KATO’S EULER SYSTEM AND RATIONAL POINTS ON ELLIPTIC CURVES I: A p-ADIC BEILINSON FORMULA

MASSIMO BERTOLINI AND HENRI DARMON

Abstract. This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.

Contents

1. Introduction 1 2. Complex L-series 2 2.1. Eisenstein series 3 2.2. Rankin’s method 4 2.3. A factorisation of critical values 6 2.4. Beilinson’s formula 7 2.5. Algebraicity 8 3. p-adic L-functions 9 3.1. Rankin’s p-adic L-functions 9 3.2. Factorisation of p-adic L-functions 10 4. p-adic regulators 11 4.1. The regulator on K_{2} 12 4.2. The general case 14 5. The p-adic Beilinson formula 15 5.1. The main results 15 5.2. Relation with the work of Brunault and Gealy 15 References 17

1. Introduction

This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. In a simple but prototypical setting, Kato’s construction [Kato], [Colz2] yields a global class κ in H^{1}(ℚ, V_{p}(E)), where V_{p}(E) is the p-adic Galois representation attached to a modular elliptic curve E/ℚ. Kato’s reciprocity law implies that κ is crystalline, and hence belongs to the p-adic Selmer group of E, precisely when the Hasse-Weil L-series L(E, s) vanishes at s = 1. In this case, Perrin-Riou [PR1] conjectures that the image res_{p}(κ) in H_{f}^{1}(ℚ_{p}, V_{p}(E)) is non-zero if and only if L′(E, 1) is non-zero, and predicts a precise relation between the logarithm of res_{p}(κ) and the formal group logarithm of a global point in E(ℚ).

The ultimate goal of this series is the proof of Perrin-Riou’s conjecture, which will be described in [BD3]. One of the cornerstones of our strategy is a proof of a p-adic Beilinson formula relating the syntomic regulators (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to a cusp form f of weight 2. This proof is independent of Kato’s reciprocity law, and will in fact be used in [BD2] to re-derive it. It is based instead on the direct evaluation (Theorems 4.4 and 5.1) of the p-adic Rankin L-function attached to a Hida family interpolating f introduced in Section 3.1. The p-adic Beilinson formula then follows from the factorisation of this p-adic Rankin L-function into a product of two Mazur-Kitagawa p-adic L-functions (Theorem 3.4).

In the complex setting, the connection between regulators and values of L(E, s) at integers ℓ ≥ 2 was described in the work of Beilinson [Bei], and in prior work of Bloch [Bl] for elliptic curves with complex multiplication. The first p-adic avatar of this formula was obtained by Coleman and de Shalit [CodS] in the CM setting considered by Bloch. The work of Brunault [Br2] for ℓ = 2 and Gealy [Ge] for ℓ ≥ 2 extended this p-adic Beilinson formula to all (modular) elliptic curves as a consequence of Kato’s general machinery. Our approach, which is somewhat more direct, relies instead on Besser’s description ([Bes1] and [Bes2]) of the Coleman-de Shalit p-adic regulator and on the techniques developed in [DR] for relating p-adic Abel-Jacobi images of diagonal cycles to values of Garrett-Rankin triple product p-adic L-functions. The results of [DR]—and, by extension, of the present work as well as of [BDR]— were inspired by the study undertaken earlier in [BDP], which the reader may consult for an analogous formula in the setting of Heegner points (resp. “generalised Heegner cycles”) on modular curves (resp. on products of Kuga-Sato varieties with powers of CM elliptic curves).

After this paper was submitted, the authors’ attention was drawn to the earlier work of Maximilian Niklas [Nik] which also provides a direct proof of the p-adic Beilinson formula based on a description of the rigid syntomic regulator given in [BK]. The principal novelty of the present work—and, arguably, its main interest—lies in the explicit connection that it draws with

(1) the results of [DR] relating the p-adic Abel-Jacobi images of diagonal cycles on triple products of Kuga-Sato varieties to special values of Harris-Tilouine’s p-adic L-functions attached to the Garrett-Rankin convolution of three Hida families of cusp forms;

(2) the results in [BDR] relating the syntomic regulators of Beilinson-Flach elements in higher Chow groups of products of two modular curves to special values of Hida’s p-adic L-functions attached to the Rankin-Selberg convolution of two cusp forms;

(3) the results in [BDP] relating the p-adic logarithms of Heegner points on modular curves to special values of the p-adic L-functions attached to the Rankin convolution of a weight two cusp form and a theta series of an imaginary quadratic field, based on a formula of Waldspurger.

The authors’ strategy for proving Perrin-Riou’s conjecture is based on a comparison between Kato’s Euler system and those arising in the above settings.

Acknowledgements: The authors thank Victor Rotger for numerous exchanges related to this article, and François Brunault and Pierre Colmez for helpful advice on improving its presentation. They are also grateful to the anonymous referee for a number of suggestions which helped them to clarify the exposition, and for drawing their attention to the related work of M. Niklas [Nik].

2. Complex L-series

A p-ADIC BEILINSON FORMULA 3

This section provides explicit formulae (see equations (20) and (23)) for the special values of the complex Rankin L-functions associated to the convolution of cusp forms and Eisenstein series, based on Rankin’s method and the reducibility of the Galois representations of Eisenstein series. These formulae are crucial in the definition and study of the Rankin p-adic L-functions of Section 3. Along the way, we briefly recall the application of Rankin’s method to the proof of the original complex Beilinson formula (cf. Proposition 2.3).

The Poincaré upper half plane of complex numbers with strictly positive imaginary part is denoted H, and the variable on H is written as z = x + iy with x ∈ ℝ and y ∈ ℝ_{>0}.

A Dirichlet character of modulus N is a homomorphism ψ : (ℤ/Nℤ)^{×} ⟶ ℂ^{×}, extended to a function on ℤ by the convention that ψ(n) = 0 whenever gcd(n, N) ≠ 1. The conductor of ψ is the smallest positive integer M for which there is a Dirichlet character of modulus M agreeing with ψ on the integers that are relatively prime to N. A Dirichlet character is said to be primitive if its conductor is equal to its modulus. If N = N_{1}N_{2} is a factorisation of N into co-prime positive integers N_{1} and N_{2}, frequent use will be made of the fact (following from the Chinese remainder theorem) that a character ψ of modulus (resp. conductor) N can be uniquely expressed as ψ = ψ_{1}ψ_{2}, where ψ_{j} is of modulus (resp. of conductor) N_{j}.

We denote by S_{k}(N, χ) ⊂ M_{k}(N, χ) the spaces of holomorphic cusp forms and modular forms of weight k, level N and character χ, and by S_{k}^{an}(N, χ) ⊂ M_{k}^{an}(N, χ) their real analytic counterparts consisting of real analytic functions on H with the same transformation properties under Γ_{0}(N), and having bounded growth (resp. rapid decay) at the cusps for elements of M_{k}^{an}(N, χ) (resp. S_{k}^{an}(N, χ)). Likewise, for any congruence subgroup Γ of SL_{2}(ℤ), the spaces S_{k}(Γ) ⊂ M_{k}(Γ) and S_{k}^{an}(Γ) ⊂ M_{k}^{an}(Γ) are given their obvious meanings. The only cases arising in this article are where Γ is one of the standard Hecke congruence groups Γ_{0}(N) or Γ_{1}(N).

2.1. Eisenstein series. The non-holomorphic Eisenstein series of weight k and level N attached to the primitive character χ : (ℤ/Nℤ)^{×} ⟶ ℂ^{×} is the function on H × ℂ defined by

(1)  Ẽ_{k,χ}(z, s) = ∑_{(m,n)∈Nℤ×ℤ}^{′} \frac{χ^{−1}(n)}{(mz + n)^{k}} · \frac{y^{s}}{|mz + n|^{2s}} ,

where the superscript ′ indicates that the sum is taken over the non-zero lattice vectors (m, n) ∈ Nℤ × ℤ.

The series defining Ẽ_{k,χ}(z, s) converges for ℜ(s) > 1 − k/2 but admits a meromorphic continuation to all s ∈ ℂ. A direct calculation shows that

Ẽ_{k,χ} (\frac{az + b}{cz + d}, s) = χ(d)(cz + d)^{k}Ẽ_{k,χ}(z, s), for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),

i.e., Ẽ_{k,χ}(z, s) transforms like a modular form of weight k and character χ on Γ_{0}(N) when viewed as a function of z, and hence belongs to M_{k}^{an}(N, χ). In particular, if k > 2

Ẽ_{k,χ}(z) := Ẽ_{k,χ}(z, 0) belongs to M_{k}(N, χ).

(The same conclusion holds for k ≥ 1, provided that χ is non-trivial.) Assume from now on that χ satisfies the parity condition

(2)  χ(−1) = (−1)^{k},

which guarantees that Ẽ_{k,χ}(z) is non-zero. We introduce (cf. [Hi93], p. 128) the normalised Eisenstein series E_{k,χ}(z), related to Ẽ_{k,χ}(z) by the equation

(3)  Ẽ_{k,χ}(z) = 2N^{−k}τ(χ^{−1})\frac{(−2πi)^{k}}{(k − 1)!} E_{k,χ}(z),

where

τ(χ) = ∑_{a=1}^{N_{χ}} χ(a)e^{2πia/N_{χ}}, N_{χ} = cond(χ)

is the Gauss sum attached to χ, and the q-expansion of E_{k,χ}(z) is given by

(4)  E_{k,χ}(z) = 2^{−1}L(χ, 1 − k) + ∑_{n=1}^{∞} σ_{k−1,χ}(n)q^{n},  σ_{k−1,χ}(n) = ∑_{d|n} χ(d)d^{k−1}.

