycles is directly analogous to the construction of the Kubota-Leopoldt (resp. Katz) p-adic L-function from the p-adic logarithms of circular (resp. elliptic) units. An application of this point of view to the Birch-Swinnerton-Dyer conjecture, obtained by considering weight one specialisations of g and h, is the implication

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

where E is an elliptic curve over ℚ and ρ_{1} and ρ_{2} are odd irreducible two-dimensional Artin representations for which ρ_{1} ⊗ ρ_{2} has determinant one.

II. Beilinson-Flach elements. The methods of the present article, transposed to the setting where h is a Hida family of Eisenstein series, lead to a proof [BDR] of a p-adic Beilinson formula relating the p-adic regulators of certain Flach elements in the Higher Chow group CH^{2}(X_{0}(N) × X_{0}(N), 1) to values of Hida’s three-variable p-adic L-function attached to the Rankin convolution of two Hida families f and g of cusp forms. A notable application of this result (when made to vary in p-adic families, as in the previous paragraph) is the implication

(9)  L(E, ρ, 1) ≠ 0 ⟹ Hom_{G_{ℚ}}(ρ, E(ℚ̄) ⊗ ℂ) = 0,

where ρ is an odd, irreducible two-dimensional Artin representation.

III. Beilinson elements and Kato’s Euler system. The point of view sketched in the two previous Remarks is also consistent with the Kato-Perrin-Riou approach to the Mazur-Swinnerton-Dyer p-adic L-function attached to a cusp form of weight 2, which one recovers when both g and h are taken to be Hida families of Eisenstein series, the role of the diagonal cycles being played by Beilinson elements in the K_{2} of modular curves, or in the higher Chow group CH^{2}(X_{0}(N), 2). Guided by this analogy and relying crucially on the techniques of the present work, the article [BD] gives a new proof of the p-adic Beilinson formula relating the p-adic regulators of the Beilinson elements to the special values at s = 2 of the Mazur-Swinnerton-Dyer p-adic L-functions. The authors’ study of diagonal cycles and Flach elements is thus a direct generalisation of the approach of Kato which underlies the proof of (9) when ρ is replaced by a Dirichlet character.

IV. The Euler system of Heegner points, revisited. In part, the authors were led to Theorem 1.3 by the analogy with the main result of [BDP], in which the images of Heegner points (or more general Heegner cycles) under p-adic Abel-Jacobi maps are related to the values of certain anticyclotomic p-adic L-functions at classical points lying outside the range of p-adic interpolation defining them. In his forthcoming PhD thesis, F. Castella uses this to construct these anticyclotomic p-adic L-functions in terms of p-adic logarithms of Heegner points, leading to a treatment of the Heegner point Euler system entirely parallel to the other examples alluded to above (namely, circular units, elliptic units, Beilinson-Kato elements, Flach elements, and diagonal cycles.)

V. p-adic calculations of Chow-Heegner points. The article [La2] and the forthcoming Ph. D thesis of M. Daub [Da] combine Theorem 1.3 with Alan Lauder’s fast algorithm [La1] for calculating ordinary projections to calculate the “Chow-Heegner points” attached to Gross-Schoen diagonal cycles (as described in [YZZ] and studied in [DRS]) by p-adic analytic methods, thus supplying the p-adic counterpart of the complex calculations carried out in [DDLR].

Notations: Throughout the article, given a power q = p^{d} of a prime p, the symbols 𝔽_{q}, ℤ_{q}, and ℚ_{q} are reserved for the finite field with q elements, its ring of Witt vectors, and the finite unramified extension of ℚ_{p} of degree d respectively. For any field extension F/ℚ, we will write G_{F} = Gal (F̄/F) for the absolute Galois group of an algebraic closure of F, and if X is any variety over F, we let X̄ := X ×_{Spec(F)} Spec(F̄) denote the base change of X to the algebraic closure. We adopt the usual conventions regarding motives and their Tate twists: for any integer j we write ℤ(j) = H^{2}(ℙ_{1})^{⊗−j}, so that a geometric frobenius element at ℓ ≠ p (resp. at ℓ = p) acts on its p-adic étale (resp. cristalline) realisation as multiplication by ℓ^{−j}. If M is a motive over ℚ, we let M(j) := M ⊗ ℤ(j) denote its j-th Tate twist, so that L(M(j), s) = L(M, s + j), where L(M, s) is the L-function attached to M.

Acknowledgements: The authors thank Ignacio Sols and Dipendra Prasad for many stimulating exchanges about the geometric aspects of diagonal cycles and the analytic properties of triple product L-functions, respectively. The work on this article began during the summer of 2010 at the special number theory semester organised by the CRM in Barcelona, and the authors also gratefully acknowledge the role of this institute in facilitating their collaboration. The first author was funded by an NSERC Discovery grant, and the second author received financial support from DGICYT Grant MTM2009-13060-C02-01 and from 2009 SGR 1220.

2. Cohomology and modular forms

2.1. The de Rham cohomology of curves over p-adic rings. Let X be an arbitrary smooth proper curve over Spec(ℤ_{p}) and write X̃ and X for its special and generic fiber, respectively. Let {P_{1}, …, P_{s}} ⊂ X̃(𝔽̄_{p}) be a non-empty G_{𝔽_{p}}-stable collection of closed points in its special fiber. Since X̃ is smooth, these points admit lifts P̃_{1}, …, P̃_{s} ∈ X (O_{ℂ_{p}}) to characteristic zero. It will be convenient (albeit not indispensable) to fix a choice of such lifts which is stable under the natural action of G_{ℚ_{p}}. This determines the affine scheme

X′ = X − {P̃_{1}, …, P̃_{s}}

over Spec(ℤ_{p}), whose special and generic fiber are denoted X̃′ and X′ respectively. Because X is proper over Spec(ℤ_{p}), there is a natural identification X(ℂ_{p}) = X (O_{ℂ_{p}}) and a resulting reduction map

red : X(ℂ_{p}) ⟶ X̃(𝔽̄_{p}).

The standard affinoid in X(ℂ_{p}) attached to X′ is defined to be

A := red^{−1}(X̃′(𝔽̄_{p})).

It is a connected affinoid region obtained by deleting from X(ℂ_{p}) a collection of s disjoint residue discs.

For each j = 1, …, s, choose a local coordinate λ_{j} at P̃_{j}, i.e., a rigid analytic isomorphism from the residue disc of P̃_{j} in X(ℂ_{p}) to the open unit disc in ℂ_{p}, sending P̃_{j} to the origin. These local parameters give rise to a family of wide open neighborhoods of A, indexed by a real parameter ϵ > 0 and defined by

W_{ϵ} := A ∪ ⋃_{j=1}^{s}{x ∈ red^{−1}(P_{j}) with ord_{p}λ_{j}(x) < ϵ}.

The region W_{ϵ} is obtained by adjoining open annuli V_{1}, …, V_{s} of “width ϵ” to A around the boundaries of each of the deleted residue discs. Note the obvious inclusions

A ⊂ W_{ϵ} ⊂ X′(ℂ_{p}), W_{ϵ_{1}} ⊂ W_{ϵ_{2}} if ϵ_{1} < ϵ_{2}.

One of the main reasons for working with the wide open neighborhoods W_{ϵ} rather than with the affinoid A itself is that their cohomology is better behaved and a more faithful reflection of algebraic de Rham cohomology over ℂ_{p}. More precisely, let O and Ω^{1} denote the structure sheaf and the sheaf of differentials on X. The restrictions of these sheaves to A and to W_{ϵ}, viewed as rigid analytic sheaves, are denoted by the same symbols by a slight abuse of notation. If K is any complete subfield of ℂ_{p} we will denote by Ω^{1}(W_{ϵ}/K) the rigid differentials on W_{ϵ} that are defined over K. Set

H_{rig}^{1}(W_{ϵ}/K) := \frac{Ω^{1}(W_{ϵ}/K)}{dO_{W_{ϵ}/K}}.

Note that, because X′ is affine, we have H_{dR}^{1}(X′/K) = \frac{Ω^{1}(X′/K)}{dO_{X′}}. The natural restriction map Ω^{1}(X′/K) ⟶ Ω^{1}(W_{ϵ}/K) sends exact forms to exact forms and therefore induces a map

comp_{ϵ} : H_{dR}^{1}(X′/K) ⟶ H_{rig}^{1}(W_{ϵ}/K).

For each annulus V_{1}, …, V_{s} appearing in the definition of W_{ϵ}, let

res_{V_{j}} : Ω^{1}(W_{ϵ}/K) ⟶ K(−1)

denote the p-adic annular residue, as it is described in [Cole94, Ch. 7] for example.

The annular residue vanishes on dO_{W_{ϵ}}, and is therefore well-defined on cohomology. It is related to the usual residue of algebraic differential forms through the following commutative diagram:

(10)  [figure: H_{dR}^{1}(X′/K); ⊕_{j}resP̃_{j}; _{comp_{ϵ}}; K(−1)^{s}; Σ; K(−1); 0; H_{rig}^{1}(W_{ϵ}/K); ⊕_{j}res_{V_{j}}; K(−1)^{s}; Σ; K(−1); 0.]

Proposition 2.1. For any ϵ > 0, the map comp_{ϵ} is an isomorphism of K-vector spaces.

Proof. This follows as in [Cole94] from the long exact Mayer-Vietoris sequence (cf. A.2. of loc.cit. for instance) associated to the admissible covering of X by wide open subsets consisting of W_{ϵ} together with the p-adic residue discs in the complement of A. □

Thanks to this proposition, the de Rham cohomology H_{dR}^{1}(X_{K}) of the base change of X to K can be identified with the space of classes of rigid analytic forms on W_{ϵ} over K with vanishing annular residues (cf. e.g. [BDP, Prop. 4.11]). This description of H_{dR}^{1}(X_{K}) is the basis for a concrete description of the action of the Frobenius operator on de Rham cohomology. Let σ ∈ Gal (K̄/ℚ_{p}) be a Frobenius automorphism and let

Φ : A ⟶ A

be any characteristic zero lift of the Frobenius morphism on the special fiber X̃′. This rigid analytic morphism extends to a morphism Φ : W_{ϵ} ⟶ W_{ϵ′}, for suitable 0 < ϵ < ϵ′, and induces linear maps

Φ : O(W_{ϵ′}/K) ⟶ O(W_{ϵ}/K), Φ : Ω^{1}(W_{ϵ′}/K) ⟶ Ω^{1}(W_{ϵ}/K).

Although the operator Φ need not preserve any of the spaces Ω^{1}(W_{ϵ}/K) for a given fixed ϵ, it does give rise to an endomorphism on H_{dR}^{1}(X′/K), which shall also be denoted Φ by abuse of notation. This Frobenius morphism is thus the unique endomorphism of H_{dR}^{1}(X′/K) that completes the top row in the following commutative diagram:

(11)  [figure: H_{dR}^{1}(X′/K); Φ; _{comp_{ϵ′}}; H_{dR}^{1}(X′/K); _{comp_{ϵ}}; H_{rig}^{1}(W_{ϵ′}/K); Φ; H_{rig}^{1}(W_{ϵ}/K).]

The endomorphism Φ, which is compatible with the annular residue in the obvious sense, preserves the subspace H_{dR}^{1}(X_{K}).

The K-vector subspace of H_{dR}^{1}(X_{K}) (resp. of H_{dR}^{1}(X′/K)) spanned by the vectors on which Φ acts via multiplication by a p-adic unit is called the unit root subspace and is denoted by H_{dR}^{1}(X_{K})^{u-r} (resp. by H_{dR}^{1}(X′/K)^{u-r}). More generally, the subspaces spanned by vectors on which Φ acts with slope t ∈ ℚ (i.e., as multiplication by a scalar λ ∈ ℂ_{p}^{×} with ord_{p}λ = t) is called the slope t subspace of H_{dR}^{1}(X_{K}) and is denoted H_{dR}^{1}(X_{K})^{Φ,t}.

The de Rham cohomology H_{dR}^{1}(X_{K}) is equipped with the usual alternating Poincaré duality

⟨ , ⟩ : H_{dR}^{1}(X_{K}) × H_{dR}^{1}(X_{K}) ⟶ H_{dR}^{2}(X_{K}) = K(−1),

which in terms of representatives ω_{1}, ω_{2} ∈ Ω_{rig}^{1}(W_{ϵ}) for cohomology classes ξ_{1} and ξ_{2} is described by the formula

⟨ξ_{1}, ξ_{2}⟩ = ∑_{j=1}^{s} res_{V_{j}}(F_{ω_{1}}^{(j)} · ω_{2}),

where res_{V_{j}} is the p-adic annular residue and F_{ω_{1}}^{(j)} denotes a local analytic primitive of ω_{1} on the annulus V_{j}.

Poincaré duality is compatible with the Frobenius endomorphism in the sense that

⟨Φξ_{1}, Φξ_{2}⟩ = Φ⟨ξ_{1}, ξ_{2}⟩ = p⟨ξ_{1}, ξ_{2}⟩.

In particular, Poincaré duality descends to a well-defined non-degenerate pairing

(12)  ⟨ , ⟩ : H_{dR}^{1}(X_{K})^{u-r} × H_{dR}^{1}(X_{K})^{Φ,1} ⟶ K(−1).

2.2. p-adic modular forms. Denote by S_{k}^{(p)}(N) the space of p-adic modular forms of weight k and tame level N (as defined in [Se] or [Kat73] for example) and let S_{k}^{oc}(N) denote its subspace of overconvergent forms. These spaces are equipped with two non-commuting operators U and V, defined on q-expansions by the rule

(13)  (Uf)(q) = ∑_{n=1}^{∞} a_{pn}q^{n},  (V f)(q) = ∑_{n=1}^{∞} a_{n}q^{pn}.

These operators satisfy

UV f = f, V Uf(q) = ∑_{n=1}^{∞} a_{np}q^{np},

so that

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

has fourier coefficients supported on the integers prime to p. Note that the operators U and V do not commute–although, as we will explain later, the operators induced on the cohomology of modular curves (with coefficients in the relevant local systems) do commute, and are inverse to each other.

The operator U on the p-adic Banach space of overconvergent modular forms is completely continuous and gives rise to a slope decomposition on this infinite-dimensional vector space. A p-adic modular form which belongs to the slope 0 subspace for U is said to be ordinary, and the space of all such p-adic modular forms is denoted by S_{k}^{ord}(N). The ordinary subspace (like all the finite slope subspaces) is finite-dimensional over ℂ_{p}. More precisely, Coleman’s classicality theorem (cf. [Cole95]) asserts that any ordinary overconvergent modular form of weight k ≥ 2 is (the p-stabilisation of) a classical modular form of weight k on Γ_{1}(N). In particular, S_{k}^{ord}(N) is naturally contained in the space S_{k}(Γ_{1}(N) ∩ Γ_{0}(p)) of classical modular forms (with fourier coefficients in ℂ_{p}). Hida’s ordinary projector

(15)  e_{ord} := lim_{n} U^{n!}

gives a Hecke-equivariant projection from S_{k}^{oc}(N) to S_{k}^{ord}(N).

Let X_{1}(N) denote the modular curve over Spec(ℤ[1/N]) classifying generalised elliptic curves equipped with an embedding of the finite flat group scheme μ_{N} of N-th roots of unity. (See [Kas99] for a more general scenario.)

Recalling that the prime p does not divide N, set

X := X_{1}(N) ×_{Spec ℤ[1/N]} ℤ_{p}, X = X × Spec(ℚ_{p})

for the smooth curve over ℤ_{p} (resp. over ℚ_{p}) obtained by change of base.

Let P_{1}, …, P_{s} ∈ X̃(𝔽_{p^{2}}) denote the supersingular points of the special fiber of X. These points are the zeroes of a distinguished mod p modular form of weight (p − 1)—the so-called Hasse invariant. As in the previous section, we may choose lifts P̃_{1}, …, P̃_{s} ∈ X (ℤ_{p^{2}}) of the supersingular points to characteristic zero. For example, when p ≥ 5, one may do this by taking the zeroes of the Eisenstein series E_{p−1}, which is the customary lift of the Hasse invariant to characteristic 0. We continue to write

X′ = X − {P̃_{1}, …, P̃_{s}}

for the resulting affine scheme over Spec(ℤ_{p}), whose special and generic fiber are denoted X̃′ and X′ respectively. The connected affinoid region obtained by deleting from X(ℂ_{p}) the s disjoint supersingular residue discs:

A = A_{ord} := red^{−1}(X̃′(𝔽̄_{p}))

is called the ordinary locus, or Hasse domain, in X(ℂ_{p}). Note that

(16)  A = {x ∈ X(ℂ_{p}) with ord_{p}E_{p−1}(x) = 0},

where

ord_{p}E_{p−1}(x) := ord_{p}E_{p−1}(A_{x}, ω_{x}),

and

(1) A_{x} is the generalised elliptic curve with Γ_{1}(N)-structure attached to x by the moduli interpretation of X;

(2) ω_{x} ∈ Ω^{1}(A_{x}/ℂ_{p}) is a regular differential on A_{x}, chosen so that it extends to a regular differential over O_{ℂ_{p}} if A_{x} has good reduction at p, or corresponds to the canonical differential on the Tate curve if x lies in the residue disc of a cusp.

(3) The notation E_{p−1}(A_{x}, ω_{x}) follows Katz’s geometric definition of modular forms as

functions on such pairs, as described in [Kat73, Ch. 1], for example. While E_{p−1}(A_{x}, ω_{x}) genuinely depends on the choice of ω_{x}, its p-adic valuation is independent of it since any two such choices differ by multiplication by an element of O_{ℂ_{p}}^{×}.

Define a system of wide open neighborhoods of A by choosing a real parameter ϵ > 0 and setting

W_{ϵ} := {x ∈ X(ℂ_{p}) with ord_{p}E_{p−1}(x) < ϵ}.

When 0 < ϵ < 1, the region W_{ϵ} does not depend on the choice of lift of the Hasse invariant to characteristic zero. As suggested by (16), the quantity ord_{p}E_{p−1}(x) is (at least, when it is not too large) a sensible measure of the “degree of supersingularity” of the elliptic curve A_{x} associated to x. If ord_{p}E_{p−}(x) < \frac{p}{p+1}, Katz has shown that the elliptic curve A_{x}, although supersingular at p, continues to admit a canonical subgroup Z_{x}, a connected subgroup scheme of A_{x} of order p generalising the canonical subgroup on an ordinary elliptic curve. This makes it possible to choose a canonical lift to characteristic zero of the Frobenius morphism, by setting, for all x ∈ W_{ϵ} with ϵ < \frac{p}{p+1},

(17)  Φ(x) = Point corresponding to A_{x}/Z_{x}.

The affinoid A and the wide opens W_{ϵ} play a key role in Katz’s geometric description of p-adic and overconvergent modular forms, which we will now briefly recall.

Modular forms of weight two. We begin with the somewhat simpler case of forms of weight k = 2, which can be treated by specialising the discussion of Section 2.1 to the case where X = X_{1}(N) with p ∤ N, so that X has good reduction at p. For any complete subfield K of ℂ_{p} we have

(18)  S_{2}^{(p)}(N; K) = Ω_{rig}^{1}(A/K),  S_{2}^{oc}(N; K) = ⋃_{ϵ>0} Ω_{rig}^{1}(W_{ϵ}/K),

that is to say, the space of rigid sections of Ω^{1}(W_{ϵ}/K) is identified with the p-adic Banach space of overconvergent modular forms of weight two with “annuli of convergence of width ϵ”, and fourier coefficients in K. Under this identification, a weight two modular form f with q-expansion f(q) = ∑_{n} a_{n}(f)q^{n} corresponds to the differential

ω_{f} = f(q)\frac{dq}{q} = (∑_{n=1}^{∞} a_{n}(f)q^{n}) \frac{dq}{q} .

By definition of the operators U and V,

(19)  f^{[p]}(q) = (1 − V U)f(q) = q \frac{d}{dq} F,  where F (q) = ∑_{p∤n} \frac{a_{n}(f)}{n}q^{n}.

The modular form F (q) is an overconvergent modular form of weight 0 and level N, i.e., an element of O_{W_{ϵ}} for a suitable ϵ > 0. It corresponds to the rigid analytic primitive of ω_{f^{[p]}} which vanishes at the cusp ∞, i.e., it is determined by the properties

dF = f^{[p]}(q)\frac{dq}{q} , F (∞) = 0.

Define S_{2}^{oc}(N; K)_{0} to be the subspace of S_{2}^{oc}(N; K) consisting of overconvergent modular forms with coefficients in K and vanishing residues at the supersingular annuli. Set also S_{2}^{ord}(N; K)_{0} = S_{2}^{oc}(N; K)_{0} ∩ S_{2}^{ord}(N; K) and write ϕ^{ord} := e_{ord}ϕ for the ordinary projection of an overconvergent modular form.

Lemma 2.2. If ϕ is an overconvergent p-adic modular form of weight two on Γ_{1}(N), then the class of ω_{ϕ} belongs to H_{rig}^{1}(W_{ϵ})^{Φ,1}. Furthermore, the assignment ϕ ↦ [ω_{ϕ}] induces isomorphisms

S_{2}^{ord}(N; K) ⟶^{∼} H_{rig}^{1}(W_{ϵ}/K)^{Φ,1}, S_{2}^{ord}(N; K)_{0} ⟶^{∼} H_{dR}^{1}(X_{K})^{Φ,1}.

Proof. The Frobenius morphism Φ : Ω^{1}(W_{ϵ}) ⟶ Ω^{1}(W_{ϵ/p}) is related to the operator V on overconvergent modular forms of weight two by Φ(ω_{f}) = pω_{V f}. The relation between the operators U and Φ on cohomology (relative to the identifications described above between differentials and weight two modular forms) is therefore given by

Φ = pV = pU^{−1}.

Hence, if ϕ belongs to the slope zero subspace for the action of U, the class of the rigid differential ω_{ϕ} lies in the slope one subspace H_{rig}^{1}(W_{ϵ})^{Φ,1} for the action of Φ. The first statement follows. The second is a well-known result of Coleman: cf. the cases k = 0 of Cor. 6.3.1. and Prop. 6.6 of [Cole95] for the injectivity and surjectivity respectively of the maps induced by restriction. □

Proposition 2.3. For any class η ∈ H_{dR}^{1}(X_{K})^{u-r} and any overconvergent modular form ϕ ∈ S_{2}^{oc}(N; K)_{0}, we have

⟨η, ω_{ϕ}⟩ = ⟨η, ω_{ϕ^{ord}}⟩,

and Poincaré duality induces a well-defined non-degenerate pairing

⟨ , ⟩ : H_{dR}^{1}(X_{K})^{u-r} × S_{2}^{ord}(N; K)_{0} ⟶ K.

