Integration on H_{p} × H and arithmetic applications

Henri Darmon^{∗}

December 5, 2000

Contents

1 Integration on (H_{p} × H)/Γ 11 1.1 Forms on T × H . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Mixed period integrals . . . . . . . . . . . . . . . . . . . . . . 19

2 Periods attached to split tori 25 2.1 Preliminary calculations . . . . . . . . . . . . . . . . . . . . . 25 2.2 Evaluation of ord_{p}(I_{Ψ}) . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Evaluation of log(I_{Ψ}) . . . . . . . . . . . . . . . . . . . . . . . 36

3 The cohomology of Γ 38 3.1 The cohomology of M-symbols . . . . . . . . . . . . . . . . . 38 3.2 Proof of theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Periods attached to real quadratic fields 45

5 Heegner points attached to real quadratic fields 47 5.1 The main conjecture . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 A Shimura Reciprocity Law . . . . . . . . . . . . . . . . . . . 50 5.3 A Gross-Zagier conjecture . . . . . . . . . . . . . . . . . . . . 55 5.4 Numerical evidence . . . . . . . . . . . . . . . . . . . . . . . 56

∗Partly supported by CICMA and by an NSERC research grant.

Abstract

This article describes a conjectural p-adic analytic construction of global points on (modular) elliptic curves, points which are defined over the ring class fields of real quadratic fields. The resulting conjectures suggest that the classical Heegner point construction, and the theory of complex multiplication on which it is based, should extend to a variety of contexts in which the underlying field is not a CM field.

Introduction

Let E be an elliptic curve over ℚ of conductor N. It is now known (cf. [Wi], [TW], [BCDT]) that E is modular, so that E(ℂ) is equipped with a non-constant analytic uniformisation

φ : H^{∗}/Γ_{0}(N) ⟶ E(ℂ),  (1)

where H^{∗} := H ∪ ℚ ∪ {i∞} is the extended Poincaré upper half-plane, and Γ_{0}(N) ⊂ PSL_{2}(ℤ) is the usual Hecke congruence group, acting on H^{∗} by Möbius transformations. The compact Riemann surface H^{∗}/Γ_{0}(N) parametrizes pairs (A, C) where A is a (generalized) elliptic curve over ℂ and C ⊂ A(ℂ) is a cyclic subgroup of order N. In this way H^{∗}/Γ_{0}(N) is identified with the complex points of an algebraic curve X_{0}(N) defined over ℚ. The map φ of equation (1) is defined over ℚ, in the sense that it arises, after extending scalars from ℚ to ℂ, from a morphism of algebraic curves defined over the rational numbers.

An important application of (1) arises from the theory of complex multiplication. More precisely, let K ⊂ ℂ be an imaginary quadratic field and let τ be any point in H ∩ K. The set O of matrices in T ∈ M_{2}(ℤ) which are upper-triangular modulo N and satisfy

T = 0 or T τ = τ  (2)

is isomorphic to an order in K (of conductor 𝔣, say) which can be identified with the endomorphism ring of the pair (A_{τ}, C_{τ}) attached to τ. The theory of complex multiplication asserts that this pair is defined over the ring class field H of K of conductor 𝔣. Hence the so-called Heegner point P_{τ} := φ(τ) ∈ E(ℂ) is defined over H as well. This remark enables the construction of a plentiful supply of algebraic points on E, points which are defined over suitable ring class fields of imaginary quadratic fields.

The Heegner point construction is consistent with the Birch and Swinnerton-Dyer conjecture, in the following sense. The complex L-function L(E/H, s) factors as a product

L(E/H, s) = ∏_{χ} L(E/K, χ, s),  (3)

where χ ranges over the complex characters Gal(H/K) ⟶ ℂ^{×}. The definition of L(E/K, χ, s) as an Euler product implies that

L(E/K, χ, s) = L(E/K, χ̄, s).  (4)

At the same time, Rankin’s method yields the analytic continuation and functional equation of the L-function L(E/K, χ, s) for each χ, relating the expressions L(E/K, χ, s) and L(E/K, χ̄, 2 − s). Assume for simplicity that 𝔣 is prime to N. In this case the sign appearing in the functional equation is −1, so that, by parity considerations,

L(E/K, χ, 1) = 0 for each χ : Gal(H/K) ⟶ ℂ^{×}.  (5)

It follows from (3) and (5) that

ord_{s=1}L(E/H, s) ≥ [H : K].  (6)

The Birch and Swinnerton-Dyer conjecture leads to the expectation that

rank(E(H)) ≥^{?} [H : K].  (7)

It is believed that Heegner points account for the bulk of the growth of rank(E(H)) as H varies over all ring class fields of K of discriminant prime to N. For example, calculations of the kind carried out by Gross and Zagier in [GZ] should prove that an equality in (6) implies that the Heegner points in E(H) generate a subgroup of rank at least [H : K]. Under this hypothesis the work of Kolyvagin establishes an equality in (7), so that the Heegner points generate a finite index subgroup of E(H).

