HIDA FAMILIES AND RATIONAL POINTS ON ELLIPTIC CURVES

MASSIMO BERTOLINI HENRI DARMON

Contents

Introduction 1 1. Modular forms and Hida families 5 1.1. Modular symbols 5 1.2. Hida theory 7 1.3. Measure-valued modular symbols 8 1.4. The Mazur-Kitagawa p-adic L-functions 11 2. Modular forms on quaternion algebras 13 2.1. Basic definitions 13 2.2. The Bruhat-Tits tree 16 2.3. The Jacquet-Langlands correspondence 18 2.4. Measure-valued forms 19 2.5. Forms of weight two and p-adic boundary measures 21 2.6. Indefinite integrals 23 2.7. Elliptic curves and the L-invariant 26 3. Hida p-adic L-functions attached to imaginary quadratic fields 27 3.1. Optimal embeddings and special values 28 3.2. Two-variable p-adic L-functions 31 3.3. The interpolation formula 32 3.4. Calculation of derivatives 35 4. Heegner points on Shimura curves 36 4.1. Shimura curves 36 4.2. Modular parametrizations 37 4.3. Heegner points 38 4.4. Heegner points and derivatives of Hida L-functions 40 5. The main result 40 5.1. A factorisation of p-adic L-functions 40 5.2. Proof of the main result 41 References 43

Introduction

Let E be an elliptic curve over ℚ of conductor N = Mp, having a prime p||N of multiplicative reduction. Because E is modular, it corresponds to a normalised weight two eigenform on Γ_{0}(N), whose q-expansion is denoted f = ∑_{n} a_{n}q^{n}.

Let X := hom(ℤ_{p}^{×}, ℤ_{p}^{×}) ≃ ℤ/(p − 1)ℤ × ℤ_{p}, which contains ℤ as a dense subset by associating to k ∈ ℤ the character x ↦ x^{k−2}. Denote by A(U) the ring of ℂ_{p}-valued p-adic analytic functions on a compact open subset U of X. Hida’s theory associates to f a neighborhood U of 2 ∈ X (which can be assumed, for simplicity, to be contained in the residue class of 2 modulo p − 1) and a formal q-expansion

f_{∞} = ∑_{n=1}^{∞} a_{n}(k)q^{n},  a_{1} = 1,  a_{n} ∈ A(U)

which is characterised by the following properties of its specialisations

f_{k} := ∑_{n=1}^{∞} a_{n}(k)q^{n}, (for fixed k ∈ U) :

(1) For all k ∈ U ∩ ℤ^{≥2}, the series f_{k} is the q-expansion of a normalised ordinary eigenform of weight k on Γ_{0}(N). (For this reason, f_{k} is sometimes referred to as the “ weight k specialisation of f_{∞}”.)

(2) The weight 2 specialisation f_{2} is equal to f.

In [M-Sw-D] and [MTT], a p-adic L-function L_{p}(g, s) is associated to any ordinary eigenform g of even weight k ≥ 2, and to a choice of complex period Ω_{g} attached to g. The article [GS] explains how the functions L_{p}(f_{k}, s) can be packaged into a single two-variable p-adic L-function: the so-called Mazur-Kitagawa p-adic L-function, denoted L_{p}(f_{∞}; k, s), whose definition is recalled in Section 1. Section 1 also describes the more general twisted p-adic L-functions L_{p}(f_{∞}, χ, k, s) attached to a (quadratic) Dirichlet character χ, which similarly interpolate the functions L_{p}(f_{k}, χ, s).

The ℂ_{p}-valued function L_{p}(f_{∞}, k, s) is p-adic analytic in a neighborhood of (k, s) = (2, 1). For each k ∈ ℤ^{≥2} ∩ U, it satisfies

(1)  L_{p}(f_{∞}; k, s) = λ(k)L_{p}(f_{k}, s),

for a suitable scalar λ(k) ∈ ℂ_{p} attached to f_{k} satisfying λ(2) = 1. In particular,

(2)  L_{p}(f_{∞}, 2, s) = L_{p}(f, s) = L_{p}(E, s).

Among the general properties of L_{p}(f_{k}, s) studied in [MTT] is a functional equation relating L_{p}(f_{∞}, k, s) to L_{p}(f_{∞}, k, k − s). The sign that appears in this functional equation does not depend on k, and will for that reason be denoted sign(f_{∞}). If E has split multiplicative reduction at p, then

(3)  sign(f_{∞}) = − sign(E, ℚ),

where sign(E, ℚ) denotes the sign in the functional equation of the classical Hasse-Weil L-function L(E, s). The discrepancy between the p-adic and classical signs reflects the fact that L_{p}(f, s) has a so-called “exceptional zero” in the sense of [MTT] at the central critical point s = 1, arising from the fact that p is a prime of split multiplicative reduction for E.

The present article is primarily interested in the behaviour of L_{p}(f_{∞}, k, s) in the presence of such an exceptional zero. One is naturally led to distinguish two cases.

Case 1: sign(E, ℚ) = 1. Equation (3) then shows that L_{p}(f_{∞}; k, s) vanishes identically on the critical line s = k/2, so that

\frac{∂}{∂s}L_{p}(f_{∞}; 2, 1) = −2\frac{∂}{∂k} L_{p}(f_{∞}; 2, 1).

In [GS], Greenberg and Stevens exploit this identity, together with a factorisation of L_{p}(f_{∞}, k, s) when restricted to the line s = 1, to study the first derivative of L_{p}(f, s) at s = 1, obtaining:

(4)  L′_{p}(f, 1) = −2a′_{p}(2)\frac{L(f, 1)}{Ω_{f}},

Ω_{f} being the same real period that enters in the definition of L_{p}(f, s). By exploiting the ordinary Λ-adic representation attached to f_{∞}, and certain properties of its restriction to a decomposition group at p established by Mazur and Wiles, the article [GS] also shows that

(5)  −2a′_{p}(2) = \frac{log(q)}{ord_{p}(q)},

where q ∈ pℤ_{p} is Tate’s p-adic period attached to E, which generates the kernel of the Tate uniformisation

Φ_{Tate} : ℂ_{p}^{×}⟶E(ℂ_{p}).

Note that formulas (4) and (5) imply that

(6)  L′_{p}(f, 1) = \frac{log(q)}{ord_{p}(q)} \frac{L(f, 1)}{Ω_{f}},

a statement that was discovered empirically and formulated as the “exceptional zero conjecture” in [MTT].

Case 2: sign(E, ℚ) = −1. In that case L(E, 1) = 0, and the Birch and Swinnerton-Dyer conjecture predicts that E(ℚ) is infinite. The analysis of L_{p}(f_{∞}, k, s) at (k, s) = (2, 1) carried out by Greenberg and Stevens (and recalled in Section 5.2) shows that

\frac{∂}{∂s}L_{p}(f_{∞}; 2, 1) = \frac{∂}{∂k} L_{p}(f_{∞}; 2, 1) = 0,

so that L_{p}(f_{∞}; k, s) vanishes to order at least 2 at (k, s) = (2, 1). On the other hand, because sign(f_{∞}) = 1, the restriction of L_{p}(f_{∞}; k, s) to the critical line s = k/2 need not vanish identically, and one is led to consider its second derivative.

Let

log_{E} : E(ℚ_{p})⟶𝔾_{a}(ℚ_{p})

be the formal group logarithm on E, defined by letting log_{q} : ℂ_{p}^{×}⟶ℂ_{p} be the branch of the p-adic logarithm which satisfies log_{q}(q) = 0, and setting

(7)  log_{E}(P) := log_{q}(Φ_{Tate}^{−1}(P)).

The main result of this paper, which follows by specialising Theorem 5.4 of Section 5.2 to the trivial character, is:

Theorem 1. Suppose that E has at least two primes of semistable reduction.

(1) There is a global point P ∈ E(ℚ) ⊗ ℚ and a scalar ℓ ∈ ℚ^{×} such that

\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}; k, k/2)_{k=2} = ℓ · log_{E}(P)^{2}.

(2) The point P is of infinite order if and only if L′(E, 1) ≠ 0.

Remark 2. As will be explained shortly, the proof of Theorem 1 exhibits P as a Heegner point arising from an appropriate Shimura curve parametrisation; Part 2 of Theorem 1 then follows from a result of Zhang [Zh1] extending the Gross-Zagier formula to these points.

Remark 3. Theorem 1 (or rather, the somewhat more general Theorem 5.4) is the main ingredient used in [BD3] to prove the rationality of Stark-Heegner points over genus fields of real quadratic fields. Independently of this application, Theorem 1 suggests a general mechanism for producing rational points on elliptic curves from values of L-series, encompassing situations where the Heegner point construction used in its proof may be unavailable. (In this connection, it is worth comparing Theorem 1 with the main theorem of [Ru] concerning elliptic curves with complex multiplication, and with the conjecture of Perrin-Riou described in [PR].) Theorem 1 also yields as a corollary the first non-trivial case of a conjecture of Greenberg [Gre]:

Corollary 4. Assume the hypotheses of Theorem 1. If L′(E, 1) ≠ 0, then there exist infinitely many k ∈ ℤ^{>2} for which L(f_{k}, k/2) ≠ 0.

Remark 5. There would be no difficulty in generalising the statement or the proof of Theorem 1 to the case where f is a normalised eigenform with not necessarily rational Fourier coefficients. In this case the second derivative of L_{p}(f_{∞}; k, k/2) at k = 2 is related to the formal group logarithm of a rational point on the modular abelian variety attached to f.

Remark 6. The hypothesis of an extra prime of semistable reduction made in Theorem 1 is automatically satisfied if E is semistable everywhere, since a semistable elliptic curve with sign(E, ℚ) = −1 must have an even number of primes of split multiplicative reduction. We expect Theorem 1 to be true without this assumption, but the proof that we offer relies on it in a crucial way. It would be of interest to dispense with it. This might be possible using Hida families attached to Hilbert modular forms over real quadratic fields.

We now give a brief overview of the proof of Theorem 1. This proof rests on the study of an auxiliary twovariable p-adic L-function attached to a suitable choice of imaginary quadratic field K in which p is inert. This L-function, introduced in Definition 3.5 and Remark 3.6 of Section 3 and denoted L_{p}(f_{∞}/K; k, s), interpolates the anticyclotomic p-adic L-functions attached to f_{k}/K defined in [BD1] and in [BDIS]. Of special interest is its restriction to the central critical line s = k/2, denoted L_{p}(f_{∞}/K, k).

In Section 4 a global point P_{K} ∈ E(ℚ) ⊗ ℚ is constructed as the trace to ℚ of a Heegner point attached to K arising from an appropriate Shimura curve parametrisation. Corollary 4.10 of Section 4 establishes a direct relationship between P_{K} and the p-adic L-function L_{p}(f_{∞}/K, k), namely,

Theorem 7.

(8)  \frac{d^{2}}{dk^{2}} L_{p}(f_{∞}/K, k)_{k=2} = 2 log_{E}(P_{K})^{2}.

This result is analogous to part 1 of Theorem 1, with the crucial difference that the point P_{K} is independently defined via a Heegner-type construction, so that the proof of an identity like (8) can be approached by building on the techniques introduced in [BD2].

Remark 8. It would be worthwhile to understand Theorem 7 in the framework of the p-adic Birch and Swinnerton-Dyer conjectures for the anticyclotomic setting, in the same way that the conjectures of [BD1] inspired the main results of [BD2].

Section 3.3 uses work of Gross, Hatcher, Zhang and Hui Xue to describe a precise interpolation property relating L_{p}(f_{∞}/K, k) to the algebraic parts of the corresponding classical central critical values L(f_{k}/K, k/2) for even integers k ≥ 2. This leads to the following factorisation formula which is proved in Section 5.1:

Theorem 9. Let ϵ_{K} be the quadratic Dirichlet character associated to K. There exists a p-adic analytic function η of k ∈ U satisfying η(2) ∈ ℚ^{×}, for which

(9)  L_{p}(f_{∞}/K, k) = η(k)L_{p}(f_{∞}, k, k/2)L_{p}(f_{∞}, ϵ_{K}, k, k/2),

for all k ∈ U.

The proof of Theorem 1, which is described in Section 5, proceeds by invoking non-vanishing results for quadratic twists of L-series (such as those that are explained in [MM]) to choose an imaginary quadratic field K in such a way that L(f, ϵ_{K}, 1) ≠ 0. Since p is inert in K, we have ϵ_{K}(p) = −a_{p}. Therefore the p-adic L-function L_{p}(f, ϵ_{K}, s) does not have an exceptional zero. In particular, L_{p}(f, ϵ_{K}, 1) belongs to ℚ^{×}.

Taking second derivatives at k = 2 in the formula (9) of Theorem 9 and applying Theorem 7 yields

η(2)\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}; k, k/2)_{k=2} · L_{p}(f, ϵ_{K}, 1) = 2 log_{E}(P_{K})^{2}.

Theorem 1 now follows by setting ℓ = 2η(2)^{−1}L_{p}(f, ϵ_{K}, 1)^{−1} (a non-zero rational number by the choice of K) and P = P_{K}. Note that both P := P_{K} and ℓ depend on the choice of K that was made, but the product ℓ log_{E}(P)^{2} is independent of this choice.

Part 2 of Theorem 1 follows from a result of Zhang generalising the original Gross-Zagier formula, which expresses the height of P_{K} as a multiple of L′(E/K, 1) by a non-zero scalar. In particular, because of the non-vanishing of L(E, ϵ_{K}, 1), the point P_{K} is of infinite order precisely when L′(E, 1) ≠ 0.

1. Modular forms and Hida families

1.1. Modular symbols. Let

g = ∑_{n=1}^{∞} a_{n}(g)q^{n} ∈ S_{k}(Γ_{0}(N))

be a normalised cusp form of even weight k ≥ 2 on Γ_{0}(N). The Fourier coefficients a_{n}(g) generate a finite algebraic extension of ℚ, denoted K_{g}. Fix embeddings of K_{g} into both ℂ and ℂ_{p}.

Let 𝔓_{k} = 𝔓_{k}(ℂ_{p}) denote the space of homogeneous polynomials in two variables of degree k − 2 with coefficients in ℂ_{p}. It is equipped with a right action of GL_{2}(ℚ_{p}) given by the rule

(10)  (P |γ)(x, y) := P (ax + by, cx + dy) , for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

Let V_{k} = V_{k}(ℂ_{p}) denote the ℂ_{p}-linear dual of 𝔓_{k}, which inherits a left action of GL_{2}(ℚ_{p}) by the usual rule

(11)  (γR)(P) = R(P |γ).

A modular symbol with values in an abelian group G is a function

I : ℙ_{1}(ℚ) × ℙ_{1}(ℚ)⟶G, denoted (r, s) ↦ I{r → s},

satisfying

I{r → s} + I{s → t} = I{r → t}, for all r, s, t ∈ ℙ_{1}(ℚ).

The space of all such modular symbols is denoted MS(G). The group GL_{2}(ℚ) acts on ℙ_{1}(ℚ) on the left by Möbius transformations in the usual way. Hence if G is equipped with a left action of GL_{2}(ℚ), then MS(G) inherits such an action by the rule

(γI){r → s} := γI{γ^{−1}r → γ^{−1}s}.

To the form g is associated the basic V_{k}(ℂ)-valued modular symbol Ĩ_{g} defined by

Ĩ_{g}{r → s}(P) := 2πi ∫_{r}^{s} g(z)P (z, 1)dz.

The invariance property of g under the group Γ_{0}(N) implies that Ĩ_{g} is fixed by the action of this group, i.e.,

Ĩ_{g}{γ^{−1}r → γ^{−1}s}(P |γ) = Ĩ_{g}{r → s}(P), for all γ ∈ Γ_{0}(N).

The matrix c = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) induces an involution on MS_{Γ_{0}(N)}(V_{k}(ℂ)), the space of V_{k}(ℂ)-valued modular

symbols which are fixed by Γ_{0}(N). Let Ĩ_{g}^{+} and Ĩ_{g}^{−} denote the plus and minus eigencomponents of Ĩ_{g} for this involution.

The modular symbols Ĩ_{g}, Ĩ_{g}^{+} and Ĩ_{g}^{−} encode certain complex line integrals of g. The following result of Shimura asserts that these functions can be rescaled so as to belong to MS(V_{k}(K_{g})).

Proposition 1.1. There exist complex periods Ω_{g}^{+} and Ω_{g}^{−} with the property that the modular symbols

I_{g}^{+} := (Ω_{g}^{+})^{−1}Ĩ_{g}^{+}, I_{g}^{−} := (Ω_{g}^{−})^{−1}Ĩ_{g}^{−}

belong to MS(V_{k}(K_{g}))). These periods can be chosen to satisfy

Ω_{g}^{+}Ω_{g}^{−} = ⟨g, g⟩,

where ⟨g, g⟩ is the Petersson scalar product of g with itself.

In proposition 1.1, the Petersson scalar product is normalised so that, if g and h are cusp forms of weight k on Γ_{0}(M), then

⟨g, h⟩ = 4π^{2} ∫∫_{Γ_{0}(M)\H} g(z)h̄(z)y^{k−2}dxdy, z = x + iy,

where the integral is taken over any fundamental region for the action of Γ_{0}(M) on H.

Choose a “sign at infinity” w_{∞} ∈ {+1, −1}, and set

Ω_{g} := \{ \begin{matrix}Ω_{g}^{+} & if w_{∞} = +1; \\ Ω_{g}^{−} & if w_{∞} = −1; \end{matrix} I_{g} := \{ \begin{matrix}I_{g}^{+} & if w_{∞} = +1; \\ I_{g}^{−} & if w_{∞} = −1. \end{matrix}

Note that the modular symbol I_{g} can be viewed as an element of MS(V_{k}(ℂ_{p})) thanks to the chosen embedding of K_{g} into ℂ_{p}. This will be frequently done in the sequel.

It is useful to record, for future reference, the explicit formula describing the action of the Hecke operator

U_{p} on the element I_{g} ∈ MS_{Γ_{0}(N)}(V_{k}(ℂ_{p})). For each 0 ≤ a ≤ p − 1, denote by γ_{a} the matrix ( \begin{matrix}1 & a \\ 0 & p \end{matrix} ).

Lemma 1.2. For all r, s ∈ ℙ_{1}(ℚ) and all P ∈ 𝔓_{k}(ℂ_{p}),

I_{g|U_{p}}{r → s}(P) = ∑_{a=0}^{p−1} I_{g}{γ_{a}r → γ_{a}s}(P |pγ_{a}^{−1}).

Proof. This follows (after a direct calculation) from the proposition in §4, Chapter I of [MTT]. □

The modular symbol I_{g} can be used to define a function of j, a, m ∈ ℤ and P ∈ 𝔓_{k}(ℂ_{p}) by the rule

I_{g}[j, a, m] := I_{g} \{∞ → \frac{a}{m}\} ((x − \frac{a}{m}y)^{j−1} y^{k−j−1}) .

The invariance properties of I_{g} under the matrix ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) imply that the value of I_{g}[j, a, m] depends

only on the value of a in ℤ/mℤ.

Let χ : (ℤ/mℤ)^{×}⟶{±1} be a primitive quadratic Dirichlet character, and let

τ(χ) := ∑_{a=1}^{m} χ(a)e^{2πia/m}

denote the usual Gauss sum. The following proposition relates I_{g} to special values of L-functions, and in particular asserts the algebraicity of these special values.

Proposition 1.3. Let 1 ≤ j ≤ k − 1 be an integer, and suppose that the character χ satisfies χ(−1) = (−1)^{j−1}w_{∞}. Then the expression

L^{∗}(g, χ, j) := \frac{(j − 1)!τ(χ)}{(−2πi)^{j−1}Ω_{g}} L(g, χ, j)

belongs to K_{g}, and

L^{∗}(g, χ, j) = ∑_{a=1}^{m} χ(a)I_{g}[j, a, m].

Proof. This well known formula of Birch and Manin expressing special values of L-series in terms of modular symbols can be found, for example, in formula (8.6) of [MTT]. □

The expression L^{∗}(g, χ, j) will sometimes be referred to as the algebraic part of the special value L(g, χ, j). Note that this expression is a multiple of L(g, χ, j) by a simple non-zero factor, and that it can be viewed as an element of ℂ_{p} thanks to the chosen embedding of K_{g} into ℂ_{p}.

1.2. Hida theory. Let

Λ̃ = ℤ_{p}[[ℤ_{p}^{×}]] := lim_{←} ℤ_{p}[(ℤ/p^{n}ℤ)^{×}], Λ = ℤ_{p}[[(1 + pℤ_{p})^{×}]]

denote the usual Iwasawa algebras. It is often useful to view Λ̃ as functions on the space of continuous ℤ_{p}-algebra homomorphisms

X := hom_{cts}(Λ̃, ℤ_{p}) = hom_{gp}(ℤ_{p}^{×}, ℤ_{p}^{×}) ≃ ℤ/(p − 1)ℤ × ℤ_{p}.