Proof. This follows directly from Lemma 2.2 in light of (12). □

Modular forms of higher weight. Turning now to the case of forms of general weight k ≥ 2, let E ⟶ Y denote the universal elliptic curve over the affine modular curve Y = Y_{1}(N)/ℚ. Let \underline{ω} := π_{∗}Ω_{E/Y}^{1} denote the line bundle of relative differentials on E over Y. It extends to a sheaf over X = X_{1}(N), also denoted by \underline{ω}, by setting

H^{0}(Spec ℚ[[q]], \underline{ω}) = ℚ[[q]] · ω_{can},

where ω_{can} := \frac{dt}{t} is the canonical differential on the Tate curve 𝔾_{m}/q^{ℤ}.

Let Ω_{X}^{1}(log cusps) be the sheaf of differentials 1-forms on X with logarithmic poles at the cusps, for which

H^{0}(Spec ℚ[[q]], Ω^{1}(log cusps)) = ℚ[[q]] · \frac{dq}{q} .

A modular form ϕ on X of weight k = r + 2 with fourier coefficients in a field K (viewed as a function on “test objects” (E, ω_{E}), following the point of view adopted in [Kat73, Ch. 1] for example) corresponds to a global section of the sheaf \underline{ω}^{r+2} over X_{K}, by sending ϕ to the global section ϕ(E, ω_{E})ω_{E}^{r+2}, and hence we can define

S_{k}(N; K) = H^{0}(X_{1}(N)_{K}, \underline{ω}^{k}).

Let

L := ℝ^{1}π_{∗}ΩE/Y^{•}

be the relative de Rham cohomology sheaf on Y. It is equipped with a filtration

(20)  0 ⟶ \underline{ω} ⟶ L ⟶ \underline{ω}^{−1} ⟶ 0

arising from the Hodge filtration on the fibers. The sheaf L and its r-th symmetric power

L_{r} := sym^{r} L

are vector bundles over Y (of rank 2 and r+1 respectively) endowed with a canonical integrable connection:

∇ : L_{r} ⟶ L_{r} ⊗ Ω_{X}^{1}(log cusps),

the so-called Gauss-Manin connection. The sheaf L_{r} extends to a sheaf on X by setting

(21)  H^{0}(Spec ℚ[[q]], L_{r}) = ℚ[[q]]η_{can}^{r} + ℚ[[q]]η_{can}^{r−1}ω_{can} + ⋯ + ℚ[[q]]ω_{can}^{r},

where η_{can} := ∇(q \frac{d}{dq})(ω_{can}) is a complementary vector to ω_{can} in H_{dR}^{1}(E/ℤ[[q]]), and similarly at the other cusps of X. The connection ∇ extends by setting

(22)  ∇ω_{can} = η_{can} ⊗ \frac{dq}{q} ,  ∇η_{can} = 0.

The filtration (20) gives rise to an (r+1)-step decreasing filtration on L_{r}, with successive quotients

(23)  L_{r}/Fil^{1}L_{r} ≃ \underline{ω}^{−r}, ⋯, Fil^{r−j}L_{r}/Fil^{r−j+1}L_{r} ≃ \underline{ω}^{r−2j}, ⋯, Fil^{r}L_{r} ≃ \underline{ω}^{r}.

The connection ∇ obeys Griffiths transversality:

∇Fil^{r−j}L_{r} ⊂ Fil^{r−j−1}L_{r} ⊗ Ω_{X}^{1}(log cusps).

Furthermore, it induces an isomorphism of O_{X}-modules:

(24)  ∇ : \frac{Fil^{r−j}L_{r}}{Fil^{r−j+1}L_{r}} ≃ \frac{Fil^{r−j−1}L_{r}}{Fil^{r−j}L_{r}} ⊗ Ω_{X}^{1}(log cusps),

which gives rise (by setting r = 1 and j = 0 for example) to the Kodaira-Spencer isomorphism

KS : \underline{ω}^{⊗2} ⟶^{∼} Ω_{X}^{1}(log cusps).

It follows that a modular form ϕ of weight r +2 can be interpreted, via this isomorphism, as a global section of \underline{ω}^{r} ⊗ Ω_{X}^{1}(log cusps), while a cusp form can be interpreted as a global section of \underline{ω}^{r} ⊗ Ω_{X}^{1}, i.e.,

S_{k}(N; K) = H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}).

At a geometric point of X attached to an elliptic curve E with Γ_{1}(N)-level structure, the section ω_{ϕ} is given by

ω_{ϕ} = ϕ(E, ω_{E})ω_{E}^{r} KS(ω_{E}^{2}).

At the test object (𝔾_{m}/q^{ℤ}, ω_{can}) corresponding to the Tate curve with its canonical differential, this leads to the formula for ω_{ϕ} in terms of the q-expansion of ϕ:

ω_{ϕ}(q) = ϕ(q)ω_{can}^{r} (\frac{dq}{q} ) .

Let (L_{r} ⊗ Ω_{X}^{1})_{par} denote the subsheaf of L_{r} ⊗ Ω^{1}(log cusps) defined in the neighborhood of the cusps by

H^{0}(Spec ℚ[[q]], (L_{r} ⊗ Ω_{X}^{1})_{par}) = ∇H^{0}(Spec ℚ[[q]], L_{r})
= (ℚ[[q]]η_{can}^{r} + ℚ[[q]]η_{can}^{r−1}ω_{can} + ⋯ + qℚ[[q]]ω_{can}^{r})\frac{dq}{q} .

Note that for r = 0 we have (L_{r} ⊗ Ω_{X}^{1})_{par} = Ω_{X}^{1}. A standard inductive argument using (24) shows that the natural inclusion \underline{ω}^{r} ⟶ L_{r} induces an isomorphism

(25)  H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) ≃ \frac{H^{0}(X_{K}, (L_{r} ⊗ Ω_{X}^{1})_{par})}{∇H^{0}(X_{K}, L_{r})}.

We write

H_{dR}^{j}(X_{K}, L_{r}, ∇) := ℍ^{j}(0 → L_{r} ⟶^{∇} L_{r} ⊗ Ω_{X}^{1}(log cusps) → 0)

for the j-th hypercohomology of the complex of sheaves on X over K associated to ∇, and H_{par}^{j}(X_{K}, L_{r}, ∇) for the parabolic cohomology:

H_{par}^{j}(X_{K}, L_{r}, ∇) := ℍ^{j}(0 → L_{r} ⟶^{∇} (L_{r} ⊗ Ω_{X}^{1})_{par} → 0).

(Cf. also Chapter 2.1 of [BDP] for example.) These algebraic de Rham cohomology groups are finite-dimensional K-vector spaces. The group H_{par}^{1}(X_{K}, L_{r}, ∇) is equipped with a two-step Hodge filtration given by

(26)  Fil^{j}H_{par}^{1}(X_{K}, L_{r}, ∇) = \{ \begin{matrix}H_{par}^{1}(X_{K}, L_{r}, ∇) & if j ≤ 0; \\ H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) & if 1 ≤ j ≤ r + 1; \\ 0 & if j ≥ r + 2, \end{matrix}

giving rise to the exact sequence

(27)  0 ⟶ H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) ⟶ H_{par}^{1}(X_{K}, L_{r}, ∇) ⟶ H^{1}(X_{K}, \underline{ω}^{−r}) ⟶ 0.

We note the Poincaré duality pairing

(28)  ⟨ , ⟩ : H_{par}^{1}(X_{K}, L_{r}, ∇) × H_{par}^{1}(X_{K}, L_{r}, ∇) ⟶ K(−1 − r).

Since H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) is its own orthogonal complement under (28), Poincaré duality descends to a perfect pairing

(29)  ⟨ , ⟩ : H^{1}(X_{K}, \underline{ω}^{−r}) × H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) ⟶ K(−1 − r),

which is denoted by the same symbol by a slight abuse of notation.

In the sequel, if L is a local system on a variety V, endowed with an integrable connection ∇ that is clear from the context, we shall write H^{⋆}(V, L) instead of H^{⋆}(V, L, ∇), and likewise for similar cohomology groups.

2.3. Nearly holomorphic modular forms and the Shimura-Maass derivative. Suppose now that K = ℂ. Hodge theory gives a canonical (real-analytic, but non-holomorphic) splitting

Spl_{hdg} : L ⟶ \underline{ω}

of the exact sequence (20) over the affine modular curve Y. This map can be viewed as a homomorphism of O_{Y (ℂ)_{an}}–modules, where O_{Y (ℂ)_{an}} is the structure sheaf of real analytic functions on Y (ℂ). We will also denote by the same symbol the associated map L^{k} ⟶ \underline{ω}^{k}, as well as the resulting map

(30)  Spl_{hdg} : H^{0}(X_{ℂ}, (L_{r} ⊗ Ω_{X}^{1})_{par}) ⟶ H^{0}(Y (ℂ)_{an}, \underline{ω}^{r} ⊗ Ω_{X}^{1})

restricted to regular (i.e., holomorphic) sections over X of the sheaf (L_{r} ⊗ Ω_{X}^{1})_{par}.

Definition 2.4. The image of Spl_{hdg} is called the space of nearly holomorphic cusp forms of weight k = r + 2 on Γ_{1}(N).

Nearly holomorphic cusp forms were introduced in [Sh86]; see also [Hi93, Ch. 10] for a more elementary description. The space of nearly holomorphic cusp forms on Γ_{1}(N) is denoted by S_{k}^{nh}(N; ℂ). It is contained in the space of real analytic functions on the upper half plane which satisfy the same transformation property under Γ_{1}(N) as (holomorphic) modular forms of weight k. The following basic facts about nearly holomorphic modular forms, which we recall without proof, will be useful in the sequel:

(i) The map Spl_{hdg} of (30) is injective, and hence induces an isomorphism of finitedimensional complex vector spaces:

Spl_{hdg} : H^{0}(X_{ℂ}, (L_{r} ⊗ Ω_{X}^{1})_{par}) ⟶^{∼} S_{k}^{nh}(N; ℂ).

(Cf. equation (5a) in §10.1 of [Hi93].) (ii) If K is any subfield of ℂ, the image of H^{0}(X_{K}, (L_{r} ⊗ Ω_{X}^{1})_{par}) under Spl_{hdg} yields a

natural K-structure on S_{k}^{nh}(N; ℂ), and is denoted by S_{k}^{nh}(N; K). (iii) If ϕ = Spl_{hdg}(ω_{ϕ}) belongs to S_{k}^{nh}(N; K), then equation (25) allows us to write

(31)  ω_{ϕ} = Π_{N}^{hol}(ϕ) + ∇s, with \{ \begin{matrix}Π_{N}^{hol}(ϕ) ∈ H^{0}(X_{K}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) = S_{k}(N; K), \\ s ∈ H^{0}(X_{K}, L_{r}). \end{matrix}

The modular form Π_{N}^{hol}(ϕ) is called the holomorphic projection of the nearly holomorphic modular form ϕ. (Cf. equation (8a) in §10.1 of [Hi93]; the fact that Π_{N}^{hol}, which is denoted by H in loc. cit., preserves K-rational structures is stated in (8b).)

(iv) The inverse of the Kodaira-Spencer isomorphism followed by the Gauss-Manin connection gives a well-defined map

(32)  ˜∇ : H^{0}(X_{K}, (L_{r} ⊗ Ω_{X}^{1})_{par}) ⟶ H^{0}(X_{K}, (L_{r+2} ⊗ Ω_{X}^{1})_{par}).

This map corresponds (under the identification Spl_{hdg}) to the weight k Shimura-Maass derivative operator δ_{k} = \frac{1}{2πi}(\frac{d}{dz} + \frac{k}{z−z̄}), i.e., the following diagram commutes:

\xymatrix{H^{0}(X_{K}, (L_{r} ⊗ Ω_{X}^{1})_{par}) \ar[r]^{^{Spl^{hdg}}} \ar[d]_{˜∇} & S_{k}^{nh}(N; K) \ar[d]^{δ_{k}} \\ H^{0}(X_{K}, (L_{r+2} ⊗ Ω_{X}^{1})_{par}) \ar[r]^{_{Spl_{hdg}}} & S_{k+2}^{nh}(N; K).}

This follows from a direct calculation based on the identities (in terms of the standard complex coordinates τ ∈ H and z ∈ ℂ/⟨1, τ⟩)

∇(2πidz) = 2πi (\frac{dz − d̄z}{τ − τ̄} ) ⊗ dτ,  ∇d̄z = 0,  KS((2πidz)^{⊗2}) = 2πidτ.

Since nearly holomorphic modular forms are closed under taking products and under applying the Shimura-Maass derivative, it follows that, if g ∈ S_{ℓ}(N; K) and h ∈ S_{m}(N; K) are classical cusp forms with fourier coefficients in K, then for all t ≥ 0, the product δ_{ℓ}^{t}g × h belongs to S_{ℓ+m+2t}^{nh}(N; K). More precisely, let

(33)  ˜∇^{t}ω_{g} ⊗ ω̃_{h} ∈ H^{0}(X_{K}, (L_{ℓ+m+2t−2} ⊗ Ω_{X}^{1})_{par})

be the global section obtained by tensoring the sections

˜∇^{t}ω_{g} ∈ H^{0}(X_{K}, (L_{ℓ+2t−2} ⊗ Ω_{X}^{1})_{par}) and ω̃_{h} := KS^{−1}(ω_{h}) ∈ H^{0}(X_{K}, L_{m}).

Then

(34)  δ_{ℓ}^{t}g × h = Spl_{hdg}( ˜∇^{t}ω_{g} ⊗ ω̃_{h}).

Hodge theory gives a canonical splitting of the exact sequence (27):

H_{par}^{1}(X_{ℂ}, L_{r}) = H^{0}(X_{ℂ}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) ⊕ \overline{H^{0}(X_{ℂ}, \underline{ω}^{r} ⊗ Ω_{X}^{1})}.

The Petersson scalar product ( , )_{N} of level N on S_{k}^{nh}(N; ℂ) is defined by the familiar rule

(35) ( , )_{N} : S_{k}^{nh}(N; ℂ) × S_{k}^{nh}(N; ℂ) ⟶ ℂ,  (f_{1}, f_{2})_{N} = ∫_{Γ_{1}(N)\ℌ} \overline{f_{1}(z)} f_{2}(z) y^{k} \frac{dxdy}{y^{2}} ,

where the integration is performed relative to the variable z = x + iy on any fundamental domain in the upper-half plane ℌ under the action of Γ_{1}(N). Note that this pairing is Hermitian-linear in the first argument and ℂ-linear in the second, in contrast with the more customary conventions.

Lemma 2.5. For all η ∈ S_{r+2}(N; ℂ), and all ϕ ∈ S_{r+2}^{nh}(N; ℂ),

(η, ϕ)_{N} = (η, Π_{N}^{hol}(ϕ))_{N}.

The equation of Lemma 2.5 gives an independent definition of the holomorphic projection, since the Petersson product is a perfect pairing on the space S_{k}(N; ℂ) of cusp forms.

2.4. Nearly overconvergent modular forms and the d operator. Recall the ordinary locus A and its system of wide open neighborhoods W_{ϵ} ⊃ A that were introduced in Section 2.2. We define

(36)  H_{rig}^{1}(W_{ϵ}, L_{r}) := \frac{H^{0}(W_{ϵ}, (L_{r} ⊗ Ω_{X}^{1})_{par})}{∇H^{0}(W_{ϵ}, L_{r})},

where H^{0}(W_{ϵ}, −) designates a space of rigid sections. The quotient on the right of (36) is related to overconvergent cusp forms of weight k = r + 2 by noting that any rigid section η ∈ H^{0}(W_{ϵ}, (L_{r} ⊗ Ω_{X}^{1})_{par}) can be written as

(37)  η = η_{0} + ∇s,  with η_{0} ∈ H^{0}(W_{ϵ}, \underline{ω}^{r} ⊗ Ω_{X}^{1}), s ∈ H^{0}(W_{ϵ}, L_{r}).

This fact follows from an inductive argument based on (24).

Similarly as in (18), the spaces of overconvergent modular (cusp) forms of weight k are

(38)  S_{k}^{oc}(N; K) = ⋃_{ϵ>0} H^{0}(W_{ϵ}/K, \underline{ω}^{r} ⊗ Ω_{rig}^{1}) ⊆ M_{k}^{oc}(Γ_{1}(N), K) = ⋃_{ϵ>0} H^{0}(W_{ϵ}/K, \underline{ω}^{k}).

It is also known (cf. [Cole95]) that any overconvergent modular form of weight −r, viewed as a section s ∈ H^{0}(W_{ϵ}, \underline{ω}^{−r}), admits a unique lift s̃ under the projection H^{0}(W_{ϵ}, L_{r}) ⟶ H^{0}(W_{ϵ}, \underline{ω}^{−r}) induced by (23) satisfying

∇s̃ ∈ H^{0}(W_{ϵ}, \underline{ω}^{r} ⊗ Ω_{X}^{1}),

and that ∇s̃ corresponds to the overconvergent modular form d^{r+1}s, where d := q \frac{d}{dq} is Serre’s operator sending p-adic modular forms of weight m to p-adic modular forms of weight m + 2. (Note that the operator d does not preserve overconvergence in general, even though d^{r+1} maps S_{−r}^{oc}(N) to S_{r+2}^{oc}(N).) Thanks to (37), equation (36) can be re-written as

(39)  H_{rig}^{1}(W_{ϵ}, L_{r}) = \frac{H^{0}(W_{ϵ}, \underline{ω}^{r} ⊗ Ω_{X}^{1})}{∇H^{0}(W_{ϵ}, L_{r}) ∩ H^{0}(W_{ϵ}, \underline{ω}^{r} ⊗ Ω_{X}^{1})} = \frac{S_{r+2}^{oc}(N)}{d^{r+1}S_{−r}^{oc}(N)}.

Just as when r = 0, there is a canonical isomorphism

(40)  H_{par}^{1}(X′_{ℂ_{p}}, L_{r}) = H_{rig}^{1}(W_{ϵ}, L_{r})

between the algebraic (parabolic) de Rham cohomology over ℂ_{p} and the rigid de Rham cohomology. As explained in [BDP, § 4.5], for each supersingular annulus V_{j}, j = 1, ..., s, there is a residue map

(41)  res : H_{rig}^{1}(W_{ϵ}, L_{r}) ⟶ (H^{0}(V_{j}, L_{r})^{∇=0})^{∨} ≃ L_{r}(P_{j}) ≃ (Sym^{r} H_{dR}^{1}(E_{j})) (−1),

where E_{j}/ℂ_{p} stands for the supersingular elliptic curve corresponding to the point P̃_{j}.

By [BDP, Prop. 4.11], the image of H_{par}^{1}(X_{ℂ_{p}}, L_{r}) in H_{par}^{1}(X′_{ℂ_{p}}, L_{r}) under the restriction map consists of those classes represented by L_{r}-valued differential forms ω on W_{ϵ} whose residues at the supersingular annuli V_{j}, j = 1, ..., s, are all zero.

Over A, the slope decomposition arising from the action of Frobenius gives a canonical splitting

(42)  Spl_{u-r} : L ⟶ \underline{ω}

of the exact sequence (20). This map can be viewed as a homomorphism of O_{A}-modules, where O_{A} is the structure sheaf of rigid analytic functions on A. We will also denote by the same symbol the associated map L^{r} ⟶ \underline{ω}^{r}, as well as the resulting map

(43)  Spl_{u-r} : ⋃_{ϵ>0} H^{0}(W_{ϵ}, (L_{r} ⊗ Ω_{X}^{1})_{par}) ⟶ H^{0}(A, \underline{ω}^{r} ⊗ Ω_{X}^{1})

on overconvergent sections. The image of Spl_{u-r} is contained in the space of p-adic modular forms of weight k = r + 2, and contains the space of overconvergent modular forms, but is not equal to it in general, because the splitting (42) does not extend to a rigid analytic splitting over any of the wide opens W_{ϵ}. The following definition arises naturally from our parallel discussion of the complex-analytic setting.

Definition 2.6. The image of the map Spl_{u-r} in (43), denoted S_{k}^{n-oc}(N; ℂ_{p}), is called the space of nearly overconvergent modular forms of weight k on Γ_{1}(N).

We refer the reader to the forthcoming work [Ur] of Urban, where this notion has also been introduced independently.

Note that the weight of a nearly overconvergent modular form always belongs to ℤ^{≥2}, by definition. The following basic facts about nearly overconvergent modular forms are analogous to those that were observed in the complex setting:

(i) The map Spl_{u-r} of (43) is injective, and induces an isomorphism of p-adic Fréchet

spaces:

Spl_{u-r} : ⋃_{ϵ>0} H^{0}(W_{ϵ}, (L_{r} ⊗ Ω_{X}^{1})_{par}) ⟶^{∼} S_{k}^{n-oc}(N; ℂ_{p}).

This is a consequence of the main theorem of [CGJ], which asserts that the p-adic modular form E_{2} is transcendental over the ring of overconvergent modular forms.

(ii) If K is any subfield of ℂ_{p}, the image of ⋃_{ϵ>0} H^{0}(W_{ϵ}/K, (L_{r} ⊗ Ω_{X}^{1})_{par}) under Spl_{u-r}

yields a natural K-vector space S_{k}^{n-oc}(N; K) ⊂ S_{k}^{n-oc}(N; ℂ_{p}). (iii) Let ϕ = Spl_{u-r}(ω_{ϕ}) be an element of S_{k}^{n-oc}(N; ℂ_{p}), where ω_{ϕ} is a global section of

(L_{r} ⊗ Ω_{X}^{1})_{par} over some W_{ϵ}. Equation (37) then allows us to write

(44)  ω_{ϕ} = Π_{N}^{oc}(ϕ) + ∇s, with \{ \begin{matrix}Π_{N}^{oc}(ϕ) ∈ H^{0}(W_{ϵ}, \underline{ω}^{r} ⊗ Ω^{1}) = S_{k}^{oc}(N; K), \\ s ∈ H^{0}(W_{ϵ}, L_{r}). \end{matrix}

The overconvergent modular form Π_{N}^{oc}(ϕ) of weight k and level N is called the overconvergent projection of the nearly overconvergent modular form ϕ. Note that Π_{N}^{oc}(ϕ) is only well-defined modulo d^{r+1}(S_{−r}^{oc}(N)), by (39).