Replacing K by a real quadratic field, one is confronted with many situations in which inequality (6) continues to hold. For example, suppose that N = pM, where p is a prime which does not divide M. Let K be a real quadratic field in which p is inert and all primes ℓ dividing M are split. If H is any ring class field of K of conductor prime to N, then an argument identical to the one sketched above carries over to establish inequality (6). This is tantalising insofar as the theory of complex multiplication provides no handle on the problem of understanding inequality (7). In fact, no extension of the theory of complex multiplication to the context of real quadratic fields is known. The problem of supplying such a theory is intimately connected to Hilbert’s 12th problem of constructing the class fields of real quadratic fields (or of more general number fields) by analytic means.

The main goal of this work is to formulate a conjectural (p-adic) analytic construction of global points in the Mordell-Weil groups of E over certain ring class fields of real quadratic fields, generalising the theory of complex multiplication presented above. For the convenience of the reader, the main steps in this construction, and the main theorems of this article, are summarised in the remainder of the introduction.

1. Integration on H_{p} × H. Suppose now that N = pM, where M is a positive integer, and p is a prime not dividing M. Choose an Eichler ℤ[1/p]- order R of level M in M_{2}(ℚ). (These are all conjugate to each other.) To fix ideas, take from now on

R = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ[1/p]) with c ≡ 0 (mod M)\} .  (8)

Denote by Γ ⊂ PSL_{2}(ℤ[1/p]) the image of the group of elements of determinant 1 in R^{×}. It acts by Möbius transformations both on the p-adic upper half-plane

H_{p} := ℙ_{1}(ℂ_{p}) − ℙ_{1}(ℚ_{p})  (9)

and on the extended Poincaré upper half-plane H^{∗}. The induced diagonal action of Γ on H_{p}×H is discrete and discontinuous. The quotient (H_{p}×H)/Γ is investigated in a series of works of Ihara (cf. for example [Ih1] and [Ih2]) where it is related to the special fiber of X_{0}(M) in characteristic p. Stark [St] remarked that this quotient is formally analogous to a Hilbert modular surface.

Section 1 develops Stark’s analogy by formulating a theory of integration of “differential two-forms” on the quotient (H_{p}×H)/Γ, and deriving a notion of p-adic periods associated to such forms. More precisely, it is explained how a newform f_{0} of weight 2 on Γ_{0}(N) can be viewed as encoding the padic residues of a form ω of weight (2, 2) on (H_{p} × H)/Γ. If f_{0} has rational Fourier coefficients (i.e., it arises from an elliptic curve E of conductor N) a so-called double multiplicative integral is defined, depending on a choice of a homomorphism from the Néron lattice of E to ℤ:

× ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω ∈ ℂ_{p}^{×}, τ_{1}, τ_{2} ∈ H_{p}, x, y ∈ ℙ_{1}(ℚ).  (10)

As stated in lemma 1.11 of section 1, this function of the variables τ_{1}, τ_{2}, x, y behaves formally “as if” ω were a Hilbert modular form of weight (2, 2) on (H_{p} × H)/Γ, even though the object ω is not defined independently of its periods.

2. Periods attached to split tori. Since E has conductor N and p divides N exactly, the curve E has multiplicative reduction at p. Let

Φ_{Tate} : ℂ_{p}^{×} ⟶ E(ℂ_{p})  (11)

be Tate’s p-adic uniformisation attached to E over ℂ_{p}, and let q ∈ pℤ_{p} denote the p-adic period attached to this uniformisation.