Given λ ∈ Λ̃, and x ∈ X, we therefore occasionally write λ(x) for x(λ). (Elements of Λ̃ are then referred to as Iwasawa functions on X.)

The set ℤ of integers embeds naturally in X by the rule

k ↦ x_{k}, with x_{k}(t) = t^{k−2} for t ∈ ℤ_{p}^{×}.

Note that with these conventions, the element 2 corresponds to the augmentation map on Λ̃ and Λ.

For each integer n ≥ 0, let Y (N, p^{n}) denote the open modular curve whose complex points are identified with H/(Γ_{0}(N) ∩ Γ_{1}(p^{n})), and let

H_{n} := H_{1}(Y (N, p^{n}), ℤ_{p}) = (Γ_{0}(N) ∩ Γ_{1}(p^{n}))_{ab} ⊗ ℤ_{p}.

The H_{n} are finitely generated ℤ_{p}-modules, and are equipped with a natural action of the Hecke operators T_{ℓ} (for ℓ ≠ p) as well as of the operator T_{p} = U_{p}. We may decompose H_{n} as a direct sum of two spaces

H_{n} = H_{n}^{ord} ⊕ H_{n}^{nil},

on which the Hecke operator U_{p} acts invertibly and topologically nilpotently, respectively. More precisely,

H_{n}^{ord} := e_{ord}H_{n},

where e_{ord} := lim_{n⟶∞} U_{p}^{n!} is Hida’s projector to the ordinary part in the Hecke algebra.

The natural homomorphisms H_{n+1}⟶H_{n} induced by the inclusions

Γ_{0}(N) ∩ Γ_{1}(p^{n+1})⟶Γ_{0}(N) ∩ Γ_{1}(p^{n})

respect the ordinary parts, and hence the modules H_{n}^{ord} are part of a projective system of modules over the Hecke algebra. Note also that the diamond operators give rise to an inclusion of the group ring ℤ_{p}[(ℤ/p^{n}ℤ)^{×}] in the ring of Hecke operators acting on H_{n}. The inverse limit

H_{∞}^{ord} := lim_{←} H_{n}^{ord}

therefore inherits a structure of a Λ-module on which the Hecke operators act Λ-linearly. Let 𝕋_{∞}^{ord} denote the Λ-algebra generated by the images of the Hecke operators acting on H_{∞}^{ord}.

Crucial to the constructions of this paper is the following structure theorem due to Hida.

Theorem 1.4 (Hida). The algebra 𝕋_{∞}^{ord} is a free Λ-module of finite rank, and is unramified over the augmentation ideal of Λ.

The normalised eigenform f of the Introduction gives rise to an algebra homomorphism

η_{f} : 𝕋_{∞}^{ord}⟶ℤ_{p}

sending the Hecke operator T_{n} to a_{n}(f), and U_{p} to a_{p}(f). The restriction of η_{f} to Λ is the augmentation map, corresponding to 2 ∈ X. Let Λ^{†} ⊃ Λ denote the ring of power series with coefficients in ℂ_{p} which converge in some neighborhood of 2 ∈ X. Since Λ^{†} is Henselian and since the augmentation ideal is unramified in 𝕋_{∞}^{ord}, the homomorphism η_{f} lifts uniquely to a Λ-algebra homomorphism

η_{f_{∞}} : 𝕋_{∞}^{ord}⟶Λ^{†}.

Following the notations of the introduction, let a_{n}(k) := η_{f_{∞}}(T_{n}). The formal q-expansion with coefficients in Λ^{†}

f_{∞} := ∑_{n=1}^{∞} a_{n}(k)q^{n}

is the Hida family alluded to in the introduction. Fix a neighborhood U of 2 ∈ X on which the elements a_{n} converge, and suppose for simplicity that U is contained in the residue class of 2 modulo p − 1.

If k belongs to U ∩ ℤ^{≥2}, the weight k specialisation

(12)  f_{k} := ∑_{n} a_{n}(k)q^{n}

is a normalised eigenform of weight k on Γ_{0}(N). It is new at the primes dividing M = N/p (see for example Theorem (2.6) of [GS]), but it is not new at p when k > 2. More precisely, if k belongs to U ∩ ℤ^{>2}, the modular form f_{k} arises from a normalised eigenform on Γ_{0}(M), denoted f_{k}^{♯} = ∑_{n} a_{n}(f_{k}^{♯})q^{n}. If (p, n) = 1, then a_{n}(f_{k}^{♯}) = a_{n}(f_{k}). Letting

1 − a_{p}(f_{k}^{♯})p^{−s} + p^{k−1−2s} = (1 − α_{p}(k)p^{−s})(1 − β_{p}(k)p^{−s})

denote the Euler factor at p that appears in the L-series of f_{k}^{♯}, we may order the roots α_{p}(k) and β_{p}(k) in such a way that

α_{p}(k) = a_{p}(f_{k}), β_{p}(k) = p^{k−1}a_{p}(f_{k})^{−1}.

With this convention, we have

(13)  f_{k}(z) = f_{k}^{♯}(z) − β_{p}(k)f_{k}^{♯}(pz).

The field K_{f_{k}} generated by the Fourier coefficients of f_{k} is an extension of ℚ of finite degree, which we view as being embedded into both ℂ and ℂ_{p}. For each k, we choose the Shimura periods Ω_{k}^{+} := Ω_{f_{k}}^{+} and Ω_{k}^{−} := Ω_{f_{k}}^{−} as in Proposition 1.1, requiring that

(14)  Ω_{2}^{+}Ω_{2}^{−} = ⟨f, f⟩,  Ω_{k}^{+}Ω_{k}^{−} = ⟨f_{k}^{♯}, f_{k}^{♯}⟩ (k > 2).

Thanks to these periods we may talk about the V_{k}(ℂ_{p})-valued modular symbols I_{f_{k}}^{+} and I_{f_{k}}^{−} associated to each f_{k}. Since the Fourier coefficients of f_{k} vary p-adically analytically with k, it is natural to ask whether the functions I_{f_{k}}, which encode the Shimura periods of f_{k}, can be likewise viewed as part of an analytically varying family (as a function of k). This question is considered in the next section.

1.3. Measure-valued modular symbols. Let L_{∗} := ℤ_{p}^{2} denote the standard ℤ_{p}-lattice in ℚ_{p}^{2}, and let L′_{∗} denote its set of primitive vectors, i.e., the vectors in L_{∗} which are not divisible by p.

The space of continuous ℂ_{p}-valued functions on L′_{∗} is equipped with the right action of GL_{2}(ℤ_{p}) defined by the rule which is compatible with (10)

(15)  (F |g)(x, y) := F (ax + by, cx + dy), for g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,

and with a topology given by the sup norm. Its continuous dual is called the space of measures on L′_{∗}. Denote by 𝔻_{∗} this latter space. The action of the group ℤ_{p}^{×} on L′_{∗} given by λ(x, y) = (λx, λy) gives rise to a natural Λ̃-module structure on 𝔻_{∗} by setting

∫_{L′_{∗}} F (x, y)d(a · μ)(x, y) := ∫_{L′_{∗}} F (ax, ay)dμ(x, y),

for all a ∈ ℤ_{p}^{×}. If X is any compact open subset of L′_{∗}, we adopt the common notation

∫_{X} F dμ := ∫_{L′_{∗}} 1_{X}F dμ,

where 1_{X} is the characteristic function of X. Note that the group GL_{2}(ℤ_{p}) also acts on 𝔻_{∗} on the left by translation, so that

∫_{X} F d(γμ) = ∫_{γ^{−1}X}(F |γ)dμ.

Denote by Γ_{0}(pℤ_{p}) the group of matrices in GL_{2}(ℤ_{p}) which are upper triangular modulo p. Our interest in the space 𝔻_{∗} lies in the fact that it is equipped, for all k ∈ ℤ^{≥2}, with a Γ_{0}(pℤ_{p})-equivariant homomorphism

ρ_{k} : 𝔻_{∗}⟶V_{k}

defined by

(16)  ρ_{k}(μ)(P) := ∫_{ℤ_{p}×ℤ_{p}^{×}} P (x, y)dμ(x, y).

(Note that ρ_{k} does not respect the full action of GL_{2}(ℤ_{p}), because the domain of integration that appears in its definition is only preserved by Γ_{0}(pℤ_{p}).) The homomorphism ρ_{k} gives rise to a homomorphism, denoted by the same letter by abuse of notation:

(17)  ρ_{k} : MS_{Γ_{0}(M)}(𝔻_{∗})⟶MS_{Γ_{0}(N)}(V_{k}).

Recall the ring Λ^{†} ⊃ Λ of power series which converge in some neighborhood of 2 ∈ X, and set

𝔻_{∗}^{†} := 𝔻_{∗} ⊗_{Λ} Λ^{†}.

If

μ = λ_{1}μ_{1} + ⋯ + λ_{t}μ_{t}, with λ_{j} ∈ Λ^{†}, μ_{j} ∈ 𝔻_{∗}

is any element of 𝔻_{∗}^{†}, then there exists a neigbourhood U_{μ} of 2 ∈ X on which all the coefficients λ_{j} converge. Call such a region U_{μ} a neighborhood of regularity for μ.

Given k ∈ U_{μ}, a continuous function F on L′_{∗} is said to be homogeneous of degree k − 2 if F (λx, λy) = λ^{k−2}F (x, y), for all λ ∈ ℤ_{p}^{×}. For any k ∈ U_{μ}, and any homogenous function F (x, y) of degree k − 2, the function F can be integrated against μ by the rule

∫_{X} F dμ := λ_{1}(k) ∫_{X} F dμ_{1} + ⋯ + λ_{t}(k) ∫_{X} F dμ_{t},

for any compact open X ⊂ L′_{∗}.

The space MS_{Γ_{0}(M)}(𝔻_{∗}) is equipped with a natural action of the Hecke operators, including an operator U_{p} which is defined so as to be compatible with the specialisation maps ρ_{k} of (17). More precisely, in the notations of Lemma 1.2, it is given by the formula

(18)  ∫_{X} F d(U_{p}μ){r → s} = ∑_{a=0}^{p−1} ∫_{p^{−1}γ_{a}X}(F |pγ_{a}^{−1})dμ{γ_{a}r → γ_{a}s}.

Let MS_{Γ_{0}(M)}^{ord}(𝔻_{∗}) denote the ordinary subspace of MS_{Γ_{0}(M)}(𝔻_{∗}). Proposition (6.1) of [GS] asserts that this module is free and of finite rank over Λ. Therefore the same is true of the Λ^{†}-module

MS_{Γ_{0}(M)}^{ord}(𝔻_{∗})^{†} := MS_{Γ_{0}(M)}^{ord}(𝔻_{∗}) ⊗_{Λ} Λ^{†} ⊂ MS_{Γ_{0}(M)}(𝔻_{∗}^{†}).

Given r, s ∈ ℙ_{1}(ℚ), and μ ∈ MS_{Γ_{0}(M)}^{ord}(𝔻_{∗})^{†}, denote by μ{r → s} the corresponding element of 𝔻_{∗}^{†}. It is possible to choose a common neighborhood of regularity U_{μ} for the measures μ{r → s}, and this makes it possible to define ρ_{k}(μ) for all k ∈ U_{μ} ∩ ℤ^{≥2}.

Recall the Hida family f_{∞} that was introduced in the previous section. The following result of Greenberg and Stevens plays a key role in the constructions of this section.

Theorem 1.5. There exists a neighborhood U of 2 ∈ X and a measure-valued symbol μ_{∗} ∈ MS_{Γ_{0}(M)}^{ord}(𝔻_{∗})^{†} which is regular on U, and satisfies

(1) ρ_{2}(μ_{∗}) = I_{f};

(2) For all k ∈ U ∩ ℤ^{≥2}, there exists a scalar λ(k) ∈ ℂ_{p} such that
ρ_{k}(μ_{∗}) = λ(k)I_{f_{k}}.

Proof. See Theorem (5.13) of [GS], whose proof is explained in Section 6 of that paper. □

Remark 1.6. Note that μ_{∗} depends on the choice of sign w_{∞}, and that there are therefore two 𝔻_{∗}^{†}-valued modular symbols, μ_{∗}^{+} and μ_{∗}^{−}, which interpolate the modular symbols I_{f_{k}}^{+} and I_{f_{k}}^{−} respectively. Likewise, there are two families of scalars, λ^{+}(k) and λ^{−}(k), which depend on the choices of Shimura periods Ω_{k}^{+} and Ω_{k}^{−} that had to be made for each f_{k} in order to define I_{f_{k}}^{+} and I_{f_{k}}^{−} respectively. There has been no attempt at making a coherent choice of these Shimura periods as k varies (and it is not even clear that such a choice can be made in a natural way) so that the function λ(k) cannot be expected to be continuous as a function of k (with respect to the topology on ℤ^{≥2} induced by its inclusion into X). It would be interesting to know whether it is possible to choose the periods Ω_{k} a priori in such a way that λ(k) extends to a continuous, or even analytic, function in a neighborhood of 2 ∈ X. An understanding of this issue is not germane to the proofs of the theorems in this article.

We note the following non-vanishing property of the scalars λ(k) of Theorem 1.5.

Proposition 1.7. There is a neighborhood U of 2 ∈ X with λ(k) ≠ 0, for all k ∈ U ∩ ℤ^{≥2}.

Proof. The set of k for which ρ_{k}(μ_{∗}) = 0 is a finite subset of ℤ which does not contain 2. □

By shrinking U if necessary, we may therefore assume that the scalars λ(k) are non-zero, for all k ∈ U ∩ ℤ^{≥2}. Note that, even when the periods Ω_{k} have been chosen, the measure-valued modular symbol μ_{∗} is only defined up to multiplication by an element α ∈ Λ^{†} satisfying α(2) = 1.

Let 𝔻 denote the module of compactly supported measures on

W := ℚ_{p}^{2} − {(0, 0)}.

The space 𝔻_{∗} is contained in 𝔻 by viewing elements of 𝔻 as measures with support contained in L′_{∗}. Write 𝔻^{†} := 𝔻 ⊗ Λ^{†}. It is convenient to attach to μ_{∗} a family μ_{L} of 𝔻^{†}-valued modular symbols, indexed by the ℤ_{p}-lattices L in ℚ_{p}^{2}. This is done by exploiting the action of the group

Γ := \{γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ[1/p]) with M|c, det(γ) > 0.\} .

Proposition 1.8. There exists a unique collection {μ_{L}} of 𝔻^{†}-valued modular symbols, indexed by the ℤ_{p}-lattices L ⊂ ℚ_{p}^{2}, and satisfying:

(1) μ_{L_{∗}} = μ_{∗};

(2) For all γ ∈ Γ, and all compact open X ⊂ W,

(19)  ∫_{γX}(F |γ^{−1})dμ_{γL}{γr → γs} = ∫_{X} F dμ_{L}{r → s}.

Proof. The group Γ acts transitively on the collection of ℤ_{p}-lattices in ℚ_{p}^{2}. The stabiliser of L_{∗} in Γ for this action is the group Γ_{0}(M). Since the modular symbol μ_{∗} is fixed under the action of Γ_{0}(M), the rule of (19) is well-defined and determines μ_{L} uniquely, for each lattice L. □

For future reference, some of the salient properties of the system μ_{L} are recorded in a sequence of lemmas.

Lemma 1.9. Let L be a lattice. The distributions μ_{L}{r → s} are supported on L′, for all r, s ∈ ℙ_{1}(ℚ).

Proof. The distributions μ_{L_{∗}}{r → s} = μ_{∗}{r → s} are supported on L′_{∗}, for all r, s ∈ ℙ_{1}(ℚ). The lemma follows from the Γ-equivariance property defining the distributions μ_{L}{r → s} at other lattices L. □

Lemma 1.10. Suppose that L_{2} ⊂ L_{1} is a sublattice of index p in L_{1}. Then for all k ∈ U ∩ ℤ^{≥2}, for all homogeneous functions F on L′_{1} ∩ L′_{2} of degree k − 2, and for all r, s ∈ ℙ_{1}(ℚ), we have

∫_{L′_{1}∩L′_{2}} F dμ_{L_{2}}{r → s} = a_{p}(k) ∫_{L′_{1}∩L′_{2}} F dμ_{L_{1}}{r → s}.

Proof. Since the group Γ acts transitively on the pairs (L_{1}, L_{2}) of lattices satisfying [L_{1} : L_{2}] = p, it is enough to prove the lemma in the case where

L_{1} = \frac{1}{p}ℤ_{p} + ℤ_{p}, L_{2} = ℤ_{p}^{2},

so that

L′_{1} ∩ L′_{2} = ℤ_{p} × ℤ_{p}^{×}.

For each a = 0, …, p − 1, let L_{(a)} denote the ℤ_{p}-lattice

L_{(a)} := {(x, y) ∈ ℤ_{p}^{2} such that p|x + ay}.

The reader will note that

L′_{2} ∩ L′_{(a)} = {(x, y) ∈ ℤ_{p} × ℤ_{p}^{×} such that x/y ≡ −a (mod p)},

and hence L′_{1} ∩ L′_{2} can be written as a disjoint union

L′_{1} ∩ L′_{2} = ⋃_{a=0}^{p−1} L′_{2} ∩ L′_{(a)}.

In particular,

(20)  ∫_{L′_{1}∩L′_{2}} F dμ_{L_{2}}{r → s} = ∑_{a=0}^{p−1} ∫_{L′_{2}∩L′_{(a)}} F dμ_{L_{2}}{r → s}.

Recall the matrices γ_{a} = ( \begin{matrix}1 & a \\ 0 & p \end{matrix} ) that were introduced earlier in describing the Hecke operator U_{p}, and

observe that

γ_{a}L_{2} = pL_{1}, γ_{a}L_{(a)} = pL_{2}, so that γ_{a}(L′_{2} ∩ L′_{(a)}) = p(L′_{1} ∩ L′_{2}).

Using the invariance under γ_{a} of the a-th term in (20), this expression can be rewritten as:

(21)  ∫_{L′_{1}∩L′_{2}} F dμ_{L_{2}}{r → s} = ∑_{a=0}^{p−1} ∫_{p(L′_{1}∩L′_{2})}(F |γ_{a}^{−1})dμ_{pL_{1}}{γ_{a}r → γ_{a}s}

(22)  = ∑_{a=0}^{p−1} ∫_{L′_{1}∩L′_{2}}(F |pγ_{a}^{−1})dμ_{L_{1}}{γ_{a}r → γ_{a}s}

(23)  = ∫_{L′_{1}∩L′_{2}} F d(U_{p}μ_{L_{1}}){r → s}
(24)  = a_{p}(k) ∫_{L′_{1}∩L′_{2}} F dμ_{L_{1}}{r → s}.

where the penultimate equality follows from (18). The result follows. □

1.4. The Mazur-Kitagawa p-adic L-functions. We use the 𝔻_{∗}^{†}-valued modular symbol μ_{∗} to define the Mazur-Kitagawa two-variable p-adic L-function attached to f and a Dirichlet character χ, as follows:

Definition 1.11. Let χ be a primitive quadratic Dirichlet character of conductor m satisfying χ(−1) = w_{∞}. The Mazur-Kitagawa two-variable p-adic L-function attached to χ is the function of (k, s) ∈ U × X defined by the rule

L_{p}(f_{∞}, χ, k, s) = ∑_{a=1}^{m} χ(ap) ∫_{ℤ_{p}^{×}×ℤ_{p}^{×}}(x − \frac{pa}{m}y)^{s−1} y^{k−s−1}dμ_{∗} \{∞ → \frac{pa}{m}\} .

The function L_{p}(f_{∞}, χ, k, s) satisfies the following interpolation property with respect to special values of the classical L-functions L(f_{k}, χ, s).

Theorem 1.12. Suppose that k belongs to U ∩ ℤ^{≥2}, and that 1 ≤ j ≤ k − 1 satisfies χ(−1) = (−1)^{j−1}w_{∞}. Then

L_{p}(f_{∞}, χ, k, j) = λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{j−1})L^{∗}(f_{k}, χ, j).