The Shimura-Maass derivative operator

δ_{k} := \frac{1}{2πi} (\frac{d}{dz} + \frac{ik}{2y} )

sends M_{k}^{an}(N, χ) to M_{k+2}^{an}(N, χ). A direct calculation (see also loc.cit., page 317, formula (13)) reveals that

(5)  δ_{k}Ẽ_{k,χ}(z, s) = −\frac{(s + k)}{4π}Ẽ_{k+2,χ}(z, s − 1).

Denoting by δ_{k}^{t} := δ_{k+2t−2} ⋯ δ_{k+2}δ_{k} the t-fold iterate of the Shimura-Maass derivative, it follows that

δ_{k}^{t}Ẽ_{k,χ}(z, s) = \frac{(−1)^{t}}{(4π)^{t}} (s + k) ⋯ (s + k + t − 1)Ẽ_{k+2t,χ}(z, s − t).

Replacing k by k − 2t and setting s = 0 in the above equation, we find

(6)  Ẽ_{k,χ}(z, −t) = \frac{(k − 2t − 1)!}{(k − t − 1)!}(−4π)^{t}δ_{k−2t}^{t}Ẽ_{k−2t,χ}(z).

In particular, if 0 ≤ t ≤ k/2−1, then the Eisenstein series Ẽ_{k,χ}(z, −t), while not holomorphic, is an example of a nearly holomorphic modular form in the sense of [Sh2].

We will also have a need for the more general Eisenstein series E_{k}(χ_{1}, χ_{2}) ∈ M_{k}(N, χ_{1}χ_{2}), attached to a pair χ_{1} and χ_{2} of Dirichlet characters of modulus N_{1} and N_{2}, respectively, with N_{1}N_{2} = N. Recall that condition (2) is in force, i.e., χ_{1}χ_{2}(−1) = (−1)^{k}. Then, for k ≥ 1 and (χ_{1}, χ_{2}) ≠ (1, 1), the q-expansion of E_{k}(χ_{1}, χ_{2}) is given by

(7)  E_{k}(χ_{1}, χ_{2})(z) = δ_{χ_{1}} L(χ−1_{1}χ_{2}, 1 − k) + ∑_{n=1}^{∞} σ_{k−1}(χ_{1}, χ_{2})(n)q^{n},

where δ_{χ_{1}} = 1/2 if N_{1} = 1 and 0 otherwise, and

σ_{k−1}(χ_{1}, χ_{2})(n) = ∑_{d|n} χ_{1}(n/d)χ_{2}(d)d^{k−1}.

Thus, E_{k}(1, χ) is equal to E_{k,χ}. Note that E_{k}(χ_{1}, χ_{2}) is a simultaneous eigenvector for all the Hecke operators, and satisfies

(8)  L(E_{k}(χ_{1}, χ_{2}), s) = L(χ_{1}, s)L(χ_{2}, s − k + 1).

2.2. Rankin’s method. Let

f := ∑_{n=1}^{∞} a_{n}(f)q^{n} ∈ S_{k}(N, χ_{f})

be a cusp form of weight k, level N and character χ_{f}, and let

g := ∑_{n=0}^{∞} a_{n}(g)q^{n} ∈ M_{ℓ}(N, χ_{g})

be a modular form of weight ℓ < k and character χ_{g}. We do not assume for now that f or g are eigenforms. Let

D(f, g, s) := ∑ a_{n}(f)a_{n}(g)n^{−s}

denote the Rankin L-series attached to f and g. Recall the Petersson scalar product defined on S_{k}^{an}(N, χ) × M_{k}^{an}(N, χ). It is given by the formula

(9)  ⟨f_{1}, f_{2}⟩_{k,N} := ∫_{Γ_{0}(N)\H} y^{k}\overline{f_{1}(z)}f_{2}(z)\frac{dxdy}{y^{2}} ,

and is hermitian linear in the first argument and ℂ-linear in the second. Let χ := χ_{f}^{−1}χ_{g}^{−1}, and denote by f^{∗} ∈ S_{k}(N, χ_{f}^{−1}) the modular form satisfying a_{n}(f^{∗}) = \overline{a_{n}}. Since the forms f^{∗}(z) and Ẽ_{k−ℓ,χ}(z, s)g(z) belong to S_{k}(N, χ_{f}^{−1}) and M_{k}^{an}(N, χ_{f}^{−1}) respectively, it is natural to consider their Petersson scalar product.

Proposition 2.1 (Shimura). For all s ∈ ℂ with ℜ(s) >> 0,

⟨f^{∗}(z) , Ẽ_{k−ℓ,χ}(z, s) · g(z)⟩_{k,N} = 2\frac{Γ(s + k − 1)}{(4π)^{s+k−1}}L(χ^{−1}, 2s + k − ℓ) × D(f, g, s + k − 1).

Proof. See formula (2.4) of [Sh1], where this result is proved by an application of Rankin’s method. See also [Hi93], page 317, formula (1), for a statement in the form given here. □

Replacing s by s − k + 1 in Proposition 2.1 and rearranging the factors, we obtain

(10) L(χ^{−1}, 2s − k − ℓ + 2)D(f, g, s) = \frac{1}{2} \frac{(4π)^{s}}{Γ(s)} ⟨f^{∗}(z), Ẽ_{k−ℓ,χ}(z, s − k + 1) · g(z)⟩_{k,N} .

Assume now that the modular forms f and g are normalised eigenforms of level N. We do not assume that they are new of this level, but we do assume that they are simultaneous eigenvectors for the Hecke operators T_{r} with gcd(r, N) = 1 as well as the operators U_{r} attached to the primes r dividing N. For each prime p, let α_{p}(f) and β_{p}(f) be the roots of the Hecke polynomial x^{2} −a_{p}(f)x+χ_{f}(p)p^{k−1}, choosing (α_{p}(f), β_{p}(f)) = (a_{p}(f), 0) when p|N. Similarly, we let α_{p}(g) and β_{p}(g) denote the roots of the polynomial x^{2} − a_{p}(g)x + χ_{g}(p)p^{ℓ−1}.

Then the coefficients of the L-series D(f, g, s) are weakly multiplicative and hence D(f, g, s) has an Euler product factorisation over the rational primes p:

(11)  D(f, g, s) = ∏_{p} D_{(p)}(f, g, s),  where D_{(p)}(f, g, s) = ∑_{n=0}^{∞} a_{p^{n}}(f)a_{p^{n}}(g)p^{−s}.

Let

L(f ⊗ g, s) := ∏_{p} L_{(p)}(f ⊗ g, s),  where
L_{(p)}(f ⊗ g, s) := (1 − α_{p}(f)α_{p}(g)p^{−s})^{−1}(1 − α_{p}(f)β_{p}(g)p^{−s})^{−1}
×(1 − β_{p}(f)α_{p}(g)p^{−s})^{−1}(1 − β_{p}(f)β_{p}(g)p^{−s})^{−1}.

The calculation of the Euler factors D_{(p)}(f, g, s) – a mildly tedious exercise in manipulation and rearranging of infinite series – shows that, for all primes p,

(12)  D_{(p)}(f, g, s) = (1 − χ^{−1}(p)p^{k+ℓ−2−2s})L_{(p)}(f ⊗ g, s),

so that

(13)  L(f ⊗ g, s) = L(χ^{−1}, 2s − k − ℓ + 2)D(f, g, s).

By combining (10) and (13), we find that

(14)  L(f ⊗ g, s) = \frac{1}{2} \frac{(4π)^{s}}{Γ(s)} ⟨f^{∗}(z), Ẽ_{k−ℓ,χ}(z, s − k + 1) · g(z)⟩_{k,N} .

Choose integers m and t satisfying

k = ℓ + m + 2t,

and set

c := \frac{k + ℓ + m − 2}{2} = k − t − 1.

By specialising equation (14) at s = c, we find

(15)  L(f ⊗ g, c) = \frac{1}{2} \frac{(4π)^{c}}{Γ(c)} ⟨f^{∗}(z), Ẽ_{k−ℓ,χ}(z, −t) · g(z)⟩_{k,N} .

If m ≥ 1 and t ≥ 0, then replacing k by k − ℓ in equation (6), we obtain

(16)  Ẽ_{k−ℓ,χ}(z, −t) = \frac{(m − 1)!}{(m + t − 1)!}(−4π)^{t}δ_{m}^{t}Ẽ_{m,χ}(z).

Combining (15) with (16) gives

(17)  L(f ⊗ g, c) = \frac{1}{2}(−1)^{t}(4π)^{c+t} \frac{(m − 1)!}{(m + t − 1)!(c − 1)!} ⟨f^{∗}(z), δ_{m}^{t}Ẽ_{m,χ}(z) · g(z)⟩_{k,N} .