Let K ≃ ℚ × ℚ be the split quadratic algebra, and let Ψ : K ⟶ M_{2}(ℚ) be a ℚ-algebra embedding. Write Ψ̄ : K^{×} ⟶ PGL_{2}(ℚ) for the natural homomorphism induced by Ψ. The torus Ψ̄(K^{×}) acting on the extended upper half plane H^{∗} has exactly two fixed points x_{Ψ}, y_{Ψ} ∈ ℙ_{1}(ℚ). The group Ψ̄(K^{×}) ∩ Γ is free of rank one, generated by an element γ_{Ψ}. A p-adic period I_{Ψ} ∈ ℂ_{p}^{×} is associated to Ψ by choosing a point z ∈ H_{p} and setting

I_{Ψ} := × ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω.  (12)

It is shown in lemma 2.3 of section 2 that I_{Ψ} does not depend on the choice of z that was made to define it, and that it belongs to ℚ_{p}^{×}.

Let ord_{p} : ℂ_{p}^{×} ⟶ ℚ be the valuation at p normalised so that ord_{p}(p) = 1, and let log : ℂ_{p}^{×} ⟶ ℂ_{p} be any choice of p-adic logarithm. In section 2 the following identity is proved

Theorem 1

log(I_{Ψ}) = \frac{log(q)}{ord_{p}(q)}ord_{p}(I_{Ψ}).

The proof of this theorem proceeds by evaluating both expressions ord_{p}(I_{Ψ}) and log(I_{Ψ}) independently. The quantity ord_{p}(I_{Ψ}) is related to special values of certain partial L-functions attached to E/ℚ, while log(I_{Ψ}) is expressed in terms of the first derivative of the corresponding p-adic L-function. Viewed in this way theorem 1 becomes a reformulation of a conjecture of Mazur, Tate and Teitelbaum [MTT] that was proved by Greenberg and Stevens [GS].

To allow for a cleaner statement of the conjectures and results, the following assumption on E is made from now on throughout the article:

Assumption 2 The elliptic curve E is unique in its ℚ-isogeny class.

The following conjecture, a multiplicative refinement of theorem 1, relates the period I_{Ψ} directly to q:

Conjecture 3 The period I_{Ψ} belongs to q^{ℤ}.

Remarks:

1. Theorem 1 implies that the group I_{Ψ}^{ℤ} ⊂ ℚ_{p}^{×} is either finite or is a lattice commensurable with q^{ℤ}. More precisely, it follows from this theorem that

I_{Ψ}^{ord_{p}(q)} = q^{ord_{p}(I_{Ψ})} (mod (ℚ_{p}^{×})_{tors})  (13)

To prove conjecture 3 in its entirety, it remains to:

• show that ord_{p}(q) divides ord_{p}(I_{Ψ}), so that

I_{Ψ} = q^{n} (mod (ℚ_{p}^{×})_{tors}), with n = ord_{p}(I_{Ψ})/ord_{p}(q).  (14)

Some theoretical evidence for this divisibility is given in [BD7], using multiplicity-one results of Mazur, the level-lowering result of Ribet, and the theory of Wiles yielding an isomorphism between certain Hecke rings and deformation rings. Alternately, as explained in section 2, much of the divisibility of ord_{p}(I_{Ψ}) by ord_{p}(q) can be derived as a consequence of the Birch and Swinnerton-Dyer conjecture.

• remove the (ℚ_{p}^{×})_{tors}-indeterminacy in formula (14). In the case where M = 1, de Shalit’s multiplicative refinement [DS] of the result of Greenberg and Stevens can be used to remove a large part of this indeterminacy. There is every reason to expect that the assumption M = 1 in [DS] is not essential, so that a full proof of conjecture 3, while not present in the literature, should lie within the scope of the methods developed in [GS] and [DS].

2. Note the analogy of conjecture 3 with the results of Oda [O] concerning periods on the Hilbert modular surface attached to a real quadratic field.

3. The Cohomology of Γ. An M-symbol with values in an abelian group C is a function m : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ C, denoted (x, y) ↦ m{x→y} and satisfying

m{x→y} + m{y→z} = m{x→z},  m{x→y} = −m{y→x},  (15)

for all x, y, z ∈ ℙ_{1}(ℚ). Denote by M the group of ℂ_{p}-valued M-symbols, and by M(C) the group of C-valued M-symbols. The group PSL_{2}(ℚ) (and hence, Γ) acts on M(C) by the rule

(γm){x→y} := m{γ^{−1}x→γ^{−1}y}.  (16)

Choose τ ∈ H_{p} and x ∈ ℙ_{1}(ℚ). The double multiplicative integral of equation (10) gives rise to a one-cocycle c̃_{f,τ} ∈ Z^{1}(Γ, M(ℂ_{p}^{×})) by the rule

c̃_{f,τ}(γ){x→y} = × ∫_{τ}^{γτ}∫_{x}^{y}ω.  (17)

The natural image c_{f} of c̃_{f,τ} in H^{1}(Γ, M(ℂ_{p}^{×})) is independent of the choice of τ that was made to define it.