Proof. Let L_{1} and L_{2} denote the ℤ_{p} lattices that are defined in the proof of Lemma 1.10. The integral that appears in the definition of L_{p}(f_{∞}, χ, k, j) can be expressed as a difference of two contributions

(25)  ∫_{ℤ_{p}^{×}×ℤ_{p}^{×}} (x − \frac{pa}{m}y)^{j−1} y^{k−j−1}dμ_{∗} \{∞ → \frac{pa}{m}y\} = V_{1} − V_{2},

where

(26)  V_{1} = ∫_{ℤ_{p}×ℤ_{p}^{×}} (x − \frac{pa}{m}y)^{j−1} y^{k−j−1}dμ_{L_{2}} \{∞ → \frac{pa}{m}\}
= λ(k)I_{f_{k}} \{∞ → \frac{pa}{m}\} ((x − \frac{pa}{m}y)^{j−1} y^{k−j−1})
(27)  = λ(k)I_{f_{k}}[j, pa, m],

and

V_{2} = ∫_{pℤ_{p}×ℤ_{p}^{×}} (x − \frac{pa}{m}y)^{j−1} y^{k−j−1}dμ_{L_{2}} \{∞ → \frac{pa}{m}\}
= p^{j−1} ∫_{ℤ_{p}×ℤ_{p}^{×}} (x − \frac{a}{m}y)^{j−1} y^{k−j−1}dμ_{L_{1}} \{∞ → \frac{a}{m}\} ,

where this last equality follows from (19) with γ = ( \begin{matrix}1/p & 0 \\ 0 & 1 \end{matrix} ). Hence by Lemma 1.10,

(28)  V_{2} = a_{p}(k)^{−1}p^{j−1} ∫_{ℤ_{p}×ℤ_{p}^{×}} (x − \frac{a}{m}y)^{j−1} y^{k−j−1}dμ_{L_{2}} \{∞ → \frac{a}{m}\}
= λ(k)a_{p}(k)^{−1}p^{j−1}I_{f_{k}} \{∞ → \frac{a}{m}\} (((x − \frac{a}{m}y)^{j−1} y^{k−j−1})
(29)  = λ(k)a_{p}(k)^{−1}p^{j−1}I_{f_{k}}[j, a, m].

Combining equations (25), (27), and (29) with Definition 1.11 yields

L_{p}(f_{∞}, χ, k, j) = λ(k) ∑_{a=1}^{m} χ(pa)I_{f_{k}}[j, pa, m]

− λ(k)a_{p}(k)^{−1}p^{j−1} ∑_{a=1}^{m} χ(ap)I_{f_{k}}[j, a, m]

= λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{j−1})L^{∗}(f_{k}, χ, j),

where the last equation follows from Proposition 1.3. □

Remark 1.13. Setting j = 1 in (26) and (28) and summing over the a ∈ ℤ/mℤ shows that

(30)  L_{p}(f_{∞}, χ, k, 1) = (1 − χ(p)a_{p}(k)^{−1})L_{p}^{∗}(f_{∞}, χ, k),

where

L_{p}^{∗}(f_{∞}, χ, k) := ∑_{a=1}^{m} χ(a) ∫_{ℤ_{p}×ℤ_{p}^{×}} y^{k−2}dμ_{∗} \{∞ → \frac{a}{m}\} .

The expression for L_{p}^{∗}(f_{∞}, χ, k) extends to a p-adic analytic function of k ∈ U, called the improved p-adic L-function in [GS]. The factorisation (30) plays an important role in the proof by Greenberg and Stevens of the exceptional zero conjecture of Mazur, Tate and Teitelbaum.

Theorem 1.12 can also be written in terms of the form f_{k}^{♯} of equation (13), so that L_{p}(f_{∞}, χ, k, j) is equal to

λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{j−1})(1 − χ(p)a_{p}(k)^{−1}p^{k−j−1})L^{∗}(f_{k}^{♯}, χ, j).

Note in particular that, after specialising at j = k/2, one finds

(31)  L_{p}(f_{∞}, χ, k, k/2) = λ(k)(1 − χ(p)a_{p}(k)^{−1}p^{\frac{k}{2} −1})^{2}L^{∗}(f_{k}^{♯}, χ, k/2).

It will be important for our later study that the Euler factor appearing in this formula is a square.

2. Modular forms on quaternion algebras

2.1. Basic definitions. Let N be a postive integer admitting a factorisation as a product of three relatively prime integers

(32)  N = pN^{+}N^{−},

where p is a prime and N^{−} is the square-free product of an odd number of primes. Let B be the definite quaternion algebra over ℚ ramified exactly at the primes dividing N^{−}. That is to say, the algebra B ⊗ ℝ is isomorphic to Hamilton’s real quaternions, and for each prime ℓ the ring B_{ℓ} := B ⊗_{ℚ} ℚ_{ℓ} is isomorphic to the matrix algebra M_{2}(ℚ_{ℓ}) if ℓ does not divide N^{−}, and to the quaternion division algebra over ℚ_{ℓ} otherwise. The assumption on N^{−} ensures the existence of this quaternion algebra.

Let ℤ̂ be the profinite completion of ℤ, and for any ring Z write

Ẑ := Z ⊗ ℤ̂.

Thus, for example, ℚ̂ is the ring of finite adèles of ℚ and

B̂^{×} = B^{×}(ℚ̂) ⊂ ∏_{ℓ} B_{ℓ}^{×}

is the group of adèlic points of B^{×} viewed as an algebraic group over ℚ. Given b ∈ B̂^{×}, denote by b_{p} ∈ B_{p}^{×} its component at p.

Since p does not divide N^{−}, we may fix an isomorphism of ℚ_{p}-algebras

ι_{p} : B_{p} = B ⊗_{ℚ} ℚ_{p}⟶M_{2}(ℚ_{p})

and denote by the same letter the corresponding identification of B_{p}^{×} with GL_{2}(ℚ_{p}). Let Σ = ∏_{ℓ} Σ_{ℓ} be a compact open subgroup of B̂^{×}, and let A be any ℚ_{p}–vector space (or sometimes, ℤ_{p}-module) equipped with a linear left action of the semigroup M_{2}(ℤ_{p}) of matrices with entries in ℤ_{p} and non-zero determinant.

Definition 2.1. An A-valued automorphic form on B^{×} of level Σ is a function

ϕ : B̂^{×}⟶A

satisfying

ϕ(gbσ) = ι_{p}(σ_{p}^{−1}) · ϕ(b),

for all g ∈ B^{×}, b ∈ B̂^{×}, and σ ∈ Σ.

Denote by S(Σ; A) the space of such forms.

Remark 2.2. Definition 2.1 differs from the one that is encountered in [BD1]—[BDIS] , where an automorphic form on B^{×} “of level Σ” is defined to be a function ϕ̃: B̂^{×}⟶A satisfying

ϕ̃(gbσ) = ι_{p}(g)ϕ̃(b), for all g ∈ B^{×}, b ∈ B̂^{×}, σ ∈ Σ.

One passes from one definition to the other by setting

ϕ̃(b) = ι_{p}(b_{p})ϕ(b).

Remark 2.3. An element of S(Σ; A) is completely determined by its values on a set of representatives for the double coset space

𝕏_{Σ} := B^{×}\B̂^{×}/Σ,

which is finite (since it is both compact, and discrete). Hence if A is finite dimensional, the same is true for S(Σ; A).

We will primarily be interested in three situations:

(1) The case where A = ℤ_{p} with trivial action. In that case the A-valued automorphic forms of level Σ are said to be of weight 2 and the ℤ_{p}-module of such forms is denoted S_{2}(Σ).

(2) Let 𝔓_{k}(ℚ_{p}) and V_{k}(ℚ_{p}) denote the spaces introduced in Section 1.1 with the left and right actions by GL_{2}(ℚ_{p}) described in equations (10) and (11) respectively. The V_{k}(ℚ_{p})-valued automorphic forms of level Σ are said to be of weight k and the ℚ_{p}–vector space of such forms is denoted S_{k}(Σ). (The reason for this terminology will become apparent in Section 2.3.)

(3) If Σ_{p} = GL_{2}(ℤ_{p}), and A is the module 𝔻_{∗} or 𝔻_{∗}^{†} defined in Section 1.3, an A-valued modular form of level Σ is referred to as a p-adic family of modular forms, and the space of such modular forms is denoted S_{∞}(Σ) or S_{∞}^{†}(Σ), depending on whether A = 𝔻_{∗} or 𝔻_{∗}^{†}. The spaces S_{∞}(Σ) and S_{∞}^{†}(Σ) are equipped with a natural action of Λ and Λ^{†} respectively, which commute with the Hecke operators. Let Σ′ be the level structure obtained from Σ by replacing Σ_{p} = GL_{2}(ℤ_{p}) by the subgroup Γ_{0}(pℤ_{p}) of matrices which are upper triangular modulo p. Recall the Γ_{0}(pℤ_{p})- equivariant “weight k specialisation” homomorphism ρ_{k} : 𝔻_{∗}⟶V_{k} defined in equation (16) of Section 1.3. In this chapter, it is convenient to adopt a slightly different definition, by setting

(33)  ρ_{k}(μ)(P) := ∫_{ℤ_{p}^{×}×pℤ_{p}} P (x, y)dμ(x, y).

Thus we are now integrating on the region which is complementary to the region ℤ_{p} × ℤ_{p}^{×} that appears in (16). Since this region is also preserved by Γ_{0}(pℤ_{p}), the specialisation map ρ_{k} is Γ_{0}(pℤ_{p})-equivariant and gives rise to homomorphisms, denoted ρ_{k} as well by abuse of notation:

ρ_{k} : S_{∞}(Σ)⟶S_{k}(Σ′).

As in Section 1.3, these specialisation maps allow us to think of S_{∞}(Σ) as a space of families of modular forms of level Σ′ and “varying weights”.

The level structures Σ that interest us will be of the form Σ = Σ_{0}(N^{+}, N^{−}), where N^{+} is any integer that is relatively prime to N^{−}. To describe this compact open subgroup precisely, choose a maximal order R in B satisfying

ι_{p}(R ⊗ ℤ_{p}) = M_{2}(ℤ_{p}).

(Note that such an order is not unique, even up to conjugation by elements of B^{×} in general, since B is a definite quaternion algebra and therefore does not satisfy the Eichler condition.) For each place ℓ of ℚ not dividing N^{−}, the ring R ⊗ ℤ_{ℓ} is isomorphic to the 2 × 2 matrix ring M_{2}(ℤ_{ℓ}). For each ℓ not dividing N^{−}, fix an isomorphism

ι_{ℓ} : B ⊗ ℚ_{ℓ}⟶M_{2}(ℚ_{ℓ})

sending R ⊗ ℤ_{ℓ} to M_{2}(ℤ_{ℓ}). The compact open subgroup Σ_{0}(N^{+}, N^{−}) = ∏_{ℓ} Σ_{ℓ} is then defined by letting Γ_{0}(N^{+}ℤ_{ℓ}) be the subgroup of GL_{2}(ℤ_{ℓ}) consisting of matrices that are upper triangular modulo N^{+}, and setting

(34)  Σ_{ℓ} = \{ \begin{matrix}(R ⊗ ℤ_{ℓ})^{×} & if ℓ|N^{−}, \\ ι_{ℓ}^{−1}(Γ_{0}(N^{+}ℤ_{ℓ})) & otherwise. \end{matrix}

Denote by

𝕏_{0}(N^{+}, N^{−}) = B^{×}\B̂^{×}/Σ_{0}(N^{+}, N^{−})

the corresponding double coset space.

The strong approximation theorem for B ([Vi], p. 61) asserts that, for any compact open subgroup Σ of B̂^{×},

B̂^{×} = B^{×}B_{p}^{×}Σ.

Hence the double coset space 𝕏_{0}(pN^{+}, N^{−}) can be rewritten as

(35)  𝕏_{0}(pN^{+}, N^{−}) = R^{×}\B_{p}^{×}/ι_{p}^{−1}(Γ_{0}(pℤ_{p})) = Γ̃\GL_{2}(ℚ_{p})/Γ_{0}(pℤ_{p}),

where

R := {x ∈ R[1/p] such that ι_{ℓ}(x) ∈ ℤ_{ℓ}[Γ_{0}(Nℤ_{ℓ})], for all ℓ|N^{+}} ,
Γ̃ := ι_{p}(R^{×}).

The ring R is a so-called “Eichler ℤ[1/p]-order of level N^{+}” in B. Such ℤ[1/p]-orders are unique, up to conjugation by elements of B^{×}. Thanks to (35), a modular form in S_{k}(Σ_{0}(pN^{+}, N^{−})) can be viewed as a function ϕ : GL_{2}(ℚ_{p})⟶V_{k} satisfying

ϕ(γbσ) = σ^{−1}ϕ(b), for all γ ∈ Γ̃, b ∈ GL_{2}(ℚ_{p}), and σ ∈ Γ_{0}(pℤ_{p}).

It is this point of view that will be taken from now on.

The spaces S(Σ_{0}(pN^{+}; N^{−}); A) are equipped with an action of Hecke operators T_{ℓ} indexed by the rational primes which do not divide N, defined as in [Gr] for example. If ℓ is a prime that does not divide pN^{+}N^{−}, then the Hecke operator T_{ℓ} on S_{k}(N^{+}, N^{−}) is defined in terms of the double coset decomposition

(36)  GL_{2}(ℤ_{ℓ}) ( \begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix} ) GL_{2}(ℤ_{ℓ}) = ⋃_{a=0}^{ℓ} σ_{a}(ℓ)GL_{2}(ℤ_{ℓ})

by the rule

(37)  (T_{ℓ}ϕ)(b)(P) = ∑_{a=0}^{ℓ} ϕ(bσ_{a}(ℓ))(P |σ_{a}(ℓ)).

The Hecke operator U_{p} plays a particularly important role in our discussion. It is defined in terms of the decomposition

(38)  Γ_{0}(pℤ_{p}) ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) Γ_{0}(pℤ_{p}) = ⋃_{a=0}^{p−1} σ_{a}Γ_{0}(pℤ_{p})

of the double coset attached to the matrix diag(1, p) ∈ GL_{2}(ℚ_{p}) as a disjoint union of left cosets. Since Γ_{0}(pℤ_{p}) is the stabiliser in GL_{2}(ℚ_{p}) of the lattices ℤ_{p}^{2} and ℤ_{p} ⊕ pℤ_{p}, the double coset appearing in (38) consists of matrices in GL_{2}(ℚ_{p}) which send the lattice ℤ_{p}^{2} to ℤ_{p} ⊕pℤ_{p}. Let L(0), L(1), …, L(p−1) denote the p sublattices of index p of ℤ_{p} ⊕ pℤ_{p} which are different from p(ℤ_{p}^{2}). The matrices σ_{a} appearing in (38) can be arranged so that

(39)  σ_{a}(ℤ_{p}^{2}) = ℤ_{p} ⊕ pℤ_{p}, σ_{a}(ℤ_{p} ⊕ pℤ_{p}) = L(a).

Note, for example, that (39) determines σ_{a} uniquely up to multiplication by elements of Γ_{0}(pℤ_{p}) on the right. The Hecke operator U_{p} acting on S(Σ_{0}(pN^{+}, N^{−}); A) is defined by

(40)  (U_{p}ϕ)(b) = ∑_{a=0}^{p−1} σ_{a}ϕ(bσ_{a}).

In particular, if ϕ belongs to S_{k}(Σ_{0}(pN^{+}, N^{−})), then

(41)  (U_{p}ϕ)(b)(P) = ∑_{a=0}^{p−1} ϕ(bσ_{a})(P |σ_{a}),

for any b ∈ GL_{2}(ℚ_{p}), and all P ∈ 𝔓_{k}, while if ϕ belongs to the space S_{∞}(Σ_{0}(N^{+}, N^{−}))^{†}, and has U as a neighborhood of regularity, then

(42)  ∫_{X} F d(U_{p}ϕ)(b) = ∑_{a=0}^{p−1} ∫_{σ_{a}^{−1}X}(F |σ_{a})dϕ(bσ_{a}),

for all compact open X ⊂ L′_{∗} and all homogeneous F of degree k − 2 with k ∈ U.

For future reference we note the following relation

(43)  ℤ_{p}^{×} × pℤ_{p} = ⋃_{a=0}^{p−1} σ_{a}(ℤ_{p}^{×} × pℤ_{p}).

This relation can be rewritten as

(ℤ_{p}^{2})′ ∩ (ℤ_{p} ⊕ pℤ_{p})′ = ⋃_{a=0}^{p−1}(ℤ_{p} ⊕ pℤ_{p})′ ∩ L(a)′,

which can be checked by specialising to any specific collection of lattices satisfying the same inclusion relations as ℤ_{p}^{2}, ℤ_{p} ⊕ pℤ_{p}, L(0), …, L(p − 1), using the fact that GL_{2}(ℚ_{p}) acts transitively on such collections.

2.2. The Bruhat-Tits tree. A modular form η ∈ S(Σ_{0}(N^{+}, N^{−}); A) gives rise to a Γ̃-equivariant Avalued function on the set of ℤ_{p}-lattices in ℚ_{p}^{2}, by setting

(44)  c_{η}(L) = g_{L}η(g_{L}),

where g_{L} is any element of GL_{2}(ℚ_{p}) satisfying g_{L}(ℤ_{p}^{2}) = L. Likewise, if η belongs to S(Σ_{0}(pN^{+}, N^{−}); A), it gives rise to a Γ̃-equivariant A-valued function on the set of pairs (L_{1}, L_{2}) of ℤ_{p}-lattices in ℚ_{p}^{2} for which L_{1} contains L_{2} with index p, by setting

(45)  c_{η}(L_{1}, L_{2}) = g_{L_{1},L_{2}}η(g_{L_{1},L_{2}}),

where g_{L_{1},L_{2}} is any element of GL_{2}(ℚ_{p}) satisfying

(46)  g_{L_{1},L_{2}}(ℤ_{p}^{2}) = L_{1},  g_{L_{1},L_{2}}(ℤ_{p} ⊕ pℤ_{p}) = L_{2}.

The Hecke operators T_{p} and U_{p} acting on S(Σ_{0}(N^{+}, N^{−}); A) and on S(Σ_{0}(pN^{+}, N^{−}); A) respectively admit a particularly simple expression in terms of the associated function on lattices, namely:

(47)  c_{T_{p}η}(L) = ∑_{L̃⊂L} c_{η}(L̃),
(48)  c_{U_{p}η}(L_{1}, L_{2}) = ∑_{L̃⊂L_{2}} c_{η}(L_{2}, L̃),

where the first sum is taken over the p + 1 sublattices of L of index p, and the second sum is taken over the p sublattices of index p of L_{2} which are different from pL_{1}. These formulae are direct consequences of the definitions; for example, (48) follows from (40) and (45).

There are two natural “degeneracy maps”

d_{1}, d_{2} : S(Σ_{0}(N^{+}, N^{−}); A)⟶S(Σ_{0}(pN^{+}, N^{−}); A)

which are described as follows in terms of the associated functions on lattices:

c_{d_{1}(η)}(L_{1}, L_{2}) := c_{η}(L_{1}), c_{d_{2}(η)}(L_{1}, L_{2}) := c_{η}(L_{2}).

These degeneracy maps are compatible with the actions of the Hecke operators T_{n} with p |̸n, but do not intertwine the actions of the operators T_{p} and U_{p} on each side.

An eigenvector ϕ^{♯} ∈ S_{k}(Σ_{0}(N^{+}, N^{−})) for the Hecke operator T_{p} is said to be ordinary if the associated eigenvalue a_{p}(ϕ^{♯}) is a p-adic unit. In that case we can define a p-adic unit a_{p} by the rule

x^{2} − a_{p}(ϕ^{♯})x + p^{k−1} = (x − a_{p})(x − a_{p}^{−1}p^{k−1}).

The form ϕ ∈ S_{k}(Σ_{0}(pN^{+}, N^{−})) defined by

(49)  c_{ϕ}(L_{1}, L_{2}) = c_{ϕ^{♯}}(L_{2}) − a_{p}^{−1}c_{ϕ^{♯}}(pL_{1})

is an eigenvector for U_{p} satisfying U_{p}ϕ = a_{p}ϕ. This can be checked by a direct calculation using (47) and (48) and the fact that c_{ϕ} and c_{ϕ^{♯}} are homogenous of degree k − 2 in the sense that

c_{ϕ^{♯}}(pL) = p^{k−2}c_{ϕ^{♯}}(L), and c_{ϕ}(pL_{1}, pL_{2}) = p^{k−2}c_{ϕ}(L_{1}, L_{2}).

A U_{p}-eigenform ϕ of level Σ_{0}(pN^{+}, N^{−}) which is obtained from an eigenform ϕ^{♯} of level Σ_{0}(N^{+}, N^{−}) as in (49) is said to be old at p.

When k = 2 the functions c_{ϕ} and c_{ϕ^{♯}} are defined on homothety classes of lattices. It is then convenient to view c_{ϕ^{♯}} and c_{ϕ} as functions on the vertices and ordered edges respectively of the Bruhat-Tits tree of GL_{2}(ℚ_{p}). This tree is the graph, denoted T , whose set T_{0} of vertices is in bijection with the homothety classes of ℤ_{p}-lattices in ℚ_{p}^{2}, two such vertices being joined by an unordered edge if the corresponding homothety classes admit representatives which are contained in each other with index p. Write T_{1} for the set of unordered edges of T .

The group GL_{2}(ℚ_{p}), which acts naturally on ℚ_{p}^{2} via left multiplication on column vectors, also acts on T , and this action preserves the adjacency relations between vertices of T .