(iv) If K is a field equipped with simultaneous embeddings into ℂ and ℂ_{p}, then there are natural identifications

\xymatrix{S_{k}^{nh}(N; K) & H^{0}(X_{K}, (L_{r} ⊗ Ω_{X}^{1})_{par}) \ar[l]_{^{Spl^{hdg}}} \ar[r]^{^{Spl^{u-r}}} & S_{k}^{n-oc}(N; K).}

It follows directly from the definitions that the holomorphic and overconvergent projections, restricted to S_{k}^{nh}(N; K) and S_{k}^{n-oc}(N; K) respectively, take values in S_{k}(N; K) and are equal (under the above identification of S_{k}^{nh}(N; K) and S_{k}^{n-oc}(N; K)).

(v) The map ˜∇ of (32) corresponds (under the identification Spl_{u-r}) to the operator d = q \frac{d}{dq} on p-adic modular forms, i.e., the following diagram commutes:

\xymatrix{⋃_{ϵ>0} H^{0}(W_{ϵ}, (L_{r} ⊗ Ω_{X}^{1})_{par}) \ar[r]^{^{Spl^{u-r}}} \ar[d]_{˜∇} & S_{k}^{n-oc}(N; ℂ_{p}) \ar[d]_{d} \\ ⋃_{ϵ>0} H^{0}(W_{ϵ}, (L_{r+2} ⊗ Ω_{X}^{1})_{par}) \ar[r]^{_{Spl_{u-r}}} & S_{k+2}^{n-oc}(N; ℂ_{p}).}

A nearly overconvergent modular form admits a q-expansion, and hence Hida’s ordinary projector e_{ord} of (15) extends formally to the space S_{k}^{n-oc}(N; ℂ_{p}). The following lemma relates this ordinary projection to the overconvergent projection Π_{N}^{oc}.

Lemma 2.7. Let ϕ be a nearly overconvergent modular form on Γ_{1}(N). Its image under Hida’s ordinary projector is overconvergent, and thus classical on Γ_{1}(N) ∩ Γ_{0}(p). More precisely,

(45)  e_{ord}ϕ = e_{ord}Π_{N}^{oc}(ϕ).

Proof. If s is any overconvergent section of L_{r}, then a direct calculation using the relations (22) shows that

Spl_{u-r}(∇s) belongs to d(ℂ_{p} ⊗ O_{ℂ_{p}}[[q]]) · ω_{can}^{r} ⊗ \frac{dq}{q} .

But e_{ord} annihilates the image under d of any q-series with bounded denominators, so

(46)  e_{ord}(Spl_{u-r}(∇s)) = 0.

Now write ϕ = Spl_{u-r}(ω_{ϕ}), with ω_{ϕ} a rigid section of (L_{r} ⊗Ω_{X}^{1})_{par} over some W_{ϵ}. By definition of the overconvergent projection,

ω_{ϕ} = ϕ_{0} + ∇s with ϕ_{0} ∈ H^{0}(W_{ϵ}, \underline{ω}^{r} ⊗ Ω_{X}^{1}) and s ∈ H^{0}(W_{ϵ}, L_{r}).

Applying the operator Spl_{u-r} to this last identity gives ϕ = Π_{N}^{oc}(ϕ) + Spl_{u-r}(∇s). The result now follows by applying e_{ord} and invoking (46). □

Let g ∈ S_{ℓ}(N; K) and h ∈ S_{m}(N; K) be classical cusp forms defined over K, and fix embeddings of K into ℂ and ℂ_{p}. The forms g and h can then be regarded simultaneously as complex and overconvergent modular forms.

Proposition 2.8. For all t ≥ 0, the modular form d^{t}g × h belongs to S_{ℓ+m+2t}^{n-oc}(N; K) and

e_{ord}(d^{t}g × h) = e_{ord}Π_{N}^{hol}(δ_{ℓ}^{t}g × h).

Proof. Recall the global section ˜∇^{t}ω_{g} ⊗ ω̃_{h} ∈ H^{0}(X_{K}, (L_{ℓ+m+2t−2} ⊗ Ω_{X}^{1})_{par}) that was introduced in (33). Since

d^{t}g × h = Spl_{u-r}( ˜∇^{t}ω_{g} ⊗ ω̃_{h}), δ_{ℓ}^{t}g × h = Spl_{hdg}( ˜∇^{t}ω_{g} ⊗ ω̃_{h}),

it follows that

Π_{N}^{oc}(d^{t}g × h) = Π_{N}^{hol}(δ_{ℓ}^{t}g × h).

The proposition follows by applying the projector e_{ord} to this identity and invoking (45). □

We next turn to the case where the exponent appearing in Proposition 2.8 is strictly negative. After replacing g by its p-depletion g^{[p]} ∈ S_{k}^{oc}(N; ℂ_{p}), the form d^{−1−t}g^{[p]} ×h (with t ≥ 0) is still a p-adic modular form of weight k := ℓ+m−2t−2. (Cf. for instance Théorème 5 of §2 of [Se].) The following proposition shows that it is nearly overconvergent, at least in certain cases where k ≥ 2.

Proposition 2.9. Assume that 0 ≤ t ≤ min(ℓ − 2, m − 2), so that k := ℓ + m − 2t − 2 ≥ 2. Then the p-adic modular form d^{−1−t}g^{[p]} × h belongs to S_{k}^{n-oc}(N; ℂ_{p}), and in particular

e_{ord}(d^{−1−t}g^{[p]} × h) ∈ S_{k}^{ord}(N; ℂ_{p}) ⊂ S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ_{p}).

Proof. Set r_{2} = ℓ − 2 and r_{3} = m − 2. Since 1 − V U annihilates H_{rig}^{1}(W_{ϵ}, L_{r_{2}}), it follows that the overconvergent section ω_{g^{[p]}} of \underline{ω}^{r_{2}} ⊗ Ω_{X}^{1} is ∇-exact, i.e., there exists G^{[p]} ∈ H^{0}(W_{ϵ}, L_{r_{2}}) satisfying ω_{g^{[p]}} = ∇G^{[p]}. The q-expansion of the section G^{[p]} can be written down explicitly in terms of the differentials ω_{can} and η_{can} on the Tate curve. Using (22), one checks that it is equal to

(47)  G^{[p]}(q) = ∑_{j=0}^{r_{2}}(−1)^{j}j!\binom{r_{2}}{j}d^{−j−1}g^{[p]}(q)ω_{can}^{r_{2}−j}η_{can}^{j}.

Set r = r_{2} + r_{3} − t, and let L^{(1)}, …, L^{(r)} denote r copies of the sheaf L_{1} over X, numbered consecutively. If S = {i_{1}, …, i_{s}} with i_{1} < i_{2} < … < i_{s} is any subset of {1, …, r}, we set

L^{S} := L^{(i_{1})} ⊗ ⋯ ⊗ L^{(i_{s})}.

Let ϵ_{S} := \frac{1}{a!} ∑_{σ∈Σ_{s}} sgn(σ)σ denote the symmetrisation projector and write L_{S} := ϵ_{S}L^{S}, viewed as a subsheaf of L^{S} in the obvious way.

Choose now two subsets B and C of {1, …, r} of cardinalities r_{2} and r_{3} respectively, whose union is equal to {1, …, r}. Such a choice is possible, since r = r_{2} + r_{3} − t ≤ r_{2} + r_{3}. Put A′ = B ∩ C and A = {1, …, r} \ A′.

Poincaré duality on the fibers of E gives rise to a duality L_{1} × L_{1} ⟶ O_{X}(−1) of sheaves. Since the cardinality of A′ is t, this in turn induces a map of sheaves

(48)  L^{B} ⊗ L^{C} ⟶ L^{A}(−t).

The natural inclusions L_{r_{2}} ⊂ L^{r_{2}} ≃ L^{B} and L_{r_{3}} ⊂ L^{r_{3}} ≃ L^{C} allow us to build certain rigid sections of L^{B} and (L^{C} ⊗ Ω_{X}^{1})_{par} out of G^{[p]} and ω_{h} respectively, denoted

G^{[p]}(B) ∈ H^{0}(W_{ϵ}, L^{B}), ω_{h}(C) ∈ H^{0}(W_{ϵ}, (L^{C} ⊗ Ω_{X}^{1})_{par}).

Taking the tensor product of these two sections and applying (48) gives an overconvergent section G^{[p]}(B) ⊗ ω_{h}(C) ∈ H^{0}(W_{ϵ}, (L^{A} ⊗ Ω_{X}^{1})_{par}), whose symmetrisation can be viewed as an element ϵ_{A}(G^{[p]}(B) ⊗ ω_{h}(C)) ∈ H^{0}(W_{ϵ}, (L_{r−t} ⊗ Ω_{X}^{1})_{par}). A direct calculation using (47) reveals that

(49)  Spl_{u-r}(ϵ_{A}(G^{[p]}(B) ⊗ ω_{h}(C))) = (−1)^{t}t!(d^{−1−t}g^{[p]} × h)ω_{can}^{r_{1}} (\frac{dq}{q} ) ,

which implies that d^{−1−t}g^{[p]} × h belongs to S_{k}^{n-oc}(N; ℂ_{p}), as desired. □

The sheaf L_{r} equipped with the Gauss-Manin connection is an overconvergent F -crystal in the sense of [Cole94, Sec. 10], i.e., the action of Frobenius on the relative de Rham cohomology L_{r} induces a horizontal morphism

Φ_{L_{r},ϵ,ϵ′} : Φ^{∗}(L_{r})_{|W_{ϵ}} ⟶ (L_{r})_{|W_{ϵ′}} ,

for suitable ϵ, ϵ′ > 0. These give rise to the Frobenius endomorphism

Φ_{r} := comp_{ϵ′}^{−1} ∘Φ_{L_{r},ϵ,ϵ′} ∘ Φ^{∗} ∘ comp_{ϵ}

of H_{dR}^{1}(X, L_{r}), which is equal to Φ when r = 0. By abuse of notation, we will continue to write Φ instead of Φ_{r}, since the context will make it clear which Frobenius endomorphism is being referred to.

The action of Φ on H_{dR}^{1}(X_{ℂ_{p}}, L_{r}) induces a decomposition of its f-isotypic part, for all ordinary modular forms f of weight k = r + 2:

H_{dR}^{1}(X_{ℂ_{p}}, L_{r}, ∇)^{f} = H^{0}(X_{ℂ_{p}}, \underline{ω}^{r} ⊗ Ω_{X}^{1})^{f} ⊕ H_{dR}^{1}(X_{ℂ_{p}}, L_{r})^{f,u-r},

where the superscript u-r denotes the unit-root subspace, i.e., the part of the cohomology on which the Frobenius endomorphism acts with slope zero. This decomposition plays a role somewhat analogous to that of the Hodge decomposition in the complex setting.

The following lemma generalizes Lemma 2.2 to higher weight.

Lemma 2.10. If ϕ = Spl_{u-r}(ω_{ϕ}) ∈ S_{k}^{n-oc}(N)_{0} is an ordinary overconvergent p-adic modular form of weight k = r + 2 > 2 on Γ_{1}(N), then the class of ω_{ϕ} belongs to H_{dR}^{1}(X_{K}, L_{r})^{Φ,k−1}. Furthermore, the assignment ϕ ↦ [ω_{ϕ}] induces an isomorphism

S_{k}^{ord}(N) ⟶^{∼} H_{dR}^{1}(X_{K}, L_{r})^{Φ,k−1}.

Proof. The Frobenius morphism Φ is related to the operator V on overconvergent modular forms of weight k by

Φ(ω_{f}) = p^{k−1}ω_{V f},

as can be seen from a computation on the Tate object (𝔾_{m}/q^{ℤ}, \frac{dt}{t}) after noting that Φ(\frac{dt}{t}) = p\frac{dt}{t}. The relation between the operators U and Φ on cohomology (relative to the identifications described above between differentials and weight two modular forms) is therefore given by

Φ = p^{k−1}V = p^{k−1}U^{−1}.

If ϕ belongs to the slope zero subspace for the action of U, the class of the rigid differential ω_{ϕ} lies in the slope k − 1 subspace of H_{rig}^{1}(W_{ϵ}, L_{r})^{Φ,k−1} for the action of Φ. Since Φ acts with slope k/2 on the target of the residue map, the class ω_{ϕ} automatically has vanishing residues at the supersingular annuli when k > 2, and the class of ω_{ϕ} can therefore be viewed as an element of H_{dR}^{1}(X_{K}, L_{r}). The injectivity and surjectivity assertions in the second statement follow from Cor. 6.3.1. and Prop. 6.6 of [Cole95], just as in the proof of Lemma 2.2. □

We also record the generalisation of Prop. 2.3 for k ≥ 2.

Proposition 2.11. Let η be any class in H_{dR}^{1}(X_{K}, L_{r})^{u-r}, and let ϕ be a nearly overconvergent modular form of weight k > 2 on Γ_{1}(N) with vanishing residues at the supersingular annuli. Then

⟨η, ω_{ϕ}⟩ = ⟨η, ω_{ϕ^{ord}}⟩, where ϕ^{ord} := e_{ord}ϕ.

In other words the expression ⟨η, ω_{ϕ}⟩ depends only on the ordinary projection of ϕ, and Poincaré duality induces a well-defined pairing

⟨ , ⟩ : H_{dR}^{1}(X_{ℂ_{p}}, L_{r})^{u-r} × e_{ord}S_{k}^{n-oc}(N) ⟶ ℂ_{p}.

Proof. The Poincaré pairing of (28) is compatible with the Frobenius endomorphism Φ and hence gives rise to a well-defined pairing

(50)  ⟨ , ⟩ : H_{dR}^{1}(X_{K}, L_{r})^{u-r} × H_{dR}^{1}(X_{K}, L_{r})^{Φ,k−1} ⟶ ℂ_{p}(−1 − r).

The proposition therefore follows from Lemma 2.10. □

2.5. Periods of modular forms. The newform f ∈ S_{k}(N_{f}, χ_{f}; K_{f}) ⊂ S_{k}(N_{f}; K_{f}) generates an automorphic representation of GL_{2}(𝔸_{ℚ}), denoted π_{f}. For any multiple N of N_{f} and any field K ⊃ K_{f}, let S_{k}(N; K)[π_{f}] denote the f-isotypic subspace of S_{k}(N; K), attached to the automorphic representation π_{f}. The space S_{k}(N; K)[π_{f}] is a finite-dimensional K-vector space of dimension σ_{0}(N/N_{f}), where σ_{0}(m) denotes as usual the number of divisors of m, and a basis for S_{k}(N; K)[π_{f}] is given by

(51)  {f(dz) = f(q^{d})}_{d|(N/N_{f})}.

Similar remarks and notations apply to the modular forms g and h.

Fix from now on a field K ⊃ K_{f}, endowed with a complex embedding K ⊂ ℂ.

Lemma 2.12. For all f̆ ∈ S_{k}(N; K)[π_{f}] and all ϕ ∈ S_{k}(N; K), the Petersson scalar product (f̆, ϕ)_{N} is a K-rational multiple of the period (f, f)_{N}. Proof. Let 𝕋_{N} be the Hecke algebra of level N generated over ℚ by the “good” Hecke operators T_{n} with gcd(n, N) = 1. The eigenform f corresponds to an idempotent e_{f} ∈ 𝕋_{N} ⊗ K_{f} which induces a projection of S_{k}(N; K) onto S_{k}(N; K)[π_{f}]. The vector f̆ is orthogonal to the kernel of e_{f} and therefore (f̆, ϕ)_{N} = (f̆, e_{f}ϕ)_{N} depends only on the projection of ϕ to S_{k}(N; K)[π_{f}]. By the remark preceeding equation (51), it is thererefore enough to show that, for all divisors d_{1} and d_{2} of N/N_{f},

(f(d_{1}z), f(d_{2}z))_{N} = ϱ(f, d_{1}, d_{2}) · (f, f)_{N}, with ϱ(f, d_{1}, d_{2}) ∈ K_{f}.

If d_{1}d_{2} = 1, this is clear. Otherwise, let q be a prime dividing d_{1}d_{2}. The compatibility beween the weight k slash operator and the Petersson scalar product shows that

(52)  (f(d_{1}z), f(d_{2}z))_{N} = q^{−k}(f(d_{1}/qz), f(d_{2}/qz))_{N},  if q|d_{1} and q|d_{2}.

Otherwise, assume without loss of generality that q divides d_{1} but not d_{2}. We then have

(53) (f(d_{1}z), f(d_{2}z))_{N} = \{ \begin{matrix}\frac{q^{1−k}a_{q}(f)}{q+1}(f(d_{1}/qz), f(d_{2}z))_{N} & if q||d_{1}, \\ \frac{q^{1−k}a_{q}(f)}{q}(f(d_{1}/qz), f(d_{2}z))_{N} \\ − χ_{f}(q)q^{−k−1}(f(d_{1}/q^{2}z), f(d_{2}z))_{N}, & if q^{2}|d_{1}. \end{matrix}

These relations imply that (54)

ϱ(f, d_{1}, d_{2}) = \{ \begin{matrix}q^{−k}ϱ(f, d_{1}/q, d_{2}/q) & if q|d_{1} and q|d_{2}; \\ \frac{q^{1−k}a_{q}(f)}{q+1}ϱ(f, d_{1}/q, d_{2}) & if q||d_{1} and q ∤ d_{2}; \\ \frac{q^{1−k}a_{q}(f)}{q}ϱ(f, d_{1}/q, d_{2}) − χ_{f}(q)q^{−1−k}ϱ(f, d_{1}/q^{2}, d_{2}), & if q^{2}|d_{1} and q ∤ d_{2}. \end{matrix}

Equation (54), together with the fact that ϱ(f, 1, 1) = 1 and ϱ(f, d_{1}, d_{2}) = ϱ̄(f, d_{2}, d_{1}) make it clear (by induction on d_{1}d_{2}) that ϱ(f, d_{1}, d_{2}) belongs to K_{f} for all d_{1}, d_{2} dividing N/N_{f}. □

The above proposition allows us to associate to any (not necessarily new) eigenform f̆ ∈ S_{k}(N; K_{f}), and to any modular form ϕ ∈ S_{k}(N; K), a K-rational period

J(f̆, ϕ) := \frac{(f̆, ϕ)_{N}}{(f, f)_{N}} ∈ K.

Let

(55)  ηa-h ; f̆ ∈ \overline{H^{0}(X_{ℂ}, \underline{ω}^{r} ⊗ Ω_{X}^{1})} ⊂ H_{dR}^{1}(X_{ℂ}, L_{r})

denote the class of the anti-holomorphic form \frac{1}{⟨\overline{ωf̆},ωf̆⟩} · \overline{ωf̆}, and let ηf̆ denote its natural image

in H^{1}(X_{ℂ}, \underline{ω}^{−r}) under the projection in (27). Corollary 2.13. The class ηf̆ belongs to H^{1}(X_{K_{f}} , \underline{ω}^{−r}).

Proof. By Lemma 2.12, for any ω_{ϕ} ∈ H^{0}(X_{K_{f}} , \underline{ω}^{r} ⊗ Ω_{X}^{1}) associated to a cusp form ϕ ∈ S_{k}(N; K_{f}) with K_{f}-rational fourier coefficients, we have

(56)  ⟨ηf̆, ω_{ϕ}⟩ = ⟨ηa-h ; f̆, ω_{ϕ}⟩ = \frac{(f̆, ϕ)_{N}}{(f, f)_{N}} belongs to K_{f}.

(Where the first occurence of ⟨ , ⟩ designates the pairing of (29) induced from Poincaré duality, with K = ℂ.) But H^{1}(X_{K_{f}} , \underline{ω}^{−r}) can be characterized as the set of η ∈ H^{1}(X_{ℂ}, \underline{ω}^{−r}) satisfying (56) for all ϕ ∈ S_{k}(N; K_{f}). Corollary 2.13 follows. □

2.6. Hida families. Fix a rational prime p > 2 and assume that the field K ⊃ K_{f} of the previous section is furnished with a p-adic embedding K ↪ ℂ_{p}. Let O denote the ring of integers of the p-adic closure of K in ℂ_{p}.

For every p ∤ N write the characteristic polynomial of a Frobenius element at p acting on the two-dimensional Galois representation over K_{f} ⊗ ℚ_{p} associated by Deligne to f as

(57)  L^{(p)}(f; T) = (1 − a_{p}(f)T + χ_{f}(p)p^{k−1}T) = (1 − α_{f,p}T)(1 − β_{f,p}T),

where χ is the nebentype character of f and α_{f,p} and β_{f,p} ∈ ℚ̄ are the reciprocal roots of the polynomial L^{(p)}(f; T).

The modular form f is said to be ordinary at p if the two reciprocal roots of (57) can be labelled in such a way that α_{f,p} is a p-adic unit, i.e., belongs to O^{×}. The modular form given by

f^{(p)}(q) := f(q) − β_{f,p}f(q^{p})

is called the ordinary p-stabilization of f; it has level pN, although it is only new at the primes dividing N, and it is again an eigenform for all the Hecke operators. To be precise,

f^{(p)}|T_{ℓ} = a_{ℓ}(f)f^{(p)}, ∀ℓ ≠ p, f^{(p)}|U_{p} = α_{f,p}f^{(p)}.

Set Γ = 1 + pNℤ_{p} and let Λ = O[[Γ]] be the completed group ring of Γ. The weight space is defined to be

Ω = Spf(Λ)(O) = Hom_{O−alg}(Λ, O),

which may naturally be identified with the space Hom_{cts}(Γ, O^{×}) of continuous characters of Γ. The subset of classical characters of Ω is defined to be

Ω_{cl} = {χ_{k} := (γ ↦ γ^{k}), with k ∈ ℤ^{≥2}}.

Given any finite flat extension Λ_{f} of Λ, let

Ω_{f} := Spf(Λ_{f})(O) = Hom(Λ_{f}, O).