Basic facts about the structure of H^{1}(Γ, M) as a module over the Hecke algebra show that the classes

ord_{p}(c_{f}) ∈ H^{1}(Γ, M(ℚ)) ⊂ H^{1}(Γ, M)  (18)

and

log(c_{f}) ∈ H^{1}(Γ, M)  (19)

belong to the same one-dimensional ℂ_{p}-vector subspace of H^{1}(Γ, M). Theorem 1 is used to conclude:

Theorem 4

log(c_{f}) = \frac{log(q)}{ord_{p}(q)}ord_{p}(c_{f}).

Conjecture 3 suggests the following multiplicative refinement of theorem 4.

Conjecture 5 Let L = ℚ_{p}(τ) be the field generated over ℚ_{p} by τ. There exists a one-cocycle ẽ_{f,τ} ∈ Z^{1}(Γ, M(ℤ)) and an M-symbol η̃_{f,τ} ∈ M(L^{×}) satisfying the relation

c̃_{f,τ}(γ){x→y} = q^{ẽ_{f,τ} (γ){x→y}} × (η̃_{f,τ}{γ^{−1}x→γ^{−1}y} ÷ η̃_{f,τ}{x→y}) , (20)

for all γ ∈ Γ and x, y ∈ ℙ_{1}(ℚ).

Remark:

1. Note that the image e_{f} of ẽ_{f,τ} in H^{1}(Γ, M(ℚ)) is determined by the property

ord_{p}(c_{f}) = ord_{p}(q)e_{f}.  (21)

2. The proofs of theorems 1 and 4 make no use of assumption 2, so these results hold without this assumption. On the other hand, computer calculations carried out by Peter Green [DG] indicate that conjecture 5 is false in general in the absence of assumption 2.

Let η_{f,τ} be the natural image of η̃_{f,τ} in M(L^{×}/q^{ℤ}). Reducing equation (20) of conjecture 5 modulo q^{ℤ} yields

× ∫_{τ}^{γτ}∫_{x}^{y}ω = η_{f,τ}{γ^{−1}x→γ^{−1}y} ÷ η_{f,τ}{x→y} (mod q^{ℤ}).  (22)

Note that this relation makes η_{f,τ} well-defined up to multiplication by elements in H^{0}(Γ, M(L^{×}/q^{ℤ})).

To avoid certain technical complications that will receive due treatment in chapter 3.3, assume for the rest of the introduction that M = 1; in this case the group H^{0}(Γ, M(L^{×}/q^{ℤ})) vanishes, so that η_{f,τ} ∈ M(L^{×}/q^{ℤ}) is uniquely determined by (22). Thus conjecture 5 makes it possible to define, for all x, y ∈ ℙ_{1}(ℚ), the indefinite multiplicative integral

× ∫ ^{τ}∫_{x}^{y}ω := η_{f,τ}{x→y} ∈ L^{×}/q^{ℤ}.  (23)

This indefinite integral satisfies the basic multiplicativity and Γ-equivariance properties stated in lemma 3.7 of section 3 (with Q = q^{ℤ}).

The last two sections (sections 4 and 5) are devoted to studying certain p-adic periods in ℂ_{p}^{×} associated to an embedding

Ψ : K ⟶ M_{2}(ℚ)  (24)

of a quadratic étale algebra K into M_{2}(ℚ), the case K = ℚ×ℚ corresponding to the situation already studied in section 2. An analogous setting in which K is an imaginary quadratic field, falling somewhat outside the scope of the machinery developed above, has been explored implicitly in [BD2], using the theory of complex multiplication and some facts about the bad reduction of modular curves. Sections 4 and 5 concentrate on the case where K is a real quadratic field.

4. Periods attached to real quadratic fields. Suppose that the prime p splits in the real quadratic field K. Choose τ ∈ H_{p} and x ∈ ℙ_{1}(ℚ). Given α, β ∈ Ψ̄(K^{×}) ∩ Γ, the quantity

⟨α, β⟩_{Ψ} := × ∫_{τ}^{α^{−1}τ}∫_{x}^{βx}ω ÷ × ∫_{τ}^{β^{−1}τ}∫_{x}^{αx}ω  (25)

is proven in section 4 to be independent of the choice of τ ∈ H_{p} and x ∈ ℙ_{1}(ℚ) that is made to define it. Moreover, ⟨ , ⟩_{Ψ} defines a ℂ_{p}^{×}-valued alternating bilinear pairing on Ψ̄(K^{×}) ∩ Γ.