Furthermore, in light of equation (3), we have

(18)  L(f ⊗ g, c) = \frac{(−1)^{t}2^{k−1}(2π)^{k+m−1}(iN)^{−m}τ(χ^{−1})}{(m + t − 1)!(c − 1)!} ⟨f^{∗}(z), δ_{m}^{t}E_{m,χ}(z) · g(z)⟩_{k,N} .

Equation (18) is equivalent to Theorem 2 of [Sh1]. Note that the normalisations of Eisenstein series used in loc.cit. are different from those adopted here.

2.3. A factorisation of critical values. Let now g be the Eisenstein series E_{ℓ}(χ_{1}, χ_{2}) of equation (7), and assume that

χ_{g} (= χ_{1}χ_{2}) = χ_{f}^{−1}χ^{−1}.

In light of (8), the left hand side of (18) becomes

(19)  L(f ⊗ E_{ℓ}(χ_{1}, χ_{2}), c) = L(f, χ_{1}, c) · L(f, χ_{2}, c − ℓ + 1).

Assumption 2.2. The following assumptions on (k, ℓ, m) and (f, χ_{1}, χ_{2}) will be enforced for the rest of the paper:

(1) ℓ = m,

(2) χ_{f} = 1, so that f^{∗} = f and χ = χ̄_{1} χ̄_{2},

(3) χ is primitive, so that |τ(χ)|^{2} = N,

(4) (N_{1}, N_{2}) = 1, so that N_{χ_{1}} = N_{1} and N_{χ_{2}} = N_{2}, and

τ(χ) = τ(χ̄_{1})τ(χ̄_{2})χ_{1}(N_{2})χ_{2}(N_{1}) = \overline{τ(χ_{1})} \overline{τ(χ_{2})}χ(−1)χ_{1}(N_{2})χ_{2}(N_{1}).

Under the above assumptions, f is an eigenform on Γ_{0}(N) of even weight k = 2ℓ + 2t. If furthermore t ≥ 0, then c = k/2+ℓ−1 is a critical point for the L-functions L(f ⊗E_{ℓ}(χ_{1}, χ_{2}), s) and L(f, χ_{1}, s), and (19) becomes

(20)  L(f ⊗ E_{ℓ}(χ_{1}, χ_{2}), k/2 + ℓ − 1) = L(f, χ_{1}, k/2 + ℓ − 1) · L(f, χ_{2}, k/2).

Note that k/2 is the central critical point for L(f, χ_{2}, s). We choose complex periods Ω_{f}^{+} and Ω_{f}^{−} as in Proposition 1.1 of [BD1]. These periods satisfy

(21)  Ω_{f}^{+}Ω_{f}^{−} = (2π)^{2} ⟨f, f⟩_{k,N} ,

and, for 1 ≤ j ≤ k − 1,

(22)  L^{∗}(f, ψ, j) := \frac{(j − 1)!τ(ψ̄)}{(−2πi)^{j−1}Ω_{f}^{ε}} L(f, ψ, j)  belongs to  ℚ_{f,ψ},

where ψ is any Dirichlet character, ε = ψ(−1)(−1)^{j−1} and ℚ_{f,ψ} is the field generated by the Fourier coefficients of f and the values of ψ. (See Proposition 1.3 of loc. cit., where ψ = ψ̄.) By combining equations (18), (20) and (22), we find

(23) L^{∗}(f, χ_{1}, k/2+ℓ−1) · L^{∗}(f, χ_{2}, k/2) = C_{f,χ_{1},χ_{2}} · \begin{matrix}⟨f, (δ_{ℓ}^{k/2−ℓ}E_{ℓ,χ}) · E_{ℓ}(χ_{1}, χ_{2})⟩ & k,N \\ ⟨f, f⟩ & k,N \end{matrix} ,

where

(24)  C_{f,χ_{1},χ_{2}} := \frac{i2^{k−1}}{N^{ℓ−1}} χ_{1}(N_{2})χ_{2}(N_{1}).

2.4. Beilinson’s formula. We now focus on the case k = ℓ = 2 (so that c = 2 and t = −1) and deduce a complex Beilinson formula for the non critical value of L(f, s) at s = 2.

By equation (15) with g = E_{2}(χ_{1}, χ_{2}),

(25)  L(f ⊗ E_{2}(χ_{1}, χ_{2}), 2) = \frac{1}{2}(4π)^{2} ⟨f(z), Ẽ_{0,χ}(z, 1) · E_{2}(χ_{1}, χ_{2})(z)⟩_{2,N} .

By specialising equation (5) to the case k = 0 and s = 1, and invoking (3), we obtain

(26)  \frac{1}{2πi} \frac{d}{dz} Ẽ_{0,χ}(z, 1) = −\frac{1}{4π} Ẽ_{2,χ}(z) = 2πN^{−2}τ(χ^{−1})E_{2,χ}(z).

Given a field F, let Eis_{ℓ}(Γ_{1}(N), F) denote the subspace of M_{ℓ}(Γ_{1}(N), F) spanned by the weight ℓ Eisenstein series with coefficients in F. Let Y_{1} = Y_{1}(N) denote the usual open modular curve of level N over ℚ whose complex points are identified with Γ_{1}(N)\H, and let Ȳ_{1} = Ȳ_{1}(N) denote its extension to ℚ̄. The logarithmic derivative

dlog(u) := \frac{1}{2πi} \frac{u′(z)}{u(z)}

gives a surjective homomorphism

(27)  \xymatrix{O(Ȳ_{1}(N))^{×} ⊗ F \ar[r]^{dlog} & Eis_{2}(Γ_{1}(N), F).}

Take F to be a finite extension of ℚ containing the values of all characters of conductor dividing N. Let u_{χ} and u(χ_{1}, χ_{2}) be units satisfying

(28)  dlog(u_{χ}) = E_{2,χ},  dlog(u(χ_{1}, χ_{2})) = E_{2}(χ_{1}, χ_{2}).

It can be shown that u(χ_{1}, χ_{2}) belongs to the χ_{1}-eigenspace (O(Ȳ_{1})^{×} ⊗ F)^{χ_{1}} for the natural action of G_{ℚ} on the space of modular units. The unit u_{χ} is only determined up to a multiplicative constant. It can be shown (see for example [Br1], Section 5) that u_{χ} can be normalised in such a way that the equality

(29)  Ẽ_{0,χ}(z, 1) = 2πN^{−2}τ(χ^{−1}) log |u_{χ}(z)|

holds. Note that equation (29) is consistent with (26). By combining (25) with (29) and (28), we obtain

(30) L(f ⊗ E_{2}(χ_{1}, χ_{2}), 2) = 16π^{3}N^{−2}τ(χ^{−1}) ⟨f(z), log |u_{χ}(z)| · dlog(u(χ_{1}, χ_{2})(z))⟩_{2,N} .

Extend definition (22) of L^{∗}(f, ψ, j) to integers j lying outside of the critical range. By the formulae in Section 2.3, we may rewrite (30) as

(31)  L^{∗}(f, χ_{1}, 2) · L^{∗}(f, χ_{2}, 1) = \frac{C_{f,χ_{1},χ_{2}}}{⟨f, f⟩_{2,N}} · ∫_{Γ_{0}(N)\H} f̄· log |u_{χ}| · dlog(u(χ_{1}, χ_{2}))dxdy.

Define the anti-holomorphic differential attached to f to be

(32)  η_{f}^{ah} := \frac{f̄(z)dz̄}{⟨f, f⟩_{2,N}} .

Up to the constant C_{f,χ_{1},χ_{2}} , the right-hand-side of (31) is the value on the class of η_{f}^{ah} of the complex regulator reg_{ℂ}{u_{χ}, u(χ_{1}, χ_{2})} attached to the symbol

{u_{χ}, u(χ_{1}, χ_{2})} ∈ K_{2}(ℂ(Ȳ_{1}(N)).

We obtain the following proposition, which generalizes slightly the explicit version of Beilinson’s theorem proved in [Br1].

Proposition 2.3.

L^{∗}(f, χ_{1}, 2) · L^{∗}(f, χ_{2}, 1) = C_{f,χ_{1},χ_{2}} · reg_{ℂ}{u_{χ}, u(χ_{1}, χ_{2})}(η_{f}^{ah}).

This is the formula whose precise p-adic counterpart is obtained in Corollary 5.2 below.

2.5. Algebraicity. The modular form

(33)  Ξ_{k,ℓ}(χ_{1}, χ_{2}) := (δ_{ℓ}^{k/2−ℓ}E_{ℓ,χ}) · E_{ℓ}(χ_{1}, χ_{2})

belongs to the space M_{k}^{nh}(N, ℚ_{χ_{1}χ_{2}} ) of nearly-holomorphic modular forms defined over ℚ_{χ_{1}χ_{2}} in the sense of Shimura (cf. Section 2.3 of [DR]). Hence, its image

(34)  Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2}) := Π_{N}^{hol}(Ξ_{k,ℓ}(χ_{1}, χ_{2}))

under the holomorphic projection Π_{N}^{hol} of loc. cit. belongs to M_{k}(N, ℚ_{χ_{1}χ_{2}} ). The ratio appearing in the right-hand side of (23) can then be re-written as

(35)  \frac{⟨f, Ξ_{k,ℓ}(χ_{1}, χ_{2})⟩_{k,N}}{⟨f, f⟩_{k,N}} = \frac{⟨f, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,N}}{⟨f, f⟩_{k,N}} ,

and hence belongs to ℚ_{f,χ_{1}χ_{2}} . One recovers Shimura’s approach [Sh1] to the algebraicity of L^{∗}(f, ψ, j), which differs from the approach based on modular symbols followed in [BD1].