This space is endowed with a natural p-adic topology and is equipped with a natural projection

κ : Ω_{f} → Ω

to weight space induced by the inclusion Λ ⊆ Λ_{f} of O-algebras. A point x ∈ Ω_{f} for which κ(x) belongs to Ω_{cl} will be referred to as a classical point of Ω_{f}, and the set of all such classical points will be denoted Ω_{f,cl}.

We will mostly work with the following definition of a Hida family of p-adic modular forms, which is slightly more restrictive than what can sometimes be found in the literature.

Definition 2.14. Let N_{f} ≥ 1 be an integer and let p be a prime not dividing N_{f}. A Hida family of tame level N_{f} is a quadruple (Λ_{f}, Ω_{f}, Ω_{f,cl}, f), where

(i) Λ_{f} is a finite flat extension of Λ; (ii) Ω_{f} is a non-empty open subset of X_{f} := Hom(Λ_{f}, ℂ_{p}) and Ω_{f,cl} is a p-adically dense

subset of Ω_{f} whose image under κ lies in Ω_{cl}; (iii) f := ∑ a_{n}q^{n} ∈ Λ_{f}[[q]] is a formal q-series with coefficients in Λ_{f} such that, for all

x ∈ Ω_{f,cl}, the power series

f_{x}^{(p)} := ∑_{n=1}^{∞} a_{n}(x)q^{n}

is the q-expansion of the ordinary p-stabilization of a normalised newform (denoted f_{x}) of weight κ(x) on Γ_{1}(N_{f}).

The collection of {f_{x}}_{x∈Ω_{f,cl}} arising from Hida theory can be thought of as a p-adically coherent collection of eigenforms on Γ_{1}(N_{f}) of varying weights. In particular, the fourier coefficients a_{n}(f_{x}) are analytic functions of x when p ∤ n, (but not when p|n, in general). The following theorem of Hida reveals the ubiquity of Hida families in the above sense.

Theorem 2.15 (Hida). [Hi86b] Let f be an ordinary newform in S_{k}(N_{f}; K_{f}). There exists a Hida family (Λ_{f}, Ω_{f}, Ω_{f,cl}, f) of tame level N_{f} and a classical point x_{0} ∈ Ω_{f} satisfying

κ(x_{0}) = k, f_{x_{0}} = f.

By shrinking Ω_{f} if necessary, we can and will assume that κ(x) = k (mod p − 1) for all x ∈ Ω_{f}; in particular the integers κ(x) and k have the same parity for all classical x ∈ Ω_{f}.

It will be also convenient, for later purposes, to dispose of a somewhat more flexible notion of p-adic families of modular forms, interpolating classical modular forms which are not necessarily new, or even Hecke eigenvectors; as well, it will be convenient to allow the fourier coefficients to belong to more general coefficients rings which are not necessarily finite over Λ. This leads to the following definition:

Definition 2.16. A Λ-adic modular form of tame level N is a quadruple (R, Ω_{ϕ}, Ω_{ϕ,cl}, ϕ), where

(i) R is a complete, finitely generated (but not necessarily finite), flat extension of Λ; (ii) Ω_{ϕ} is an open subset of Hom(R, ℂ_{p}) and Ω_{ϕ,cl} is a dense subset of Ω_{ϕ}; (ii) ϕ := ∑ a_{n}q^{n} ∈ R[[q]] is a formal q-series with coefficients in R such that, for all

x ∈ Ω_{ϕ,cl}, the power series

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

is the q-expansion of a classical ordinary cusp form in S_{κ(x)}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ_{p}) :=

S_{κ(x)}(Γ_{1}(N) ∩ Γ_{0}(p); ℚ) ⊗ ℂ_{p}.

The following examples of Λ-adic families of modular forms are of importance in our discussion.

1. Families of old forms. Let f ∈ S_{k}(N_{f}; K_{f}) be a newform of level N_{f} and let N be some multiple of N_{f} with p ∤ N. Let f̆ ∈ S_{k}(N; K_{f})[π_{f}] be an element of the old class of level N associated to f. By the remark preceding equation (51), there are (unique) scalars λ_{d} ∈ K indexed by the divisors of N/N_{f} and satisfying

f̆ = ∑_{d|(N/N_{f})} λ_{d} · f(q^{d}).

The p-stabilisation of the modular form f(q^{d}) is the weight k specialisation of the formal q-series

f(q^{d}) := ∑_{n} a_{n}q^{dn},

where (Λ_{f}, Ω_{f}, Ω_{f,cl}, f) is the Hida family of tame level N_{f} attached to f via Theorem 2.15. We can then set

(58)  f̆(q) := ∑_{d|(N/N_{f})} λ_{d}f(q^{d}).

The triple (Λ_{f}, Ω_{f}, Ω_{f,cl}, f̆) is a Λ-adic modular form whose classical specialisations are eigenvectors for the good Hecke operators T_{n} with gcd(n, N) = 1, and for the U operator, but they are not new at the primes dividing N/N_{f} and specialise to the old-form f̆ at the weight k point x_{0} ∈ Ω_{f,cl} alluded to in Theorem 2.15.

2. Products of modular forms. Let (Λ_{g}, Ω_{g}, g) and (Λ_{h}, Ω_{h}, h) be Λ-adic modular forms of tame level N (for example, Hida families of eigenforms arising from Theorem 2.15 or families of oldforms constructed as in the previous paragraph). Let Λ_{gh} := Λ_{g} ⊗_{O} Λ_{h} be the finitely generated (but not finite) Λ-algebra equipped with the natural diagonal embedding Λ ⟶ Λ_{g} ⊗ Λ_{h} sending the group-like element [a] ∈ Λ to [a] ⊗ [a]. Set

Ω_{gh} := Ω_{g} × Ω_{h}, Ω_{gh,cl} := Ω_{g,cl} × Ω_{h,cl}.

Then the quadruple (Λ_{gh}, Ω_{gh}, Ω_{gh,cl}, e_{ord}(g × h)), where the power series g × h is viewed as an element of Λ_{gh}[[q]] in the natural way, and e_{ord} is Hida’s ordinary projection operator, is an example of a Λ-adic family of modular forms with Fourier coefficients in Λ_{gh}.

3. Derivatives of modular forms. Recall the operators U, V defined in (13), and the differential operator d := q \frac{d}{dq}, which induces a map from p-adic modular forms of weight k to p-adic modular forms of weight k + 2. Recall that

Ω := Spf(Λ)(O) = Hom_{O−alg}(Λ, O) = Hom(Γ, O^{×})

and notice that for any n ∈ ℤ such that p ∤ n, the group-like element [n] gives rise to a function

[n] : Ω ⟶ O

whose value at a character k ∈ Ω represented by γ ↦ γ^{k}, is simply [n](k) = n^{k}.

Let g and h be Λ-adic families of eigenforms of tame level N (but not necessarily new of that level). Assume that p is a prime that does not divide N, and let a_{p}(g) ∈ Λ_{g}, a_{p}(h) ∈ Λ_{h} denote the Hecke eigenvalues associated to the Hecke operator T_{p}. Let

g^{[p]} := (1 − V U)g = (1 − a_{p}(g)V + [p]p^{−1}V^{2})g = ∑_{p∤n}a_{n}(g)q^{n}

be the Λ-adic counterpart of the modular form defined in (14). The specialisation g_{y}^{[p]} of g^{[p]} at a classical point y ∈ Ω_{g,cl} can either be viewed as a p-adic modular form of tame level N as in (14), or as a classical modular form of level Np^{2}. Note that g^{[p]} is not ordinary; in fact it lies in the kernel of the U operator. But the fact that g^{[p]} has fourier coefficients supported on the integers prime to p allows the formal q-series

(59)  d^{•}g^{[p]} := ∑_{p∤n}[n]a_{n}(g)q^{n}

to be viewed as an element of Λ ⊗_{O} Λ_{g}[[q]]. The specialization of this q-series at a classical point (t, y) ∈ Ω_{cl} × Ω_{g,cl} is simply

(d^{•}g^{[p]})_{(t,y)} = d^{t}g_{y}^{[p]}.

Define R_{gh} := Λ ⊗_{O} Λ_{g} ⊗_{O} Λ_{h}, regarded as a Λ-algebra by mapping the group-like element [a] ∈ Γ to [a^{2}] ⊗ [a] ⊗ [a], so that the map from Hom(R_{gh}, ℂ_{p}) = Ω × Ω_{g} × Ω_{h} to the weight space sends the classical point (t, y, z) ∈ ℤ^{≥0} × Ω_{g,cl} × Ω_{h,cl} to κ(y) + κ(z) + 2t.

Let d^{•}g^{[p]} ×h denote the product of d^{•}g^{[p]} and h, viewed as a formal series with coefficients in R_{gh}. Let e_{ord}(d^{•}g × h) denote its image under the ordinary projection operator. The specialisation of e_{ord}(d^{•}g^{[p]} × h) at (t, y, z) is equal to

(60)  e_{ord}(d^{•}g^{[p]} × h)_{t,y,z} = e_{ord}(d^{t}g_{y}^{[p]} × h_{z}^{(p)}) = e_{ord}(d^{t}g_{y}^{[p]} × h_{z}),

where the last equality is a consequence of the following simple but extremely useful lemma:

Lemma 2.17. If g and h are p-adic modular forms of tame level N, then g^{[p]} × (V h) is in the kernel of the U operator, and in particular

e_{ord}(g^{[p]} × V h) = 0.

Proof. This follows from the fact that a_{n}(g^{[p]} × (V h)) = 0 whenever p|n. □

The above discussion is summarised in the following proposition, which plays a key role in the construction of the triple Garrett-Rankin p-adic L-function described in Chapter 4.

Proposition 2.18. Let

Ω_{gh,cl} := {(t, y, z) ∈ ℤ × Ω_{g,cl} × Ω_{h,cl}, t > − min(κ(y), κ(z)).}

The quadruple

e_{ord}(d^{•}g^{[p]} × h) := (R_{gh}, Ω × Ω_{g} × Ω_{h}, Ω_{gh,cl}, e_{ord}(d^{•}g^{[p]} × h))

is an ordinary Λ-adic modular form of tame level N. For all (t, y, z) ∈ Ω_{gh,cl}, the specialisation of this family at (t, y, z) is the classical modular form (with coefficients in ℂ_{p})

e_{ord}(d^{t}g_{y}^{[p]} × h_{z}) ∈ S_{k}(Γ_{1}(N) ∩ Γ_{0}(p); ℂ_{p}).

This specialisation has algebraic fourier coefficients (lying in K = K_{g_{y}}K_{h_{z}}) when t ≥ 0.

We conclude this section by describing the Λ-adic interpolation of the periods that arise in Lemma 2.12. Write S^{ord}(N; R) for the space of Λ-adic modular forms with coefficients in the Λ-algebra R. A Hida family (Λ_{f}, Ω_{f}, f) of eigenforms in the sense of definition 2.14 gives rise to a subspace

S^{ord}(N; Λ_{f})[π_{f}] := {f̆ ∈ S^{ord}(N; Λ_{f}) such that T_{n}f̆ = a_{n}f̆, for all (n, N) = 1} .

Letting Λ′_{f} denote the fraction field of the integral domain Λ_{f}, the vector space S^{ord}(N; Λ′_{f})[π_{f}] is finite-dimensional over Λ′_{f} and has for basis the set {f(q^{d})}_{d|(N/N_{f})} of Λ-adic forms. Let f ∈ S^{ord}(N_{f}, Λ_{f}) be a Hida family of ordinary eigenforms, and let f̆ ∈ S^{ord}(N, Λ_{f})[π_{f} ] be an associated test vector of tame level N.

Let ϕ = (R, Ω_{ϕ}, Ω_{ϕ,cl}, ϕ) ∈ S^{ord}(N; R) be a Λ-adic modular form, and let (x, y) ∈ Ω_{f,cl} × Ω_{ϕ,cl} be a pair of points with κ(x) = κ(y). The specialisation ϕ_{y}^{(p)} of ϕ at y ∈ Ω_{ϕ,cl} need not be the p-stabilisation of a classical modular form, but its projection e_{f_{x}}ϕ_{y}^{(p)} to the f_{x}-isotypic component is the p-stabilisation of a classical modular form, which shall be denoted ϕ_{x,y}.

If R is any flat Λ-algebra with associated analytic space Ω_{R} = Hom(R, O), the elements of the ring Λ′_{f} ⊗_{Λ} R can be viewed as “rational functions” on the fiber product Ω_{f} ×_{Ω} Ω_{R} with poles at finitely many x ∈ Ω_{f}. Given J ∈ Λ′_{f} ⊗_{Λ} R and (x, y) ∈ Ω_{f} ×_{Ω} Ω_{R}, we write J(x, y) for the value of J at (x, y), when it is defined. Let ( , )_{N,p} denote the Petersson scalar product on modular forms attached to the group Γ_{1}(N) ∩ Γ_{0}(p).

Lemma 2.19. For all f̆ ∈ S^{ord}(N; Λ_{f})[π_{f}] and all ϕ = (R, Ω_{ϕ}, Ω_{ϕ,cl}, ϕ) ∈ S^{ord}(N; R), there exists (a unique) J(f̆, ϕ) ∈ Λ′_{f} ⊗_{Λ} R such that, for all classical points (x, y) ∈ Ω_{f,cl} ×_{Ω_{cl}} Ω_{ϕ,cl},

(61)  J(f̆, ϕ)(x, y) = \frac{(f̆_{x}^{(p)}, ϕ_{x,y}^{(p)})_{N,p}}{(f_{x}^{(p)}, f_{x}^{(p)})_{N,p}} = \frac{(f̆_{x}, ϕ_{x,y})_{N}}{(f_{x}, f_{x})_{N}} = ⟨η_{f_{x}}, ϕ_{x,y}⟩,

where the last pairing is the Poincaré duality between H^{1}(X_{ℂ_{p}}, \underline{ω}^{−r}) and H^{0}(X_{ℂ_{p}}, \underline{ω}^{r} ⊗ Ω_{X}^{1}).

Proof. Let \underline{𝕋}_{N} be the Λ-adic Hecke algebra of tame level N generated by the Hecke operators T_{n} with gcd(n, Np) = 1 and U acting on S^{ord}(N; Λ). The Hida family f corresponds to an idempotent e_{f} ∈ \underline{𝕋}_{N} ⊗_{Λ}Λ′_{f} which induces a projection of S^{ord}(N; Λ) to S^{ord}(N; Λ′_{f})[π_{f}]. Hence e_{f} ϕ is a (Λ′_{f} ⊗_{Λ} R)-linear combination of the forms f(q^{d}) with d|N/N_{f}, while the same is of course true for f̆. It is therefore enough to show that, for all divisors d_{1} and d_{2} of N/N_{f},

\frac{(f_{x}^{(p)}(q^{d_{1}}), f_{x}^{(p)}(q^{d_{2}}))_{N,p}}{(f_{x}^{(p)}, f_{x}^{(p)})_{N,p}} = ϱ(f, d_{1}, d_{2})(x), for some ϱ(f, d_{1}, d_{2}) ∈ Λ_{f} ⊗ ℚ.

But the proof of Lemma 2.12 with f replaced by f_{x}^{(p)} and Γ_{1}(N) by Γ_{1}(N) ∩ Γ_{0}(p) shows that ϱ(f_{x}^{(p)}, d_{1}, d_{2}) (defined in the obvious way) is a polynomial involving the expressions q^{κ(x)}, a_{q}(f_{x}), \frac{1}{q} and \frac{1}{q+1} as q ranges over the divisors of d_{1}d_{2}. Since the primes q that arise are different from p, these expressions all belong to Λ_{f} ⊗ ℚ and hence the same is true of ϱ(f, d_{1}, d_{2}). It follows that the expression J(f̆, ϕ) defined by

J(f̆, ϕ)(x, y) := \frac{(f̆_{x}^{(p)}, ϕ_{y}^{(p)})_{N,p}}{(f_{x}^{(p)}, f_{x}^{(p)})_{N,p}}

belongs to Λ′_{f} ⊗_{Λ} R, and the first equality in (61) follows. The second equality follows from a direct calculation, and the third from the definition of η_{f_{x}} and the familiar expression for the Poincaré pairing in terms of the Petersson scalar product. Lemma 2.19 follows. □

3. Diagonal cycles and p-adic Abel-Jacobi maps

3.1. Generalised Gross-Kudla-Schoen cycles. The main aim of this section is to introduce, for each triplet (k, ℓ, m) of balanced weights, a distinguished algebraic cycle Δ_{k,ℓ,m} on the product of three Kuga-Sato varieties fibered over the modular curve X = X_{1}(N). When the triplet is (k, ℓ, m) = (2, 2, 2), the cycle Δ_{2,2,2} is the one introduced by Gross and Kudla in [GrKu] and studied in detail by Gross and Schoen in [GrSc].

Let E denote the universal elliptic curve fibered over X, as constructed in [Sc]. It is a projective smooth algebraic surface defined over ℚ equipped with a proper regular (but not smooth) fibration

π : E ⟶ X,

whose fiber E_{x} at a point x ∈ Y outside the finite set of cusps is the elliptic curve corresponding to x under the moduli interpretation.

Fix a base point o ∈ X(ℚ), say the cusp at infinity. Write u for the unique automorphism on E extending the involution −1 on the fibers with respect to the zero section σ_{0} of π.

For any r ≥ 0, write E^{r} = E_{1} ×_{X} ... ×_{X} E_{r} for the r-th fibered product of E over X. A generic point in E^{r} is (x; P_{1}, ..., P_{r}) where x ∈ X and P_{i} are points in the fiber E_{x}. Let

ϵ_{sym} = \frac{1}{r!} ∑_{σ∈Σ_{r}} sgn(σ)σ ∈ Corr(E^{r}) ⊗ ℚ

denote the projector in which a permutation σ acts on E^{r} by permuting the factors in the fibration π : E^{r} ⟶ X. Let also

ϵ_{inv} := (\frac{1 − u_{1}}{2} ) ⊗ ⋯ ⊗ (\frac{1 − u_{r}}{2} ) ∈ Corr(E^{r}) ⊗ ℚ

denote the idempotent in the ring of correspondences from E^{r} to itself, in which u_{j} denotes the involution on the j-th factor in the fibration E^{r} ⟶ X. Write

(62)  ϵ_{r} = ϵ_{sym} · ϵ_{inv} ∈ Corr(E^{r}) ⊗ ℚ

for the composition of the two idempotents.

Put (k, ℓ, m) = (r_{1} + 2, r_{2} + 2, r_{3} + 2) with r_{3} ≥ r_{2} ≥ r_{1} ≥ 0 and set

r = \frac{r_{1} + r_{2} + r_{3}}{2} ≥ 0.

Let us now define a generalized Gross-Kudla-Schoen cycle Δ_{k,ℓ,m} of codimension r + 2 in the (2r + 3)-dimensional variety

W = E^{r_{1}} × E^{r_{2}} × E^{r_{3}}

introduced in (1) of the introduction. We shall regard it as an element in the group

CH^{r+2}(W) := Z^{r+2}(W) ⊗ ℚ/Z_{rat}^{r+2}(W) ⊗ ℚ

of rational equivalence classes of cycles of codimension r + 2 with coefficients in ℚ. We treat the three cases (k, ℓ, m) = (2, 2, 2), (2, ℓ, ℓ) with ℓ > 2, and k, ℓ, m > 2 separately.

Definition 3.1. Assume (k, ℓ, m) = (2, 2, 2). For any non-empty subset I ⊆ {1, 2, 3}, let

X_{I} := {(P_{1}, P_{2}, P_{3}) ∈ X × X × X : P_{i} = P_{j} for all {i, j} ⊂ I, P_{j} = o for all j ∉ I}.

Then the Gross-Kudla-Schoen diagonal cycle is defined to be

Δ_{2,2,2} = X_{123} − X_{12} − X_{13} − X_{23} + X_{1} + X_{2} + X_{3} ∈ CH^{2}(X_{1} × X_{2} × X_{3}).

To treat the remaining cases, choose three subsets

(63)  A = {a_{1}, …, a_{r_{1}}},  B = {b_{1}, …, b_{r_{2}}},  C = {c_{1}, …, c_{r_{3}}}

of {1, …, r} of cardinalities r_{1}, r_{2} and r_{3} respectively, such that A ∩ B ∩ C = ∅. If some of the r_{i} is 0, we take the corresponding subset to be the empty set. One readily checks that such subsets exist, and that they satisfy A = B ∪ C \ B ∩ C, B = C ∪ A \ C ∩ A and C = A ∪ B \ A ∩ B. The choice of the triplet (A, B, C) is unique up to permutations in S_{r}. Since the union of any two of the sets A, B, C is equal to {1, ..., r}, the maps

(64)  \begin{matrix}φ_{ABC} : E^{r} ⟶ E^{r_{1}} × E^{r_{2}} × E^{r_{3}}, (x; P_{1}, ..., P_{r}) ↦ ((x; P_{a_{i}}), (x; P_{b_{i}}), (x; P_{c_{i}})) \\ and φ_{BC} : E^{r} ⟶ E^{r_{2}} × E^{r_{3}}, (x; P_{1}, ..., P_{r}) ↦ ((x; P_{b_{i}}), (x; P_{c_{i}})) \end{matrix}

are closed embeddings of E^{r} into W (and likewise for the analogous maps φ_{AB} and φ_{AC}).

Definition 3.2. If (k, ℓ, m) = (2, ℓ, ℓ) for some ℓ = r + 2 > 2, the generalised Gross-Kudla-Schoen cycle is defined by

Δ_{2,ℓ,ℓ} = (Id, ϵ_{r_{2}}, ϵ_{r_{3}})(φ_{ABC}(E^{r}) − {o} × φ_{BC}(E^{r})) ∈ CH^{r+2}(X × E^{r} × E^{r}).

Definition 3.3. If k, ℓ, m > 2, the generalised Gross-Kudla-Schoen cycle is defined by

Δ_{k,ℓ,m} = (ϵ_{r_{1}}, ϵ_{r_{2}}, ϵ_{r_{3}})φ_{ABC}(E^{r}) ∈ CH^{r+2}(W).

By examining the image of these cycles by the cycle class map in each of the Künneth components of the complex de Rham cohomology group H_{dR}^{2r+4}(W/ℂ) of the variety W, it follows from [BDP, Lemma 2.2] that the cycles Δ_{k,ℓ,m} we defined for each triple of balanced weights (k, ℓ, m) are null-homologous, that is to say:

Δ_{k,ℓ,m} ∈ CH^{r+2}(W)_{0} := ker (cl : CH^{r+2}(W) → H_{dR}^{2r+4}(W)) .