The group Ψ̄(K^{×}) ∩ Γ is a free ℤ-module of rank two. Let γ_{1} and γ_{2} ∈ Γ be ℤ-module generators for this group. The period I_{Ψ} ∈ ℂ_{p}^{×} attached to Ψ is defined by setting

I_{Ψ} := ⟨γ_{1}, γ_{2}⟩_{Ψ}.  (26)

Note that {I_{Ψ}, I_{Ψ}^{−1}} is independent of the choice of basis (γ_{1}, γ_{2}). Theorem 6 below expresses I_{Ψ} in terms of the Tate period for E. More precisely, theorem 4 is shown to imply the analogue of theorem 1 for the real quadratic field K:

Theorem 6 If I_{Ψ} is the period attached to Ψ as in (26), then

log(I_{Ψ}) = \frac{log(q)}{ord_{p}(q)}ord_{p}(I_{Ψ}).

Furthermore, if conjecture 5 holds, then I_{Ψ} belongs to q^{ℤ}.

This identity should be viewed as the counterpart of the Greenberg-Stevens formula for real quadratic fields. See [BD6], where the connection of I_{Ψ} with special values of L-functions attached to E/K is briefly discussed.

5. Heegner points attached to real quadratic fields. The most intriguing application of the formalism of p-adic period integrals arises when the prime p is inert in the real quadratic field K, so that the p-adic completion K_{p} of K is isomorphic to the quadratic unramified extension of ℚ_{p}. In that case certain periods J_{Ψ} ∈ K_{p}^{×}/q^{ℤ} attached to Ψ are conjectured to give rise to global points on E defined over the ring class fields of K, in a manner analogous to the classical Heegner point construction when K is imaginary quadratic.

To describe J_{Ψ} precisely, note first that the torus Ψ(K^{×}) acting on H_{p} has precisely two fixed points z_{Ψ} and z̄_{Ψ}, which belong to ℙ_{1}(K_{p}) − ℙ_{1}(ℚ_{p}) ⊂ H_{p} and are interchanged by the action of Gal(K_{p}/ℚ_{p}). The group Ψ̄(K^{×}) ∩ Γ is free of rank one, with a generator γ_{Ψ} of the form Ψ̄(u), where u is an appropriate power of the fundamental unit attached to K.

The period J_{Ψ} ∈ K_{p}^{×}/q^{ℤ} is defined by choosing a base point x ∈ ℙ_{1}(ℚ) and using the indefinite integral of (23) to set

J_{Ψ} := × ∫ ^{z_{Ψ}}∫_{x}^{γ_{Ψ}x}ω ∈ K_{p}^{×}/q^{ℤ}.  (27)

Section 5 shows that J_{Ψ} does not depend on the choice of x ∈ ℙ_{1}(ℚ) that was made to define it, and that it depends in fact only on the Γ-conjugacy class of Ψ.

The algebra Ψ(K) ∩ R is isomorphic to a ℤ[1/p]-order in K. Let 𝔣_{Ψ} be the conductor of this order, and denote by H^{+} the narrow ring class field of K of conductor 𝔣_{Ψ}, defined as in section 5.2. Since p is inert in K/ℚ and does not divide 𝔣_{Ψ}, it splits completely in H^{+}/K. Choose an embedding of H^{+} into K_{p}. The main conjecture of section 5 (and, indeed, of the entire paper) is

Conjecture 7 The local point

P_{Ψ} := Φ_{Tate}(J_{Ψ}) ∈ E(K_{p})

is a global point in E(H^{+}).

Conjecture 7 extends the repertoire of modular constructions of rational points on elliptic curves beyond the currently known methods, which are all based on the theory of complex multiplication. The possibility of this construction, foreshadowed in [Da2], is directly inspired by the main theorem of [BD3]. But a proof would appear to fall beyond the scope of the methods used in that article, where the theory of complex multiplication and the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves play a central role.

Section 5 refines conjecture 7 by formulating a conjectural generalized Shimura Reciprocity Law describing the action of Gal(H^{+}/K) on P_{Ψ}. After a brief discussion of the expected relation between the points P_{Ψ} and derivatives of L-series, in the spirit of the formula of Gross and Zagier, the article concludes by presenting some numerical evidence for conjecture 7 and its refinements.