For the purposes of making the connection with p-adic regulators, it is useful to describe the right-hand side of (35) more algebraically, in terms of the Poincaré duality on the deRham cohomology of the open modular curve with values in appropriate sheaves with connection (as described in Section 2.2 of [DR]). More precisely, let Y, resp. X be the open modular curve Y_{0}(N), resp. the complete modular curve X_{0}(N), and let K be any field containing ℚ_{f,χ_{1}χ_{2}} . Denote by E ⟶ Y the universal elliptic curve over Y, and by ω the sheaf of relative differentials on E over Y, extended to X = X_{0}(N) as in Section 1.1 of [BDP]. Recall the Kodaira-Spencer isomorphism ω^{2} = Ω_{X}^{1}(log cusps), where Ω_{X}^{1}(log cusps) is the sheaf of regular differentials on Y with log poles at the cusps. A modular form ϕ on Γ_{0}(N) of weight k = r + 2 with Fourier coefficients in K corresponds to a global section of the sheaf ω^{r+2} = ω^{r} ⊗ Ω_{X}^{1}(log cusps) over X_{K}.

The sheaf ω^{r} can be viewed as a subsheaf of L_{r} := Sym^{r} L, where

L := R^{1}π_{∗}(E ⟶ Y)

is the relative de Rham cohomology sheaf on Y, suitably extended to X, equipped with the filtration

(36)  0 ⟶ ω ⟶ L ⟶ ω^{−1} ⟶ 0

arising from the Hodge filtration on the fibers. The sheaf L_{r} is a coherent sheaf over X of rank r+1, endowed with the Gauss-Manin connection

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

Let H_{dR}^{1}(Y_{K}, L_{r}, ∇), resp. H_{dR,c}^{1}(Y_{K}, L_{r}, ∇) be the de Rham cohomology, resp. the de Rham cohomology with compact support of L_{r}. These two groups are related by the perfect Poincaré pairing

(37)  ⟨ , ⟩_{k,Y} : H_{dR,c}^{1}(Y_{K}, L_{r}, ∇) × H_{dR}^{1}(Y_{K}, L_{r}, ∇) ⟶ K.

There are exact sequences

(38)  0 ⟶ H^{0}(X_{K}, ω^{r} ⊗ Ω_{X}^{1}) ⟶ H_{dR,c}^{1}(Y_{K}, L_{r}, ∇) ⟶ H^{1}(X_{K}, ω^{−r} ⊗ I) ⟶ 0,
(39) 0 ⟶ H^{0}(X_{K}, ω^{r} ⊗ Ω_{X}^{1}(log cusps)) ⟶ H_{dR}^{1}(Y_{K}, L_{r}, ∇) ⟶ H^{1}(X_{K}, ω^{−r}) ⟶ 0,

where I is the ideal sheaf of the cusps (cf. Sections 2 and 3 of [Col94]). The left-most terms of these two sequences are mutually isotropic, and hence (37) induces a perfect pairing

(40)  ⟨ , ⟩_{k,Y} : H^{1}(X_{K}, ω^{−r} ⊗ I) × H^{0}(X_{K}, ω^{r} ⊗ Ω_{X}^{1}(log cusps)) ⟶ K,

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

The antiholomorphic differential η_{f}^{ah} of equation (32) gives rise to a class in H_{dR,c}^{1}(Y_{ℂ}, L_{r}, ∇), whose image η_{f} in H^{1}(X_{ℂ}, ω^{−r} ⊗ I) belongs to H^{1}(X_{K}, ω^{−r} ⊗ I) (cf. Corollary 2.13 of [DR]). Recalling the concrete definition of (40) via complex integration given in (9), we find that the right-hand side of (35) is equal to

(41)  \begin{matrix}⟨f, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩ & k,N \\ ⟨f, f⟩ & k,N \end{matrix} = ⟨η_{f}, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,Y} .

3. p-adic L-functions

This section defines the Rankin p-adic L-function associated to the convolution of two Hida families of cusp forms and Eisenstein series (cf. equation (46)). Furthermore, it shows that this p-adic L-function factors as a product of two Mazur-Kitagawa p-adic L-functions (cf. Theorem 3.4).

A similar p-adic L-function, associated to the convolution of two Hida families of cusp forms, has been constructed by Hida [Hi93]. For the sake of brevity, we follow here the approach of [DR], which constructs the p-adic L-function associated to a triple product of Hida families of cusp forms, referring to the calculations of this article whenever possible.

3.1. Rankin’s p-adic L-functions. Let p ≥ 3 be a prime, and fix an embedding of K into ℂ_{p}. From now on we will be working under the following

Assumption 3.1. The eigenform f is ordinary at p, and p ∤ N.

The f-isotypic part of the exact sequence (38) with K = ℂ_{p} then admits a canonical unit root splitting, arising from the action of Frobenius on the de Rham cohomology. Let η_{f}^{ur} be the lift of η_{f} to the unit root subspace H_{dR,c}^{1}(Y_{ℂ_{p}}, L_{r}, ∇)^{f,ur}. The right-hand side of (41) is then equal to

(42)  ⟨η_{f}, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,Y} = ⟨η_{f}^{ur}, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,Y} .

After viewing Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2}) as an overconvergent p-adic modular form, Lemma 2.10 of [DR] identifies its ordinary projection e_{ord}Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2}) with a cohomology class in H_{dR}^{1}(Y_{K}, L_{r}, ∇)^{ord}. By Proposition 2.11 of loc.cit., the right-hand side of (42) can be re-written as

(43)  ⟨η_{f}^{ur}, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,Y} = ⟨η_{f}^{ur}, e_{ord}Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,Y} .

By Proposition 2.8 of loc.cit.,

(44)  Ξ_{k,ℓ}^{ord}(χ_{1}, χ_{2}) := e_{ord}Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2}) = e_{ord}((d^{k/2−ℓ}E_{ℓ,χ}) · E_{ℓ}(χ_{1}, χ_{2})),

where d = q \frac{d}{dq} is Serre’s derivative operator on p-adic modular forms.

Note that the ordinary p-stabilisation of E_{ℓ}(χ_{1}, χ_{2}) is the weight ℓ specialisation of a Hida family of Eisenstein series denoted E(χ_{1}, χ_{2}). Likewise, let f be a Hida family of eigenforms on Γ_{0}(N), indexed by a weight variable k in a suitable neighborhood U_{f} of (ℤ/(p−1)ℤ)×ℤ_{p}, which is contained in a single residue class modulo p−1. For k ∈ U_{f} ∩ℤ_{≥2}, let f_{k} ∈ S_{k}(N) = S_{k}(N, 1) be the classical modular form whose p-stabilisation is the weight k specialisation of f.

Given a p-adic modular form g = ∑ b_{n}q^{n}, let

g^{[p]} := ∑_{p∤n}b_{n}q^{n}

denote its “p-depletion”. The family of p-adic modular forms

(45)  Ξ_{k,ℓ}^{ord,p}(χ_{1}, χ_{2}) := e_{ord}((d^{k/2−ℓ}E_{ℓ,χ}^{[p]}) · E_{ℓ}(χ_{1}, χ_{2}))

has Fourier coefficients which extend analytically to U_{f} × (ℤ/(p − 1)ℤ × ℤ_{p}), as functions in k and ℓ. See for example [Hi93] and [DR], Section 2.6.

Proposition 3.2. Let e_{f_{k}} be the projector to the f_{k}-isotypic subspace H_{dR}^{1}(Y_{K}, L_{r}, ∇)^{f_{k}}. For all k ≥ 2 and 2 ≤ ℓ ≤ k/2,

e_{f_{k}}Ξ_{k,ℓ}^{ord,p}(χ_{1}, χ_{2}) = \frac{E(f_{k}, χ_{1}, χ_{2}, ℓ)}{E(f_{k})} · e_{f_{k}}Ξ_{k,ℓ}^{ord}(χ_{1}, χ_{2}),

where

E(f_{k}, χ_{1}, χ_{2}, ℓ) = (1 − β_{p}(f_{k})χ_{1}(p)p^{−k/2−(ℓ−1)})(1 − β_{p}(f_{k})χ̄_{1}(p)p^{−k/2+(ℓ−1)})
×(1 − β_{p}(f_{k})χ_{2}(p)p^{−k/2})(1 − β_{p}(f_{k})χ̄_{2}(p)p^{−k/2}),
E(f_{k}) = 1 − β_{p}(f_{k})^{2}p^{−k},

Proof. This follows from Corollary 4.13 of [DR], in light of Proposition 2.8 of loc. cit. □

Set

E^{∗}(f_{k}) := 1 − β_{p}(f_{k})^{2}p^{1−k}.