Fix a prime p ∤ N and

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

denote the p-adic Abel-Jacobi map, as introduced e.g. in [Ne2, (1.2)], [Bes00].

There are several equivalent definitions of (65). Given that Fil^{r+2}H_{dR}^{2r+3}(W)^{∨} is naturally isomorphic to the group Ext^{1}(ℚ_{p}, H_{dR}^{2r+3}(W)) of isomorphism classes of extensions of the trivial filtered frobenius module ℚ_{p} by H_{dR}^{2r+3}(W), the map AJ_{p} sends Δ to the extension class given by

(66)  [figure: 0; H_{dR}^{2r+3}(W); V_{Δ}; ℚ_{p}; _{cl_{dR}}; 0; 0; H_{dR}^{2r+3}(W); H_{dR}^{2r+3}(W \ |Δ|); H_{|Δ|}^{2r+4}(W); 0,]

where the lower row is the short excision exact sequence associated with the pair (W, Δ) and the upper row is obtained from the lower by pull-back under the cycle class map cl_{dR} in de Rham cohomology.

3.2. A formula for AJ_{p}(Δ_{k,ℓ,m}) in terms of Coleman integration. The goal of this section is to give a p-adic analytic description of (a piece of) the functional AJ_{p}(Δ_{k,ℓ,m}), involving p-adic integration of differential forms, à la Coleman.

Let (k, ℓ, m) = (r_{1} + 2, r_{2} + 2, r_{3} + 2) with r_{3} ≥ r_{2} ≥ r_{1} ≥ 0 be a triplet of balanced weights and recall that we set r = \frac{r_{1}+r_{2}+r_{3}}{2} .

Our description of AJ_{p}(Δ_{k,ℓ,m}) rests on the p-adic analytic description of H_{dR}^{1}(X_{ℂ_{p}}, L_{r}) that was given in Sections 2.1 and 2.2. In order to keep track of the factors it will be convenient to write X^{2} = X_{2} × X_{3} and likewise write (X′)^{2} = X′_{2} × X′_{3}, where X′ denotes the affine subvariety of X fixed in Section 2.1. By Proposition 2.1 and equation (40) we have

H_{dR}^{1}(X′, L_{r_{2}}) ⊗ H_{dR}^{1}(X′, L_{r_{3}}) ≃ H_{rig}^{1}(W_{ϵ}, L_{r_{2}}) ⊗ H_{rig}^{1}(W_{ϵ}, L_{r_{3}}).

We fix a choice of lift of Frobenius to W_{ϵ} × W_{ϵ} by setting Φ := Φ_{2} × Φ_{3}, where Φ_{2} and Φ_{3} denote the canonical lift of the Frobenius endomorphism on W_{ϵ} described in (17), viewed as acting on the first and second factors of X_{2} × X_{3} respectively. This choice yields a linear transformation

Φ := Φ_{2} ⊗ Φ_{3} : Ω_{rig}^{1}(W_{ϵ}, L_{r_{2}}) ⊗ Ω_{rig}^{1}(W_{ϵ}, L_{r_{3}}) ⟶ Ω_{rig}^{1}(W_{ϵ}, L_{r_{2}}) ⊗ Ω_{rig}^{1}(W_{ϵ}, L_{r_{3}})

and induces an endomorphism of H_{rig}^{1}(W_{ϵ}, L_{r_{2}})⊗H_{rig}^{1}(W_{ϵ}, L_{r_{3}}) which we denote with the same letter. View L_{r_{2}} ⊗ L_{r_{3}} as a sheaf on X × X. The Gauss-Manin connection on L_{r_{1}} and on L_{r_{2}} gives rise to a connection

∇ : L_{r_{2}} ⊗ L_{r_{2}} ⟶ L_{r_{2}} ⊗ L_{r_{3}} ⊗ Ω_{X×X}^{1}

on sheaves on X × X, denoted again by ∇ by abuse of notation, and defined on sections by the rule

∇(s_{2} ⊗ s_{3}) = ∇(s_{2}) ⊗ s_{3} + s_{2} ⊗ ∇(s_{3}).

Given a class

ω_{2} ⊗ ω_{3} ∈ H^{0}(X, Ω_{X}^{1} ⊗ \underline{ω}^{r_{2}}) ⊗ H^{0}(X, Ω_{X}^{1} ⊗ \underline{ω}^{r_{3}})
= Fil^{r_{2}+r_{3}+2}(H_{dR}^{1}(X, L_{r_{2}}) ⊗ H_{dR}^{1}(X, L_{r_{3}}))
⊂ H_{rig}^{1}(W_{ϵ}, L_{r_{2}}) ⊗ H_{rig}^{1}(W_{ϵ}, L_{r_{3}}) ⊂ H_{rig}^{2}(W_{ϵ} × W_{ϵ}, L_{r_{2}} ⊗ L_{r_{3}}),

choose a polynomial P ∈ ℂ_{p}[x] satisfying (i) P (Φ) = P (Φ_{2}Φ_{3}) annihilates the class of ω_{2} ⊗ ω_{3} in H_{rig}^{2}(W_{ϵ} × W_{ϵ}, L_{r_{2}} ⊗ L_{r_{3}}). (ii) For each supersingular annulus V_{j}, j = 1, ..., s, P (Φ) annihilates the target of the

residue map (41), namely (Sym^{r} H_{dR}^{1}(E_{j}))(−1). (iii) None of the roots of P (x) are of complex absolute value p^{\frac{r_{2}+r_{3}+1}{2}} . The existence of such a polynomial follows from the fact that the eigenvalues of the geo-

metric Frobenius Φ acting on H_{dR}^{1}(X_{ℂ_{p}}, L_{r}) (resp. on (Sym^{r} H_{dR}^{1}(E_{j}))(−1)) are algebraic numbers with complex absolute value p^{\frac{r+1}{2}} (resp. p^{\frac{r+2}{2}} ), and hence a polynomial P satisfying (i) and (ii) can be chosen so that all its roots have complex absolute value either p^{\frac{r_{2}+r_{3}+2}{2}} or p^{\frac{r+2}{2}} ; such a choice of P automatically satisfies (iii) because the triplet of weights is assumed throughout to be balanced.

See the discussion right after Proposition 3.7 below for a justification of the need of conditions (ii) and (iii). As for (i), note that a direct consequence of it is the existence of a rigid analytic primitive of P (Φ)(ω_{2} ∧ ω_{3}), as we quote in the following statement.

Lemma 3.4. There exists a real ϵ > 0 and an L_{r_{2}} ⊗ L_{r_{3}}-valued rigid one-form ρ(P, ω_{2}, ω_{3}) on W_{ϵ} × W_{ϵ} satisfying

(67)  ∇ρ(P, ω_{2}, ω_{3}) = P (Φ)(ω_{2} ∧ ω_{3}).

This one-form is well-defined up to rigid ∇-closed one-forms on W_{ϵ} × W_{ϵ}.

Recall the base point o ∈ X that was chosen to define the cycle Δ_{2,2,2}. This choice of base point determines the horizontal and vertical inclusions

(68)  ι_{2} : X = X × {o} ↪ X × X,  ι_{3} : X = {o} × X ↪ X × X.

Let

(69)  ι_{23} : X ↪ X × X

denote the diagonal morphism. By abuse of notation, we will denote by the same symbols the resulting maps

ι_{2}, ι_{3}, ι_{23} : X′ ↪ X′ × X′, ι_{2}, ι_{3}, ι_{23} : W_{ϵ} ↪ W_{ϵ} × W_{ϵ}.

Finally write

φ_{23}^{∗} := ι_{23}^{∗} − ι_{2}^{∗} − ι_{3}^{∗} : Ω_{rig}^{1}(W_{ϵ} × W_{ϵ}) ⟶ Ω^{1}(W_{ϵ})

for the maps obtained by combining the pullbacks of these three morphisms.

Lemma 3.5. The map H_{rig}^{1}(W_{ϵ} × W_{ϵ}) ⟶ H_{rig}^{1}(W_{ϵ}) induced by φ_{23}^{∗} is the zero map.

Proof. This is a direct consequence of the following two facts:

(1) The map

φ_{23}^{∗} : H_{dR}^{1}(X′_{2} × X′_{3}/ℚ_{p}) ⟶ H_{dR}^{1}(X′/ℚ_{p})

is 0. This can be checked by choosing a complex embedding of ℚ_{p} and noting that the map φ_{23} : H_{1}(X′/ℂ) ⟶ H_{1}(X′_{2} × X′_{3}/ℂ) vanishes, using topological methods.

(2) There is a commutative diagram

[figure: H_{dR}^{1}(X′_{2} × X′_{3}/ℚ_{p}); φ_{23}^{∗}; _{comp_{ϵ}}; H_{dR}^{1}(X′/ℚ_{p}); _{comp_{ϵ}}; H_{rig}^{1}(W_{ϵ} × W_{ϵ}/ℚ_{p}); φ_{23}^{∗}; H_{rig}^{1}(W_{ϵ}/ℚ_{p}).]

□

To formulate the counterpart of Lemma 3.5 for general weights (k, ℓ, m), recall the subsets A, B and C of {1, …, r} of cardinalities r_{1}, r_{2} and r_{3} ≥ 0 that were chosen in §3.1, and denote by A′, B′ and C′ their respective complements. Note that

A′ = B ∩ C, B′ = A ∩ C and C′ = A ∩ B.

Recall also the diagonal inclusions

φ_{ABC} : E^{r} ⟶ E^{r_{1}} × E^{r_{2}} × E^{r_{3}} and φ_{BC} : E^{r} ⟶ E^{r_{2}} × E^{r_{3}}

introduced in (64). As in (48), the map φ_{BC} together with Poincaré duality on the fibers of the projection E^{r} → E^{r_{1}} induced by A, give rise to a pullback operation on sheaves

(70)  L_{r_{2}} ⊗ L_{r_{3}} ⟶ L_{A}(−t)

where t := |A′| = r − r_{1} satisfies k = ℓ + m − 2 − 2t. We hence obtain a map

φ_{A,BC}^{∗} : Ω_{rig}^{1}(W_{ϵ} × W_{ϵ}, L_{r_{2}} ⊗ L_{r_{3}}) ⟶ Ω_{rig}^{1}(W_{ϵ}, L_{A}(−t)).

Set

(71)  φ^{∗} := \begin{cases}φ_{23}^{∗} & if (k, ℓ, m) = (2, 2, 2), \\ φ_{A,BC}^{∗} & otherwise. \end{cases}

Corollary 3.6. If σ ∈ Ω^{1}(W_{ϵ} × W_{ϵ}, L_{r_{2}} ⊗ L_{r_{3}}) is ∇-closed, then φ^{∗}(σ) is ∇-exact on W_{ϵ}.

Proof. If (r_{1}, r_{2}, r_{3}) = (0, 0, 0), this is just Lemma 3.5. If r_{3}, r_{2} > 0, by the Künneth decomposition we have

H_{dR}^{1}(X′ × X′, L_{r_{2}} ⊗ L_{r_{3}}) ≃ H_{dR}^{1}(X′, L_{r_{2}}) ⊗ H_{dR}^{0}(X′, L_{r_{3}})
⊕ H_{dR}^{0}(X′, L_{r_{2}}) ⊗ H_{dR}^{1}(X′, L_{r_{3}}),

and these modules vanish because the sheaves L_{r_{2}} and L_{r_{3}} have no global horizontal sections (cf. [BDP, Lemma 2.1]). In light of (40) we deduce that in fact σ is already exact, and a fortiori so is φ^{∗}(σ). □

Proposition 3.7. The element

ξ(P, ω_{2}, ω_{3}) := class of φ^{∗}ρ(P, ω_{2}, ω_{3}) in H_{rig}^{1}(W_{ϵ}, L_{A}(−t))

does not depend on the choice of rigid differential ρ(P, ω_{2}, ω_{3}) satisfying (67) of Lemma 3.4 and has vanishing annular residues. In particular, ξ(P, ω_{2}, ω_{3}) belongs to H_{dR}^{1}(X, L_{A}(−t)).

Proof. The first assertion is a direct consequence of Corollary 3.6. In light of (41) and the discussion following it, the second assertion follows the condition (ii) satisfied by P. □

The Frobenius endomorphism Φ acts on H_{dR}^{1}(X, L_{A}(−t)) with eigenvalues of complex absolute value \sqrt{p}^{1+r_{2}+r_{3}}. Therefore, since the roots of P have absolute value either p^{r+1} or \sqrt{p}^{2+r_{2}+r_{3}}, the endomorphism P (Φ) acts invertibly on H_{dR}^{1}(X, L_{A}(−t)). In particular, for all η ∈ H_{dR}^{1}(X, L_{A}(−t)), the class P (Φ)^{−1}η is well-defined. Furthermore, the class

(72)  ξ(ω_{2}, ω_{3}) := P (Φ)^{−1}ξ(P, ω_{2}, ω_{3}) ∈ H_{rig}^{1}(W_{ϵ}, L_{A}(−t))

does not depend on the choice of polynomial P. More precisely, replacing P by a polynomial P Q satisfying the conditions before the statement of Lemma 3.4, one sees that

ρ(P Q, ω_{2}, ω_{3}) = Q(Φ)ρ(P, ω_{2}, ω_{3}), ξ(P Q, ω_{2}, ω_{3}) = Q(Φ)ξ(P, ω_{2}, ω_{3}).

We will now describe (part of) the image of the generalized Gross-Kudla-Schoen cycle Δ_{k,ℓ,m} under the p-adic Abel-Jacobi map. More precisely, we will describe the restriction of the functional AJ_{p}(Δ_{k,ℓ,m}) to the summand

(73)  H_{dR}^{1}(X, L_{r_{1}}) ⊗ H^{0}(X, \underline{ω}^{r_{2}} ⊗ Ω_{X}^{1}) ⊗ H^{0}(X, \underline{ω}^{r_{3}} ⊗ Ω_{X}^{1}) ⊆ H_{dR}^{2r+3}(W),

where the inclusion arises from the Künneth decomposition as in (5).

Let ι_{A} : E^{A} ⟶ E^{r_{1}} denote the natural isomorphism between the varieties E^{A} := E_{a_{1}} ×_{X} ×... ×_{X} E_{a_{r_{1}}} and E^{r_{1}}, giving rise to an isomorphism of sheaves between L_{r_{1}} and L_{A}.

Theorem 3.8. Let η ⊗ ω_{2} ⊗ ω_{3} be any class in (73). Then

(74)  AJ_{p}(Δ_{k,ℓ,m})(η ⊗ ω_{2} ⊗ ω_{3}) = ⟨ι_{A}^{∗}η, ξ(ω_{2}, ω_{3})⟩,

where

⟨ , ⟩ : H_{dR}^{1}(X, L_{A}) × H_{dR}^{1}(X, L_{A}(−t)) ⟶ ℂ_{p}(−1 − r)

arises from Poincaré duality.

Remark 3.9. The reader mainly interested in the case (k, ℓ, m) = (2, 2, 2) will notice that Theorem 3.8 asserts that the image of the Gross-Kudla-Schoen cycle Δ_{2,2,2} in X^{3} under the p-adic Abel-Jacobi map satisfies

(75)  AJ_{p}(Δ_{2,2,2})(η ⊗ ω_{2} ⊗ ω_{3}) = ⟨η, ξ(ω_{2}, ω_{3})⟩,

where ⟨ , ⟩ is the usual Poincaré duality pairing on H_{dR}^{1}(X).

3.3. Proof of the formula: Besser’s finite polynomial cohomology. The goal of this section is proving Theorem 3.8, and to do that we first set some notation. Let V be a smooth, proper, irreducible variety of dimension d over ℚ_{p} which admits a smooth proper, flat model V over ℤ_{p}. There are syntomic and finite polynomial cohomology groups

H_{syn}^{i}(V, n) ⊆ H_{fp}^{i}(V, n)

for every i, n ≥ 0 (cf. e.g. [Bes00] and the references therein). These groups are related to the p-adic de Rham cohomology of V by the following diagram (cf. [Bes00, Prop. 2.5 (1-2)]):

(76)  [figure: 0; H_{dR}^{i−1}(V)/Fil^{n}H_{dR}^{i−1}(V); i_{syn}; H_{syn}^{i}(V, n); p_{syn}; Fil^{n}H_{dR}^{i}(V); 0; H_{dR}^{i−1}(V)/Fil^{n}H_{dR}^{i−1}(V); i_{fp}; H_{fp}^{i}(V, n); p_{fp}; Fil^{n}H_{dR}^{i}(V); 0,]

in which the rows are exact. In addition there are compatible cycle class maps

\xymatrix{CH^{i}(V) \ar[d] \ar[r]^{^{cl^{syn}}} \ar[rr]^{^{cl^{fp}}} & H_{syn}^{2i}(V, i) \ar[dr]^{p_{syn}} \ar[r] & H_{fp}^{2i}(V, i) \ar[d]^{p_{fp}} \\ CH^{i}(V) \ar[rr]^{_{cl_{dR}}} &  & Fil^{i}H_{dR}^{2i}(V)}

which give rise (by the formalism of e.g. [Ne2, (1.2)]) to the p-adic Abel-Jacobi maps

(77) AJ_{p} : CH^{i}(V)_{0} ⟶ ker(p_{syn}) = H_{dR}^{2i−1}(V)/Fil^{i}(H_{dR}^{2i−1}(V)) ≃ Fil^{d−i+1}H_{dR}^{2d−2i+1}(V)^{∨}

that we already introduced in (65). Besides the definition of AJ_{p} given in (66), Besser proved in [Bes00, Theorem 1,2] that the p-adic Abel-Jacobi map may be expressed purely in terms of Coleman integration.

Namely, let Δ = ∑ a_{j}Δ_{j} be a representative of a class in CH^{i}(V)_{0}, with a_{j} ∈ ℚ and Δ_{j} irreducible, smooth proper subschemes of V over ℤ_{p}. Besser showed that AJ_{p} sends the null-homological cycle Δ to the functional

(78)  AJ_{p}(Δ) : Fil^{d−i+1}H_{dR}^{2d−2i+1}(V) ⟶ ℚ_{p}, ω ↦ ∫_{Δ} ω := ∑ a_{j}tr(ι_{j}^{∗}(ω̃)),

where ω̃ ∈ H_{fp}^{2d−2i+1}(V, d−i+1) is a lift of ω under the map p_{fp} appearing in (76), ι_{j} : Δ_{j} ↪ V is the natural inclusion and

tr : H_{fp}^{2d−2i+1}(Δ_{j}, d − i + 1) ⟶^{∼} ℚ_{p}

is the canonical trace isomorphism ([Bes00, Prop. 2.5 (4)]).

It is Besser’s formula (78) which we shall apply in order to prove Theorem 3.8. As a final piece of notation, let

(79)  ⟨ , ⟩_{fp} : H_{fp}^{i}(V, n) × H_{fp}^{2d+1−i}(V, d + 1 − n) ⟶ ℚ_{p}

denote the cup product in finite polynomial cohomology described in [Bes00, Prop. 2.5 (4)]. We first focus in the particular case (k, ℓ, m) = (2, 2, 2). Set

φ_{dR,∗} := ι_{23,dR,∗} − ι_{2,dR,∗} − ι_{3,dR,∗} : H_{dR}^{1}(X) ⟶ Fil^{1}H_{dR}^{3}(X^{2}),
φ_{fp,∗} := ι_{23,fp ∗} − ι_{2,fp ∗} − ι_{3,fp ∗} : H_{fp}^{1}(X , 0) ⟶ H_{fp}^{3}(X^{2}, 1)

to be the maps induced by push-forward by the three embeddings ι_{2}, ι_{3}, ι_{23} introduced in (68) and (69) on de Rham and finite polynomial cohomology, respectively.

By [Bes00, Proposition 2.5 (1) and Lemma 5.1], there is a commutative diagram

(80)  [figure: 0; i_{1}; H_{fp}^{1}(X , 0); p_{1}; φ_{fp,∗}; H_{dR}^{1}(X); φ_{dR,∗}; 0; 0; \frac{H_{dR}^{2}(X^{2})}{Fil^{1}H_{dR}^{2}(X^{2})}; i_{2}; H_{fp}^{3}(X^{2}, 1); p_{2}; Fil^{1}H_{dR}^{3}(X^{2}); 0]

with exact rows arising from (76).

Let η̃ ∈ H_{fp}^{1}(X , 0) be a preimage of η under p_{1} and ω̃_{2}, ω̃_{3} ∈ H_{fp}^{1}(X , 1) be preimages of ω_{2} and ω_{3} respectively, so that ω̃_{2} ⊗ ω̃_{3} is a preimage in H_{fp}^{2}(X^{2}, 2) of ω_{2} ⊗ ω_{3} ∈ Fil^{2}H_{dR}^{2}(X^{2}).

Lemma 3.10. The following equality holds in ℚ_{p}:

⟨cl_{fp}(Δ_{2,2,2}), η̃ ⊗ ω̃_{2} ⊗ ω̃_{3}⟩_{fp} = ⟨φ_{fp ∗}(η̃), ω̃_{2} ⊗ ω̃_{3}⟩_{fp}.

Proof. The class of the cycle X_{123} − X_{12} − X_{13} − X_{23} + X_{1} + X_{2} + X_{3} is an integral, smooth proper model of Δ_{2,2,2} in CH^{2}(X^{3})_{0}. Hence

⟨cl_{fp}(Δ_{2,2,2}), η̃ ⊗ ω̃_{2} ⊗ ω̃_{3}⟩_{fp} = − ∑_{I}(−1)^{|I|}tr_{X} (ι_{I}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3}))

where I ranges through the seven non-empty subsets of {1, 2, 3} and tr_{X} : H_{fp}^{3}(X , 2) ⟶^{∼} ℚ_{p} is Besser’s trace isomorphism. For I = 123, 12 and 13, it follows from the very definitions that the above traces may be recast as the cup-products

tr_{X} (ι_{123}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) = ⟨η̃, (ω̃_{2} ⊗ ω̃_{3})_{|X_{23}}⟩_{fp}
tr_{X} (ι_{12}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) = ⟨η̃, (ω̃_{2} ⊗ ω̃_{3})_{|X_{2}}⟩_{fp}
tr_{X} (ι_{13}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) = ⟨η̃, (ω̃_{2} ⊗ ω̃_{3})_{|X_{3}}⟩_{fp}

on X, while tr_{X} (ι_{I}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) vanishes for the remaining four choices of I.