Acknowledgements. Several insights gleaned over the years in exchanges with Massimo Bertolini and Adrian Iovita have been instrumental in developing the point of view that is formulated here. It is a pleasure to acknowledge a considerable debt to these two collaborators. The author also thanks the participants of the undergraduate seminar on complex multiplication and real quadratic fields held at McGill in the summer of 2000: Derrick Chung, Jack Fearnley, Peter Green, Chiu-Fan Lee, Ivo Panyatov and Dan Segal. In particular, the implementation by Green, with the assistance of Fearnley, Lee and Panyatov, of a package of routines in Pari and C for calculating the p-adic integrals defined in section 1 has yielded a wealth of numerical evidence for conjecture 7. Some corrections and refinements of the conjectures formulated in a first draft of this article emerged in later discussions with Peter Green. While the fruit of these exchanges will be discussed more fully in [DG], some of Green’s observations have found their way into the revision of this article. Finally, the author is grateful to the referee for a thorough review of the original manuscript which has led to significant improvements in the exposition.

The author’s research was supported by CICMA and by grants from NSERC and FCAR. The participants of the McGill seminar on complex multiplication and real quadratic fields were supported by NSERC USR (Undergraduate Summer Research) awards.

1 Integration on (H_{p} × H)/Γ

1.1 Forms on T × H

Let T be the Bruhat-Tits tree of PGL_{2}(ℚ_{p}). Its set V(T ) of vertices is identified with the set of ℚ_{p}^{×}-homothety classes of rank two ℤ_{p}-modules in ℚ_{p}^{2}, two vertices being joined by an edge if the corresponding homothety classes have representatives which are contained in each other with index p. Write E(T ) for the set of ordered edges of T , and denote by s(e) and t(e) respectively the source and target vertex of e ∈ E(T ).

The group PGL_{2}(ℚ_{p}) acts naturally on T by isometries. The stabiliser of a vertex is conjugate to PGL_{2}(ℤ_{p}) while the stabiliser of an edge is conjugate to the group

Γ_{0}(pℤ_{p}) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ PGL_{2}(ℤ_{p}) such that p|c\} .  (28)

Let v_{∗} be the distinguished vertex corresponding to the homothety class of ℤ_{p}^{2} ⊂ ℚ_{p}^{2}, whose stabiliser is equal to PGL_{2}(ℤ_{p}). Let e_{∗} be the edge whose source is v_{∗} and whose stabiliser is equal to Γ_{0}(pℤ_{p}). Note that the stabiliser of v_{∗} in Γ is equal to Γ_{0}(M), while the stabiliser of e_{∗} in Γ is equal to Γ_{0}(N).

Recall (cf. [GvdP]) the reduction map

red : H_{p} ⟶ T  (29)

from H_{p} to the Bruhat-Tits tree T of PGL_{2}(ℚ_{p}). The inverse image of each open edge e ∈ E(T ) is called a basic wide open annulus in H_{p}, while the inverse image of a vertex is an example of an affinoid subdomain of H_{p}. If e is an edge of T , the orientation on it determines an orientation of the associated annulus V_{e}, i.e., an “interior” B_{0} and an “exterior” B_{∞}. (The reversed edge ē obtained from e by interchanging source and target gives rise to the same annulus V_{e}, but the interior and exterior p-adic discs attached to e and ē are exchanged.) Choose a coordinate function z_{e} on ℙ_{1}(ℂ_{p}) which induces the identifications

V_{e} →^{≃} {z ∈ H_{p} : 1/p < |z|_{p} < 1}, B_{0} →^{≃} {z ∈ ℂ_{p} : |z|_{p} ≤ 1/p}.  (30)

This coordinate function is well-defined, up to multiplication by an element of O_{ℂ_{p}}^{×}. (For further background on T and its relation to H_{p}, the reader is invited to consult [Kl] or [GvdP] for example.)

Digression on p-adic modular forms. For this paragraph, replace Γ temporarily by a subgroup of PSL_{2}(ℚ_{p}) acting discretely on H_{p}. (Arithmetic groups of this type can be obtained from the unit groups of ℤ[1/p] orders of definite quaternion algebras B which are split at p, after choosing an identification of B ⊗ ℚ_{p} with M_{2}(ℚ_{p}); see for example [BD1] or [BDIS].) Following [Sch], a rigid-analytic modular form of weight 2 on H_{p}/Γ is defined to be a rigid-analytic function f on H_{p} satisfying

f(\frac{az + b}{cz + d}) = (cz + d)^{2}f(z), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ, z ∈ H_{p}. (31)

The coefficient of dz_{e}/z_{e} in the Mittag-Leffler expansion of f on the affinoid B_{0} is called the residue of f at e and is denoted res_{e}(f(z)dz). (Cf. [Sch], p. 224.)