A vertex in T_{0} is said to be even if its distance from the standard vertex v_{∗} := [ℤ_{p}^{2}] is even, and is said to be odd otherwise. Denoting by E(T ) the set of ordered edges of T (i.e., the set of ordered pairs of adjacent vertices in T ), we say that such an edge is even if its origin is even, and is odd otherwise. Thus we have decompositions

T_{0} = T_{0}^{+} ∪ T_{0}^{−}, E(T ) = E(T )^{+} ∪ E(T )^{−}

of the sets T_{0} and E(T ) respectively into a disjoint union of the subsets of even and odd elements. The group GL_{2}(ℚ_{p}) acts transitively on T_{0} and on T_{1}, while the group SL_{2}(ℚ_{p}) preserves the subsets T_{0}^{+} and E(T )^{+}.

The Petersson scalar product on S_{k}(Σ_{0}(N^{+}, N^{−})).

Let U := ad^{0}(B) be the representation of B^{×} consisting of the elements of B of reduced trace zero, equipped with the right action of B^{×} given by

u · b := b^{−1}ub.

There is a ℚ-valued perfect symmetric pairing

⟨ , ⟩ : U × U⟶ℚ,

defined by ⟨u, v⟩ := (1/2)tr(uv̄), where tr denotes the reduced trace on B and v̄ is the image of v by the canonical involution on B. This pairing is B^{×}-invariant, in the sense that

⟨u · b, v · b⟩ = ⟨u, v⟩

for all b ∈ B^{×}. The map sending u ∈ U to

(50)  P_{u}(x, y) := Trace(ι_{p}(u) ( \begin{matrix}−xy & x^{2} \\ −y^{2} & xy \end{matrix} ) ) ∈ 𝔓_{4}

defines a surjective homomorphism from U ⊗ ℂ_{p} to 𝔓_{4}, which intertwines the right actions of B_{1}^{×}, the group of elements of B^{×} of reduced norm one, and SL_{2}(ℚ_{p}) on the first and second space, respectively. This yields a surjection from Sym^{k/2−1}(U) ⊗ ℂ_{p} to 𝔓_{k}. The pairing ⟨ , ⟩ induces a perfect symmetric pairing ⟨ , ⟩_{k} on Sym^{k/2−1}(U) by setting

(51)  ⟨(u_{1}, …, u_{k/2−1}), (v_{1}, …, v_{k/2−1})⟩_{k} := ∑_{σ∈S_{k/2−1}}⟨u_{1}, v_{σ(1)}⟩ ⋯ ⟨u_{k/2−1}, v_{σ(k/2−1)}⟩,

S_{k/2−1} being the symmetric group on k/2 − 1 letters.

By dualizing this map, and using ⟨ , ⟩_{k} to identify Sym^{k/2−1}(U) ⊗ ℂ_{p} with its ℂ_{p}-dual, we obtain an inclusion of V_{k} into Sym^{k/2−1}(U) ⊗ ℂ_{p}. By an abuse of notation, we denote by ⟨ , ⟩_{k} also the perfect symmetric pairing on V_{k} obtained by restriction.

Given η ∈ S_{k}(Σ_{0}(N^{+}, N^{−})), let c_{η} be the Γ̃-equivariant function on lattices defined earlier in this section. Write |L| for the generalised index of the lattice L in ℤ_{p}^{2}, defined as the power of p which exactly divides the determinant of any element of GL_{2}(ℚ_{p}) sending ℤ_{p}^{2} to L. Since c_{η} is homogenous of degree k − 2, the quantity |L|^{1−k/2}c_{η}(L) depends only on the homothety class v ∈ T_{0} of the lattice L, and will be denoted c̄_{η}(v). Define the Petersson norm of η to be

(52)  ⟨η, η⟩ := ∑_{v∈Γ\T_{0}} w_{v} · ⟨c̄_{η}(v), c̄_{η}(v)⟩_{k},

where w_{v} denotes the order of the stabiliser of v in Γ. Note that ⟨η, η⟩ does not depend on the choice of representatives for the finite set Γ\T_{0}.

For ϕ ∈ S_{k}(Σ_{0}(N^{+}p, N^{−}), it is possible to define its Petersson norm ⟨η, η⟩ in a similar way, with the set Γ\E(T ) replacing the role of Γ\T_{0}. In particular, given ϕ ∈ S_{2}(Σ_{0}(N^{+}p, N^{−}), one has

⟨ϕ, ϕ⟩ := ∑_{e∈Γ\E(T )} w_{e} · ⟨c_{ϕ}(e), c_{ϕ}(e)⟩_{2},

where w_{e} denotes the order of the stabiliser of e in Γ.

2.3. The Jacquet-Langlands correspondence. Recall the Hecke congruence group Γ_{0}(N) of SL_{2}(ℤ), and let S_{k}(Γ_{0}(N)) denote the vector space of classical cusp forms of weight k for Γ_{0}(N) with rational Fourier expansion, tensored with ℚ_{p}. Recall the factorisation N = pM = pN^{+}N^{−} of N that was introduced earlier. A modular form in S_{k}(Γ_{0}(N)) is said to be old at N^{−} if it can be expressed as a linear combination of modular forms of the form g(dz) where g(z) is of level not divisible by N^{−}. The orthogonal complement of this space of N^{−}-old forms is called the space of N^{−}-new forms on Γ_{0}(N), and is denoted by S_{k}^{new−N^{−}}(Γ_{0}(N)).

The following theorem, a special case of the Jacquet-Langlands correspondence, explains the interest of modular forms on definite quaternion algebras in our study.

Theorem 2.4. There exist Hecke-equivariant isomorphisms

S_{k}(Σ_{0}(N^{+}, N^{−})) ⟶ S_{k}^{new−N^{−}}(Γ_{0}(M))
S_{k}(Σ_{0}(pN^{+}, N^{−})) ⟶ S_{k}^{new−N^{−}}(Γ_{0}(N)).

The assumption that the elliptic curve E of the Introduction has at least one prime ≠ p of multiplicative reduction allows us to write its conductor as N = pN^{+}N^{−}, whose factors satisfy the hypotheses of (32). Hence, the normalised eigenform f_{k}^{♯} ∈ S_{k}^{new−N^{−}}(Γ_{0}(M)) of weight k ∈ U ∩ ℤ^{>2} associated to f via Hida’s theory in (12) and (13) corresponds to a modular form ϕ_{k}^{♯} ∈ S_{k}(Σ_{0}(N^{+}, N^{−})) which is an eigenvector for the Hecke operators with the same associated eigenvalues. The form ϕ_{k}^{♯} is only determined up to multiplication by a non-zero scalar. We normalise it in such a way that

(53)  ⟨ϕ_{k}^{♯}, ϕ_{k}^{♯}⟩ = 1,

so that ϕ_{k}^{♯} is now determined up to sign. Let ϕ_{k} ∈ S_{k}(Σ_{0}(pN^{+}, N^{−})) be the eigenvector for U_{p} which is obtained from ϕ_{k}^{♯} via (49), and let c_{k} and c_{k}^{♯} denote the corresponding functions on lattices.

When k = 2, let ϕ_{2} ∈ S_{2}(Σ_{0}(pN^{+}, N^{−})) be a modular form associated to f via Theorem 2.4. Since ϕ_{2} is new at p, we set ϕ_{2}^{♯} := 0. We do not impose a normalising condition on ϕ_{2}, but assume instead that it takes its values in ℤ. Note then that the quantity ⟨ϕ_{2}, ϕ_{2}⟩ is well-defined up to multiplication by elements of (ℚ^{×})^{2}. Occasionally we will write ϕ instead of ϕ_{2}.

The following result plays the role of Theorem 1.5 in the context of automorphic forms on definite quaternion algebras.

Theorem 2.5. There exists a family ϕ_{∞} ∈ S_{∞}^{†}(Σ_{0}(N^{+}, N^{−})) such that (1) ρ_{2}(ϕ_{∞}) = ϕ_{2};

(2) Let U be a neighborhood of regularity for the measures ϕ_{∞}(g) ∈ 𝔻_{∗}^{†}. For all k ∈ U ∩ ℤ^{≥2}, there exists a scalar λ_{B}(k) ∈ ℂ_{p} such that

ρ_{k}(ϕ_{∞}) = λ_{B}(k)ϕ_{k}.

In other words, for all P ∈ 𝔓_{k} and for all b ∈ GL_{2}(ℚ_{p}),

(54)  ∫_{ℤ_{p}^{×}×pℤ_{p}} P (x, y)dϕ_{∞}(b) = λ_{B}(k)ϕ_{k}(b)(P).

Proof. The proof of Theorem (5.13) of [GS] adapts to the present context mutatis mutandis, after replacing “modular symbols” by “functions on B̂^{×}”. See Section 6 of [GS] for more details. Further details on the case of definite quaternion algebras are given in some (as yet unpublished) notes of Glenn Stevens. □

Remark 2.6. Theorem 2.5 is a consequence of the more general “Jacquet-Langlands correspondence for families” established in [Ch] which applies also to non-ordinary eigenforms. The methods developed in Section 6 of [GS] is sufficient to handle the case of interest in this paper, which is only concerned with Hida families of ordinary eigenforms, and this allows us to work with measures (elements of 𝔻^{†}) rather than the locally analytic functions and distributions that appear in [Bu] and [Ch].

2.4. Measure-valued forms. Recall the space 𝔻 of compactly supported measures on W, which contains 𝔻_{∗} in a natural way and is also equipped with a natural left action of the group GL_{2}(ℚ_{p}) extending the action of M_{2}(ℤ_{p}) on 𝔻_{∗}.

Guided by the constructions of Section 1.3, (cf. in particular Proposition 1.8), we associate to the form ϕ_{∞} of the previous section a collection of measures μ_{L} ∈ 𝔻^{†} indexed by the ℤ_{p}-lattices in ℚ_{p}^{2}, by setting

(55)  μ_{L} := g_{L}ϕ_{∞}(g_{L}),

where g_{L} ∈ GL_{2}(ℚ_{p}) is any matrix satisfying g_{L}(ℤ_{p}^{2}) = L. The measures μ_{L} satisfy the basic property

(56)  ∫_{X} F dμ_{L} = ∫_{g_{L}^{−1}X}(F |g_{L})dϕ_{∞}(g_{L}),

for all compact open X ⊂ W and for all homogeneous F of degree k with k ∈ U ∩ ℤ^{≥2}.

We note the following properties of the μ_{L}:

Lemma 2.7. The measures μ_{L} are supported on L′, for all L.

Proof. Property (56) implies that μ_{L} is supported on g_{L}((ℤ_{p}^{2})′) = L′ since the measures ϕ_{∞}(g) are supported on (ℤ_{p}^{2})′. □

Lemma 2.8. For all γ ∈ Γ̃, and all homogeneous functions F (x, y) of degree k − 2 with k ∈ U,

∫_{γX}(F |γ^{−1})dμ_{γL} = ∫_{X} F dμ_{L}.

In particular, if P is a homogeneous polynomial of degree k − 2 with k ∈ U ∩ ℤ^{≥2},

∫_{pL′} P dμ_{pL} = p^{k−2} ∫_{L′} P dμ_{L}.

Proof. This follows from the invariance property of ϕ_{∞} under left multiplication by Γ̃. □

The following is an analogue of Lemma 1.10 of Section 1.3.

Lemma 2.9. Suppose that L_{2} ⊂ L_{1} is a sublattice of index p in L_{1}. Then for all k ∈ U, and for all homogenous functions F (x, y) of degree k − 2, we have

∫_{L′_{1}∩L′_{2}} F dμ_{L_{2}} = a_{p}(k) ∫_{L′_{1}∩L′_{2}} F dμ_{L_{1}}.

Proof. Let 𝔻′ be the Λ-module of measures on ℤ_{p}^{×} × pℤ_{p}, and, as a natural extension of the notations adopted previously, write (𝔻′)^{†} = 𝔻′ ⊗ Λ^{†}. The restriction map from 𝔻_{∗} to 𝔻′ is Γ_{0}(pℤ_{p})-equivariant and induces an isomorphism

S_{∞}(Σ_{0}(N^{+}, N^{−}))⟶S(Σ_{0}(pN^{+}, N^{−}); 𝔻′).

The same holds, of course, for the corresponding modules tensored with Λ^{†}. Let

ϕ′_{∞} ∈ S(Σ_{0}(pN^{+}, N^{−}); 𝔻′) ⊗ Λ^{†}

be the image of ϕ_{∞} under this isomorphism. Note that the matrices σ_{a} that appear in formula (40) defining the action of U_{p} satisfy

σ_{a}^{−1}(ℤ_{p}^{×} × pℤ_{p}) ⊃ ℤ_{p}^{×} × pℤ_{p}.

By formula (42) and the fact that ϕ′_{∞} is an eigenvector for U_{p} with eigenvalue a_{p}(k), we have

∑_{a=0}^{p−1} ∫_{ℤ_{p}^{×}×pℤ_{p}}(F |σ_{a})dϕ′_{∞}(bσ_{a}) = a_{p}(k) ∫_{ℤ_{p}^{×}×pℤ_{p}} F dϕ′_{∞}(b),

for all b ∈ GL_{2}(ℚ_{p}), for all k ∈ U and all homogenous F of degree k − 2. The same relation holds with ϕ′_{∞} replaced by ϕ_{∞}. By the definition of the action of GL_{2}(ℚ_{p}) on 𝔻, this equation can be rewritten as

∑_{a=0}^{p−1} ∫_{bσ_{a}(ℤ_{p}^{×}×pℤ_{p})}(F |b^{−1})d(bσ_{a}ϕ_{∞}(bσ_{a})) = a_{p}(k) ∫_{b(ℤ_{p}^{×}×pℤ_{p})}(F |b^{−1})d(bϕ_{∞}(b)).

Now let b be any matrix in GL_{2}(ℚ_{p}) such that b(ℤ_{p}^{2}) = L_{1} and b(ℤ_{p} ⊕ pℤ_{p}) = L_{2}, so that b(ℤ_{p}^{×} × pℤ_{p}) = L′_{1} ∩ L′_{2}. Then the previous displayed equation becomes:

∑_{a=0}^{p−1} ∫_{bσ_{a}(ℤ_{p}^{×}×pℤ_{p})}(F |b^{−1})dμ_{L_{2}} = a_{p}(k) ∫_{L′_{1}∩L′_{2}}(F |b^{−1})dμ_{L_{1}}.

Multiplying equation (43) by b on the left gives the following expression for L′_{1} ∩ L′_{2} as a disjoint union:

L′_{1} ∩ L′_{2} = ⋃_{a=0}^{p−1} bσ_{a}(ℤ_{p}^{×} × pℤ_{p}).

Hence it follows that

∫_{L′_{1}∩L′_{2}}(F |b^{−1})dμ_{L_{2}} = a_{p}(k) ∫_{L′_{1}∩L′_{2}}(F |b^{−1})dμ_{L_{1}}.

Since this is true for any homogenous F of degree k − 2, the result follows. □

Lemma 2.10. For any L_{1} ⊃ L_{2} with index p, and for any polynomial P of degree k − 2 with k ∈ U ∩ ℤ^{≥2},

∫_{L′_{1}∩L′_{2}} P dμ_{L_{1}} = λ_{B}(k)c_{k}(L_{1}, L_{2})(P).

Proof. The defining property of ϕ_{∞} states that

∫_{ℤ_{p}^{×}×pℤ_{p}} P dϕ_{∞}(g) = λ_{B}(k)ϕ_{k}(g)(P).

This implies that

∫_{g(ℤ_{p}^{×}×pℤ_{p})}(P |g^{−1})d(gϕ_{∞}(g)) = λ_{B}(k)gϕ_{k}(g)(P |g^{−1}).

Taking g to be the element g_{L_{1},L_{2}} of equation (46), this gives

∫_{L′_{1}∩L′_{2}}(P |g^{−1})dμ_{L_{1}} = λ_{B}(k)c_{k}(L_{1}, L_{2})(P |g^{−1}).

The result follows. □

Proposition 2.11. For any lattice L, for any k ∈ U ∩ ℤ^{≥2}, and any polynomial P ∈ 𝔓_{k}, the equality

∫_{L′} P (x, y)dμ_{L}(x, y) = λ_{B}(k)a_{p}(k)(1 − a_{p}(k)^{−2}p^{k−2}) · c_{k}^{♯}(L)(P)

holds.

Proof. Write L_{1} for L, and let L_{2} be any lattice which is contained in L_{1} with index p. Then since L′_{1} can be expressed as the disjoint union of L′_{1} ∩ L′_{2} and \frac{1}{p}L′_{2} ∩ L′_{1}, we have

∫_{L′_{1}} P dμ_{L_{1}} = ∫_{L′_{1}∩L′_{2}} P dμ_{L_{1}} + ∫_{\frac{1}{p} L′_{2}∩L′_{1}} P dμ_{L_{1}}

= ∫_{L′_{1}∩L′_{2}} P dμ_{L_{1}} + a_{p}(k) ∫_{\frac{1}{p} L′_{2}∩L′_{1}} P dμ_{\frac{1}{p} L_{2}},

where the last equality follows from Lemma 2.9. By Lemma 2.10,

(57)  ∫_{L′_{1}} P dμ_{L_{1}} = λ_{B}(k) (c_{k}(L_{1}, L_{2})(P) + a_{p}(k)c_{k}(\frac{1}{p}L_{2}, L_{1})(P)) .

Equation (49) allows us to rewrite c_{k} in terms of c_{k}^{♯}, on any pair (L_{1}, L_{2}) of lattices with L_{1} ⊃ L_{2} with index p:

(58)  c_{k}(L_{1}, L_{2}) = c_{k}^{♯}(L_{2}) − a_{p}(k)^{−1}c_{k}^{♯}(pL_{1}).

Proposition 2.11 now follows from (57) and (58), by a direct calculation. □

2.5. Forms of weight two and p-adic boundary measures. In this section, let c_{ϕ} : E(T )⟶ℚ_{p} denote the Γ̃-invariant function on E(T ) associated to ϕ = ϕ_{2} ∈ S_{2}(Σ_{0}(pN^{+}, N^{−})). Recall that the form ϕ has been normalised so that ϕ (and therefore also c_{ϕ}) take values in ℤ.

Let

(59)  π : W⟶ℙ_{1}(ℚ_{p})

denote the natural projection which sends the vector (x, y) to the element x/y ∈ ℙ_{1}(ℚ_{p}). If L is any ℤ_{p}- lattice in ℚ_{p}^{2}, the restriction of π to L′ has compact fibers and therefore the pushforward of the measure μ_{L} by π is defined, by the usual rule

∫_{X} h(t)d(π_{∗}μ_{L})(t) := ∫_{π^{−1}(X)} h(x/y)dμ_{L}(x, y),

for any compact open X ⊂ ℙ_{1}(ℚ_{p}) and any continuous function h on ℙ_{1}(ℚ_{p}).

Choose any lattice L whose homothety class corresponds to an even vertex of T , and consider the pushforward

μ_{ϕ} := π_{∗}(μ_{L}).

Lemma 2.12. The measure μ_{ϕ} is independent of the choice of (even) L that was made to define it.

Proof. Suppose that L_{1} contains L_{2} with index p, and that L_{1} corresponds to an even vertex of T , so that L_{2} corresponds to an odd vertex. The region ℙ_{1}(ℚ_{p}) can be written as a disjoint union

ℙ_{1}(ℚ_{p}) = π(L′_{1} ∩ L′_{2}) ∪ π (\frac{1}{p}L′_{2} ∩ L′_{1}) .

Hence by Lemma 2.9, we have

π_{∗}(μ_{L_{2}}) = a_{p}(2)π_{∗}(μ_{L_{1}}) on π(L′_{1} ∩ L′_{2});
π_{∗}(μ_{L_{2}}) = a_{p}(2)^{−1}π_{∗}(μ_{L_{1}}) on π(\frac{1}{p}L′_{2} ∩ L′_{1}).

Since ϕ_{2} corresponds to a form which is new at p, we have a_{p}(2) = ±1, i.e., a_{p}(2) = a_{p}(2)^{−1}. Hence π_{∗}(μ_{L_{1}}) = a_{p}(2)π_{∗}(μ_{L_{2}}). The result follows after noting that π_{∗}(μ_{L}) depends only on the homothety class of L, and that T is connected. □

Remark 2.13. The parity restriction in the definition of μ_{ϕ} is only needed when a_{p}(2) = −1, which occurs when the modular forms f and ϕ correspond to an elliptic curve over ℚ with non-split multiplicative reduction at p.

Let Γ ⊂ SL_{2}(ℚ_{p}) denote the subgroup of Γ̃ of elements of determinant 1.

Corollary 2.14. The measure μ_{ϕ} is invariant under the action of Γ, and satisfies

μ_{ϕ}(ℙ_{1}(ℚ_{p})) = 0.