On the other hand, the push-forward map φ_{fp ∗} is dual under cup-product (79) to the pull-back homomorphism φ_{fp}^{∗} = ι_{23,fp}^{∗} − ι_{2,fp}^{∗} − ι_{3,fp}^{∗} : H_{fp}^{2}(X^{2}, 2) → H_{fp}^{2}(X , 2), and hence

⟨φ_{fp ∗}(η̃), ω̃_{2} ⊗ ω̃_{3}⟩_{fp} = ⟨η̃, (ω̃_{2} ⊗ ω̃_{3})_{|X_{23}}⟩_{fp} − ⟨η̃, (ω̃_{2} ⊗ ω̃_{3})_{|X_{2}}⟩_{fp} − ⟨η̃, (ω̃_{2} ⊗ ω̃_{3})_{|X_{3}}⟩_{fp}.

This proves the lemma. □

Lemma 3.10 combined with (78) yields

(81)  AJ_{p}(Δ_{2,2,2})(η ⊗ ω_{2} ⊗ ω_{3}) = ⟨φ_{fp ∗}(η̃), ω̃_{2} ⊗ ω̃_{3}⟩_{fp}.

By the functoriality of [Bes00, (8)], there is another commutative diagram with exact rows

(82)  \xymatrix{0 \ar[r] & H_{dR}^{1}(X^{2}) \ar[r]^{i} \ar[d]_{φ_{dR}^{∗}} & H_{fp}^{2}(X^{2}, 2) \ar[r]^{p} \ar[d]_{φ_{fp}^{∗}} & Fil^{2}H_{dR}^{2}(X^{2}) \ar[r] \ar[d]_{φ_{dR}^{∗}} & 0 \\ 0 \ar[r] & H_{dR}^{1}(X) \ar[r]^{i} & H_{fp}^{2}(X , 2) \ar[r]^{p} & Fil^{2}H_{dR}^{2}(X) = 0.}

In light of [Bes00, Def. 2.2 and eq. (6)], a pair

(α, β) ∈ Ω_{rig}^{1}(W_{ϵ}^{2}) ⊕ Fil^{2}H_{dR}^{2}(X^{2}) = Ω_{rig}^{1}(W_{ϵ}^{2}) ⊕ Ω^{2}(X^{2})

satisfying dα = P (Φ)(β) gives rise to a class in H_{fp}^{2}(X^{2}, 2). Since the map p of the upper row of (82) is the projection [α, β] ↦ [β] to the second factor, the class ω̃_{2} ⊗ ω̃_{3} in H_{fp}^{2}(X^{2}, 2) may be represented by the pair

(ρ, ω_{2} ⊗ ω_{3}) ∈ Ω_{rig}^{1}(W_{ϵ}^{2}) ⊕ Ω^{2}(X^{2}),

where ρ := ρ(P, ω_{2}, ω_{3}) is as in Lemma 3.4. Since Fil^{2}H_{dR}^{2}(X) = 0, the class φ_{fp}^{∗}(ω̃_{2} ⊗ ω̃_{3}) has a preimage in H_{dR}^{1}(X) under the map i of the lower row of (82). Such preimage is the class of ξ := ξ(ω_{2}, ω_{3}) ∈ H_{dR}^{1}(X), as

(83)  i(ξ) := [P (Φ)ξ, 0] = [φ^{∗}ρ, 0] = φ_{fp}^{∗}(ω̃_{2} ⊗ ω̃_{3}) ∈ H_{fp}^{2}(X , 2).

From this we find, as desired, that

⟨φ_{fp ∗}(η̃), ω̃_{2} ⊗ ω̃_{3}⟩_{fp} = ⟨η̃, φ_{fp}^{∗}(ω̃_{2} ⊗ ω̃_{3})⟩_{fp} = ⟨η̃, i(ξ)⟩_{fp} = ⟨η, ξ⟩,

where the first equality follows from the functoriality of the cup product, the second is (83) and the third is a direct consequence of [Bes00, (14)].

This concludes the proof of Theorem 3.8 in the case where all three weights are equal to 2. We turn now to show how a similar approach yields Theorem 3.8 in the remaining cases, that is to say, when either (r_{1}, r_{2}, r_{3}) = (0, r, r) with r > 0 or (r_{1}, r_{2}, r_{3}) with r_{3} ≥ r_{2} ≥ r_{1} > 0 and r_{3} ≤ r_{1} + r_{2}.

By a slight abuse of notation, let us still denote W the base change of the triple product E^{r_{1}} × E^{r_{2}} × E^{r_{3}} to ℚ_{p}, which recall has dimension r_{1} + r_{2} + r_{3} + 3 = 2r + 3.

Let η ∈ H_{dR}^{1}(X, L_{r_{1}}) ⊂ H_{dR}^{r_{1}+1}(E^{r_{1}}) and

ω_{2} ∈ H_{dR}^{1}(X, ω^{r_{2}} ⊗ Ω_{X}^{1}) ⊂ Fil^{r_{2}+1}H_{dR}^{r_{2}+1}(E^{r_{2}}), ω_{3} ∈ H_{dR}^{1}(X, ω^{r_{3}} ⊗ Ω_{X}^{1}) ⊂ Fil^{r_{3}+1}H_{dR}^{r_{3}+1}(E^{r_{3}})

be classes as in (73), which we may interchangeably regard either as L_{r_{i}}-valued 1-forms on X or as (r_{i} + 1)-forms on E^{r_{i}}. We thus have

η ⊗ ω_{2} ⊗ ω_{3} ∈ Fil^{r_{2}+r_{3}+2}H_{dR}^{3}(X^{3}, L_{r_{1}} ⊗ L_{r_{2}} ⊗ L_{r_{3}}) ⊂ Fil^{r_{2}+r_{3}+2}H_{dR}^{2r+3}(W).

Denote 𝔈^{r} the canonical integral, smooth and proper model of E^{r} over ℤ_{p}. Let

η̃ ∈ H_{fp}^{r_{1}+1}(𝔈^{r_{1}}, −t), ω̃_{2} ∈ H_{fp}^{r_{2}+1}(𝔈^{r_{2}}, r_{2} + 1) and ω̃_{3} ∈ H_{fp}^{r_{3}+1}(𝔈^{r_{3}}, r_{3} + 1)

be, respectively, lifts of η, ω_{2} and ω_{3} in finite polynomial cohomology, and write η̃⊗ ω̃_{2} ⊗ ω̃_{3} ∈ H_{fp}^{2r+3}(W, r + 2) for their tensor product.

Similarly as for φ_{BC} and φ_{ABC}, write φ_{A} : E^{r} ⟶ E^{r_{1}}, (x; P_{1}, ..., P_{r}) ↦ (x; P_{a_{1}}, ..., P_{a_{r_{1}}} ) for the natural projection induced by the choice of the subset A of {1, ..., r}.

Lemma 3.11. We have

(84)  AJ_{p}(Δ_{k,ℓ,m})(η ⊗ ω_{2} ⊗ ω_{3}) = ⟨φ_{A,fp}^{∗}(η̃), φ_{BC,fp}^{∗}(ω̃_{2} ⊗ ω̃_{3})⟩_{𝔈^{r},fp}.

Proof. Cycle Δ_{k,ℓ,m} is a linear combination ∑_{j} a_{j}Δ_{j}, where a_{j} ∈ ℚ and Δ_{j} are irreducible subvarieties of W, all of which are the image of E^{r} under a closed embedding ι_{j} : E^{r} ↪ W; see Definitions 3.2 and 3.3 for more details. These embeddings lift in a natural way to proper maps ι_{j} : 𝔈^{r} ↪ W, and ∑ a_{j}ι_{j}(𝔈^{r}) is thus a representative of Δ_{k,ℓ,m} in CH^{r+2}(W)_{0}.

By Besser’s formula (78),

(85) AJ_{p}(Δ_{k,ℓ,m})(η ⊗ ω_{2} ⊗ ω_{3}) = ⟨cl_{fp}(Δ_{k,ℓ,m}), η̃ ⊗ ω̃_{2} ⊗ ω̃_{3}⟩_{fp} = ∑_{j} a_{j}tr_{𝔈^{r}}(ι_{j}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})).

If k = 2 (and thus ℓ = m ≥ 3), according to Definition 3.2 then half of the terms in the above sum correspond to the image of {o} × φ_{BC}(E^{r}) under some automorphism of E^{r_{1}} × E^{r_{2}} × E^{r_{3}} that acts as the identity on the copy of E^{r_{1}}. Here o is some choice of a base point in E^{r_{1}}. For such embeddings, it follows that ι_{j}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) vanishes, and hence those terms do not contribute to the sum in (85).

In view of this and Definition 3.3, for any balanced triple (k, ℓ, m) ≠ (2, 2, 2) we conclude that AJ_{p}(Δ_{k,ℓ,m})(η ⊗ ω_{2} ⊗ ω_{3}) is equal to a linear combination of the form

(86)  ∑ a_{α_{1},α_{2},α_{3}}tr_{𝔈^{r}}(φ_{ABC}^{∗} ∘ (α_{1} ⊗ α_{2} ⊗ α_{3})^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3}))

where α_{i} are automorphisms of 𝔈^{r_{i}} for i = 1, 2, 3 and the scalars a_{α_{1},α_{2},α_{3}} are each equal to ±\frac{1}{2^{n}} for some n ≥ 0. We refer to §3.1 for the explicit description of the automorphisms α_{i} that intervene here; all them satisfy α_{i}^{∗}(ω̃) = ±ω̃ ∈ H_{fp}^{r_{i}+1}(𝔈^{r_{i}}, ·) for the classes in play, so that tr_{𝔈^{r}}(φ_{ABC}^{∗} ∘ (α_{1} ⊗ α_{2} ⊗ α_{3})^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) = ±tr_{𝔈^{r}}(φ_{ABC}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})). It is a simple combinatorial exercise to check that the terms in (86) sum up to tr_{𝔈^{r}}(φ_{ABC,fp}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})).

By Besser’s definitions of trace and cup-product in finite polynomial cohomology, it directly follows that tr_{𝔈^{r}}(φ_{ABC,fp}^{∗}(η̃ ⊗ ω̃_{2} ⊗ ω̃_{3})) = ⟨φ_{A,fp}^{∗}(η̃), φ_{BC,fp}^{∗}(ω̃_{2} ⊗ ω̃_{3})⟩_{𝔈^{r},fp}. □

The class φ_{BC,dR}^{∗}(ω_{2}⊗ω_{3}) in de Rham cohomology vanishes, because it lies in ϵ_{r}H_{dR}^{r_{2}+r_{3}+2}(E^{r}) and the idempotent ϵ_{r} annihilates H_{dR}^{j}(E^{r}) for all j ≠ r+1; note that indeed r_{2}+r_{2}+2 ≠ r+1 because the triplet (k, ℓ, m) is balanced. Hence the pair (φ_{BC}^{∗}(ρ(P, ω_{2}, ω_{3})), 0) is a representative of φ_{BC,fp}^{∗}(ω̃_{2} ⊗ ω̃_{3}) in finite polynomial cohomology.

It follows exactly as in (83) that P (Φ)^{−1}φ_{BC}^{∗}(ρ(P, ω_{2}, ω_{3})) is a preimage of the class φ_{BC,fp}^{∗}(ω̃_{2} ⊗ ω̃_{3}) under Besser’s map

i : H_{dR}^{r_{2}+r_{3}+1}(E^{r})/Fil^{r_{2}+r_{3}+2}(H_{dR}^{r_{2}+r_{3}+1}(E^{r})) ⟶ H_{fp}^{r_{2}+r_{3}+2}(𝔈^{r}, r_{2} + r_{3} + 2).

It thus again follows that

(87) ⟨φ_{A,fp}^{∗}(η̃), φ_{BC,fp}^{∗}(ω̃_{2} ⊗ ω̃_{3})⟩_{E^{r},fp} = ⟨φ_{A,dR}^{∗}(η), P (Φ)^{−1}φ_{BC}^{∗}(ρ(P, ω_{2}, ω_{3}))⟩_{E^{r},dR}.

Pulling-back under the natural immersion E^{r_{1}} ↪ E^{r} induced by A, we obtain that the right-hand side of (87) equals

⟨ι_{A}^{∗}η, P (Φ)^{−1}ξ(P, ω_{2}, ω_{3})⟩_{E^{r_{1}},dR} = ⟨ι_{A}^{∗}η, ξ(ω_{2}, ω_{3})⟩_{E^{r_{1}},dR}.

Theorem 3.8 follows.

3.4. A formula for AJ_{p}(Δ_{k,ℓ,m}) in terms of p-adic modular forms. Let g and h denote modular forms of weights ℓ = r_{2} + 2 and m = r_{3} + 2, respectively, on Γ_{1}(N). For the calculations of this section, it is only necessary to assume that g and h are eigenvectors for the Hecke operator T_{p}; in particular it will not be assumed that they are new of level N. Let α_{g} and β_{g} be the roots of the Hasse polynomial x^{2} − a_{p}(g)x + χ_{g}(p)p^{ℓ−1} for g at p, ordered in such a way that ord_{p}(α_{g}) ≤ ord_{p}(β_{g}), and let

(88)  g_{α} = g − β_{g}V g,  g_{β} = g − α_{g}V g

denote the respective p-stabilisations, on which the U_{p} operator acts with eigenvalue α_{g} and β_{g} respectively. Similar notational conventions are adopted for h. Note that with these conventions, when g is ordinary then g_{α} is the ordinary p-stabilisation of g that was previously denoted g^{(p)}, but that no ordinariness assumption on g or h are made in this section.

Let f denote a modular form of weight k = r_{1} + 2 on Γ_{1}(N) which is an eigenvector for all the good Hecke operators, and is ordinary at p (but is not necessarily new of level N) and let η_{f}^{u-r} ∈ H_{par}^{1}(X_{ℂ_{p}}, L_{r_{1}})^{u-r} be the unique lift to the unit root subspace of the cohomology class in H^{1}(X_{K_{f}} , \underline{ω}^{−r_{1}}) attached to f as in Corollary 2.13. Recall that e_{ord} denotes Hida’s ordinary projector. Let e_{f^{∗}} denote the commuting idempotent in the Hecke algebra giving the projection onto the f^{∗}-isotypic part, where f^{∗} is the modular form which is dual to f, obtained by applying complex conjugation to the Fourier coefficients of f, and set

e_{f^{∗},ord} = e_{f^{∗}}e_{ord}.

The goal of the next two theorems is to give an explicit expression for the class ξ(ω_{g}, ω_{h}) ∈ H_{dR}^{1}(X, L_{r_{1}}) that appears in Theorem 3.8, or rather, its image under e_{f^{∗},ord}.

Theorem 3.12 below treats the setting where k = ℓ = m = 2, in which the cycle Δ_{2,2,2} is simply the Gross-Kudla-Schoen modified diagonal on X_{1}(N)^{3}. This setting is notationally simpler and therefore easier to follow, but already brings out the key ideas needed to handle the general case, which will then be treated in Theorem 3.13.

Theorem 3.12. Suppose that k = ℓ = m = 2, so that ξ(ω_{g}, ω_{h}) can be viewed as an overconvergent modular form of weight 2. The ordinary projection e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) is equal to the classical modular form

e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) = −\frac{E_{1}(f)}{E(f, g, h)}e_{f^{∗},ord}(d^{−1}g^{[p]} × h),

where E_{1}(f) and E(f, g, h) are defined as in Theorem 1.3 of the introduction.

Proof. Let

P_{g}(x) = 1 − a_{p}(g)p^{−1}x + χ_{g}(p)p^{−1}x^{2} = (1 − α_{g}p^{−1}x)(1 − β_{g}p^{−1}x),
P_{h}(x) = 1 − a_{p}(h)p^{−1}x + χ_{h}(p)p^{−1}x^{2} = (1 − α_{h}p^{−1}x)(1 − β_{h}p^{−1}x),

denote the Hasse polynomials attached to the forms g and h, and set

P_{gh}(x) = (1 − α_{g}α_{h}p^{−2}x)(1 − α_{g}β_{h}p^{−2}x)(1 − β_{g}α_{h}p^{−2}x)(1 − β_{g}β_{h}p^{−2}x),
P_{αα}(x) = (1 − α_{g}α_{h}p^{−2}x)^{−1}P_{gh}(x), P_{αβ}(x) = (1 − α_{g}β_{h}p^{−2}x)^{−1}P_{gh}(x),
P_{βα}(x) = (1 − β_{g}α_{h}p^{−2}x)^{−1}P_{gh}(x), P_{ββ}(x) = (1 − β_{g}β_{h}p^{−2}x)^{−1}P_{gh}(x).

Recall that Φ_{2} and Φ_{3} denote the canonical lift Φ of frobenius operating on the Hasse domains, viewed as contained in the second and third factors respectively of the three-fold of X_{1}(N)^{3}. The operator Φ is related to the V operator on (nearly) overconvergent p-adic modular forms of weight two by the rule Φ = pV. After writing Φ_{2} = pV_{2} and Φ_{3} = pV_{3}, the operator Φ := Φ_{2}Φ_{3} = p^{2}V_{2}V_{3} corresponds to the the canonical lift of Frobenius acting on Ω_{rig}^{2}(W_{ϵ}×W_{ϵ}).

The operators P_{g}(Φ_{2}) and P_{h}(Φ_{3}) annihilate the classes of ω_{g} and ω_{h} in H_{rig}^{1}(W_{ϵ}). More precisely, as in (19),

(89)  P_{g}(Φ_{2})ω_{g} = (1 − a_{p}(g)V_{2} + χ_{g}(p)pV_{2}^{2})g(q) = g^{[p]}(q) = dG^{[p]},
P_{h}(Φ_{3})ω_{h} = (1 − a_{p}(h)V_{3} + χ_{h}(p)pV_{3}^{2})h(q) = h^{[p]}(q) = dH^{[p]},

where G^{[p]} and H^{[p]} ∈ O_{W_{ϵ}} are given by

G^{[p]}(q) = d^{−1}g^{[p]} = ∑_{p∤n} \frac{a_{n}(g)}{n}q^{n}, H^{[p]}(q) = d^{−1}h^{[p]} = ∑_{p∤n} \frac{a_{n}(h)}{n}q^{n}.

These primitives are overconvergent modular forms of level N and weight 0. Note that they have been normalized so that they vanish at the cusp ∞.

Equation (89) indicates that P_{gh}(Φ_{2}Φ_{3}) annihilates the class of the rigid differential ω_{g}ω_{h} in H_{dR}^{2}(X_{1}(N)^{2}) . Hence, there is a rigid one-form ρ(P_{gh}, ω_{g}, ω_{h}) satisfying

(90)  dρ(P_{gh}, ω_{g}, ω_{h}) = P_{gh}(Φ_{2}Φ_{3})ω_{g}ω_{h}.

To describe ρ(P_{gh}, ω_{g}, ω_{h}) more explicitly, we exploit the identities

g = (α_{g} − β_{g})^{−1}(α_{g}g_{α} − β_{g}g_{β}), h = (α_{h} − β_{h})^{−1}(α_{h}h_{α} − β_{h}h_{β})

expressing g and h as linear combinations of the stabilisations appearing in (88). This leads to an expression for the right-hand side of (90) as a sum of four contributions:

P_{gh}(Φ_{2}Φ_{3})ω_{g}ω_{h} = θ_{αα} − θ_{αβ} − θ_{βα} + θ_{ββ},

where θ_{αα} is the rigid two-form on W_{ϵ} × W_{ϵ} (for a suitable ϵ > 0 given by

θ_{αα} = \frac{α_{g}α_{h}P_{gh}(Φ_{2}Φ_{3})(g_{α} ⊗ h_{α})}{(α_{g} − β_{g})(α_{h} − β_{h})}
(91)  = \frac{α_{g}α_{h}P_{αα}(Φ_{2}Φ_{3})(1 − α_{g}α_{h}V_{2}V_{3})(g_{α} ⊗ h_{α})}{(α_{g} − β_{g})(α_{h} − β_{h})},

and θ_{αβ} (resp. θ_{βα} and θ_{ββ}) are defined in the identical way, after replacing (α_{g}, α_{h}) by (α_{g}, β_{h}) (resp. (β_{g}, α_{h}), (β_{g}, β_{h}).) We next observe that

(1 − α_{g}α_{h}V_{2}V_{3})(g_{α} ⊗ h_{α}) = \frac{1}{2} ((1 − α_{g}V_{2})(1 + α_{h}V_{3}) + (1 + α_{g}V_{2})(1 − α_{h}V_{3})) g_{α} ⊗ h_{α}
(92)  = \frac{1}{2} (g^{[p]} ⊗ (1 + α_{h}V_{3})h_{α} + (1 + α_{g}V_{2})g_{α} ⊗ h^{[p]}) .

It follows from (91) and (92), together with the fact that dΦ = Φd, that

(93)  θ_{αα} = dρ_{αα},

where

(94)  ρ_{αα} = −\frac{α_{g}α_{h}P_{αα}(Φ_{2}Φ_{3}) (G^{[p]} ⊗ (1 + α_{h}V_{3})h_{α} − (1 + α_{g}V_{2})g_{α} ⊗ H^{[p]})}{2(α_{g} − β_{g})(α_{h} − β_{h})}.

The rigid one-forms ρ_{αβ}, ρ_{βα}, and ρ_{ββ} are defined similarly, after replacing (α_{g}, α_{h}) as before by (α_{g}, β_{h}), (β_{g}, α_{h}), and (β_{g}, β_{h}) respectively, and the differential one-form ρ(P_{gh}, ω_{g}, ω_{h}) of (90) can be chosen to be

ρ(P_{gh}, ω_{g}, ω_{h}) = ρ_{αα} − ρ_{αβ} − ρ_{βα} + ρ_{ββ}.

Let ξ_{αα}, ξ_{αβ}, ξ_{βα}, and ξ_{ββ} denote the pullbacks to the modified diagonal of the one-forms ρ_{αα}, ρ_{αβ}, ρ_{βα}, and ρ_{ββ}, so that for example

(95)  ξ_{αα} = −\frac{α_{g}α_{h}P_{αα}(Φ) (G^{[p]} × (1 + α_{h}V)h_{α} − (1 + α_{g}V)g_{α} × H^{[p]})}{2(α_{g} − β_{g})(α_{h} − β_{h})}.