Proposition 4.6 of loc. cit. shows that the expression

(46)  L_{p}(f, E(χ_{1}, χ_{2}))(k, ℓ) := \frac{1}{E^{∗}(f_{k})} ⟨η_{f_{k}}^{ur}, Ξ_{k,ℓ}^{ord,p}(χ_{1}, χ_{2})⟩_{k,Y} ,

defined for k in U_{f} ∩ ℤ_{≥2} and 2 ≤ ℓ ≤ k/2 extends to an analytic function L_{p}(f, E(χ_{1}, χ_{2})) on U_{f} × (ℤ/(p − 1)ℤ × ℤ_{p}), which we refer to as the p-adic Rankin L-function attached to f and E(χ_{1}, χ_{2}).

3.2. Factorisation of p-adic L-functions. Let L_{p}(f_{k}, ψ, s) be the Mazur-Swinnerton-Dyer p-adic L-function attached to (f_{k}, ψ), with ψ equal to χ_{1} or χ_{2} (cf. Section 14 of [MTT]). We normalise L_{p}(f_{k}, ψ, s) so that it satisfies the interpolation property for 1 ≤ j ≤ k − 1:

(47)  L_{p}(f_{k}, ψ, j) = (1 − \overline{ψ}(p)β_{p}(f_{k})p^{−j}) × (1 − ψ(p)β_{p}(f_{k})p^{−(k−j)}) × L^{∗}(f_{k}, ψ, j).

Note that the values L_{p}(f_{k}, χ_{1}, j) and L_{p}(f_{k}, χ_{2}, j + ℓ − 1) depend on the choice of periods Ω_{f}^{±} and Ω_{f}^{∓} that was made in Section 2.3, but their product does not, in light of the normalising condition imposed in (21), since χ_{1}χ_{2}(−1) = (−1)^{ℓ}.

Proposition 3.3. For all k ∈ U_{f} ∩ ℤ_{≥2} and for all 2 ≤ ℓ ≤ k/2, we have

L_{p}(f, E(χ_{1}, χ_{2}))(k, ℓ) = (C_{f_{k},χ_{1},χ_{2}} E(f_{k})E^{∗}(f_{k}))^{−1} × L_{p}(f_{k}, χ_{1}, k/2+ℓ−1) × L_{p}(f_{k}, χ_{2}, k/2).

Proof. We have the sequence of equalities:

L_{p}(f, E(χ_{1}, χ_{2}))(k, ℓ) = \frac{1}{E^{∗}(f_{k})} ⟨η_{f_{k}}^{ur}, Ξ_{k,ℓ}^{ord,p}(χ_{1}, χ_{2})⟩_{k,Y}  by (46)
= \frac{E(f_{k}, χ_{1}, χ_{2}, ℓ)}{E(f_{k})E^{∗}(f_{k})} ⟨η_{f_{k}}^{ur}, Ξ_{k,ℓ}^{ord}(χ_{1}, χ_{2})⟩_{k,Y}  by Prop. 3.2
= \frac{E(f_{k}, χ_{1}, χ_{2}, ℓ)}{E(f_{k})E^{∗}(f_{k})} ⟨η_{f_{k}}, Ξ_{k,ℓ}^{hol}(χ_{1}, χ_{2})⟩_{k,Y}  by (43) and (42)

= \frac{E(f_{k}, χ_{1}, χ_{2}, ℓ)}{E(f_{k})E^{∗}(f_{k})} \frac{⟨f_{k}, Ξ_{k,ℓ}(χ_{1}, χ_{2})⟩_{k,N}}{⟨f_{k}, f_{k}⟩_{k,N}}  by (41) and (35).

By (23), the last term can be re-written as

\frac{E(f_{k}, χ_{1}, χ_{2}, ℓ)}{E(f_{k})E^{∗}(f_{k})C_{f_{k},χ_{1},χ_{2}}} L^{∗}(f_{k}, χ_{1}, k/2+ℓ−1) · L^{∗}(f_{k}, χ_{2}, k/2),

so that Theorem 3.3 follows by combining (47) with the exact shape of E(f_{k}, χ_{1}, χ_{2}, ℓ) stated in Proposition 3.2. □

Fix k_{0} ∈ U_{f} ∩ℤ_{≥2}. Recall the Mazur-Kitagawa two-variable p-adic L-function L_{p}(f, ψ)(k, s) defined in [Ki]. It is related to L_{p}(f_{k}, ψ, s) by the equation

(48)  L_{p}(f, ψ)(k, s) = λ^{±}(k) · L_{p}(f_{k}, ψ, s),  k ∈ U_{f} ∩ ℤ_{≥2},

where λ^{±}(k) ∈ ℂ_{p} is a p-adic period, arising from the p-adic interpolation of modular symbols, such that λ^{±}(k_{0}) = 1 (cf. Section 1.4 of [BD1]). In particular, while we will see that λ^{±}(k) need not extend to a p-adically continuous function of k ∈ U_{f} , we nonetheless know that λ^{±}(k) ≠ 0 for k in a neighborhood of k_{0}.

By combining (48) with Proposition 3.3, we obtain:

Theorem 3.4. There exists an analytic function η(k) on a neighborhood U_{f,k_{0}} of k_{0}, such that for all (k, ℓ) in U_{f,k_{0}} × (ℤ/(p − 1)ℤ × ℤ_{p}),

L_{p}(f, E(χ_{1}, χ_{2}))(k, ℓ) = η(k) × L_{p}(f, χ_{1})(k, k/2+ℓ−1) × L_{p}(f, χ_{2})(k, k/2).

The function η(k) satisfies

(49)  η(k) = (C_{f_{k},χ_{1},χ_{2}} E(f_{k})E^{∗}(f_{k})λ^{+}(k)λ^{−}(k))^{−1}

for (k, ℓ) in (U_{f,k_{0}} ∩ ℤ_{≥2}) × (ℤ/(p − 1)ℤ × ℤ_{p}), and in particular

η(k_{0}) = (C_{f_{k_{0}},χ_{1},χ_{2}} E(f_{k_{0}})E^{∗}(f_{k_{0}}))^{−1}.

This is crucial for the calculations of the next sections, where both sides of (20) will be evaluated at points outside the range of classical interpolation.

Remark 3.5. Equation (49) gives insight into the question raised in Remark 1.6 of [BD1] about the behavior of the periods λ^{±}. See also Proposition 5.2 of loc. cit., where the product λ^{+}λ^{−} is compared with a less explicit p-adic period arising from the Jacquet-Langlands correspondence to forms on definite quaternion algebras.

4. p-adic regulators

We will now describe the values of L_{p}(f, E(χ_{1}, χ_{2})) at integer points (k_{0}, ℓ_{0}) outside the range of classical interpolation in terms of certain p-adic regulators in K-theory. Recall that the range of classical interpolation defining L_{p}(f, E(χ_{1}, χ_{2})) is k_{0} ∈ U_{f} ∩ℤ_{≥2} and 2 ≤ ℓ_{0} ≤ k_{0}/2.

Proposition 4.1. For all k_{0} ∈ U_{f} ∩ ℤ_{≥2} and ℓ_{0} > k_{0}/2,

(50)  L_{p}(f, E(χ_{1}, χ_{2}))(k_{0}, ℓ_{0}) = \frac{1}{E^{∗}(f_{k_{0}})} ⟨η_{f_{k_{0}}}^{ur} , Ξ_{k_{0},ℓ_{0}}^{ord,p}(χ_{1}, χ_{2})⟩_{k_{0},Y} .

Proof. The terms in the defining expression (46) for L_{p}(f, E(χ_{1}, χ_{2})) vary analytically with ℓ ∈ (ℤ/(p − 1)ℤ) × ℤ_{p}. □

We will be particularly interested in the case where k_{0} = 2 and f_{k_{0}} corresponds to an elliptic curve A/ℚ. Let X_{1} = X_{1}(N) denote the complete modular curve over ℚ of level N obtained by adjoining to Y_{1} = Y_{1}(N) the finite set of cusps, and write X̄_{1} = X̄_{1}(N) for its extension to an algebraic closure of ℚ.

4.1. The regulator on K_{2}. Given modular units u_{1} and u_{2} in O(Ȳ_{1})^{×}, let

{u_{1}, u_{2}} ∈ K_{2}(Ȳ_{1})

be the associated Steinberg symbol.