Proof. Any γ ∈ Γ sends the even lattice L to another even lattice γL. The first part of the corollary is therefore a direct consequence of Lemmas 2.8 and 2.12. The second follows from Proposition 2.11 in the case k = 2 and P = 1. □

Let H_{p} := ℙ_{1}(ℂ_{p})−ℙ_{1}(ℚ_{p}) denote the p-adic upper half-plane equipped with its rigid analytic structure. Following [Te], we associate to μ_{ϕ} the rigid analytic function on H_{p} defined by

(60)  f_{ϕ}(z) := ∫_{ℙ_{1}(ℚ_{p})} \frac{dμ_{ϕ}(t)}{(t − z)}.

The fact that μ_{ϕ} is invariant under the action of Γ implies that the rigid analytic differential form ω_{ϕ} := f_{ϕ}(z)dz is also invariant under this group, i.e., that f_{ϕ} is a modular form of weight two on H_{p}/Γ in the sense of [Da], Section 5.2.

As in Section 5.3. of [Da], we define the Coleman line integral attached to ω_{ϕ} by the rule

(61)  ∫_{τ_{1}}^{τ_{2}}ω_{ϕ} := ∫_{ℙ_{1}(ℚ_{p})}log (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{ϕ}(t),

where log : ℂ_{p}^{×}⟶ℂ_{p} is Iwasawa’s branch of the p-adic logarithm, satisfying log(p) = 0. (Note that one could equally well have chosen any other branch of the p-adic logarithm, and in Section 2.7 we will see that another choice is in fact more appropriate.) This integral satisfies the following properties:

∫_{τ_{1}}^{τ_{2}}ω_{ϕ} + ∫_{τ_{2}}^{τ_{3}}ω_{ϕ} = ∫_{τ_{1}}^{τ_{3}}ω_{ϕ},

as well as

∫_{γτ_{1}}^{γτ_{2}}ω_{ϕ} = ∫_{τ_{1}}^{τ_{2}}ω_{ϕ}, for all γ ∈ Γ.

Since c_{ϕ} takes values in ℤ, Coleman’s line integral (61) admits a multiplicative refinement which is more precise and has the virtue of not depending on the choice of a p-adic logarithm. It is defined by formally exponentiating the expression for the Coleman line integral given in (61),

(62)  × ∫_{τ_{1}}^{τ_{2}}ω_{ϕ} := × ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{ϕ}(t),

where the expression on the right is a “multiplicative integral”, defined as a limit of Riemann products rather than sums:

(63)  × ∫_{ℙ_{1}(ℚ_{p})} g(t)dμ(t) := lim_{{U_{α}}} ∏_{α} g(t_{α})^{μ(U_{α})}.

In this formula, the limit is taken over increasingly fine coverings of ℙ_{1}(ℚ_{p}) by mutually disjoint compact open subsets U_{α}, and the t_{α} are sample points in U_{α}. We emphasize that (63) only makes sense if μ is a ℤ-valued measure on ℙ_{1}(ℚ_{p}).

Let ℚ_{p^{2}} denote the quadratic unramified extension of ℚ_{p}, and let O_{p^{2}} be its ring of integers. Any τ in ℚ_{p^{2}} − ℚ_{p} gives rise to a ℤ_{p}-lattice

L_{τ} := {(x, y) ∈ ℚ_{p}^{2} such that x − τy belongs to O_{p^{2}}}.

A direct calculation shows that, for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℚ_{p}),

(64)  L_{γτ} = p^{ord_{p}(cτ+d)}γL_{τ} ,

so that in particular L_{γτ} and γL_{τ} are homothetic lattices. In other words, if v_{τ} ∈ T_{0} denotes the vertex of T associated to L_{τ}, we have

(65)  v_{γτ} = γv_{τ} , for all γ ∈ GL_{2}(ℚ_{p}).

The set of τ ∈ ℚ_{p^{2}} − ℚ_{p} for which v_{τ} is an even vertex of T is called the quadratic unramified upper halfplane, and is denoted H′_{p}. Equation (65) shows that PSL_{2}(ℚ_{p}) acts on H′_{p} by Möbius transformations.

We now record a useful formula for the p-adic valuation of the multiplicative Coleman line integral. We do this in the case where the endpoints τ_{1} and τ_{2} belong to H′_{p}. Given an ordered edge e ∈ E(T ), define the parity function |e| by the rule

|e| = \{ \begin{matrix}0 & if e is even; \\ 1 & if e is odd, \end{matrix}

and recall that a_{p} := a_{p}(2) = ±1.

Proposition 2.15. For all τ_{1}, τ_{2} ∈ H′_{p},

ord_{p} (× ∫_{τ_{1}}^{τ_{2}}ω_{ϕ}) = ∑_{e:v_{τ_{1}} →v_{τ_{2}}} a_{p}^{|e|}c_{ϕ}(e),

where the sum is taken over the ordered edges in the path joining v_{τ_{1}} to v_{τ_{2}}.

Proof. See Proposition 2.5 of [BDG]. □

2.6. Indefinite integrals. Given x ∈ ℚ_{p^{2}}^{×}, write ⟨x⟩ for the principal unit attached to x, defined as the unique element of 1 + pO_{p^{2}} satisfying

x = p^{ord_{p}(x)}ζ_{x}⟨x⟩,

where ζ_{x} is a root of unity.

Let τ be any element of H′_{p}. We make the following key definition.

Definition 2.16. The indefinite integral attached to τ and ϕ is defined to be

∫ ^{τ}ω_{ϕ} := ∫_{L′_{τ}} log(x − τy)dμ_{L_{τ}} (x, y) := \frac{d}{dk} (∫_{L′_{τ}} ⟨x − τy⟩^{k−2}dμ_{L_{τ}} )_{k=2} .

We now collect some properties of this indefinite integral. The first of these is a basic Γ-invariance property.

Proposition 2.17. For all γ ∈ Γ, and all τ ∈ H′_{p},

∫ ^{γτ}ω_{ϕ} = ∫ ^{τ}ω_{ϕ}.

Proof. Write t = ord_{p}(cτ + d), where γ := ( \begin{matrix}a & b \\ c & d \end{matrix} ). By definition,

(66)  ∫ ^{γτ}ω_{ϕ} = ∫_{L′_{γτ}} log(x − γτy)dμ_{L_{γτ}} (x, y)
(67)  = ∫_{p^{t}γL′_{τ}} − log(cτ + d) + log(x − τy|γ^{−1})dμ_{p^{t}γL_{τ}} (x, y).

Note that the first function in this integral is a constant, while the second is periodic under the substitution (x, y) ↦ (px, py). Hence

(68)  ∫ ^{γτ}ω_{ϕ} = ∫_{γL′_{τ}} log(x − τy|γ^{−1})dμ_{γL_{τ}} (x, y)
(69)  = ∫_{L′_{τ}} log(x − τy)dμ_{L_{τ}} (x, y) = ∫ ^{τ}ω_{ϕ},

where the last line follows from Lemma 2.8 (which, although it is stated only for integrands which are homogeneous of weight k − 2 with k ∈ U, extends to the logarithmic integrand appearing in (68) thanks to Definition 2.16). □

Recall the Hecke operator T_{ℓ} (for ℓ a prime, say, not dividing N) of equation (37), which gives rise to a correspondence of degree ℓ + 1 on the group Div(H_{p}/Γ) of divisors on H_{p}/Γ. We extend the indefinite integral by additivity to Div(H_{p}/Γ).

Proposition 2.18. For all primes ℓ which do not divide pN^{+}N^{−},

∫ ^{T_{ℓ}τ}ω_{ϕ} = a_{ℓ} ∫ ^{τ}ω_{ϕ}.

Proof. By replacing U by a smaller neighborhood, we may assume that its elements are ≡ 2 (mod p^{2} − 1), so that ⟨x − τy⟩^{k−2} = (x − τy)^{k−2} on L′_{τ}. Hence, by definition of the indefinite integral

(70)  ∫ ^{τ}ω_{ϕ} = \frac{d}{dk} (∫_{L′_{τ}} (x − τy)^{k−2}dμ_{L_{τ}} (x, y)) .

On the other hand, the definition of the Hecke operator T_{ℓ}, combined with the Hecke-equivariance of the specialisation map ρ_{k}, shows that

(71)  ∫ ^{T_{ℓ}τ}ω_{ϕ} = \frac{d}{dk} (λ_{B}(k)(T_{ℓ}c_{k}^{♯})(L_{τ})((x − τy)^{k−2})) .

Since c_{k}^{♯} is an eigenvector for T_{ℓ} with eigenvalue a_{ℓ}(k),

∫ ^{T_{ℓ}τ}ω_{ϕ} = \frac{d}{dk} (λ_{B}(k)a_{ℓ}(k)c_{k}^{♯}(L_{τ})((x − τy)^{k−2})

(72)  = \frac{d}{dk} (a_{ℓ}(k) ∫_{L′_{τ}} (x − τy)^{k−2}dμ_{L_{τ}} (x, y)) .

The result follows from (70) and (72) by applying the product rule, after noting that the integral appearing in (72) vanishes at k = 2. □

The next proposition justifies the notation of the indefinite integral by relating it to the Coleman line integral of ω_{ϕ} introduced earlier.

Proposition 2.19. For all τ_{1}, τ_{2} ∈ H′_{p}, with v_{j} := v_{τ_{j}}, we have

∫ ^{τ_{2}}ω_{ϕ} − ∫ ^{τ_{1}}ω_{ϕ} = ∫_{τ_{1}}^{τ_{2}}ω_{ϕ} + 2a′_{p}(2) ∑_{e:v_{1}→v_{2}} a_{p}^{|e|}c_{ϕ}(e).

Remark 2.20. Since c_{ϕ} takes values in ℤ, this formula can be restated in terms of the multiplicative integral attached to ϕ, in light of Proposition 2.15, as

(73)  ∫ ^{τ_{2}}ω_{ϕ} − ∫ ^{τ_{1}}ω_{ϕ} = (log +2a′_{p}(2) ord_{p}) (× ∫_{τ_{1}}^{τ_{2}}ω_{ϕ}) .

Before giving the proof of Proposition 2.19 it will be useful to record the following two lemmas.

Lemma 2.21. For all α ∈ Λ^{†}, for all τ ∈ H′_{p}, and for all compact open X ⊂ L′,

∫_{X} log(x − τy)d(αμ_{L}) = α′(2)μ_{L}(X) + α(2) ∫_{X} log(x − τy)dμ_{L}.

Proof. By definition

∫_{X} log(x − τy)d(αμ_{L}) = \frac{d}{dk} (α(k) ∫_{X}(x − τy)^{k−2}dμ_{L})_{k=2} .

The result follows from the product formula for the derivative appearing in this expression. □

Corollary 2.22. If L_{2} is an index p sublattice of L_{1}, and e = ([L_{1}], [L_{2}]) denotes the corresponding ordered edge of T , then

∫_{L′_{1}∩L′_{2}} log(x − τy)d(μ_{L_{2}} − μ_{L_{1}}) = a′_{p}(2)c_{ϕ}(e)

if ϕ is split multiplicative, and

∫_{L′_{1}∩L′_{2}} log(x − τy)d(μ_{L_{2}} − μ_{L_{1}}) = a′_{p}(2)c_{ϕ}(e) − 2 ∫_{L′_{1}∩L′_{2}} log(x − τy)dμ_{L_{1}}

if ϕ is non-split multiplicative.

Proof. By Lemma 2.9,

∫_{L′_{1}∩L′_{2}}log(x − τy)d(μ_{L_{2}} − μ_{L_{1}}) = ∫_{L′_{1}∩L′_{2}}log(x − τy)d((a_{p}(k) − 1)μ_{L_{1}}).

The result now follows from Lemma 2.21. □

Lemma 2.23. Let L_{1} and L_{2} be even lattices in ℚ_{p}^{2} and let v_{1} and v_{2} denote the corresponding vertices of T . Then

∫_{W} log(x − τy)d(μ_{L_{2}} − μ_{L_{1}})(x, y) = 2a′_{p}(2) ∑_{e:v_{1}→v_{2}} a_{p}^{|e|}c_{ϕ}(e).

Proof. We treat the case where ϕ is split multiplicative so that a_{p}(2) = 1. Assume first that L_{1} contains L_{2} with index p, and that v_{1} is even (so that the adjacent vertex v_{2} is odd.) Let e be the ordered edge of T having v_{1} as source and v_{2} as target. The measure μ_{L_{1}} − μ_{L_{2}} is supported on L′_{1} ∪ L′_{2}, which can be expressed as a disjoint union

L′_{1} ∪ L′_{2} = (L′_{1} ∩ L′_{2}) ∪ (\frac{1}{p}L′_{2} ∩ L′_{1}) ∪ (L′_{2} ∩ pL′_{1}).

By Corollary 2.22,

(74)  ∫_{L′_{1}∩L′_{2}} log(x − τy)d(μ_{L_{2}} − μ_{L_{1}})(x, y) = a′_{p}(2)c_{ϕ}(e).

Furthermore, we note that the sets \frac{1}{p}L′_{2} ∩L′_{1} and (L′_{2} ∩pL′_{1}) are disjoint from the supports of the measures μ_{L_{2}} and μ_{L_{1}} respectively. Therefore

(75)  ∫_{\frac{1}{p} L′_{2}∩L′_{1}} log(x − τy)d(μ_{L_{2}} − μ_{L_{1}}) = − ∫_{\frac{1}{p} L′_{2}∩L′_{1}} log(x − τy)dμ_{L_{1}}

(76)  = − ∫_{L′_{2}∩pL′_{1}} log(x − τy)dμ_{pL_{1}}
(77)  ∫_{L′_{2}∩pL′_{1}} log(x − τy)d(μ_{L_{2}} − μ_{L_{1}}) = ∫_{L′_{2}∩pL′_{1}} log(x − τy)dμ_{L_{2}}

The result now follows by adding together the expressions for each of the three contributions given in (74), (76), and (77), invoking Corollary 2.22 to further simplify the sum of (76) and (77). □

We are now ready to prove Proposition 2.19.

Proof. Set L_{j} := L_{τ_{j}} for j = 1, 2. Observe that

(78)  ∫ ^{τ_{2}}ω_{ϕ} − ∫ ^{τ_{1}}ω_{ϕ} = ∫_{W} log (\frac{x − τ_{2}y}{x − τ_{1}y} ) dμ_{L_{1}}(x, y)
+ ∫_{W} log(x − τ_{2}y)d(μ_{L_{2}} − μ_{L_{1}})(x, y).

The first integral appearing in the right hand side of (78) involves a function which is constant along the fibers of π. This integral is therefore equal to

∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − τ_{2}}{t − τ_{1}} ) dμ_{ϕ}(t) = ∫_{τ_{1}}^{τ_{2}}ω_{ϕ}.

As for the second integral appearing in the right hand side of (78), Lemma 2.23 shows that

∫_{W} log(x − τ_{2}y)d(μ_{L_{2}} − μ_{L_{1}})(x, y) = 2a′_{p}(2) ∑_{e:v_{1}→v_{2}} a_{p}^{|e|}c_{ϕ}(e).

The proof of Proposition 2.19 follows. □

2.7. Elliptic curves and the L-invariant. This section presents an exposition of the material developed in [BDI] in the more general setting of modular forms of arbitrary even weight, specialised to the setting of weight two. This is done for completeness, and for the convenience of the reader, because the weight two setting can be treated with less machinery. Notably the “locally analytic distributions” required in [BDI] are avoided in the more elementary treatment of the weight two situation given here.

The assumption that the elliptic curve E of the Introduction has at least one prime ≠ p of multiplicative reduction allows us to write its conductor as N = pN^{+}N^{−}, whose factors satisfy the hypotheses of (32).

We now give a formula for the L-invariant of E in terms of the multiplicative line integrals attached to ω_{ϕ}, arising from the p-adic uniformisation theory of Cerednik and Drinfeld. Let Q be an infinite discrete subgroup of ℚ_{p}^{×}. The slope of Q is defined by choosing an element q ∈ Q with ord_{p}(q) ≠ 0, and setting

L_{Q} := \frac{log(q)}{ord_{p}(q)}.

Note that

(1) The slope of Q does not depend on the choice of element q that was made to define it, and in fact depends only on the commensurability class of the lattice Q.

(2) If E is an elliptic curve having multiplicative reduction at p, and Q_{E} is its lattice of Tate periods,

then by definition

L_{E} = L_{Q_{E}}

is the L-invariant attached to E.

Theorem 2.24. Let Q_{ϕ} be the multiplicative subgroup of periods of the rigid analytic differential ω_{ϕ}, defined by

Q_{ϕ} := {× ∫_{τ}^{γτ}ω_{ϕ} such that τ ∈ H′_{p}, and γ ∈ Γ} .

Then Q_{ϕ} is a lattice in ℚ_{p}^{×} which is commensurable with Q_{E}. In particular,

L_{Q_{ϕ}} = L_{E}.

Proof. This follows from the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves, as it is explained for example in [GvdP]. Some more details are also given in Section 4.2. □

We use the notions that have been developed so far to give a new proof of equation (5) of the introduction. This equation was originally proved by Greenberg and Stevens, by a different approach.

Theorem 2.25. L_{E} = −2a′_{p}(2).

Proof. Choose an element γ ∈ Γ in such a way that the associated multiplicative period

q_{γ} := × ∫_{τ}^{γτ}ω_{ϕ}

generates a lattice in ℚ_{p}^{×}. By Theorem 2.24,

(79)  L_{E} = \frac{log(q_{γ})}{ord_{p}(q_{γ})}.

Now we observe that

(1) By Proposition 2.17,

∫ ^{γτ}ω_{ϕ} − ∫ ^{τ}ω_{ϕ} = 0.

(2) On the other hand, by equation (73),

(80)  ∫ ^{γτ}ω_{ϕ} − ∫ ^{τ}ω_{ϕ} = log(q_{γ}) + 2a′_{p}(2) ord_{p}(q_{γ}).

By combining these two facts it follows that

(81)  \frac{log(q_{γ})}{ord_{p}(q_{γ})} = −2a′_{p}(2).

Theorem 2.25 now follows from (79) and (81). □

Corollary 2.26. If log_{q} denotes the branch of the p-adic logarithm which vanished on q, where Q_{E} = ⟨q⟩ modulo torsion. Then

∫ ^{τ_{2}}ω_{ϕ} − ∫ ^{τ_{1}}ω_{ϕ} = log_{q} (× ∫_{τ_{1}}^{τ_{2}}ω_{ϕ}) .

Proof. This follows from equations (73) and (81). □

3. Hida p-adic L-functions attached to imaginary quadratic fields

This chapter attaches a p-adic L-function to the Hida family ϕ_{∞} and to an imaginary quadratic field K, using the system of measures {μ_{L}} constructed in Section 2.

3.1. Optimal embeddings and special values. Let N = pN^{+}N^{−} be the factorization introduced in equation (32). Recall that the factors p, N^{+} and N^{−} are pairwise relatively prime, and that N^{−} is a square-free product of an odd number of prime factors. We maintain the notations of Section 2. Thus B denotes the definite quaternion algebra of discriminant N^{−}∞, and R is a fixed Eichler ℤ[1/p]-order of level N^{+} in B. We also fix an Eichler ℤ-order \underline{R} of B of level N^{+}, choosing it in such a way that

\underline{R}[1/p] = R.

As usual let ˆ \underline{R} := \underline{R} ⊗ ℤ̂ denote the adelisation of \underline{R}.

Definition 3.1. An imaginary quadratic field K is said to be admissible relative to the factorisation (32) if it satisfies

(1) All primes dividing N^{+}, resp. N^{−} are split, resp. inert in K.

(2) The prime p is unramified in K.

Fix an admissible imaginary quadratic field K, of discriminant D < 0 say. Let O_{K} and O := O_{K}[1/p] denote its ring of integers and p-integers respectively. Denote by hom(K, B) the set of ℚ-algebra homomorphisms from K to B.

Definition 3.2. An optimal embedding of K into B of level N^{+} is a pair (Ψ, b) ∈ hom(K, B) × (B̂^{×}/ ˆ\underline{R}^{×}) satisfying

(82)  Ψ(K) ∩ (b ˆ\underline{R}b^{−1}) = Ψ(O_{K}).

The assumption that K is admissible implies the existence of such optimal embeddings. More precisely, the condition in Definition 3.1 that all the primes dividing N^{−} are inert in K implies that the set hom(K, B) is non-empty. Given Ψ ∈ hom(K, B), the possibility of producing b ∈ B̂^{×} satisfying (82) is then guaranteed by the theory of local embeddings, in light of the assumption that the primes dividing N^{+} are split in K. See Section 3 of [Gr] for more details.

Occasionally we commit an abuse of notation and use the symbol Ψ to denote the optimal embedding (Ψ, b), omitting the extra datum of b ∈ B̂^{×} when this results in no ambiguity. It should be kept in mind that such a b is always part of the data associated to an optimal embedding, even if it is suppressed from the notation in order to lighten it.

Any optimal embedding Ψ = (Ψ, b) gives rise to a pair (Ψ, \underline{R}_{b}) where \underline{R}_{b} is the Eichler order of level N^{+} in B determined from b by the rule

\underline{R}_{Ψ} = (b\underline{R̂}b^{−1}) ∩ B,

so that Ψ is an embedding of O_{K} into \underline{R}_{b}.