The frobenius operator Φ acts on e_{f^{∗},ord}(H_{rig}^{1}(W_{ϵ})) with eigenvalue pα_{f^{∗}}^{−1} = β_{f}, and hence, by Lemma 2.17,

e_{f^{∗},ord} (P_{αα}(Φ)(G^{[p]}(1 + α_{h}V)h_{α})) = P_{αα}(β_{f})e_{f^{∗},ord}(G^{[p]}h),
e_{f^{∗},ord} (P_{αα}(Φ)((1 + α_{g}V)g_{α}H^{[p]})) = P_{αα}(β_{f})e_{f^{∗},ord}(gH^{[p]}).

Invoking Lemma 2.17 once again, we find

e_{f^{∗},ord}(gH^{[p]}) = e_{f^{∗},ord}(g^{[p]}H^{[p]}) = −e_{f^{∗},ord}(G^{[p]}h^{[p]}) = −e_{f^{∗},ord}(G^{[p]}h),

where the second equality follows by noting that g^{[p]}H^{[p]} + G^{[p]}h^{[p]} = d(G^{[p]}H^{[p]}) is exact and invoking the fact that exact rigid differentials are in the kernel of the ordinary projection. Hence applying the projector e_{f^{∗},ord} to (95) gives

(96)  e_{f^{∗},ord}(ξ_{αα}) = −\frac{α_{g}α_{h}P_{αα}(β_{f})}{(α_{g} − β_{g})(α_{h} − β_{h})}e_{f^{∗},ord}(G^{[p]}h).

The definition of ξ(ω_{g}, ω_{h}) given in (72) therefore implies that

(97)  e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) = ξ′_{αα} − ξ′_{αβ} − ξ′_{βα} + ξ′_{ββ},

where

ξ′_{αα} = −\frac{α_{g}α_{h}}{(α_{g} − β_{g})(α_{h} − β_{h})(1 − α_{g}α_{h}β_{f}p^{−2})}e_{f^{∗},ord}(G^{[p]}h),

and likewise for the other three contributions to e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})). A direct calculation reveals that

ξ′_{αα} − ξ′_{αβ} = −\frac{α_{g}}{(α_{g} − β_{g})(1 − α_{g}α_{h}β_{f}p^{−2})(1 − α_{g}β_{h}β_{f}p^{−2})}e_{f^{∗},ord}(G^{[p]}h),
ξ′_{βα} − ξ′_{ββ} = −\frac{β_{g}}{(α_{g} − β_{g})(1 − β_{g}α_{h}β_{f}p^{−2})(1 − β_{g}β_{h}β_{f}p^{−2})}e_{f^{∗},ord}(G^{[p]}h).

Subtracting these two equations and invoking (97) gives

e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) = −\frac{(1 − χ_{f}(p)^{−1}β_{f}^{2}p^{−2})}{E(f, g, h)}e_{f^{∗},ord}(d^{−1}g^{[p]} × h),

as was to be shown. □

We now turn to the case of general weights (k, ℓ, m). Before stating the result, recall that the triple of weights (k, ℓ, m) is still assumed to be balanced, and that, following notations similar to those in the proof of Proposition 2.9, we have set

k = ℓ + m − 2 − 2t, with t ≥ 0, c = (k + ℓ + m − 2)/2,

so that ξ(ω_{g}, ω_{h}) corresponds to a class in H_{dR}^{1}(X, L_{r_{1}}(−t)).

Theorem 3.13. The projection e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) is represented by the classical modular form

e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) = −\frac{(−1)^{t} · t! · E_{1}(f)}{E(f, g, h)}e_{f^{∗},ord}(d^{−1−t}g^{[p]} × h),

where E_{1}(f) and E(f, g, h) are defined as in Theorem 1.3 of the introduction.

Proof. Let ω_{g} and ω_{h} be the global sections of \underline{ω}^{r_{2}} ⊗ Ω_{X}^{1} and of \underline{ω}^{r_{3}} ⊗ Ω_{X}^{1} over X = X_{1}(N) attached to g and h respectively. Since these sections are algebraic, they can also be viewed as L_{r_{2}} and L_{r_{3}}-valued rigid differentials on W_{ϵ}/ℂ_{p} for any ϵ > 0. Modifying slightly the definitions of the weight two setting, we define

P_{g}(x) = (1 − α_{g}p^{−r_{2}−1}x)(1 − β_{g}p^{−r_{2}−1}x), P_{h}(x) = (1 − α_{h}p^{−r_{3}−1}x)(1 − β_{h}p^{−r_{3}−1}x).

The operators P_{g}(Φ_{2}) and P_{h}(Φ_{3}) annihilate the classes of ω_{g} and ω_{h} in H_{rig}^{1}(W_{ϵ}, L_{r_{2}}) and H_{rig}^{1}(W_{ϵ}, L_{r_{3}}) respectively, and in terms of q-expansions,

P_{g}(Φ_{2})(ω_{g}) = (1 − α_{g}V_{2})(1 − β_{g}V_{2})g = g^{[p]},
P_{h}(Φ_{3})(ω_{h}) = (1 − α_{h}V_{3})(1 − β_{h}V_{3})g = h^{[p]}.

Let G^{[p]} and H^{[p]} denote the overconvergent sections of L_{r_{2}} and L_{r_{3}} satisfying

∇G^{[p]} = ω_{g^{[p]}}, ∇H^{[p]} = ω_{h^{[p]}}.

The q-expansions of these sections are given in equation (47) of Section 2.4, i.e.,

(98)  G^{[p]}(q) = ∑_{j=0}^{r_{2}}(−1)^{j}j!\binom{r_{2}}{j}d^{−1−j}g^{[p]}(q)ω_{can}^{r_{2}−j}η_{can}^{j},

(99)  H^{[p]}(q) = ∑_{j=0}^{r_{3}}(−1)^{j}j!\binom{r_{3}}{j}d^{−1−j}h^{[p]}(q)ω_{can}^{r_{3}−j}η_{can}^{j}.

As in the weight two setting, we let

P_{gh}(x) = (1 − α_{g}α_{h}p^{−r_{2}−r_{3}−2}x)(1 − α_{g}β_{h}p^{−r_{2}−r_{3}−2}x)(1 − β_{g}α_{h}p^{−r_{2}−r_{3}−2}x)
×(1 − β_{g}β_{h}p^{−r_{2}−r_{3}−2}x).

Just as before, the operator P_{gh}(Φ_{2}Φ_{3}) annihilates the class of ω_{g} ⊗ω_{h} in the hypercohomology group H_{dR}^{2}(X_{1}(N)^{2}, L_{r_{2}} ⊗ L_{r_{3}}), so there is a rigid L_{r_{2}} ⊗ L_{r_{3}}-valued one-form ρ(P_{gh}, ω_{g}, ω_{h}) on W_{ϵ} × W_{ϵ} satisfying

(100)  ∇ρ(P_{gh}, ω_{g}, ω_{h}) = P_{gh}(Φ_{2}Φ_{3})ω_{g}ω_{h}.

The same manipulations as in the weight two case allow us to express ρ(P_{gh}, ω_{g}, ω_{h}) as a sum of four contributions:

ρ(P_{gh}, ω_{g}, ω_{h}) = ρ_{αα} − ρ_{αβ} − ρ_{βα} + ρ_{ββ},

where for instance

ρ_{αα} = −\frac{α_{g}α_{h}P_{αα}(Φ_{2}Φ_{3}) (G^{[p]} ⊗ (1 + α_{h}V_{3})ω_{h_{α}} − (1 + α_{g}V_{2})ω_{g_{α}} ⊗ H^{[p]})}{2(α_{g} − β_{g})(α_{h} − β_{h})},

just as in equations (93) and (94). Let ξ_{αα}, ξ_{αβ}, ξ_{βα}, and ξ_{ββ} denote the pullbacks to the generalised diagonal cycle E^{r} of the one-forms ρ_{αα}, ρ_{αβ}, ρ_{βα}, and ρ_{ββ} via the map φ_{BC}, viewed as overconvergent sections of L_{A}(−t) ⊗ Ω^{1} on W_{ϵ}. Write

ξ_{αα}^{n-oc} = Spl_{u-r}(ξ_{αα}), ξ_{αα}^{oc} = Π_{N}^{oc}(ξ_{αα}^{n-oc}),

for the nearly overconvergent and overconvergent modular forms attached to ξ_{αα}. By equation (44) defining Π_{N}^{oc}, the elements ξ_{αα} and ξ_{αα}^{oc} have the same image in cohomology, and therefore the following equality holds in H_{dR}^{1}(X, L_{r_{1}}(−t)):

ξ(P_{gh}, ω_{g}, ω_{h}) = ξ_{αα}^{oc} − ξ_{αβ}^{oc} − ξ_{βα}^{oc} + ξ_{ββ}^{oc}.

Applying e_{f^{∗},ord} to this equation and invoking Lemma 2.7 gives

e_{f^{∗},ord}(ξ(P_{gh}, ω_{g}, ω_{h})) = e_{f^{∗},ord}(ξ_{αα}^{n-oc} − ξ_{αβ}^{n-oc} − ξ_{βα}^{n-oc} + ξ_{ββ}^{n-oc}),

and the same argument as in the proof of Proposition 2.9 (cf. in particular equation (49)) shows that

(101)  ξ_{αα}^{n-oc} = −\frac{α_{g}α_{h}P_{αα}(Φ) (d^{−1−t}g^{[p]} × (1 + α_{h}V)h_{α} − (1 + α_{g}V)g_{α} × d^{−1−t}h^{[p]})}{2(α_{g} − β_{g})(α_{h} − β_{h})}.

By exactly the same reasoning used to obtain (96) in the weight two case, and using the fact that Φ acts on e_{f^{∗},ord}(H_{dR}^{1}(X, L_{r_{1}}(−t))) with eigenvalue p^{r_{1}+1+t}α_{f^{∗}}^{−1} = β_{f}p^{t}, we find after applying the projector e_{f^{∗},ord} to (101) that

(102)  e_{f^{∗},ord}(ξ_{αα}^{n-oc}) = −\frac{(−1)^{t} · t! · α_{g}α_{h}P_{αα}(β_{f}p^{t})}{(α_{g} − β_{g})(α_{h} − β_{h})}e_{f^{∗},ord}(d^{−1−t}g^{[p]} × h).

It now follows from (72) that

(103)  e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})) = ξ′_{αα} − ξ′_{αβ} − ξ′_{βα} + ξ′_{ββ},

where

ξ′_{αα} = −\frac{(−1)^{t}t!α_{g}α_{h}}{(α_{g} − β_{g})(α_{h} − β_{h})(1 − α_{g}α_{h}β_{f}p^{−c})}e_{f^{∗},ord}(d^{−1−t}g^{[p]} × h),

and likewise for the other three contributions to e_{f^{∗},ord}(ξ(ω_{g}, ω_{h})). The result now follows from a simple direct calculation, as in the conclusion of the proof of Theorem 3.12. □

We can now conclude this section with a (partial) formula for the image of the generalised Gross-Kudla Schoen cycle Δ_{k,ℓ,m} under the p-adic Abel-Jacobi map.

Theorem 3.14. With notations as in the statement of Theorems 3.12 and 3.13,

AJ_{p}(Δ_{k,ℓ,m})(η_{f}^{u-r}ω_{g}ω_{h}) = (−1)^{t+1} \frac{t! · E_{1}(f)}{E(f, g, h)}⟨η_{f}^{u-r}, d^{−1−t}g^{[p]} × h⟩,

for all η_{f}^{u-r} ∈ H_{dR}^{1}(X, L_{r_{1}})^{u-r}[π_{f^{∗}}].

Proof. Assume first that (k, ℓ, m) = (2, 2, 2). By Proposition 2.3, the class η_{f}^{u-r} is orthogonal to the kernel of e_{f^{∗},ord}, and hence

⟨η_{f}^{u-r}, ξ(ω_{g}, ω_{h})⟩ = ⟨η_{f}^{u-r}, e_{f^{∗},ord}ξ(ω_{g}, ω_{h})⟩.

Theorem 3.14 therefore follows from Theorem 3.12. The case of general weights (k, ℓ, m) follows from an identical argument using Theorem 3.13 instead of Theorem 3.12. □

4. Garrett-Rankin triple product L-functions

4.1. Classical L-functions. As in the introduction, let

f ∈ S_{k}(N_{f}, χ_{f}), g ∈ S_{ℓ}(N_{g}, χ_{g}), h ∈ S_{m}(N_{h}, χ_{h})

be a triplet of normalized primitive cuspidal eigenforms such that χ_{f} · χ_{g} · χ_{h} = 1, and let

ℚ_{f,g,h} = ℚ_{f} · ℚ_{g} · ℚ_{h} = ℚ({a_{n}(f), a_{n}(g), a_{n}(h)}_{n≥1})

denote the field generated by the fourier coefficients of f, g and h. Write also N = lcm(N_{f}, N_{g}, N_{h}). The Garrett-Rankin triple product L-function L(f, g, h; s) is defined by an Euler product

L(f, g, h; s) = ∏_{p} L^{(p)}(f, g, h; p^{−s})^{−1},

where, for p ∤ N the local factor L^{(p)}(f, g, h; T) is the degree 8 polynomial

(104)  L^{(p)}(f, g, h; T) = (1 − α_{f,p}α_{g,p}α_{h,p}T) × (1 − α_{f,p}α_{g,p}β_{h,p}T)
×(1 − α_{f,p}β_{g,p}α_{h,p}T) × (1 − α_{f,p}β_{g,p}β_{h,p}T)
×(1 − β_{f,p}α_{g,p}α_{h,p}T) × (1 − β_{f,p}α_{g,p}β_{h,p}T)
×(1 − β_{f,p}β_{g,p}α_{h,p}T) × (1 − β_{f,p}β_{g,p}β_{h,p}T).

Piatetski-Shapiro and Rallis have given a precise recipe [PSR] for the local Euler factors L^{(p)}(f, g, h; s) at the bad primes p | N and have shown that there exists an archimedean factor L^{(∞)}(f, g, h; s) for which the completed L-function

Λ(f, g, h; s) = L(f, g, h; s) · L^{(∞)}(f, g, h; s)

satisfies the functional equation

(105)  Λ(f, g, h; s) = ε(f, g, h)Λ(f, g, h; k + ℓ + m − 2 − s),

where ε(f, g, h) ∈ {±1}. The sign in this functional equation determines the parity of the order of vanishing of L(f, g, h, s) at the center of symmetry

(106)  c = c_{f,g,h} = \frac{k + ℓ + m − 2}{2},

at which there is no pole (cf. [PSR, Theorem 5.2]). The sign ε(f, g, h) can be expressed as a product

ε(f, g, h) = ∏_{q} ε_{q}(f, g, h)

of local root numbers ε_{q}(f, g, h) ∈ {±1} indexed by the places q ≤ ∞ of ℚ. As already anticipated in the introduction, we assume throughout that ε_{q}(f, g, h) = +1 for all finite primes. According to e. g. [Pr90] and the references therein, we then have

(107)  ε(f, g, h) = ε_{∞}(f, g, h) = \begin{cases}−1 & if (k, ℓ, m) are balanced. \\ +1 & if (k, ℓ, m) are unbalanced. \end{cases}

We now recall a fundamental result which relates the central critical value of L(f, g, h, s) to certain trilinear period integrals. For this, assume that (k, ℓ, m) is unbalanced, namely (without loss of generality, by eventually permuting f, g and h), that k = ℓ + m + 2t with t ≥ 0. Recall also the Shimura-Maass operators δ_{ℓ} defined in §2.3 and the space S_{k}(N)[π_{f}] introduced in §2.5.

For any eigenform ϕ ∈ S_{k}(N) with character χ (not necessarily new at N), we write ϕ^{∗} for the dual form obtained by twisting ϕ by the character χ^{−1}. The eigenform ϕ^{∗} has nebentype character χ^{−1}, and its Hecke eigenvalues are complex conjugate to the Hecke eigenvalues attached to ϕ.

Definition 4.1. Assume that k = ℓ + m + 2t with t ≥ 0. The trilinear period attached to

(f̆, ğ, h̆) ∈ S_{k}(N)[π_{f}] × S_{ℓ}(N)[π_{g}] × S_{m}(N)[π_{h}]

is the expression

I(f̆, ğ, h̆) := (f̆^{∗}, δ_{ℓ}^{t}ğ × h̆)_{N}.

One of the main results of M. Harris and S. Kudla [HaKu], conjectured by H. Jacquet and refined recently by A. Ichino [I] and T. C. Watson [Wa] is:

Theorem 4.2. Let (f, g, h) be a triple of modular forms of unbalanced weights (k, ℓ, m) with k = ℓ + m + 2t and t ≥ 0. Then there exist

• holomorphic modular forms f̆ ∈ S_{k}(N)[π_{f}], ğ ∈ S_{ℓ}(N)[π_{g}], and h̆ ∈ S_{m}(N)[π_{h}],

• for each prime q | N∞, a constant C_{q} ∈ ℚ_{f,g,h} which depends only on the local components at q of the vectors f̆, ğ and h̆ in the admissible representations of GL_{2}(ℚ_{q}) associated to π_{f}, π_{g} and π_{h};

such that

(108)  \frac{∏_{q|N∞} C_{q}}{π^{2k}} · L(f, g, h, c) = |I(f̆, ğ, h̆)|^{2}.

Furthermore, there is a choice of (f̆, ğ, h̆) for which the constants C_{q} are all non-zero.

Remark 4.3. If N_{f} = N_{g} = N_{h} = N, then the space S_{k}(N)[π_{f}] is one-dimensional, and likewise for g and h. In that case there is only one choice (up to scaling) for the vectors f̆, ğ and h̆. Otherwise, the triplet (f̆, ğ, h̆) has to be chosen more carefully: see [GrPr] and [DiNy] for explicit recipes when at least one of π_{f}, π_{g} or π_{h} is not supercuspidal at the primes q | gcd(N_{f}, N_{g}, N_{h}). It could be tempting to choose f̆, ğ and h̆ to be the new vectors in S_{k}(N_{f}), S_{ℓ}(N_{g}) and S_{m}(N_{h}) attached to f, g and h respectively, but such a choice may not always be suitable. For example, if there is a prime q dividing N_{f} but not N_{g} or N_{h}, then C_{q} = 0 for this choice of test vectors.

4.2. p-adic L-functions. Let ℚ̄ denote the algebraic closure of ℚ in ℂ and fix throughout an odd prime p ∤ N, an algebraic closure ℚ̄_{p} of ℚ_{p} and an embedding ι_{p} : ℚ̄ ↪ ℚ̄_{p}. Let ord_{p} : ℚ̄_{p}^{×} → ℚ be the valuation on ℚ̄_{p} normalized so that ord_{p}(p) = 1.

Fix a triple (f, g, h) of newforms and keep the notations of the previous section. As before we assume that χ_{f}χ_{g}χ_{h} = 1, ε_{q}(f, g, h) = +1 for all finite primes q | N, and the triple (k, ℓ, m) of weights is unbalanced.

Let f̆ ∈ S_{k}(N; K_{f})[π_{f}], ğ ∈ S_{ℓ}(N; K_{g})[π_{g}] and h̆ ∈ S_{m}(N; K_{h})[π_{h}] be test vectors for which the local constants C_{q} of Theorem 4.2 are non-zero and the central critical value of L(π_{f}, π_{g}, π_{h}, s) is (up to elementary non-zero fudge factors) equal to the square of I(f̆, ğ, h̆).

Assume in addition that f, g and h are ordinary with respect to ι_{p} and let

f̆ = (Λ_{f}, Ω_{f}, Ω_{f,cl}, f̆), ğ = (Λ_{g}, Ω_{g}, Ω_{g,cl}, ğ), h̆ = (Λ_{h}, Ω_{h}, Ω_{h,cl}, h̆)

be the Hida families of (not necessarily new) forms on Γ_{1}(N) interpolating f̆, ğ and h̆ in weights k, ℓ and m respectively, as described in (58). Adapting the notations of the introduction, we write

Σ := {(x, y, z) ∈ Ω_{f,cl} × Ω_{g,cl} × Ω_{h,cl}},

and set

Σ_{f} = {(x, y, z) ∈ Σ, such that 2t := κ(x) − κ(y) − κ(z) ≥ 0},
Σ_{bal} = {(x, y, z) ∈ Σ, such that (κ(x), κ(y), κ(z)) is balanced.}.

For each (x, y, z) ∈ Σ_{f}, we may consider the algebraic number

J(f̆_{x}, ğ_{y}, h̆_{z}) := \frac{I(f̆_{x}, ğ_{y}, h̆_{z})}{(f_{x}^{∗}, f_{x}^{∗})_{N}} ∈ ℚ̄,

which we view in ℂ_{p} via ι_{p}. In light of the discussion in the previous section, our approach to defining the p-adic L-function attached to π_{f} ⊗ π_{g} ⊗ π_{h} will be to interpolate (suitable multiples of) the ratios J(f̆_{x}, ğ_{y}, h̆_{z}) as (x, y, z) ranges over Σ_{f}. In order to do this, we let ϕ := e_{ord}(d^{•}ğ^{[p]} × h̆) be the ordinary family of modular forms attached to ğ and h̆ following the recipe in Proposition 2.18.

Furthermore, define the Hida family f̆^{∗} = f̆ ⊗ χ_{f}^{−1}; note that (f̆^{∗})_{x} = (f̆_{x})^{∗} for all x ∈ Ω_{f,cl}. Recall that according to the notations settled in §2.6, a classical point corresponds to a character of the form γ ↦ γ^{k} for some k ∈ ℤ_{≥2}. We warn the reader that the equality (f̆^{∗})_{x} = (f̆_{x})^{∗} does not hold true for arithmetic points corresponding to characters γ ↦ ψ(γ)γ^{k} for some Dirichlet character ψ of non-trivial p-power conductor.

Definition 4.4. The Garrett-Rankin triple product p-adic L-function attached to the triple (f̆, ğ, h̆) of Λ-adic modular forms is the element

ℒ_{p} ^{f}(f̆, ğ, h̆) := J(f̆^{∗}, e_{ord}(d^{•}ğ^{[p]} × h̆)) ∈ Λ′_{f} ⊗_{Λ} (Λ_{g} ⊗ Λ_{h} ⊗ Λ)

attached to the families f̆ and ϕ = e_{ord}(d^{•}ğ^{[p]} × h̆) following the notations of Lemma 2.19.