Lemma 1.1 The function κ_{f} : E(T ) ⟶ ℂ_{p} defined by κ_{f}(e) = res_{e}(f(z)dz) is harmonic, i.e.,

κ_{f}(ē) = −κ_{f}(e) for all e ∈ E(T ), ∑_{s(e)=v} κ_{f}(e) = 0, for all v ∈ V(T ).

This lemma follows from a rigid analytic analogue of the classical residue theorem of complex analysis, and is explained in [Sch], p. 225.

Returning to the case where Γ is the subgroup of SL_{2}(ℚ_{p}) of the introduction, the objects of definition (31) become trivial, since Γ acts on H_{p} with dense orbits. Motivated by an analogy with the theory of Hilbert modular forms, it seems desirable to replace rigid analytic modular forms on H_{p}/Γ by an appropriate notion of “form of weight (2, 2) on (H_{p} × H)/Γ”. Such an object should be a Γ-invariant expression of the form ω = f(z_{p}, z)dz_{p}dz, where z_{p} is a p-adic and z a complex variable. The function f would be rigid analytic in the first variable and holomorphic in the second. While it is unclear from the outset how to supply a sensible definition of ω, it is still possible to intuit a rigorous definition for its p-adic residues. More precisely, imagine taking the Mittag-Leffler expansion of ω on affinoid subdomains of H_{p} (leaving aside for the moment the problem that ω has not been defined!). The coefficient of dz_{e}/z_{e} in such an expansion – the residue res_{e}(ω) – should be interpreted as a holomorphic differential form on H. This informal discussion leads to the following precise definition, motivated by lemma 1.1 and tailored to capture the notion of the “p-adic residues of ω”.

Definition 1.2 A cusp form of weight 2 on (T × H)/Γ is a function

f : E(T ) × H ⟶ ℂ

satisfying

1. f(γe, γz) = (cz + d)^{2}f(e, z), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ.

2. The function f is harmonic, i.e., for each vertex v of T ,

∑_{s(e)=v} f(e, z) = 0,

and for each edge e of T , f(ē, z) = −f(e, z).

3. For each edge e of T , the function f_{e}(z) := f(e, z) is a cusp form of weight 2 (in the usual sense) on the group Γ_{e} := Stab_{Γ}(e).

Since Γ acts transitively on the unoriented edges of T , the group Γ_{e} is conjugate in Γ to Γ_{e_{∗}} = Γ_{0}(N), for each e. Property 1 is suggested by the desired Γ-invariance of ω, and property 2 by lemma 1.1 arising from the padic residue theorem. Note that an element of the space S_{2}(T , Γ) of cusp forms of weight 2 on (T × H)/Γ is completely described by a collection {f_{e}} of cusp forms on Γ_{e}, indexed by the edges e of T , satisfying the compatibility relation

f_{γe}(γz)d(γz) = f_{e}(z)dz, for all γ ∈ Γ,  (32)

together with the harmonicity condition 2.

In analysing the structure of S_{2}(T , Γ) it is convenient to introduce other spaces of modular forms defined in an analogous way. Let Γ̃ be the image in PGL_{2}(ℚ_{p}) of R_{+}^{×}, the group of invertible matrices in R whose determinant is positive. The group Γ consists of all the elements in Γ̃ such that

|γ| := ord_{p}(det(γ)) = 0 (in ℤ/2ℤ),  (33)

so that Γ̃ contains Γ with index 2.

Let S_{2}(E, Γ̃) denote the space of cusp forms on (E(T ) × H)/Γ̃, defined as the functions

f : E(T ) × H ⟶ ℂ

satisfying the analogues of properties 1 and 3 of definition 1.2, but not necessarily property 2, and with Γ replaced by Γ̃:

1. f(γe, γz) = \frac{(cz+d)^{2}}{det(γ)} f(e, z), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ̃.

2. For each edge e of T , the function f_{e}(z) := f(e, z) is a cusp form of weight 2 (in the usual sense) on the group Γ_{e} := Γ̃ ∩ Stab(e).