The group B^{×} acts on hom(K, B) on the left by conjugation, and on (B̂^{×}/ ˆ\underline{R}^{×}) by left multiplication. The resulting diagonal action of B^{×} on the cartesian product hom(K, B) × (B̂^{×}/ ˆ\underline{R}^{×}) defined by

g · (Ψ, b) = (gΨg^{−1}, gb)

preserves the collection of optimal embeddings of level N^{+}. Let

Emb(O_{K}, N^{+}, N^{−}) ⊂ B^{×}\ (hom(K, B) × (B̂^{×}/\underline{R̂}^{×}))

be the set of orbits for this action. Given an optimal embedding (Ψ, b), the notation [Ψ, b], or sometimes just [Ψ], will be used to denote the associated element of Emb(O_{K}, N^{+}, N^{−}).

The class group G_{D} of K, which is described adelically as the quotient

G_{D} := K̂^{×}/K^{×}Ô_{K}^{×},

acts naturally on Emb(O_{K}, N^{+}, N^{−}) by the rule

(83)  σ · [Ψ, b] = [Ψ, Ψ̂(σ)b],

where Ψ̂ denotes the map from K̂ to B̂ induced from Ψ by tensoring with ℤ̂.

Associated to any Ψ ∈ hom(K, B) is the binary quadratic form Q_{Ψ}(x, y) of discriminant 4D with coefficients in ℚ_{p} defined by

(84)  Q_{Ψ}(x, y) = cx^{2} + (d − a)xy − by^{2}, where ( \begin{matrix}a & b \\ c & d \end{matrix} ) := ι_{p}Ψ(\sqrt{D}).

Note that Q_{Ψ} coincides with the polynomial P_{Ψ(\sqrt{D})} attached to the trace zero element Ψ(\sqrt{D}), defined in equation (50). This quadratic form is determined, up to sign, by its discriminant and the fact that

Q_{Ψ}(τ_{Ψ}, 1) = Q_{Ψ}(τ′_{Ψ}, 1) = 0,

where τ_{Ψ} and τ′_{Ψ} are the fixed points for the action of ι_{p}Ψ(K_{p}^{×}) on ℙ_{1}(ℂ_{p}) by Möbius transformations. Thus, Q_{Ψ}(x, y) = 0 if and only if (x, y) is either the zero vector or an eigenvector for this torus action. This remark makes it apparent that for any g ∈ B^{×}, we have

Q_{gΨg^{−1}} = det(g)(Q_{Ψ}|g^{−1}),

where det(g) := det(ι_{p}(g)) is the reduced norm on B.

The fixed points τ_{Ψ} and τ′_{Ψ} of ι_{p}Ψ(K^{×}) acting on ℙ_{1}(ℂ_{p}) play an important role, and it is convenient to make a consistent ordering of these by fixing an embedding of K into ℂ_{p} and requiring that

(85)  ι_{p}Ψ(α) ( \begin{matrix}τ_{Ψ} \\ 1 \end{matrix} ) = α ( \begin{matrix}τ_{Ψ} \\ 1 \end{matrix} ) .

The assignment (84) makes it possible to associate to the weight k newform ϕ_{k}^{♯} ∈ S_{k}(N^{+}, N^{−}) and to each optimal embedding Ψ = (Ψ, b) of level N^{+} a numerical invariant ϕ_{k}^{♯}[Ψ] by the rule

(86)  ϕ_{k}^{♯}[Ψ] := det(b_{p})^{1−k/2}ϕ_{k}^{♯}(b)(QΨ^{\frac{k−2}{2}} |b_{p}),

where b_{p} ∈ B_{p} denotes the component at p of the idèle b. A direct verification shows that ϕ_{k}^{♯}[Ψ] depends only on the class of Ψ in Emb(O_{K}, N^{+}, N^{−}) and hence is a well-defined function on this quotient set.

Likewise, if p is split in K, then the set Emb(O_{K}, pN^{+}, N^{−}) is non-empty and we can thus associate to [Ψ, b] and ϕ_{2} the well-defined numerical invariant

(87)  ϕ_{2}[Ψ] := ϕ_{2}(b).

We now introduce further notations that are better adapted to the p-adic calculations developed in the next two sections.

The ℤ[1/p]-order R in B satisfies the Eichler condition, and hence, by strong approximation, any optimal embedding of K of level N^{+} can be translated on the left by an element of B^{×} so that it lies in hom(O, R) × (R̂^{×}/ ˆ\underline{R}^{×}). This makes it possible to identify Emb(O_{K}, N^{+}, N^{−}) with a subset of

(88)  R^{×}\ (hom(O, R) × (GL_{2}(ℚ_{p})/GL_{2}(ℤ_{p}))) .

Given [Ψ] ∈ Emb(O_{K}, N^{+}, N^{−}), denote by [Ψ, g] with g ∈ GL_{2}(ℚ_{p}) any representative of [Ψ] lying in (88). The resulting pair (Ψ, g) is entirely determined by the embedding Ψ together with the lattice L_{Ψ} := g(ℤ_{p}^{2}) so that it is convenient to label [Ψ] by such data and write [Ψ, L_{Ψ}] for [Ψ]. The optimality condition satisfied by Ψ implies that the lattice L_{Ψ} is preserved under the action of ι_{p}Ψ(O_{K}). Note also that, since the scalar p belongs to R^{×}, we have [Ψ, L_{Ψ}] = [Ψ, pL_{Ψ}].

A similar discussion applies for Emb(O_{K}, pN^{+}, N^{−}), which can be identified with a subset of

(89)  R^{×}\ (hom(O, R) × GL_{2}(ℚ_{p})/Γ_{0}(pℤ_{p})) .

An element [Ψ] = [Ψ, g] of this space can thus be labelled by a triple [Ψ, L_{1}, L_{2}], where the ℤ_{p}-lattices L_{1} and L_{2}, defined by L_{1} := g(ℤ_{p}^{2}) and L_{2} := g(ℤ_{p} ⊕ pℤ_{p}), are both preserved under the action ι_{p}Ψ(Γ_{0}(pℤ_{p})). It shall sometimes be convenient to describe optimal embeddings of level pN^{+} in this way.

The invariants ϕ_{k}^{♯}[Ψ] and ϕ_{2}[Ψ] can be expressed in terms of the lattice functions c_{k}^{♯}(L) and c_{2}(L_{1}, L_{2}) of Sections 2.2 and 2.3 by the formulae

(90)  ϕ_{k}^{♯}[Ψ, L_{Ψ}] = |L_{Ψ}|^{1−k/2}c_{k}^{♯}(L_{Ψ})(QΨ^{\frac{k−2}{2}} ),
(91)  ϕ_{2}[Ψ, L_{1}, L_{2}] = c_{2}(L_{1}, L_{2}),

where |L| is the generalised index of the lattice L in ℤ_{p}^{2}.

Let h denote the class number of K and let

G_{D} := {σ_{1}, …, σ_{h}}

be a complete list of elements of G_{D}, with σ_{1} = 1. Choose any element [Ψ] := [Ψ, b] ∈ Emb(O_{K}, N^{+}, N^{−}) and write

(92)  [Ψ^{σ_{j}}] = [Ψ^{σ_{j}}, b^{σ_{j}}] := σ_{j} · [Ψ, b].

The importance of the optimal embeddings and the invariants ϕ_{k}^{♯}[Ψ] lies in their relation with the central critical values of L-series attached to f_{k}^{♯} and to abelian characters of Gal(K̄/K). We restrict our attention to the quadratic characters of the class group G_{D}. These are called genus characters. If χ is a non-trivial genus character, it cuts out a quadratic extension of K = ℚ(\sqrt{D}) which is a biquadratic extension of the form

K_{χ} = ℚ(\sqrt{D_{1}}, \sqrt{D_{2}}), with D = D_{1}D_{2}.

In fact, the genus characters of K are in bijection with the factorisations of D into a product of two fundamental discriminants D_{1} and D_{2}, with the trivial character corresponding to the factorisation D_{1} = 1, D_{2} = D. Let χ_{1}, χ_{2} and ϵ_{K} denote the Dirichlet characters attached to the discriminants D_{1}, D_{2} and D respectively. Then χ_{1}χ_{2} = ϵ_{K}. The L-series L(f_{k}^{♯}/K, χ, s) is defined by the usual Euler product expansion whose Euler factors at the good primes are of degree 4. This definition makes it apparent that

L(f_{k}^{♯}/K, χ, s) = L(f_{k}^{♯}, χ_{1}, s)L(f_{k}^{♯}, χ_{2}, s).

Assume from now on in the paper that D ≠ −3, −4, so that O_{K}^{×} = {±1}. This condition can always be ensured in the arguments of the next sections. The algebraic part of L(f_{k}^{♯}/K, χ, k/2) is defined as follows:

(93)  L^{∗}(f_{k}^{♯}/K, χ, k/2) := \frac{(k/2 − 1)!^{2}D^{\frac{k−1}{2}}}{(2π)^{k−2}⟨f_{k}^{♯}, f_{k}^{♯}⟩} L(f_{k}^{♯}/K, χ, k/2),

where ⟨f_{k}^{♯}, f_{k}^{♯}⟩ denotes as before the Petersson scalar product of f_{k}^{♯} with itself. Likewise for k = 2 we write

(94)  L^{∗}(f/K, χ, 1) := \frac{\sqrt{D}}{⟨f, f⟩}L(f/K, χ, 1),

Proposition 3.3 (Hatcher, Hui Xue). For all k ∈ U ∩ ℤ^{>2},

L^{∗}(f_{k}^{♯}/K, χ, k/2) = (∑_{j=1}^{h} χ(σ_{j})ϕ_{k}^{♯}[Ψ^{σ_{j}}])^{2} .

Proof. This result follows from the one that is given in [Ha1] and [Ha2] in the special case where N^{−} is prime and N^{+} = 1. For the convenience of the reader, we briefly explain how to translate the current formalism into the one used in [Ha1]; see also [BDIS], pp. 435-6. Hatcher considers the finite dimensional Hecke-module

V = \{ν : ˆ \underline{R}^{×}\B̂^{×}⟶𝔓_{k} : ν(bg) = ν(b) · ι_{p}(g), for all g ∈ B^{×}\}.

To any optimal embedding (Ψ, b) of level one, we attach the natural image of the pair (b, Q_{Ψ}(x, y)^{k/2−1}) in V, say ν_{Ψ}. Set

ν_{χ} := ∑_{j=1}^{h} χ(σ_{j})ν_{Ψ_{j}^{σ}} ∈ V.

Then, Hatcher’s result states the equality

L^{∗}(f_{k}^{♯}/K, χ, k/2) = ⟨ν_{χ,f_{k}^{♯}}, ν_{χ,f_{k}^{♯}}⟩_{k},

where ν_{χ,f_{k}^{♯}} denotes the f_{k}^{♯}-isotypic projection of ν_{χ}, and ⟨ , ⟩_{k} denotes the natural perfect pairing on V induced by the scalar product of equation (51). In view of equation (90), and of the normalisation condition ⟨f_{k}^{♯}, f_{k}^{♯}⟩ = 1, ν_{χ,f_{k}^{♯}} can be written as (∑_{j=1}^{h} χ(σ_{j})ϕ_{k}^{♯}[Ψ^{σ_{j}}]) · ν_{f_{k}^{♯}}, where ν_{f_{k}^{♯}} is a normalised eigenvector for

the action of the Hecke algebra on V via the character attached to f_{k}^{♯}. This explains our special value formula in the setting of Hatcher’s paper. Work of Hui Xue provides an extension of Hatcher’s formula to the level of generality that is required. Cf. Theorem 3 of [HX1] (and also [HX2]), generalizing the results of [Gr] and [Ha1], [Ha2]. □

When k = 2, it is necessary to work with the newforms f_{2} and ϕ_{2} of level pN^{+}N^{−}. The set Emb(O_{K}, pN^{+}, N^{−}) is non-empty if and only if p is split in K. In this case, choose the elements [Ψ^{σ_{j}}] as in (92), but starting this time with [Ψ, b] ∈ Emb(O_{K}, pN^{+}, N^{−}).

Proposition 3.4. If p is inert in K, then L^{∗}(f/K, χ, 1) = 0. If p is split in K,

L^{∗}(f/K, χ, 1) = ⟨ϕ_{2}, ϕ_{2}⟩^{−1} (∑_{j=1}^{h} χ(σ_{j})ϕ_{2}[Ψ^{σ_{j}}])^{2} .

Proof. The first statement follows from the fact that the sign in the functional equation for L(f/K, χ, s) is −1 when p is inert in K. The formula for p split follows from Proposition 3.3 by replacing N^{+} by pN^{+} and specialising to k = 2, noting that the form ϕ_{2}, unlike ϕ_{k}^{♯}, has not been normalised to be of length 1. □

3.2. Two-variable p-adic L-functions. Let [Ψ] be a class of optimal embeddings of level N^{+}, represented by an element [Ψ, L_{Ψ}] of the space (88).

If p is inert in K, then the lattice L_{Ψ} is determined up to homothety by the requirement that it be preserved under the action of the non-split torus ι_{p}Ψ(K_{p}^{×}).

If p is split in K, then the lattice L_{Ψ} is stable under the action of the split quadratic algebra ι_{p}Ψ(O_{K} ⊗ ℤ_{p}), and hence admits a ℤ_{p}-basis (v_{1}, v_{2}) consisting of eigenvectors for the action of this algebra.

Define a region L′′_{Ψ} in L′_{Ψ} by the rule

L′′_{Ψ} := \{ \begin{matrix}L′_{Ψ} & if p is inert in K; \\ ℤ_{p}^{×}v_{1} × ℤ_{p}^{×}v_{2} & if p is split in K. \end{matrix}

Definition 3.5. (1) The partial p-adic L-function associated to f_{∞} and [Ψ] is the analytic function of the variable k ∈ U defined by

L_{p}(f_{∞}/K, Ψ, k) := ∫_{L′′_{Ψ}}⟨Q_{Ψ}(x, y)⟩^{\frac{k−2}{2}} dμ_{L_{Ψ}}(x, y).

(2) The p-adic L-function attached to f_{∞} and the quadratic character χ of G_{D} is the analytic function

of k ∈ U defined by

L_{p}(f_{∞}/K, χ, k) = (L_{p}(f_{∞}/K, χ, k))^{2},

where

L_{p}(f_{∞}/K, χ, k) := ∑_{j=1}^{h} χ(σ_{j})L_{p}(f_{∞}/K, Ψ^{σ_{j}}, k).

Remark 3.6. In this remark, assume for simplicity that h = 1. Motivated by Definition 1.11, it is tempting to associate to f_{∞} and K a two-variable p-adic L-function of the variables (k, s) ∈ U × ℤ_{p} by introducing the factorisation

Q_{Ψ}(x, y) = A(x − τ_{Ψ}y)(x − τ′_{Ψ}y)

of Q_{Ψ} into a product of two linear forms, and setting

L_{p}(f_{∞}/K, k, s) = A^{\frac{k−2}{2}} ∫_{L′′_{Ψ}}⟨x − τ_{Ψ}y⟩^{s−1}⟨x − τ′_{Ψ}y⟩^{k−s−1}dμ_{L_{Ψ}}(x, y)
L_{p}(f_{∞}/K, k, s) = L_{p}(f_{∞}/K, k, s)L_{p}(f_{∞}/K, k, k − s)

This two-variable L-function specialises to the function L_{p}(f_{∞}/K, χ_{triv}, k) when restricted to the central critical line s = k/2. Its restriction to the “weight two” line k = 2 is the anticyclotomic p-adic L-function explored in [BD1] and [BD2], which interpolates the special values of the L-function of E over K at s = 1, twisted by ring class characters of K of p-power conductor. It is only the function L_{p}(f_{∞}/K, χ, k) that plays a role in the constructions of the present paper, although it is clear that a systematic study of the leading term of L_{p}(f_{∞}/K, k, s) at (k, s) = (2, 1) would be worthwhile. For instance, in the case where p is inert, the methods of [BD2] are exploited in Section 3.4 to relate this leading term to Heegner points. This strengthens the main result of [BD2] concerning the anticyclotomic p-adic L-function L_{p}(f_{∞}/K, 2, s), and places this result in a broader and more natural setting.

We close this section by recording an elementary property of the subsets L′′_{Ψ}.

Lemma 3.7. The function ord_{p}(Q_{Ψ}(x, y)) is constant on L′′_{Ψ}. More precisely,

(95)  ord_{p}(Q_{Ψ}(x, y)) = ord_{p}(|L_{Ψ}|) on L′′_{Ψ}.

Proof. If p is split in K, then the quadratic form Q_{Ψ} can be factored as a product of two linear forms q_{1}(x, y) and q_{2}(x, y) satisfying

q_{1}(v_{1}) = q_{2}(v_{2}) = 0.

For any v = av_{1} + bv_{2} ∈ L′′_{Ψ}, with a, b ∈ ℤ_{p}^{×}, we have

Q_{Ψ}(av_{1} + bv_{2}) = abq_{1}(v_{2})q_{2}(v_{1}).

The independence of ord_{p}(Q_{Ψ}) follows. The more precise formula (95) can be checked by noting that it is invariant under conjugating the embedding ι_{p}, and then verifying it in the case where ι_{p}Ψ gives rise to the diagonal embedding of K_{p} = ℚ_{p} × ℚ_{p} into M_{2}(ℚ_{p}) and L_{Ψ} = ℤ_{p}^{2}, so that Q_{Ψ} = 2\sqrt{D}xy, and (v_{1}, v_{2}) can be chosen to be the standard basis (1, 0), (0, 1). The proof when p is inert in K proceeds along similar, but simpler, lines, and is left to the reader. □

3.3. The interpolation formula. This section further justifies the designation of p-adic L-function that is given to L_{p}(f_{∞}/K, χ, k) by giving an interpolation formula relating this p-adic L-function to classical L-values.

Recall equation (93) defining the algebraic part of the special value of the L-series L(f_{k}^{♯}/K, χ, k/2).

Theorem 3.8. Assume that p is inert in K. Then L_{p}(f_{∞}/K, χ, 2) = 0, and for all k ∈ U ∩ ℤ^{>2},

L_{p}(f_{∞}/K, χ, k) = λ_{B}(k)^{2}a_{p}(k)^{2}(1 − a_{p}(k)^{−2}p^{k−2})^{2}L^{∗}(f_{k}^{♯}/K, χ, k/2).

Proof. The statement for k = 2 follows from the fact that

(96)  L_{p}(f_{∞}/K, Ψ, 2) = ∫_{L′_{Ψ}} 1dμ_{L_{Ψ}}(x, y) = 0.

Suppose now that k belongs to U ∩ ℤ^{>2}. By eventually translating (Ψ, L_{Ψ}) by an appropriate element of R^{×}, we may assume that |L_{Ψ}| = 1. Once this normalisation is made, Lemma 3.7 implies that

⟨Q_{Ψ}(x, y)⟩ = Q_{Ψ}(x, y) on L′_{Ψ}.

Definition 3.5 then implies that, for all k ∈ U ∩ ℤ^{≥2},

L_{p}(f_{∞}/K, Ψ, k) = ∫_{L′_{Ψ}} Q_{Ψ}(x, y)^{\frac{k−2}{2}} dμ_{L_{Ψ}}(x, y).

Proposition 2.11 then yields

L_{p}(f_{∞}/K, Ψ^{σ_{j}}, k) = λ_{B}(k)a_{p}(k)(1 − a_{p}(k)^{−2}p^{k−2}) · c_{k}^{♯}(L_{Ψ^{σ_{j}}})(QΨ^{σ_{j}}^{\frac{k−2}{2}} ).

The result now follows from Proposition 3.3, in light of equation (90). □

We need a similar interpolation formula in the case where p is split in K. Assume for this rest of this section that p satisfies this condition.

Fix an optimal embedding (Ψ, L) and assume that |L| = 1, after eventually translating (Ψ, L) by an appropriate element of R^{×}. Write L = ℤ_{p}v_{1} ⊕ ℤ_{p}v_{2}, where (v_{1}, v_{2}) is a ℤ_{p}-basis of L consisting of eigenvectors for the action of ι_{p}Ψ(K_{p}^{×}). For each j ∈ ℤ, define the lattice L_{j} by

(97)  L_{j} := p^{j}ℤ_{p}v_{1} ⊕ ℤ_{p}v_{2}.

Note that

[Ψ, L_{j}] ∈ Emb(O_{K}, N^{+}, N^{−}), [Ψ, L_{j}, L_{j+1}] ∈ Emb(O_{K}, pN^{+}, N^{−}),

since the L_{j} are preserved under the action of ι_{p}Ψ(O_{K}).