Remark 4.5. The definition of ℒ_{p} ^{f}(f̆, ğ, h̆) depends not just on the p-adic families of automorphic representations interpolating π_{f}, π_{g}, and π_{h}, but also on the choice of a triple (f̆, ğ, h̆) of test vectors in these automorphic representations. Note also that while

ℒ_{p} ^{f}(f̆, ğ, h̆) = ℒ_{p} ^{f}(f̆, h̆, ğ),

the Λ-adic family f̆ plays an essentially different role from the other two in the definition of ℒ_{p} ^{f}(f̆, ğ, h̆). There are thus three a priori distinct p-adic L-functions attached to (f̆, ğ, h̆), namely, ℒ_{p} ^{f}(f̆, ğ, h̆), ℒ_{p} ^{g}(f̆, ğ, h̆) := ℒ_{p} ^{g}(ğ, f̆, h̆), and ℒ_{p} ^{h}(f̆, ğ, h̆).

Any element ℒ ∈ Λ′_{f} ⊗_{Λ} (Λ_{g} ⊗ Λ_{h} ⊗ Λ) as in Definition 4.4 has poles at (x, y, z) for only finitely many x ∈ Ω_{f}, and hence is completely determined by its values at the points of Ω_{f,cl}×_{Ω}(Ω_{g,cl}×Ω_{h,cl}×Ω) where it is defined. Furthermore, it is always defined at (x, y, z) when x belongs to Ω_{f,cl}. By definition, for all (x, y, z) ∈ Σ_{f}, after setting κ(x) = κ(y) + κ(z) + 2t,

(109)  ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = \frac{(f̆_{x}^{∗(p)}, e_{ord}(d^{t}ğ_{y}^{[p]} × h̆_{z}))_{N,p}}{(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}} .

In particular, the special value ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) is algebraic and belongs to the field K generated by α_{f_{x}} and by the fourier coefficients of f̆_{x}, ğ_{y} and h̆_{z}.

The following proposition supplements (109) with a formula for the p-adic special value ℒ_{p} ^{f}(f̆, ğ, h̆) at a point (x, y, z) ∈ Σ_{bal}.

Proposition 4.6. For all (x, y, z) ∈ Σ_{bal}, let (f, g, h) := (f_{x}, g_{y}, h_{z}) and define (k, ℓ, m, t) by setting

(k, ℓ, m) = (κ(x), κ(y), κ(z)), k = ℓ + m − 2 − 2t.

Then

ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = E_{0}(f)^{−1}⟨ηu-r ; f̆_{x}, e_{ord}(d^{−1−t}ğ_{y}^{[p]} × h̆_{z})⟩,

where • E_{0}(f) is the local factor given in (7);

• ηu-r ; f̆_{x} ∈ H_{par}^{1}(X_{ℂ_{p}}, L_{r})^{u-r} is the unique lift to the unit root subspace of the cohomology

class in H^{1}(X_{K_{fx}} , \underline{ω}^{−r}) attached to f̆_{x}^{∗} as in Corollary 2.13.

• the classical modular form e_{ord}(d^{t}ğ_{y}^{[p]} × h̆_{z}) on Γ_{1}(N) ∩ Γ_{0}(p) is viewed as a class in H^{0}(W_{ϵ}, L_{r} ⊗ Ω_{X}^{1}) ⊂ H_{par}^{1}(X_{ℂ_{p}}, L_{r});

• ⟨ , ⟩_{X} denotes the Poincaré duality on H_{par}^{1}(X_{ℂ_{p}}, L_{r}).

Proof. Let e_{f_{x}^{∗}} denote the idempotent attached to the eigenform f_{x}^{∗}, set

ϕ_{xyz}^{(p)} := e_{f_{x}^{∗},ord}(d^{t}g_{y}^{[p]} × h_{z}),

and let ϕ_{xyz} ∈ S_{k}(N; ℂ_{p})[π_{f_{x}^{∗}}] be the classical modular form whose p-ordinary stabilisation is equal to ϕ_{xyz}^{(p)}. The definition of ℒ_{p} ^{f}(f̆, ğ, h̆), in view of the last equality in equation (61) of Lemma 2.19, implies that, for all (x, y, z) ∈ Σ_{f} ∪ Σ_{bal},

(110)  ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = ⟨ηf̆_{x}, ϕ_{xyz}⟩.

Since ϕ_{xyz} belongs to Fil^{r+1}H_{par}^{1}(X_{ℂ_{p}}, L_{r}), we can also express this identity in terms of the Poincaré pairing on H_{par}^{1}(X_{ℂ_{p}}, L_{r}) by replacing ηf̆_{x} by its lift to H_{par}^{1}(X_{ℂ_{p}}, L_{r}) under the unit root splitting:

(111)  ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = ⟨ηu-r ; f̆_{x}, ϕ_{xyz}⟩.

By Proposition 2.11, we then have

(112)  ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = ⟨ηu-r ; f̆_{x}, e_{ord}ϕ_{xyz}⟩.

On the other hand, applying the ordinary projector to the identity ϕ_{xyz}^{(p)} = ϕ_{xyz} − β_{f_{x}^{∗}}V ϕ_{xyz} shows that

(113)  ϕ_{xyz}^{(p)} = (1 − \frac{β_{f_{x}^{∗}}}{α_{f_{x}^{∗}}})e_{ord}ϕ_{xyz} = (1 − \frac{β_{f_{x}}}{α_{f_{x}}})e_{ord}ϕ_{xyz} = E_{0}(f_{x})e_{ord}ϕ_{xyz},

and therefore

ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = E_{0}(f_{x})^{−1}⟨ηu-r ; f̆_{x}, ϕ_{xyz}^{(p)}⟩ = E_{0}(f_{x})^{−1}⟨ηu-r ; f̆_{x}, e_{f_{x}^{∗},ord}(d^{t}ğ_{y}^{[p]} × h̆_{z})⟩
= E_{0}(f_{x})^{−1}⟨ηu-r ; f̆_{x}, e_{ord}(d^{t}ğ_{y}^{[p]} × h̆_{z})⟩.

The proposition follows. □

The next theorem evaluates ℒ_{p} ^{f}(f̆, ğ, h̆) at points (x, y, z) ∈ Σ_{f} and relates these values to the complex periods I(f̆_{x}, ğ_{y}, h̆_{z}).

Theorem 4.7. Let (x, y, z) be a point of Σ_{f} and set

(f, g, h) := (f_{x}, g_{y}, h_{z}),  (f̆, ğ, h̆) := (f̆_{x}, ğ_{y}, h̆_{z}),  (k, ℓ, m) = (κ(x), κ(y), κ(z)).

Then

(114)  ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = \frac{E(f, g, h)}{E_{0}(f)E_{1}(f)} × \frac{I(f̆, ğ, h̆)}{(f^{∗}, f^{∗})_{N}} ,

where E(f, g, h), E_{0}(f) and E_{1}(f) are given in equations (6), (7) and (8) of Theorem 1.3.

Remark 4.8. Theorems 4.2 and 4.7 above justify the designation of ℒ_{p} ^{f}(f̆, ğ, h̆) as a p-adic Garrett-Rankin triple product L-function in that it interpolates the square-roots of the central critical values of the complex L-series L(f_{x}, g_{y}, h_{z}; s) for (x, y, z) ∈ Σ_{f}. Indeed, assume that

L(f, g, h; c) ≠ 0,

so that Theorems 4.2 and 4.7 ensure that ℒ_{p} ^{f}(f̆, ğ, h̆) is non-zero in L_{f} ⊗ Λ_{g} ⊗ Λ_{h}. By the Weierstrass preparation theorem, ℒ_{p} ^{f}(f̆, ğ, h̆) vanishes at a finite number of points and therefore ⟨f̆_{x}, ğ_{y} · δ^{n}(h̆_{z})⟩_{N}^{2} ≠ 0 for all but finitely many (x, y, z) ∈ Σ_{f}. In addition, Jacquet’s conjecture as formulated and proved in [HaKu], imply that the hypothesis ε_{q}(f_{x}, g_{y}, h_{z}) = +1 for all finite primes q still holds true for all such (x, y, z). Hence Theorem 4.2 can again be applied for the triple (f_{x}, g_{y}, h_{z}), which implies that (114) expresses ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) as a non-zero multiple of the square-root of L(f_{x}, g_{x}, h_{x}; c_{f_{x},g_{y},h_{z}}). □

We now turn to the proof of Theorem 4.7. As before, p is a prime which does not divide N, and S_{k}^{n-oc}(N; ℂ_{p}) denotes the space of nearly overconvergent p-adic modular forms of level N, as described in Section 2.4. Given a classical newform f = ∑_{n} a_{n}q^{n} of some level N_{f}|N, recall that

e_{f^{∗}} : S_{k}^{ord}(N; ℂ_{p}) ⟶ S_{k}^{ord}(N; ℂ_{p})[π_{f^{∗}}]
e_{f^{∗},ord} = e_{f^{∗}}e_{ord} : S_{k}^{n-oc}(N; ℂ_{p}) ⟶ S_{k}^{ord}(N; ℂ_{p})[π_{f^{∗}}]

denote the associated projectors to the f^{∗}-isotypic component of the ordinary subspace in the space of overconvergent p-adic modular forms. The operators U and V whose action on q-expansions is given in (13) also act on the spaces of nearly overconvergent modular forms of a given level.

Lemma 4.9. Let g ∈ S_{ℓ}^{n-oc}(N; ℂ) and h ∈ S_{m}^{n-oc}(N; ℂ) be nearly overconvergent modular forms of weights ℓ, m ≥ 2 which are eigenvectors for all the good Hecke operators T_{q} with q ∤ N. (They are not assumed to be new or to be eigenvectors for the operators T_{q} with q|N.) Let

ϕ_{fgh} := e_{f^{∗},ord}(gh) ∈ S_{k}^{ord}(N; ℂ)[π_{f^{∗}}], (with k = ℓ + m).

For each r ≥ 0, the modular form

e_{f^{∗},ord}((V^{r}g) × h) ∈ S_{k}^{ord}(N; ℂ)[π_{f^{∗}}]

is a multiple of ϕ_{fgh}.

Proof. By Lemma 2.17,

(115) 0 = e_{ord}(g^{[p]} × V h) = e_{ord}(g × V h) − a_{p}(g)e_{ord}(V g × V h) + χ_{g}(p)p^{ℓ−1}e_{ord}(V^{2}g × V h).

But since

e_{ord}V = U^{−1}e_{ord}, e_{f^{∗},ord}V = α_{f^{∗}}^{−1}e_{f^{∗},ord},

it follows after applying the projector e_{f^{∗}} to (115) that

(116)  e_{f^{∗},ord}(g × V h) + χ_{g}(p)p^{ℓ−1}α_{f^{∗}}^{−1}e_{f^{∗},ord}(V g × h) = a_{p}(g)α_{f^{∗}}^{−1}ϕ_{fgh}.

The same argument with the roles of g and h interchanged shows that

(117)  χ_{h}(p)p^{m−1}α_{f^{∗}}^{−1}e_{f^{∗},ord}(g × V h) + e_{f^{∗},ord}(V g × h) = a_{p}(h)α_{f^{∗}}^{−1}ϕ_{fgh}.

The case r = 1 of the proposition is obtained by simultaneously solving equations (116) and (117) above. The case of general r > 1 can then be treated by induction on r using the identity

(118) e_{f^{∗},ord}(V^{r}g × h) = α_{f^{∗}}^{−1}a_{p}(g)e_{f^{∗},ord}(V^{r−1}g × h) − α_{f^{∗}}^{−2}χ_{h}(p)p^{m−1}e_{f^{∗},ord}(V^{r−1}g × h),

which follows from Lemma 2.17, more precisely from the fact that e_{ord}(V^{r}g × h^{[p]}) = 0. □

Lemma 4.10. The following identities, as well as their counterparts with (α_{g}, α_{h}) replaced by (α_{g}, β_{h}), (β_{g}, α_{h}, and (β_{g}, β_{h}), are satisfied:

(1) e_{f^{∗},ord}((V g_{α}) × (V h_{α})) = α_{f^{∗}}^{−1}e_{f^{∗},ord}(g_{α}h_{α});

(2) e_{f^{∗},ord}(g_{α} × V h_{α}) = α_{g}α_{f^{∗}}^{−1}e_{f^{∗},ord}(g_{α}h_{α});

(3) e_{f^{∗},ord}(V g_{α} × h_{α}) = α_{h}α_{f^{∗}}^{−1}e_{f^{∗},ord}(g_{α}h_{α});

(4) e_{f^{∗},ord}(g^{[p]} × h) = e_{f^{∗},ord}(g × h^{[p]}) = (1 − α_{g}α_{h}α_{f^{∗}}^{−1})e_{f^{∗},ord}(g_{α}h_{α}).

Proof. To prove the first assertion, note that

e_{ord}((V g_{α}) × (V h_{α})) = e_{ord} ∘ V (g_{α}h_{α}),

and that

e_{ord} ∘ V = lim U_{p}^{n!} ∘ V = lim U_{p}^{n!−1} = U_{p}^{−1}e_{ord}.

The first assertion now follows from the fact that U_{p} operates via multiplication by α_{f^{∗}} on S_{k}^{ord}(N; ℂ_{p})[π_{f}^{∗}]. For the second assertion, we invoke Lemma 2.17 to assert that

(g^{[p]} × (V h_{α})) = (g − a_{p}(g)V g) × (V h_{α})

is in the kernel of the ordinary projection to conclude that

e_{ord}(g × V h_{α}) = a_{p}(g)e_{f^{∗},ord}((V g) × (V h)),

and the second assertion now follows directly from the first. The third assertion follows from the identical argument with the roles of g and h interchanged. The last identity then follows from the previous ones by a direct calculation using the fact that g^{[p]} = g_{α} − α_{g}V g_{α}. □

Proposition 4.11. Let f ∈ S_{k}(N) be a holomorphic form of weight k and let g ∈ S_{ℓ}^{n-oc}(N) and h ∈ S_{m}^{n-oc}(N) be nearly holomorphic modular forms, with k = ℓ + m. Assume that f, g and h are eigenvectors for the good Hecke operators of level N (but are not necessarily new of this level.) Then for all primes p ∤ N,

(119)  e_{f^{∗},ord}(g^{[p]}h) = \frac{E(f, g, h)}{E_{1}(f)}e_{f^{∗},ord}(gh),

where E(f, g, h) is as in equation (6) of Theorem 1.3 (with c = k − 1).

Proof. The last identity of Lemma 4.10 shows, after setting ϕ_{fg^{[p]}h} := e_{f^{∗},ord}(g^{p]}h), that

(120)  e_{f^{∗},ord}(g_{α}h_{α}) = (1 − α_{g}α_{h}α_{f^{∗}}^{−1})^{−1}ϕ_{fg^{[p]}h};
(121)  e_{f^{∗},ord}(g_{α}h_{β}) = (1 − α_{g}β_{h}α_{f^{∗}}^{−1})^{−1}ϕ_{fg^{[p]}h};
(122)  e_{f^{∗},ord}(g_{β}h_{α}) = (1 − β_{g}α_{h}α_{f^{∗}}^{−1})^{−1}ϕ_{fg^{[p]}h};
(123)  e_{f^{∗},ord}(g_{β}h_{β}) = (1 − β_{g}β_{h}α_{f^{∗}}^{−1})^{−1}ϕ_{fg^{[p]}h}.

Combining (120) and (121) with the identity

h = (α_{h} − β_{h})^{−1}(α_{h}h_{α} − β_{h}h_{β}),

and similarly with (122) and (123), we obtain

(124)  e_{f^{∗},ord}(g_{α}h) = ((1 − α_{g}α_{h}α_{f^{∗}}^{−1})(1 − α_{g}β_{h}α_{f^{∗}}^{−1}))^{−1}ϕ_{fg^{[p]}h};
(125)  e_{f^{∗},ord}(g_{β}h) = ((1 − β_{g}α_{h}α_{f}^{−1})(1 − β_{g}β_{h}α_{f^{∗}}^{−1}))^{−1}ϕ_{fg^{[p]}h}.

The result now follows from these two equations by a direct calculation using the fact that

g = (α_{g} − β_{g})^{−1}(α_{g}g_{α} − β_{g}g_{β}).

□ Remark 4.12. A formula similar to Proposition 4.11 can be found in Proposition 2.2.2. of [HaTi]. The approach followed there is conceptually sound but more complicated than the one followed above; possibly for this reason, an error seems to have crept into the roughly five pages of elementary but tedious calculations in the proof of Prop. 2.2.2. of loc.cit, and the factor denoted there by E_{P} (f, g, h) does not agree with the factor E(f, g, h) of Proposition 4.11. We note that the factor E(f, g, h) occuring in Proposition 4.11 is consistent with the conjectural recipe for the “correction term at p” arising in the theory of p-adic L-functions, as described in e. g. in [Pan, p. 285].

Corollary 4.13. Let f ∈ S_{k}(N), g ∈ S_{ℓ}(N) and h ∈ S_{m}(N) be holomorphic forms, and assume that k = ℓ + m + 2t with t ∈ ℤ^{≥0}. Assume that f, g and h are eigenvectors for the good Hecke operators in level N (but are not necessarily new in this level.) Then for all primes p ∤ N,

e_{f^{∗},ord}(d^{t}g^{[p]} × h) = \frac{E(f, g, h)}{E_{1}(f)}e_{f^{∗},ord}(d^{t}g × h).

Proof. This follows from Proposition 4.11 with (f, g, h) replaced by

(f, d^{t}g, h) ∈ S_{k}(N) × S_{ℓ+2t}^{n-oc}(N) × S_{m}(N),

in light of the fact that d^{t}(g^{[p]}) = (d^{t}g)^{[p]}, α_{d^{t}g} = p^{t}α_{g} and β_{d^{t}g} = p^{t}β_{g}. □

Proof of Theorem 4.7. By (109), the value of ℒ_{p} ^{f}(f̆, ğ, h̆) at (x, y, z) ∈ Σ_{f} is

(126)  ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = \frac{(f̆_{x}^{∗(p)}, e_{f_{x}^{∗},ord}(d^{t}ğ_{y}^{[p]} × h̆_{z}))_{N,p}}{(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}}

(127)  = \frac{E(f_{x}, g_{y}, h_{z})}{E_{1}(f_{x})} \frac{(f̆_{x}^{∗(p)}, e_{f_{x}^{∗},ord}(d^{t}ğ_{y} × h̆_{z}))_{N,p}}{(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}} ,

where the second equality follows from Corollary 4.13. But then by Proposition 2.8,

(128)  \frac{(f̆_{x}^{∗(p)}, e_{f_{x}^{∗},ord}(d^{t}ğ_{y} × h̆_{z}))_{N,p}}{(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}} = \frac{(f̆_{x}^{∗(p)}, e_{f_{x}^{∗},ord}Π_{N}^{hol}(δ^{t}ğ_{y} × h̆_{z}))_{N,p}}{(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}} .

On the other hand, by equation (113),

\frac{(f̆_{x}^{∗(p)}, e_{f_{x}^{∗},ord}Π_{N}^{hol}(δ_{ℓ}^{t}ğ_{y} × h̆_{z}))_{N,p}}{(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}} = \frac{(f̆_{x}^{∗(p)}, (e_{f_{x}^{∗}}Π_{N}^{hol}(δ_{ℓ}^{t}ğ_{y} × h̆_{z}))^{(p)})_{N,p}}{E_{0}(f_{x})(f_{x}^{∗(p)}, f_{x}^{∗(p)})_{N,p}}

(129)  = \frac{(f̆_{x}^{∗}, e_{f_{x}^{∗}}Π_{N}^{hol}(δ_{ℓ}^{t}ğ_{y} × h̆_{z}))_{N}}{E_{0}(f_{x})(f_{x}^{∗}, f_{x}^{∗})_{N}}

(130)  = \frac{(f̆_{x}^{∗}, δ_{ℓ}^{t}ğ_{y} × h̆_{z})_{N}}{E_{0}(f_{x})(f_{x}^{∗}, f_{x}^{∗})_{N}}
(131)  = \frac{I(f̆_{x}, ğ_{y}, h̆_{z})}{E_{0}(f_{x})(f_{x}^{∗}, f_{x}^{∗})_{N}} .

Theorem 4.7 follows.

5. The p-adic Gross-Zagier formula

Recall the class ηf̆ ∈ H^{1}(X_{K_{f}} , \underline{ω}^{−r}) attached to the test vector f̆ ∈ S_{k}(N; K_{f})[π_{f}] following the discussion after (55). Thanks to the chosen embedding of K_{f} into ℂ_{p}, the class ηf̆ can be viewed as belonging to H^{1}(X_{ℂ_{p}}, \underline{ω}^{−r})[π_{f}]. Let

ηu-r ; f̆ ∈ H_{dR}^{1}(X_{ℂ_{p}}, L_{r})

denote its unique lift to the unit root subspace. We can now state and prove the p-adic Gross-Zagier formula of this paper, relating the image of the generalised diagonal cycles under the p-adic Abel-Jacobi map to the value of the Garrett-Rankin triple product p-adic L-function at points corresponding to balanced triples of classical modular forms.

Theorem 5.1. Given (x, y, z) ∈ Σ_{bal}, set

(f, g, h) = (f_{x}, g_{y}, h_{z}),  (f̆, ğ, h̆) = (f̆_{x}, ğ_{y}, h̆_{z}),  (k, ℓ, m) = (κ(x), κ(y), κ(z)),

and let Δ := Δ_{k,ℓ,m} ⊂ W be the generalised diagonal cycle attached to the weights (k, ℓ, m). Then after writing k = ℓ + m − 2 − 2t with t ≥ 0,

ℒ_{p} ^{f}(f̆, ğ, h̆)(x, y, z) = (−1)^{t} \frac{E(f, g, h)}{t! · E_{0}(f)E_{1}(f)} × AJ_{p}(Δ)(ηu-r ; f̆ ⊗ ωğ ⊗ ωh̆).

Proof. This follows directly from Theorem 3.14 and Proposition 4.6. □

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H. D.: McGill University, Montreal, Canada

   E-mail address: darmon@math.mcgill.ca

V. R.: Universitat Politècnica de Catalunya, Barcelona, Spain

   E-mail address: victor.rotger@upc.edu