We recall Besser’s description [Bes2] of the p-adic regulator reg_{p}{u_{1}, u_{2}} ∈ H_{dR}^{1}(Y_{1}) of Coleman-de Shalit [CodS]. Let Φ_{Y_{1}} denote the canonical lift of Frobenius on Y_{1}. It is a rigid morphism on a system {W_{ϵ}} of wide open neighborhoods of the ordinary locus A ⊂ Y_{1} obtained by deleting from Y_{1} both the supersingular and the cuspidal residue discs. (See Section 4.5 of [BDP], for instance, for a brief review of the relevant definitions.) Let Φ_{Y_{1}×Y_{1}} = (Φ_{Y_{1}}, Φ_{Y_{1}}) be the corresponding lift of Frobenius on Y_{1} × Y_{1}, and let P ∈ ℚ[x] be any polynomial satisfying

(1) P (Φ_{Y_{1}×Y_{1}}) annihilates the class of \frac{du_{1}}{u_{1}} ⊗ \frac{du_{2}}{u_{2}} in H_{rig}^{2}(W_{ϵ} × W_{ϵ}),

(2) P (Φ_{Y_{1}}) acts invertibly on H_{rig}^{1}(W_{ϵ}). The choice of P gives rise to a rigid 1-form ρ_{P} (u_{1}, u_{2}) on W_{ϵ} × W_{ϵ} satisfying

(51)  dρ_{P} (u_{1}, u_{2}) = P (Φ_{Y_{1}×Y_{1}}) (\frac{du_{1}}{u_{1}} ⊗ \frac{du_{2}}{u_{2}} ) ,

which is well-defined up to closed rigid 1-forms on W_{ϵ} × W_{ϵ}. After choosing a base point x ∈ W_{ϵ}, let

δ : W_{ϵ} ↪ W_{ϵ} ×W_{ϵ},  i_{x} : W_{ϵ} = W_{ϵ} ×{x} ↪ W_{ϵ} ×W_{ϵ},  j_{x} : W_{ϵ} = {x}×W_{ϵ} ↪ W_{ϵ} ×W_{ϵ},

denote the diagonal, horizontal, and vertical inclusions respectively, and set

ξ̃_{P,x}(u_{1}, u_{2}) := (δ^{∗} − i_{x}^{∗} − j_{x}^{∗})(ρ_{P} (u_{1}, u_{2})) ∈ Ω_{rig}^{1}(W_{ϵ}).

It follows from the Künneth formula (cf. also the argument in the proof of Lemma 3.5 of [DR]) that δ^{∗} − i_{x}^{∗} − j_{x}^{∗} induces the zero map from H_{rig}^{1}(W_{ϵ} × W_{ϵ}) to H_{rig}^{1}(W_{ϵ}), and therefore that it sends closed one-forms to exact one-forms. In particular, the natural image of ξ̃_{P,x}(u_{1}, u_{2}) in H_{rig}^{1}(W_{ϵ}), denoted ξ_{P,x}(u_{1}, u_{2}), does not depend on the choice of one-form ρ_{P} (u_{1}, u_{2}) satisfying equation (51). Condition (2) imposed in the choice of the polynomial P then allows us to define the class

ξ_{x}(u_{1}, u_{2}) := P (Φ_{Y_{1}})^{−1}ξ_{P,x}(u_{1}, u_{2}) ∈ H_{rig}^{1}(W_{ϵ}),

which is independent of the choice of P as above.

The exact sequence

(52)  0 ⟶ H_{dR}^{1}(X_{1}) ⟶ H_{rig}^{1}(W_{ϵ}) ⟶ ℂ_{p}(−1)^{σ−1} ⟶ 0

admits a canonical splitting that respects the Frobenius action. Let ξ(u_{1}, u_{2}) ∈ H_{dR}^{1}(X_{1}) denote the image of ξ_{x}(u_{1}, u_{2}) under this splitting.

Lemma 4.2. The class ξ(u_{1}, u_{2}) does not depend on the choice of base point x.

Proof. Write Φ_{Y_{1}×Y_{1}} = Φ_{h} × Φ_{v}, where Φ_{h} (resp. Φ_{v}) is the rigid endomorphisms of W_{ϵ} × W_{ϵ} acting as the canonical lift of Frobenius on the horizontal (resp. vertical) factor of the product and trivially on the other factor. Following the proof of Proposition 3.3 of [Bes2], let t ≥ 1 be such that (Φ/p)^{t} fixes the classes of \frac{du_{1}}{u_{1}} and \frac{du_{2}}{u_{2}} , and choose

(53)  P (x) = (1 − x^{t}/p^{2t}),

so that, after setting q := p^{t},

P (Φ_{Y_{1}×Y_{1}}) = P (Φ_{h}^{t}Φ_{v}^{t}) = (1 − \frac{Φ_{h}^{t}}{q} ) \frac{Φ_{v}^{t}}{q} + (1 − \frac{Φ_{v}^{t}}{q} ) .

After writing

u_{j}^{(0)} := u_{j}^{q}/Φ^{t∗}(u_{j}),

we find that the rigid one-form on W_{ϵ} × W_{ϵ} defined by

ρ_{P} (u_{1}, u_{2}) = \frac{1}{q^{2}} log u_{1}^{(0)}Φ^{∗t} (\frac{du_{2}}{u_{2}} ) − \frac{1}{q} \frac{du_{1}}{u_{1}} log u_{2}^{(0)}

satisfies equation (51). (Cf. for instance equation (3.2) of loc. cit.). With this choice of primitive, we observe that

i_{x}^{∗}(ρ_{P} (u_{1}, u_{2})) = −\frac{1}{q} \frac{du_{1}}{u_{1}} log u_{2}^{(0)}(x), j_{x}^{∗}(ρ_{P} (u_{1}, u_{2})) = \frac{1}{q^{2}} log u_{1}^{(0)}(x)Φ^{∗t} (\frac{du_{2}}{u_{2}} ) .

The cohomology classes of these one-forms, being multiples of the classes attached to the logarithmic derivatives of modular units, are in the kernel of the Frobenius splitting of (52) used to define ξ(u_{1}, u_{2}), and the result follows. □

It follows from the proof of Lemma 4.2 above that the differential P_{1}(Φ_{Y_{1}})η_{0}(f, g), where η_{0}(f, g) (with f = u_{1}, g = u_{2}) is the differential that appears in Proposition 3.3. of [Bes2], represents the class ξ_{P,x}(u_{1}, u_{2}), up to the addition of logarithmic derivatives of modular units which are in the kernel of the splitting (52), and therefore that the image of the class η(f, g) appearing in Proposition 3.3 of [Bes2] in H_{dR}^{1}(X_{1}) agrees with the class of ξ(u_{1}, u_{2}). We can therefore define, following [Bes2],

(54)  reg_{p}{u_{1}, u_{2}} := ξ(u_{1}, u_{2}).

In parallel with the complex notation of Proposition 2.3, we therefore have

reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}(η_{f}^{ur}) := ⟨η_{f}^{ur}, reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}⟩_{2,Y}

(where reg_{p}{u_{χ}, u(χ_{1}, χ_{2})} is viewed as a class in the de Rham cohomology of Y). We are now ready to state one of the main results of our paper.

Theorem 4.3. For all χ_{1} and χ_{2} as in Assumption 2.2, and setting f = f_{2},

L_{p}(f, E(χ_{1}, χ_{2}))(2, 2) = \frac{E(f, χ_{1}, χ_{2}, 2)}{E(f)E^{∗}(f)}reg_{p}\{u_{χ}, u(χ_{1}, χ_{2})\}(η_{f}^{ur}).

Proof. By the recipe for the p-adic regulator described before, we have

reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}(η_{f}^{ur}) = ⟨η_{f}^{ur}, ξ(u_{χ}, u(χ_{1}, χ_{2}))⟩_{2,Y}
(55)  = ⟨η_{f}^{ur}, P (Φ_{Y})^{−1}ξ_{P,x}(u_{χ}, u(χ_{1}, χ_{2}))⟩_{2,Y} .

For any ξ ∈ H_{dR}^{1}(Y), note that

(56) ⟨η_{f}^{ur}, Φ_{Y} ξ⟩_{Y,2} = α_{p}(f)^{−1} ⟨Φ_{Y} η_{f}^{ur}, Φ_{Y} ξ⟩_{Y,2} = pα_{p}(f)^{−1} ⟨η_{f}^{ur}, ξ⟩_{Y,2} = β_{p}(f) · ⟨η_{f}^{ur}, ξ⟩_{Y,2} .

Combining (55) and (56), we find

⟨η_{f}^{ur}, reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}⟩_{2,Y} = P (β_{p}(f))^{−1} ⟨η_{f}^{ur}, ξ_{P,x}(u_{χ}, u(χ_{1}, χ_{2}))⟩_{2,Y}
(57)  = P (β_{p}(f))^{−1} ⟨η_{f}^{ur}, e_{f}e_{ord}ξ_{P,x}(u_{χ}, u(χ_{1}, χ_{2}))⟩_{2,Y} .

Set

(58)  P (x) := p^{−4}(x − χ_{1}(p))(x − χ̄_{1}(p)p^{2})(x − χ_{2}(p)p)(x − χ̄_{2}(p)p).

Following Besser as in the proof of Lemma 4.2 above, a more optimal choice of P would have been to take (x − χ̄_{1}(p)p^{2}), as (Φ_{Y} − χ̄_{1}(p)p^{2}) already annihilates the class of E_{2,χ} ⊗ E_{2}(χ_{1}, χ_{2}) in cohomology. However, the above choice of P allows us to directly invoke the calculations that were already carried out in Section 3.4 of [DR], in the setting where E_{2,χ} and E_{2}(χ_{1}, χ_{2}) are replaced by cusp forms. (In such a setting, it became necessary to work with a degree 4 polynomial.) With our choice of P, and setting x = ∞, we have

(59) e_{f}e_{ord}ξ_{P,x}(u_{χ}, u(χ_{1}, χ_{2})) = E^{∗}(f)e_{f}e_{ord}(d^{−1}E_{2,χ}^{[p]} × E_{2}(χ_{1}, χ_{2})) = E^{∗}(f)e_{f}Ξ^{ord,p}(χ_{1}, χ_{2}).