Lemma 3.9. The value of L_{p}(f_{∞}/K, Ψ, k) at k = 2 is given by

L_{p}(f_{∞}/K, Ψ, 2) = a_{p}ϕ_{2}[Ψ, L_{−1}, L_{0}] − ϕ_{2}[Ψ, L_{0}, L_{1}],

while for k ∈ U ∩ ℤ^{>2},

L_{p}(f_{∞}/K, Ψ, k) = λ_{B}(k) ((a_{p}(k) + a_{p}(k)^{−1}p^{k−2})ϕ_{k}^{♯}[Ψ, L_{0}]
−p^{\frac{k−2}{2}} (ϕ_{k}^{♯}[Ψ, L_{−1}] + ϕ_{k}^{♯}[Ψ, L_{1}])) .

Proof. Since

L′_{−1} ∩ L′_{0} = ℤ_{p}v_{1} × ℤ_{p}^{×}v_{2}, and L′_{0} ∩ L′_{1} = pℤ_{p}v_{1} × ℤ_{p}^{×}v_{2},

we have

L′′_{0} = (L′_{−1} ∩ L′_{0}) − (L′_{0} ∩ L′_{1}).

Therefore, since ⟨Q_{Ψ}⟩^{k/2−1} = Q_{Ψ}^{k/2−1} on L′′_{0}, we have

L_{p}(f_{∞}/K, Ψ, k) = ∫_{L′_{−1}∩L′_{0}} QΨ^{\frac{k−2}{2}} dμ_{L_{0}} − ∫_{L′_{0}∩L′_{1}} QΨ^{\frac{k−2}{2}} dμ_{L_{0}}.

By Lemma 2.9, this expression can be rewritten as

L_{p}(f_{∞}/K, Ψ, k) = a_{p}(k) ∫_{L′_{−1}∩L′_{0}} QΨ^{\frac{k−2}{2}} dμ_{L_{−1}} − ∫_{L′_{0}∩L′_{1}} QΨ^{\frac{k−2}{2}} dμ_{L_{0}}.

By Lemma 2.10,

(98)  L_{p}(f_{∞}/K, Ψ, k) = λ_{B}(k) (a_{p}(k)c_{k}(L_{−1}, L_{0})(QΨ^{\frac{k−2}{2}} ) − c_{k}(L_{0}, L_{1})(QΨ^{\frac{k−2}{2}} )) .

This implies Lemma 3.9 for k = 2, using (91). The case k > 2 follows from (98), (90) and (58) by a direct calculation. □

Choose a prime 𝔭 of K above p, and let σ_{𝔭} ∈ G_{D} be its associated Frobenius element. Let H_{0} be the subfield of H which is fixed by σ_{𝔭}, i.e., the maximal subfield of H in which 𝔭 splits completely. Its Galois group over K is identified with

Δ := K̂^{×}/Ô^{×}K^{×} = G_{D}/⟨σ_{𝔭}⟩.

We set t =: #⟨σ_{𝔭}⟩, and n := #Δ, so that h = tn.

Let j_{p} : K⟶K_{𝔭} = ℚ_{p} denote the natural map of K into its completion at 𝔭. The eigenbasis (v_{1}, v_{2}) of L for the action of ι_{p}Ψ(K_{p}) can be ordered in such a way that ι_{p}Ψ(α) acts on v_{1} by multiplication by j_{p}(α). With this convention, we have Lemma 3.10.

σ_{𝔭} · [Ψ, L_{j}] = [Ψ, L_{j+1}], σ_{𝔭} · [Ψ, L_{j}, L_{j+1}] = [Ψ, L_{j+1}, L_{j+2}].

Proof. By equation (83),

σ_{𝔭} · [Ψ, L_{j}] = [Ψ, ι_{p}Ψ̂(γ_{p})L_{j}],

where γ_{p} ∈ K̂^{×} is any idele which is 1 at all places except 𝔭, where it is a uniformiser π_{p} of K_{𝔭}^{×} = ℚ_{p}^{×} satisfying ord_{p}(π_{p}) = 1. The matrix ι_{p}Ψ̂(γ_{p}) sends v_{1} to π_{p}v_{1}, and multiplies v_{2} by a scalar in ℤ_{p}^{×}. Hence

ι_{p}Ψ̂(γ_{p})(p^{j}ℤ_{p}v_{1} ⊕ ℤ_{p}v_{2}) = (p^{j+1}ℤ_{p}v_{1} ⊕ ℤ_{p}v_{2}).

The result follows. □

Write [Ψ_{j}] = [Ψ, L_{j}]. Recall that the genus character χ corresponds to a pair of Dirichlet characters χ_{1} and χ_{2}, one of which is even and the other odd, and which are determined by the condition that for all rational primes ℓ = λλ̄ which split in K,

χ(λ) = χ(λ̄) = χ_{1}(ℓ) = χ_{2}(ℓ).

In particular, we have χ(σ_{𝔭}) = χ_{1}(p) = χ_{2}(p).

Lemma 3.11. For all [Ψ, L] ∈ Emb(O_{K}, N^{+}, N^{−}),

∑_{j=0}^{t−1} χ(σ_{𝔭}^{j})L_{p}(f_{∞}/K, Ψ_{j}, 2) = a_{p}(1 − a_{p}^{−1}χ_{1}(p)) ∑_{j=0}^{t−1} ϕ_{2}[Ψ, L_{j}, L_{j+1}],

and for all k ∈ U ∩ ℤ^{>2},

∑_{j=0}^{t−1} χ(σ_{𝔭}^{j})L_{p}(f_{∞}/K, Ψ_{j}, k) = λ_{B}(k)a_{p}(k)(1 − a_{p}(k)^{−1}χ_{1}(p)p^{\frac{k−2}{2}} )^{2} ∑_{j=0}^{t−1} ϕ_{k}^{♯}[Ψ_{j}].

Proof. Suppose that k > 2. Lemma 3.10 implies that [Ψ_{j}] = [Ψ_{j+t}], so that the function on ℤ which sends j to ϕ_{k}^{♯}[Ψ_{j}] is periodic with period t. Lemma 3.11 is therefore a consequence of Lemma 3.9 by summing over the [Ψ_{j}] with 0 ≤ j ≤ t − 1. The proof for k = 2 is similar. □

We now come to the analogue of Theorem 3.8 in the case where p is split in K.

Theorem 3.12. Assume that p is split in K. Then

L_{p}(f_{∞}/K, χ, 2) = (1 − χ_{1}(p)a_{p}^{−1})^{2}⟨ϕ_{2}, ϕ_{2}⟩L^{∗}(f/K, χ, 1),

and for any k ∈ U ∩ ℤ^{>2},

L_{p}(f_{∞}/K, χ, k) = λ_{B}(k)^{2}a_{p}(k)^{2}(1 − χ_{1}(p)a_{p}(k)^{−1}p^{\frac{k−2}{2}} )^{4}L^{∗}(f_{k}^{♯}/K, χ, k/2)
= λ_{B}(k)^{2}a_{p}(k)^{2}(1 − χ_{1}(p)a_{p}(k)^{−1}p^{\frac{k−2}{2}} )^{2}L^{∗}(f_{k}/K, χ, k/2).

Proof. We only give the details in the case k > 2 since the case where k = 2 is treated by a similar method but is a bit simpler. Let

(99)  L_{p}(f_{∞}/K, χ, k) = ∑_{j=1}^{h} χ(σ_{j})L_{p}(f_{∞}/K, Ψ^{σ_{j}}, k)

denote the square root of the quantity to be evaluated in Theorem 3.12. It is convenient to rewrite the sum appearing in (99) by choosing lifts δ_{1}, …, δ_{n} of the n distinct elements of Δ to the group G_{D}, and setting

[Ψ^{(i)}, L^{(i)}] = δ_{i} · [Ψ, L].

After normalising these elements so that |L^{(i)}| = 1, we define [Ψ_{j}^{(i)}] = [Ψ^{(i)}, L_{j}^{(i)}], with L_{j}^{(i)} described as in equation (97) with L replaced by L^{(i)}. By Lemma 3.10,

[Ψ_{j}^{(i)}, L_{j}^{(i)}] = δ_{i}σ_{𝔭}^{j} · [Ψ, L],

and hence we may rewrite L_{p}(f_{∞}/K, χ, k) as a double sum:

L_{p}(f_{∞}/K, χ, k) = ∑_{i=1}^{n} χ(δ_{i}) (∑_{j=0}^{t−1} χ(σ_{𝔭}^{j})L_{p}(f_{∞}/K, Ψ_{j}^{(i)}, k)) .

The inner sum can be evaluated using Lemma 3.11. This gives, after setting

Θ := λ_{B}(k)a_{p}(k)(1 − a_{p}(k)^{−1}χ_{1}(p)p^{\frac{k−2}{2}} )^{2},

L_{p}(f_{∞}/K, χ, k) = Θ × (∑_{i=1}^{n} χ(δ_{i}) ∑_{j=0}^{t−1} χ(σ_{𝔭}^{j})ϕ_{k}^{♯}[Ψ_{j}^{(i)}])

= Θ × (∑_{j=1}^{h} χ(σ_{j})ϕ_{k}^{♯}[Ψ^{σ_{j}}]) .

The result follows from this last identity after invoking Proposition 3.3. □

3.4. Calculation of derivatives. Assume throughout this section that p is inert in K.

For any [Ψ] ∈ Emb(O_{K}, N^{+}, N^{−}), we have L_{p}(f_{∞}/K, Ψ; 2) = 0 by (96). One is therefore led to consider the first derivative of this function at k = 2. Given Ψ = [Ψ, L_{Ψ}] ∈ Emb(O_{K}, N^{+}, N^{−}), use the indefinite integral of Definition 2.16 to write

J_{Ψ} := ∫ ^{τ_{Ψ}}ω_{ϕ},  J̄_{Ψ} := ∫ ^{τ′_{Ψ}}ω_{ϕ}.

The reader may check that the periods J_{Ψ} and J̄_{Ψ} are well-defined and represent genuine invariants attached to elements of Emb(O_{K}, N^{+}, N^{−}). Note that J_{Ψ} belongs to ℚ_{p^{2}} and that J̄_{Ψ} is its conjugate under the action of Gal(ℚ_{p^{2}}/ℚ_{p}).

Proposition 3.13. For all Ψ ∈ Emb(O_{K}, N^{+}, N^{−}),

\frac{d}{dk} L_{p}(f_{∞}/K, Ψ; k)_{k=2} = \frac{1}{2}(J_{Ψ} + J̄_{Ψ}).

Proof. By definition of the p-adic L-function,

\frac{d}{dk} L_{p}(f_{∞}/K, Ψ; k)_{k=2} = \frac{1}{2} ∫_{L′_{Ψ}} log(A(x − τ_{Ψ}y)(x − τ′_{Ψ}y))dμ_{L_{Ψ}}(x, y)

= \frac{1}{2} (∫_{L′_{Ψ}} log(x − τ_{Ψ}y)dμ_{L_{Ψ}} + ∫_{L′_{Ψ}} log(x − τ′_{Ψ}y)dμ_{L_{Ψ}}) .

The claim follows from the definition of the indefinite integral and of J_{Ψ}. □

The interest of Proposition 3.13 is revealed in Section 4, where the period J_{Ψ} is equated with the formal group logarithm of certain Heegner points on E(H), which are defined in Section 4.3.

Remark 3.14. Assume, as in Remark 3.6, that h = 1 and consider the two-variable p-adic L-function L_{p}(f_{∞}/K, k, s) defined in that remark. Given k ∈ U ∩ ℤ^{≥2}, the arguments of the classical L-function L(f_{k}, s) that are critical in the sense of Deligne are the integers in the interval [1, k − 1]. The restriction of L_{p}(f_{∞}/K, k, s) to the boundary lines of this critical region admit particularly simple first derivatives:

J̄_{Ψ} = \frac{d}{dk} L_{p}(f_{∞}/K, k, 1)_{k=2},
J_{Ψ} = \frac{d}{dk} L_{p}(f_{∞}/K, k, k − 1)_{k=2}.

In the same vein, we note that

(100)  \frac{d}{ds}L_{p}(f_{∞}/K, 2, s)_{s=1} = ∫_{τ′_{Ψ}}^{τ_{Ψ}}ω_{ϕ} = J_{Ψ} − J̄_{Ψ}.

This last result, expressing the leading term at s = 1 of the “square root” of the anticyclotomic p-adic L-function of E/K in terms of certain periods of ω_{ϕ} attached to CM points, was the main result of [BD2].

We conclude with a formula for the first derivative of L_{p}(f_{∞}/K, χ, k) at k = 2. Set

J_{χ} := ∑_{σ∈G_{D}} χ(σ)J_{Ψ^{σ}},

and let J̄_{χ} denote its conjugate over ℚ_{p^{2}}.

Corollary 3.15. For all genus characters χ of K,

\frac{d}{dk} L_{p}(f_{∞}/K, χ, k)_{k=2} = \frac{1}{2}(J_{χ} + J̄_{χ}).

Proof. This follows from Proposition 3.13. □

4. Heegner points on Shimura curves

The goal of this section is to prove that the period J_{Ψ} appearing in Proposition 3.13 is the formal group logarithm of a Heegner point on E, arising from a parametrization of E by a Shimura curve.

This is done in a sequence of steps. The first one reviews the theory of p-adic uniformisation of Shimura curves. Then, an explicit description of the modular parametrisation of E by a Shimura curve is given in terms of the theory of p-adic integration. Using this theory, the periods J_{Ψ} are interpreted as the formal logarithm of certain local points on E. Finally, the theory of complex multiplication is used to show that these local points are global points corresponding to moduli of certain abelian surfaces with complex multiplication.

4.1. Shimura curves. The weight two rigid analytic modular form f_{ϕ}(z) attached to E in section 2 defines a Γ-invariant differential ω_{ϕ} on the p-adic upper half plane or, equivalently, a differential on the rigid analytic space X_{Γ} := H_{p}/Γ. Mumford’s theory of p-adic uniformisation of nodal curves implies that X_{Γ} is the rigid analytic space attached to the ℂ_{p}-points of a projective curve X_{ℚ_{p}} defined over ℚ_{p}: see for example [GvdP].

The Cerednik-Drinfeld theorem states that X_{ℚ_{p}} has a model over ℚ. More precisely, let B be the indefinite quaternion algebra of discriminant N^{−}p obtained from the quaternion algebra B of Section 2 by interchanging the invariants ∞ and p. Write S for a fixed Eichler order in B of level N^{+}. Fix an embedding

ι_{∞} : B⟶M_{2}(ℝ),

and set

Γ_{∞} := ι_{∞}(S_{1}^{×}) ⊂ SL_{2}(ℝ),

where S_{1}^{×} denotes the group of units in S of reduced norm one. Let H_{∞} denote the Poincaré upper half plane, and define the compact Riemann surface

X_{Γ_{∞}} := H_{∞}/Γ_{∞}.

Shimura’s work shows that X_{Γ_{∞}} is the complex analytic space attached to a projective curve X defined over ℚ. This is done by identifying X_{Γ_{∞}} with the complex points of the coarse moduli space for triples (A, j, C), where:

(1) A is an abelian scheme of relative dimension 2 over a ℚ-scheme U;

(2) j : \underline{S}⟶End_{U}(A) is the action on A of a fixed maximal order \underline{S} of B containing S;

(3) C is a subgroup scheme of A, which is locally isomorphic to ℤ/N^{+}ℤ and which is stable and locally cyclic for the action of S on A induced by j.

Using a variant of the above definition, it is possible to extend X to a scheme over ℤ.

Theorem 4.1 (Cerednik-Drinfeld). The space X_{Γ} is the rigid analytic space attached to the ℂ_{p}-points of X. More precisely, there is an isomorphism defined over ℚ_{p^{2}} between the curves X and X_{ℚ_{p}}, where ℚ_{p^{2}} denotes the quadratic unramified extension of ℚ_{p}.

Proof. See [BC]. □

Remark 4.2. The paper [BC] contains a detailed exposition of Drinfeld’s proof of Theorem 4.1. This proof relies on an interpretation of the p-adic upper half plane H_{p} as the classifying space of certain formal groups of dimension 2 and height 4, equipped with an action of the local quaternionic order S_{p} := S ⊗ ℤ_{p}. The identification of these formal groups with the formal groups at p of the abelian surfaces parametrised by the Shimura curve X yields a uniformisation of X by H_{p}.

The arithmetic structure carried by the groups Γ, resp. Γ_{∞} gives rise to a family of Hecke correspondences T_{n}^{(p)}, resp. T_{n}^{(∞)} on X_{Γ}, resp. X_{Γ_{∞}}, indexed by the positive integers. The correspondences T_{n}^{(p)} and T_{n}^{(∞)} induce the same correspondence on X, denoted by T_{n}. The operator T_{n} is defined over ℚ.

4.2. Modular parametrizations. Let

J := Div^{0}(X)/P (X)

denote the Jacobian of the Shimura curve X introduced in the previous section, whose points over an extension F of ℚ correspond to F -rational classes of degree zero divisors on X modulo the subgroup P (X) of principal divisors. The Jacquet-Langlands correspondence combined with a version of the Eichler-Shimura theory for the Shimura curve X shows that J has a simple factor A_{ϕ} satisfying

L(A_{ϕ}, s) = L(E, s).

By the isogeny theorem, the elliptic curves A_{ϕ} and E are isogenous over ℚ. Hence there exists a nonconstant morphism

φ_{E} : J⟶E

defined over ℚ, which is called a modular parametrisation attached to X and E. (One might want to normalise φ_{E} by requiring, for example, that it not factor through any non-trivial endomorphism of E, but this is not really necessary.)

Let Div^{0}(X_{Γ}) denote the groups of degree zero divisors supported on X_{Γ} := H_{p}/Γ, and let

J_{Γ} := Div^{0}(X_{Γ})/P (X_{Γ}).

Theorem 4.1 identifies J(ℂ_{p}) with J_{Γ}. We wish to give an explicit p-adic analytic description of φ_{E} in terms of this identification.

Let ω_{E} be a Néron differential on E, and let φ_{E}^{∗}(ω_{E}) denote its pullback to J. This differential is proportional to ω_{ϕ}. After possible replacing ω_{ϕ} by a non-zero rational multiple, we may assume that

ω_{ϕ} = φ_{E}^{∗}(ω_{E}).

The p-adic lattice Q_{ϕ} defined in Theorem 2.24 is then contained in the Tate lattice of E, so that the function from Div^{0}(H_{p}/Γ) to E(ℂ_{p}) defined by

d ↦ Φ_{Tate} (× ∫_{d} ω_{ϕ})

is well-defined. The next result is proved in [GvdP].

Theorem 4.3. The modular parametrisation φ_{E} : Div^{0}(X)⟶E can be described over ℚ_{p^{2}} by the formula

φ_{E}(d) = Φ_{Tate} (× ∫_{d} ω_{ϕ}) .

4.3. Heegner points. Recall that H denotes the Hilbert class field of K, whose Galois group over K is identified by class field theory with the class group G_{D} of K. Since p is inert in K, it splits completely in the extension H/K. Fix a prime 𝔭 of H above p; this determines an embedding of H into ℚ_{p^{2}} and we can thus view H as a subfield of this p-adic field.

Let [Ψ] = [Ψ, b] = [Ψ, L] ∈ Emb(O_{K}, N^{+}, N^{−}) be a class of optimal embeddings, normalised so that b belongs to R̂^{×} and so that L is an even lattice in ℚ_{p}^{2}. Let τ_{Ψ} ∈ H′_{p} denote the fixed point of ι_{p}Ψ(K_{p}^{×}) on H_{p} normalised as in (85). Since the embedding Ψ is determined by [Ψ] up to conjugation by Γ, the image of τ_{Ψ} in H_{p}/Γ depends only on [Ψ]. Let P̃_{Ψ} ∈ X_{Γ}(ℚ_{p^{2}}) = X(ℚ_{p^{2}}) be the point on X associated to τ_{Ψ}.

The next result is the main theorem of complex multiplication.

Proposition 4.4. For all [Ψ] ∈ Emb(O_{K}, N^{+}, N^{−}), the point P̃_{Ψ} belongs to X(H), and for all σ ∈ G_{D},

P̃_{Ψ^{σ}} = σ(P̃_{Ψ}).

Sketch. The proof of this result rests on Drinfeld’s theory of p-adic uniformisation of Shimura curves alluded to in Remark 4.2. This theory shows that the image of τ_{Ψ} in H_{p}/Γ corresponds to an abelian surface with quaternionic multiplication, having complex multiplication by the ring of integers of K. Such a surface is isogenous to a product of elliptic curves with complex multiplication by O_{K}. The claim then follows from the theory of complex multiplication for abelian varieties. See Theorem 5.3 of [BD2] for more details. □

Given a quadratic character χ of G_{D}, let H_{χ} = ℚ(\sqrt{D_{1}}, \sqrt{D_{2}}) denote as before the biquadratic field cut out by χ (or the quadratic field K, if χ is the trivial character). Define a Heegner divisor of degree 0 attached to χ by choosing an auxiliary prime ℓ |̸N if χ = 1 and setting

P̃_{χ} = \{ \begin{matrix}(ℓ + 1 − a_{ℓ})^{−1}(ℓ + 1 − T_{ℓ}) ( ∑_{σ∈G_{D}} P̃_{Ψ^{σ}}) & if χ = 1, \\ ∑_{σ∈G_{D}} χ(σ)P̃_{Ψ^{σ}} & if χ ≠ 1. \end{matrix}

By Proposition 4.4, the element P̃_{χ} belongs to Div^{0}(X)(H_{χ})^{χ} if χ ≠ 1, and to Div^{0}(X)(K) ⊗ ℚ if χ is the trivial character. Furthermore, the Galois group of H_{χ} over K acts on P̃_{χ} via the character χ.