This follows by replacing the cusp forms ğ and h̆ by the Eisenstein series E_{2,χ} and E_{2}(χ_{1}, χ_{2}) in Theorem 3.12 of loc. cit. Combining (57) and (59), and observing that P (β_{p}(f)) = E(f, χ_{1}, χ_{2}, 2), we obtain

(60)  ⟨η_{f}^{ur}, reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}⟩_{2,Y} = E(f, χ_{1}, χ_{2}, 2)^{−1}E^{∗}(f) ⟨η_{f}^{ur}, e_{f}Ξ^{ord,p}(χ_{1}, χ_{2})⟩_{2,Y} .

Theorem 4.3 now follows from Proposition 4.1. □

4.2. The general case. Set ℓ_{0} = r + 2, and assume in this section that r > 0. Let E_{1}, E_{2} be weight ℓ_{0} Eisenstein series in Eis_{ℓ_{0}}(Γ_{1}(N), ℚ). Besser’s description [Bes2] of the p-adic regulator reg_{p}{u_{1}, u_{2}} ∈ H_{dR}^{1}(Y_{1}) admits a natural generalisation to the setting in which the logarithmic derivatives of u_{1} and u_{2} are replaced by the Eisenstein series E_{1} and E_{2}. More precisely, note that (L_{r}, ∇) is equipped with the structure of an overconvergent Frobenius isocrystal in the sense of Definition 4.15 of [BDP]. In particular, the cohomology groups H_{dR}^{1}(Y_{1}, L_{r}, ∇) are endowed with an action of the Frobenius lift Φ_{Y_{1}}. Let P ∈ ℚ[x] be any polynomial satisfying

(1) P (Φ_{Y_{1}×Y_{1}}) annihilates the class of E_{1} ⊗ E_{2} in H_{rig}^{2}(W_{ϵ} × W_{ϵ}, L_{r} ⊗ L_{r}, ∇),

(2) P (Φ_{Y_{1}}) acts invertibly on H_{rig}^{1}(W_{ϵ})(r).

The choice of P gives rise to a L_{r}^{⊗2}-valued rigid 1-form ρ_{P} (E_{1}, E_{2}) on W_{ϵ} × W_{ϵ} satisfying

ρ_{P} (E_{1}, E_{2}) = P (Φ_{Y_{1}×Y_{1}})(E_{1} ⊗ E_{2}).

Note that ρ_{P} (E_{1}, E_{2}) is well-defined only up to closed forms on W_{ϵ} × W_{ϵ}. Let ξ_{P} (E_{1}, E_{2}) ∈ H_{rig}^{1}(W_{ϵ})(r) denote the class of the restriction of ρ_{P} (E_{1}, E_{2}) to the diagonal, composed with the pairing

L_{r} × L_{r} ⟶ O_{Y_{1}}(r),

and set

ξ_{0}(E_{1}, E_{2}) := P (Φ_{Y_{1}})^{−1}ξ_{P} (E_{1}, E_{2}) ∈ H_{rig}^{1}(W_{ϵ})(r).

As before, let ξ(E_{1}, E_{2}) denote the natural image of ξ_{0}(E_{1}, E_{2}) in H_{dR}^{1}(X_{1})(r) under the Frobenius-equivariant splitting of the exact sequence (52). The p-adic regulator attached to (E_{1}, E_{2}) is then defined to be

(61)  reg_{p}{E_{1}, E_{2}} = ξ(E_{1}, E_{2}).

The following extends Theorem 4.3 to general ℓ = ℓ_{0}, where as before we have set

reg_{p}{E_{ℓ,χ}, E_{ℓ}(χ_{1}, χ_{2})}(η_{f}^{ur}) := ⟨η_{f}^{ur}, reg_{p}{E_{ℓ,χ}, E_{ℓ}(χ_{1}, χ_{2})}⟩_{2,Y} .

Theorem 4.4. For all χ_{1} and χ_{2} as in Assumption 2.2, and setting f = f_{2},

L_{p}(f, E(χ_{1}, χ_{2}))(2, ℓ) = \frac{E(f, χ_{1}, χ_{2}, ℓ)}{E(f)E^{∗}(f)}reg_{p}\{E_{ℓ,χ}, E_{ℓ}(χ_{1}, χ_{2})\}(η_{f}^{ur}).

Proof. The proof is the same as the proof of Theorem 4.3, using the calculations of Section 3.4 of [DR]. □

5. The p-adic Beilinson formula

5.1. The main results. We can now state the main results of this article, which apply to any pair (χ_{1}, χ_{2}) of primitive Dirichlet characters with relatively prime conductors N_{1} and N_{2}, satisfying

N = N_{1}N_{2}, χ^{−1} := χ_{1}χ_{2} is even.

Theorem 5.1. For all ℓ ≥ 2,

L_{p}(f, χ_{1}, ℓ) · L^{∗}(f, χ_{2}, 1) = C_{f,χ_{1},χ_{2}} (1 − β_{p}(f)χ_{1}(p)p^{−ℓ})(1 − β_{p}(f)χ̄_{1}(p)p^{−(2−ℓ)})
× reg_{p}{E_{ℓ,χ}, E_{ℓ}(χ_{1}, χ_{2})}(η_{f}^{ur}).

Proof. By comparing Theorem 3.4 with k_{0} = 2 and Theorem 4.4, we obtain

L_{p}(f, χ_{1}, ℓ) × L_{p}(f, χ_{2}, 1) = C_{f,χ_{1},χ_{2}} E(f, χ_{1}, χ_{2}, ℓ) × reg_{p}{E_{ℓ,χ}, E_{ℓ}(χ_{1}, χ_{2})}(η_{f}^{ur}).

The theorem now follows from equation (47), giving the interpolation properties of the Mazur-Swinnerton-Dyer p-adic L-function, and the definition of E(f, χ_{1}, χ_{2}, ℓ) given in Proposition 3.2. □

For ℓ = 2, Theorem 5.1 relates the value L_{p}(f, χ_{1}, 2) to the p-adic regulator

reg_{p}{E_{2,χ}, E_{2}(χ_{1}, χ_{2})} := reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}

previously defined in terms of modular units.

Corollary 5.2.

L_{p}(f, χ_{1}, 2) · L^{∗}(f, χ_{2}, 1) = C_{f,χ_{1},χ_{2}} (1 − β_{p}(f)χ_{1}(p)p^{−2})(1 − β_{p}(f)χ̄_{1}(p))
× reg_{p}{u_{χ}, u(χ_{1}, χ_{2})}(η_{f}^{ur}).

Note the strong analogy between Corollary 5.2 and the complex Beilinson formula, as stated in Proposition 2.3. The factor L^{∗}(f, χ_{2}, 1) belongs to the field ℚ(f, χ_{2}), and χ_{2} can be chosen so that this factor does not vanish. Theorem 5.1 then expresses the value L_{p}(f, χ_{1}, ℓ) of the Mazur-Swinnerton-Dyer p-adic L-function at a point outside the range of classical interpolation as the p-adic regulator attached to two Eisenstein series of weight ℓ, times a non-zero algebraic number.

5.2. Relation with the work of Brunault and Gealy. We conclude by explaining the relation between the main results of this paper and the p-adic Beilinson formulae proved in [Br2] (for the value at ℓ = 2) and [Ge] (for the value at general ℓ ≥ 2).

The syntomic regulators of Coleman-de Shalit and Besser have counterparts in p-adic étale cohomology, whose definition we first recall for K_{2}(Y_{1}) and for the eigenspaces (K_{2}(Ȳ_{1})⊗F)^{χ_{1}} under the action of G_{ℚ}, where F is an extension of ℚ_{p} large enough to contain the values of χ_{1} and χ_{2}. Kummer theory gives connecting homomorphisms

δ : O(Y_{1})^{×} ⟶ H_{et}^{1}(Y_{1}, ℚ_{p}(1)), δ : (O(Ȳ_{1})^{×} ⊗ F)^{χ_{1}} ⟶ H_{et}^{1}(Y_{1}, F (1)(χ_{1})).

The p-adic étale regulator of {u_{1}, u_{2}} is defined to be

reg_{et}{u_{1}, u_{2}} := δ(u_{1}) ∪ δ(u_{2}) ∈ H_{et}^{2}(Y_{1}, F (2)(χ_{1})) = H^{1}(ℚ, H_{et}^{1}(Ȳ_{1}, F (2)(χ_{1}))),

where the last identification follows from the Hochschild-Serre spectral sequence (cf. equation (28) of [Br2]). The restriction of reg_{et}{u_{1}, u_{2}} to G_{ℚ_{p}} = Gal (ℚ̄_{p}/ℚ_{p}) yields an element res_{p}(reg_{et}{u_{1}, u_{2}}) of

H^{1}(ℚ_{p}, H_{et}^{1}(Ȳ_{1}, F (2)(χ_{1})) = Ext_{Rep_{ℚp}}^{1}(ℚ_{p}, H_{et}^{1}(Ȳ_{1}, F (2)(χ_{1}))),

where the group of extensions is taken in the category of continuous p-adic representations of G_{ℚ_{p}} which are cristalline. On the other hand, reg_{p}{u_{1}, u_{2}} belongs to

H_{dR}^{1}(Y_{1}) = H_{dR}^{1}(Y_{1})/Fil^{2}H_{dR}^{1}(Y_{1}) = Ext_{ffm}^{1}(ℚ_{p}, H_{dR}^{1}(Y_{1})(2)),

where the group of extensions is taken in the category of admissible filtered Frobenius modules. Fontaine’s comparison functor sets up an isomorphism

(62)  \xymatrix{Ext_{Rep_{ℚp}}^{1}(ℚ_{p}, H_{et}^{1}(Ȳ_{1}, F (2)(χ_{1}))) \ar[r]^{comp} & Ext_{ffm}^{1}(ℚ_{p}, H_{dR}^{1}(Y_{1})(2)).}

More generally, the Eisenstein series E_{1} and E_{2} of weight ℓ_{0} = r + 2 introduced in Section 4.2 give rise to classes

δ_{1} ∈ H_{et}^{1}(Y_{1}, L_{r}^{et}(1)), δ_{2} ∈ H_{et}^{1}(Y_{1}, L_{r}^{et}(1)(χ_{1})),

where L_{r}^{et} is the étale p-adic sheaf associated to the local system (L_{r}, ∇) of Section 2.5 (tensored with the field F containing the values of the characters χ_{1} and χ_{2}). Imitating the complex treatment of Beilinson [Bei], we define the p-adic étale regulator of (E_{1}, E_{2}) to be

reg_{et}{E_{1}, E_{2}} := δ_{1} ∪ δ_{2} ∈ H_{et}^{2}(Y_{1}, F (r + 2)(χ_{1})) = H^{1}(ℚ, H_{et}^{1}(Ȳ_{1}, F (r + 2)(χ_{1}))),

where we have used the pairing L_{r}^{et} ×L_{r}^{et} ⟶ F (r), and as before the last identification follows from the Hochschild-Serre spectral sequence. Note that when r = 0, reg_{et}{E_{1}, E_{2}} is equal to reg_{et}{u_{1}, u_{2}}, with E_{j} = dlog(u_{j}). As in the case r = 0, there is an isomorphism

(63)  \xymatrix{Ext_{Rep_{ℚp}}^{1}(ℚ_{p}, H_{et}^{1}(Ȳ_{1}, F (r + 2)(χ_{1}))) \ar[r]^{comp} & Ext_{ffm}^{1}(ℚ_{p}, H_{dR}^{1}(Y_{1})(r + 2)),}

and we have

Proposition 5.3. For all u_{1}, u_{2} ∈ O(Y_{1})^{×},

reg_{p}{u_{1}, u_{2}} = comp(res_{p}(reg_{et}{u_{1}, u_{2}})).

More generally, for all E_{1}, E_{2} ∈ Eis_{ℓ_{0}}(Γ_{1}(N), ℚ),

reg_{p}{E_{1}, E_{2}} = comp(res_{p}(reg_{et}{E_{1}, E_{2}})).

Proof. See Proposition 9.11 and Corollary 9.10 of [Bes1], and the references therein. □

Proposition 5.3 leads to an alternate definition of the p-adic regulator, which is the one that enters in the p-adic Beilinson formulae of [Br2] and [Ge]. Theorem 5.1 and Corollary 5.2 thus give a different proof of the main results of [Ge] and [Br2] respectively. The strategy followed in loc. cit. builds on the work of Kato, in which a collection of norm-compatible elements in the K_{2} of a tower of modular curves is used to construct a Λ-adic cohomology class κ ∈ H^{1}(ℚ, 𝕍_{p}(E)) with values in the Λ-adic representation 𝕍_{p}(E) of G_{ℚ} interpolating the Tate twists V_{p}(E)(j) for all j ∈ ℤ. Kato’s reciprocity law relates the image of κ in H^{1}(ℚ_{p}, 𝕍_{p}(E)) to the p-adic L-function attached to E. Both [Br2] and [Ge] exploit deep local results of Perrin-Riou ([PR2], [Colz1]) to parlay this relation into a precise connection between the p-adic étale regulator of the Beilinson elements and the special values at ℓ ≥ 2 of the Mazur -Swinnerton-Dyer p-adic L-function. The proof proposed in the present work can be viewed as somewhat more direct, insofar as it does not rely on Kato’s Λ-adic classes or on any facts about the behaviour of the Bloch-Kato logarithm and dual exponential maps in p-adic families.

References

[Bei] Beilinson, A. A. Higher regulators of modular curves. Applications of algebraic K-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 1-34, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986.

[BDP] M. Bertolini, H. Darmon, and K. Prasanna. Generalised Heegner cycles and p-adic Rankin L-series, Duke Math. J. 162, no. 6, (2013) 1033–1148.

[BD1] Bertolini, Massimo; Darmon, Henri. Hida families and rational points on elliptic curves. Invent. Math. 168 (2007), no. 2, 371-431.

[BD2] Bertolini, Massimo; Darmon, Henri Kato’s Euler system and rational points on elliptic curves II: The explicit reciprocity law, in preparation.

[BD3] Bertolini, Massimo; Darmon, Henri Kato’s Euler system and rational points on elliptic curves III: The conjecture of Perrin-Riou, in preparation.

[BDR] Bertolini, Massimo; Darmon, Henri; Rotger, Victor, Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series, submitted.

[Bes1] Besser, Amnon Syntomic regulators and p-adic integration. I. Rigid syntomic regulators. Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998) Israel J. Math. 120 (2000), part B, 291-334.

[Bes2] Besser, Amnon Syntomic regulators and p-adic integration. II. K_{2} of curves. Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998). Israel J. Math. 120 (2000), part B, 335-359.

[BK] Bannai, Kenichi; Kings, Guido. p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure. Amer. J. Math. 132 (2010), no. 6, 1609 – 1654.

[Bl] Bloch, Spencer J. Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series, 11. American Mathematical Society, Providence, RI, 2000. x+97 pp.

[Br1] Brunault, François Valeur en 2 de fonctions L de formes modulaires de poids 2: théorème de Beilinson explicite. Bull. Soc. Math. France 135 (2007), no. 2, 215-246.

[Br2] Brunault, François Régulateurs p-adiques explicites pour le K_{2} des courbes elliptiques. Actes de la Conférence“Fonctions L et Arithmétique”, 29-57, Publ. Math. Besançon Algèbre Théorie Nr., Lab. Math. Besançon, Besançon, 2010.

[CodS] R. F. Coleman, E. de Shalit, p-adic regulators on curves and special values of p-adic L-functions, Inventiones Math. 93 (1988), 239-266.

[Col94] R.F. Coleman, A p-adic Shimura isomorphism and p-adic periods of modular forms. p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 21–51, Contemp. Math., 165, Amer. Math. Soc., Providence, RI, 1994.

[Colz1] Colmez, Pierre. Fonctions L p-adiques. Séminaire Bourbaki, Vol. 1998/99. Astérisque No. 266 (2000), Exp. No. 851, 3, 21–58.

[Colz2] Colmez, Pierre La conjecture de Birch et Swinnerton-Dyer p-adique. (French) Astérisque No. 294 (2004), ix, 251-319.

[DR] H. Darmon and V. Rotger. Diagonal cycles and Euler systems I: A p-adic Gross-Zagier formula, Annales Scientifiques de l’Ecole Normale Supérieure, to appear.

[Ge] M. Gealy, On the Tamagawa number conjecture for motives attached to modular forms, PhD Thesis, California Institute of Technology, 2006.

[Hi93] H. Hida, Elementary theory of L-functions and Eisenstein series. London Mathematical Society Student Texts 26, Cambridge Univerisity Press, 1993.

[Kato] Kato, Kazuya, p-adic Hodge theory and values of zeta functions of modular forms. Cohomologies p-adiques et applications arithmétiques. III. Astérisque No. 295 (2004), ix, 117-290.

[Ki] K. Kitagawa. On standard p-adic L-functions of families of elliptic cusp forms, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 81-110, Contemp. Math. 165, Amer. Math. Soc., Providence, RI, 1994.

[Nik] Niklas, Maximilian, Rigid syntomic regulators and the p-adic L-function of a modular form, Regensburg PhD Thesis, 2010, available at http://epub.uni-regensburg.de/19847/

[MTT] Mazur, B.; Tate, J.; Teitelbaum, J. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), no. 1, 1-48.

[PR1] Perrin-Riou, Bernadette Fonctions L p-adiques d’une courbe elliptique et points rationnels. Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 945-995.

[PR2] Perrin-Riou, Bernadette. Théorie d’Iwasawa des représentations p-adiques sur un corps local. With an appendix by Jean-Marc Fontaine. Invent. Math. 115 (1994), no. 1, 81–161.

[Sh1] Shimura, Goro, The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804.

[Sh2] G. Shimura, On a class of nearly holomorphic automorphic forms, Annals Math. 123 (1986), 347–406.

M.B.: Milano, Italy

E-mail address: massimo.bertolini@unimi.it

H. D.: Montreal, Canada

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