We define

P_{χ} := φ_{E}(P̃_{χ}) ∈ \{ \begin{matrix}E(H_{χ})^{χ} & if χ ≠ 1; \\ E(K) ⊗ ℚ & if χ = 1. \end{matrix}

The global point P_{χ} is called the Heegner point attached to Ψ and χ.

Recall the formal group logarithm log_{E} defined in equation (7) in the Introduction, and let frob_{p} denote the Frobenius element attached to the rational prime p in the abelian extension H_{χ}/ℚ.

Proposition 4.5. For all genus characters χ of G_{D},

J_{χ} = log_{E}(P_{χ}), J̄_{χ} = a_{p} log_{E}(frob_{p} P_{χ}).

Proof. If χ ≠ 1, then P̃_{χ} is a degree zero divisor on X and therefore by Theorem 4.3,

P_{χ} = Φ_{Tate} (× ∫P̃_{χ}ω_{ϕ}) .

Applying log_{E} to both sides gives

log_{E}(P_{χ}) = ∫P̃_{χ}ω_{ϕ}.

But this last expression is equal to J_{χ}, by Corollary 2.26 and the definition of the period J_{χ} in terms of the indefinite integral. The proof for χ = 1 follows in a similar way, using the compatibility of the indefinite integral with Hecke operators asserted in Proposition 2.18. The formula for J̄_{χ} is obtained by applying the generator s_{p} of Gal(ℚ_{p^{2}}/ℚ_{p}) to both sides, and noting that Φ_{Tate} and log_{E} do not commute with this element when E has non-split multiplicative reduction at p, and that, more precisely, for any P ∈ E(ℚ_{p^{2}}),

s_{p}(log_{E}(P)) = a_{p} log_{E}(frob_{p} P).
□

We note the following facts about the Heegner point P_{χ}.

Theorem 4.6. The point P_{χ} is of infinite order if and only if L′(E/K, χ, 1) ≠ 0. In that case, the rank of E(H_{χ})^{χ} is equal to one.

Proof. The first statement follows from the work of Zhang [Zh2] extending the formula of Gross and Zagier to Shimura curve parametrisations (attached to a quaternion algebra over a totally real field F, although only the case F = ℚ is needed here.) The second statement follows from the methods of Kolyvagin and Logachev [KL] (cf. Theorem A of [Zh1]). □

For each integer m|N satisfying gcd(m, N/m) = 1, let w_{m} ∈ {−1, 1} denote the eigenvalue of the Fricke involution W_{m} acting on f. The key facts about w_{m} that will be used are:

(1) The sign in the functional equation for L(f, s) is equal to −w_{N}. More generally, if χ_{1} is a quadratic Dirichlet character of conductor prime to N, the sign in the functional equation for L(f, χ_{1}, s) is −w_{N}χ_{1}(−N).

(2) The element w_{p} is equal to −a_{p}. In particular, L_{p}(f, χ_{1}, s) has an exceptional zero at s = 1 if and only if χ_{1}(p) = −w_{p}, i.e., χ_{1}(p) = a_{p}.

Let χ_{1} and χ_{2} denote the quadratic Dirichlet characters associated to χ. Since

χ_{1}(−N)χ_{2}(−N) = ϵ_{K}(−N) = −1,

we may order (χ_{1}, χ_{2}) in such a way that

χ_{1}(−N) = w_{N}, χ_{2}(−N) = −w_{N}.

With this fixed ordering, we have the following result which gives a more precise information about the location of the Heegner point P_{χ}.

Theorem 4.7. The image of P_{χ} in E(H_{χ}) ⊗ ℚ belongs to (E(ℚ(\sqrt{D_{1}})) ⊗ ℚ)^{χ_{1}}.

Proof. If P_{χ} is of finite order, its image in E(H_{χ}) ⊗ ℚ is trivial and there is nothing to prove. Otherwise, Theorem 4.6 asserts that

(101)  L(E/K, χ, s) = L(E, χ_{1}, s)L(E, χ_{2}, s)

vanishes to order 1 at s = 1. By the chosen ordering of the characters χ_{1} and χ_{2}, the function L(E, χ_{j}, s) vanishes to odd order when j = 1, and to even order when j = 2. Hence the factors on the right in (101) vanish to order 1 and 0 respectively. Kolyvagin’s theorem implies that

dim_{ℚ}(E(ℚ(\sqrt{D_{1}})) ⊗ ℚ)^{χ_{1}} = 1; dim_{ℚ}(E(ℚ(\sqrt{D_{2}})) ⊗ ℚ)^{χ_{2}} = 0.

The result follows. □

Corollary 4.8. frob_{p}(P_{χ}) = χ_{1}(p)P_{χ}.

Proof. This follows directly from Theorem 4.7. □

4.4. Heegner points and derivatives of Hida L-functions. Assume as in the previous section that the pair (χ_{1}, χ_{2}) of Dirichlet characters associated to the genus character χ of K has been ordered in such a way that χ_{1}(−N) = w_{N}.

Theorem 4.9.

\frac{d}{dk} L_{p}(f_{∞}/K, χ, k)_{k=2} = \frac{1}{2}(1 + χ_{1}(p)a_{p}) log_{E}(P_{χ})

.

Proof. Applying Corollary 3.15 followed by Proposition 4.5,

\frac{d}{dk} L_{p}(f_{∞}/K, χ, k)_{k=2} = \frac{1}{2}(J_{χ} + J̄_{χ})
= \frac{1}{2} (log_{E}(P_{χ}) + a_{p} log_{E}(frob_{p} P_{χ})) .

The theorem now follows from Corollary 4.8. □

Corollary 4.10.

\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}/K, χ; k)_{k=2} = \{ \begin{matrix}2 log_{E}(P_{χ})^{2} & if χ_{1}(p) = a_{p} \\ 0 & if χ_{1}(p) = −a_{p} . \end{matrix}

5. The main result

5.1. A factorisation of p-adic L-functions. Define the function

(102)  η(k) := \{ \begin{matrix}⟨ϕ_{2}, ϕ_{2}⟩ & if k = 2; \\ a_{p}(k)^{2}D^{k/2−1} \frac{λ_{B}(k)^{2}}{λ^{+}(k)λ^{−}(k)}, & for k ∈ U ∩ ℤ^{>2}. \end{matrix}

Note that the scalar η(k) ∈ ℂ_{p} depends only on k, and on the factorisation (32), not on K.

Proposition 5.1. For all k ∈ U ∩ ℤ^{≥2}, and all genus characters χ,

L_{p}(f_{∞}/K, χ; k) = η(k)L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2).

Proof. Assume first that k > 2. The following formula follows from comparing the Euler products on both sides:

L(f_{k}^{♯}/K, χ, k/2) = L(f_{k}^{♯}, χ_{1}, k/2)L(f_{k}^{♯}, χ_{2}, k/2).

By definition (cf. Proposition 1.3)

L^{∗}(f_{k}^{♯}, χ_{1}, k/2) := \frac{(k/2 − 1)!τ(χ_{1})}{(−2πi)^{k/2−1}Ω_{k}^{±}}L(f_{k}^{♯}, χ_{1}, k/2)
L^{∗}(f_{k}^{♯}, χ_{2}, k/2) := \frac{(k/2 − 1)!τ(χ_{2})}{(−2πi)^{k/2−1}Ω_{k}^{∓}}L(f_{k}^{♯}, χ_{2}, k/2).

The choice of complex periods that was made in (14) implies that

Ω_{k}^{±}Ω_{k}^{∓} = ⟨f_{k}^{♯}, f_{k}^{♯}⟩.

Hence by referrring to (93), we get

(103)  L^{∗}(f_{k}^{♯}/K, χ, k/2) = D^{k/2−1}L^{∗}(f_{k}^{♯}, χ_{1}, k/2)L^{∗}(f_{k}^{♯}, χ_{2}, k/2).

By comparing the Euler factors that appear in equation (31) following Theorem 1.12 (with χ = χ_{1} and χ = χ_{2}) with the Euler factor that occurs in Theorem 3.12 (if p is split in K) and Theorem 3.8 (if p is inert in K), we find

L_{p}(f_{∞}/K, χ; k) = η(k)L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2),

as was to be shown. The proof for k = 2 is identical, and is therefore omitted. □

Although Hida theory gives little control on the variation with k of the quantities λ^{+}(k), λ^{−}(k), and λ_{B}(k) taken individually, it is still possible to prove the following result.

Proposition 5.2. The function η(k) extends to a p-adic analytic function of k ∈ U.

Proof. Consider the set of primitive quadratic Dirichlet characters χ_{1} of conductor prime to N, satisfying

(104)  χ_{1}(−N) = −w_{N}; χ_{1}(p) = −a_{p}.

The condition on χ_{1}(−N) implies that L(f, χ_{1}, s) vanishes to even order at s = 1. By the main result of [MM], there are infinitely many such χ_{1} satisfying

(105)  L(f, χ_{1}, 1) ≠ 0.

The second condition in (104) imposed on χ_{1} implies that L_{p}(f, χ_{1}, s) does not have an exceptional zero, so that

L_{p}(f, χ_{1}, 1) belongs to ℚ^{×}.

Fix one character χ_{1} satisfying (104) as well as the non-vanishing condition (105). Consider now the set of Dirichlet characters χ_{2} of conductor prime to N and to the conductor of χ_{1}, satisfying

(1) χ_{2}(ℓ) = χ_{1}(ℓ), for all ℓ|pN^{+};

(2) χ_{2}(ℓ) = −χ_{1}(ℓ), for all ℓ|N^{−} and ℓ = −1;

Note that, since N^{−} has an odd number of prime factors, χ_{2}(−N) = χ_{1}(−N), so that the L-series L(f, χ_{2}, s) also vanishes to even order at s = 1. Invoking the non-vanishing result of [MM] one more time, and noting that the restriction placed on χ_{2}(p) prevents L_{p}(f, χ_{2}, s) from acquiring an extra zero at s = 1, one obtains a character χ_{2} such that

(106)  L(f, χ_{2}, 1) ≠ 0,  L_{p}(f, χ_{2}, 1) ≠ 0.

It follows from these choices of χ_{1} and χ_{2} that the functions

L_{p}(f_{∞}/K, χ, k), L_{p}(f_{∞}, χ_{1}, k, k/2), and L_{p}(f_{∞}, χ_{2}, k, k/2)

are analytic functions on U that do not vanish at k = 2. The result now follows from Proposition 5.1. □

In particular, we have proved:

Corollary 5.3. For all k ∈ U,

L_{p}(f_{∞}/K, χ, k) = η(k)L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2),

where η(k) is a p-adic analytic function of k satisfying η(2) = ⟨ϕ_{2}, ϕ_{2}⟩.

5.2. Proof of the main result. We can now state and prove the main result of this paper, which implies Theorem 1 of the Introduction when specialised to the trivial character.

Theorem 5.4. Suppose that E has at least two distinct primes of semistable reduction. Let χ_{1} be a Dirichlet character of conductor prime to N, satisfying χ_{1}(−N) = w_{N}, and χ_{1}(p) = a_{p}. Then

(1) The p-adic L-function L_{p}(f_{∞}, χ_{1}, k, k/2) vanishes to order at least 2 at k = 2.

(2) There exists a global point P_{χ_{1}} ∈ (E(ℚ(\sqrt{D_{1}})) ⊗ ℚ)^{χ_{1}}, and ℓ ∈ ℚ^{×}, such that

\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)_{k=2} = ℓ log^{2}(P_{χ_{1}}).

(3) The point P_{χ_{1}} is of infinite order if and only if L′(E/ℚ, χ_{1}, 1) ≠ 0.

(4) The image of ℓ in ℚ^{×}/(ℚ^{×})^{2} is equal to that of L^{∗}(f, ψ, 1), where ψ is any quadratic Dirichlet character satisfying

(a) ψ(ℓ) = χ_{1}(ℓ) for all primes ℓ dividing M = N/p;

(b) ψ(p) = −χ_{1}(p);

(c) L(f, ψ, 1) ≠ 0.

Proof. 1. Since χ_{1}(−N) = w_{N}, the classical L-function L(f, χ_{1}, s) has sign −1 in its functional equation, and in particular vanishes at s = 1. Furthermore, the condition χ_{1}(p) = a_{p} implies that L_{p}(f, χ_{1}, s) has an exceptional zero at s = 1, so that the sign in the functional equation in this p-adic L-function is 1. In particular, the order of vanishing of L_{p}(f, χ_{1}, s) at s = 1 is at least 2:

(107)  ord_{s=1} L_{p}(f_{∞}, χ_{1}, 2, s) ≥ 2.

On the other hand, formula (30) expresses L_{p}(f_{∞}, χ_{1}, k, 1) as a product of two analytic functions of k which both vanish at k = 2:

L_{p}(f, χ_{1}, k, 1) = (1 − a_{p}(k)^{−1}χ_{1}(p))L_{p}^{∗}(f_{∞}, χ_{1}, k).

It follows that

(108)  ord_{k=2} L_{p}(f_{∞}, χ_{1}, k, 1) ≥ 2.

Equations (107) and (108) together imply that L_{p}(f_{∞}, χ_{1}, k, s) vanishes to order at least 2 at (k, s) = (2, 1), and part 1 of Theorem 5.4 follows.

2. Choose an auxiliary quadratic Dirichlet character χ_{2} of conductor prime to the conductor of χ_{1}, and satisfying the following conditions, by invoking the non-vanishing result of [MM], much as in the proof of Proposition 5.2.

(1) χ_{2}(ℓ) = χ_{1}(ℓ), for all ℓ|N^{+};

(2) χ_{2}(ℓ) = −χ_{1}(ℓ), for all ℓ|pN^{−} or ℓ = −1;
(3) L(f, χ_{2}, 1) ≠ 0.

The conditions placed on χ_{2} imply that the quadratic field K attached to the character ϵ_{K} := χ_{1}χ_{2} is admissible relative to N = pN^{+}N^{−}, and that p is inert in K. In particular, the p-adic L-function L_{p}(f, χ_{2}, s) does not have an exceptional zero at s = 1, so that L_{p}(f_{∞}, χ_{2}, 2, 1) belongs to ℚ^{×}.

Let χ be the genus character of K associated to the pair of quadratic characters χ_{1} and χ_{2}. By Corollary 5.3, we have

L_{p}(f_{∞}/K, χ, k) = η(k)L_{p}(f_{∞}, χ_{1}, k, k/2)L_{p}(f_{∞}, χ_{2}, k, k/2).

Taking second derivatives at k = 2, and applying Corollary 4.10, we find

2 log_{E}(P_{χ})^{2} = η(2)\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)_{k=2}L_{p}(f, χ_{2}, 1).

It follows that

\frac{d^{2}}{dk^{2}} L_{p}(f_{∞}, χ_{1}, k, k/2)_{k=2} = ℓ log_{E,p}(P_{χ})^{2}.

where

(109)  ℓ^{−1} = η(2)L^{∗}(f, χ_{2}, 1) = ⟨ϕ_{2}, ϕ_{2}⟩L^{∗}(f, χ_{2}, 1).

This proves part 2, setting P_{χ_{1}} := P_{χ}.

3. To show part 3, recall that Theorem 4.6 states that the Heegner point P_{χ_{1}} is of infinite order if and only if the derivative of the classical L-series

L′(E/K, χ, 1) = L′(E, χ_{1}, 1)L(E, χ_{2}, 1)

is non-zero. Since the second factor in the expression on the right is non-vanishing by the choice of χ_{2}, the result follows.

4. Finally, equation (109) shows that

(110)  ℓ = ⟨ϕ_{2}, ϕ_{2}⟩L^{∗}(f, χ_{2}, 1) (mod (ℚ^{×})^{2}).

Choose a character ψ satisfying the three conditions of part 4 of Theorem 1, and having conductor prime to that of χ_{2}. Then the quadratic field K̃ cut out by the character ϵ = ψχ_{2} is admissible relative to the factorisation pN^{+}N^{−}, and p is split in K̃. Let χ̃ be the genus character attached to (ψ, χ_{2}). Then it follows from Proposition 3.4 that

(111)  ⟨ϕ_{2}, ϕ_{2}⟩ = L^{∗}(f/K̃, χ̃, 1) = L^{∗}(f, ψ, 1)L^{∗}(f, χ_{2}, 1) (mod (ℚ^{×})^{2}).

The result follows by combining (110) and (111). □

References

[BC] J.-F. Boutot and H. Carayol. Uniformisation p-adique des courbes de Shimura: les théorèmes de Cerednik et de Drinfeld. Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque No. 196-197 (1991), 7, 45–158 (1992).

[BD1] M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves. Invent. Math. 126 (1996) 413–456.

[BD2] M. Bertolini and H. Darmon. Heegner points, p-adic L-functions and the Cerednik-Drinfeld uniformization. Invent. Math. 131 (1998), no. 3, 453–491.

[BD3] M. Bertolini and H. Darmon. The rationality of Stark-Heegner points over genus fields of real quadratic fields. Annals of Mathematics, to appear.

[BDG] M. Bertolini, H. Darmon, and P. Green. Periods and points attached to quadratic algebras. MSRI Publ. 49, Cambridge Univ. Press, 323-367, 2004.

[BDIS] M. Bertolini, H. Darmon, A. Iovita, and M. Spiess. Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting, American Journal of Mathematics 124 (2002) 411-449.

[BDI] M. Bertolini, H. Darmon, and A. Iovita. Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture. Submitted.

[Bu] K. Buzzard. On p-adic families of automorphic forms. Modular curves and abelian varieties, 23–44, Progr. Math., 224, Birkhäuser, Basel, 2004.

[Ch] G. Chenevier. Une correspondance de Jacquet-Langlands p-adique. Duke Math. J. 126 (2005), no. 1, 161–194.

[Da] H. Darmon. Rational points on modular elliptic curves. CBMS Regional Conference Series in Mathematics, 101. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.

[Gre] R. Greenberg. Elliptic curves and p-adic deformations. Elliptic curves and related topics, 101–110, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994.

[Gr] B. H. Gross. Heights and the special values of L-series. Number theory (Montreal, Que., 1985), 115–187, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, RI, 1987.

[GS] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111 (1993), no. 2, 407–447.

[GvdP] L. Gerritzen, M. van der Put, Shottky Groups and Mumford Curves. Lecture Notes in Mathematics, 817. Springer, Berlin, 1980.

[Ha1] R. Hatcher, Heights and L-series. Canad. J. Math. 42 (1990), no. 3, 533–560.

[Ha2] R. Hatcher, Special values of L-series. Proc. Amer. Math. Soc. 114 (1992), no. 2, 337–343.

[HX1] Xue Hui, Central values of Rankin L-series, preprint.

[HX2] Xue Hui, A central value formula for Rankin L-functions, preprint.

[JL] H. Jacquet; R.P. Langlands. Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970.

[KL] V.A Kolyvagin; D. Yu Logachev. Finiteness of � over totally real fields. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 4, 851–876; translation in Math. USSR-Izv. 39 (1992), no. 1, 829–853

[Man] J.I. Manin, Parabolic Points and Zeta Functions of Modular Curves. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), no. 1, 19–66.

[MM] Murty, M. Ram; Murty, V. Kumar. Non-vanishing of L-functions and applications. Progress in Mathematics, 157. Birkhäuser Verlag, Basel, 1997.

[M-Sw-D] Mazur, B.; Swinnerton-Dyer, P. Arithmetic of Weil curves. Invent. Math. 25 (1974), 1–61.

[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.

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

[Ru] K. Rubin. p-adic L-functions and rational points on elliptic curves with complex multiplication. Invent. Math. 107 (1992), no. 2, 323–350.

[Se] J.P. Serre, Trees. Springer, Berlin, 1980.

[Te] J.T. Teitelbaum. Values of p-adic L-functions and a p-adic Poisson kernel. Invent. Math. 101 (1990), no. 2, 395–410.

[Vi] M.-F. Vignéras. Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics, 800. Springer, Berlin, 1980.

[Wa] Waldspurger, J.-L. Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54 (1985), no. 2, 173–242.

[W] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995) 443-551.

[Zh1] Zhang, S., Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001), no. 1, 27–147.

[Zh2] S. Zhang. Gross-Zagier formula for GL_{2}. Asian J. Math. 5 (2001), no. 2, 183–290.