Likewise, let S_{2}(V, Γ̃) denote the space of forms on (V(T ) × H)/Γ̃, defined as the functions

f : V(T ) × H ⟶ ℂ

satisfying:

1. f(γv, γz) = \frac{(cz+d)^{2}}{det(γ)} f(v, z), for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ̃.

2. For each vertex v of T , the function f_{v}(z) := f(v, z) is a cusp form of weight 2 (in the usual sense) on the group Γ_{v} := Γ̃ ∩ Stab(v).

A vertex of T is said to be even if its distance from the distinguished vertex v_{∗} is even, and is said to be odd otherwise. Likewise, an edge e ∈ E(T ) is said to be even if its source vertex is even, and odd if its source vertex is odd. The group Γ preserves the parity of vertices and edges of T , while the elements in Γ̃ − Γ are parity-reversing. Given f ∈ S_{2}(T , Γ), define a form f̃ on E(T ) × H by choosing an element α ∈ Γ̃ − Γ and setting

f̃(e, z)dz = f(e, z)dz if e is even;  (34)
f̃(e, z)dz = f(αe, αz)dαz if e is odd.  (35)

Note that

1. The definition of f̃ does not depend on the choice of α that was made to define it, since any two such choices differ by left multiplication by an element of Γ, and f(e, z)dz is Γ-invariant.

2. The function f̃ belongs to S_{2}(E, Γ̃). (To check that

f̃(γe, γz)dγz = f̃(e, z)dz for all γ ∈ Γ̃,

it is easiest to separate four cases, depending on the parity of e and on whether γ belongs to Γ or Γ̃ − Γ.)

3. The assignment i : f ↦ f̃ is an injective homomorphism from S_{2}(T , Γ) to S_{2}(E, Γ̃). For if f̃ = 0, the function f(e, z) vanishes identically on all even edges, and hence on all edges by the harmonicity of f.

The normaliser of Γ_{0}(N) in Γ̃ consists of the elements of Γ̃ that fix the unordered edge attached to e_{∗}. Hence Γ_{0}(N) has index two in this normaliser. It will be convenient for later calculations to assume that α belongs to the normaliser of Γ_{0}(N) in Γ̃, so that

αe_{∗} = ē_{∗}.  (36)

There are two natural degeneracy maps

π_{s}, π_{t} : S_{2}(E, Γ̃) ⟶ S_{2}(V, Γ̃)  (37)

defined by

π_{s}(f)(v, z) := ∑_{s(e)=v} f(e, z),  π_{t}(f)(v, z) := ∑_{t(e)=v} f(e, z).  (38)

The exactness of the following sequence follows directly from the definitions:

0 ⟶ S_{2}(T , Γ) ⟶^{i} S_{2}(E, Γ̃) ⟶^{π_{s}⊕π_{t}} S_{2}(V, Γ̃)⊕S_{2}(V, Γ̃).  (39)

Let S_{2}(Γ_{0}(N)) denote the usual complex vector space of holomorphic cusp forms of weight 2 on H/Γ_{0}(N). Let

φ_{s} : X_{0}(N) ⟶ X_{0}(M)

be the natural projection arising from the inclusion Γ_{0}(N) ⊂ Γ_{0}(M), and let φ_{t} = φ_{s}W_{p} where W_{p} is the Atkin-Lehner involution at p acting on X_{0}(N), defined by

W_{p}f_{0}(z)dz = f_{0}(αz)d(αz).  (40)

Making an abuse of notation, denote by the same symbols φ_{s} and φ_{t} the two degeneracy maps from S_{2}(Γ_{0}(N)) to S_{2}(Γ_{0}(M)) induced from φ_{s} and φ_{t} by pushforward of differential forms. More precisely, choose a system of coset representatives for Γ_{0}(N) in Γ_{0}(M):

Γ_{0}(M) = γ_{1}Γ_{0}(N) ∪ ⋯ ∪ γ_{p+1}Γ_{0}(N).  (41)

One then has

φ_{s}(f)(z)dz = ∑_{j=1}^{p+1} f(γ_{j}^{−1}z)d(γ_{j}^{−1}z), φ_{t}(f)(z)dz = ∑_{j=1}^{p+1} f(αγ_{j}^{−1}z)d(αγ_{j}^{−1}z). (42)

The kernel of

φ_{s}⊕φ_{t} : S_{2}(Γ_{0}(N)) ⟶ S_{2}(Γ_{0}(M))⊕S_{2}(Γ_{0}(M))  (43)

is called the subspace of p-new forms, denoted S_{2}^{new−p}(Γ_{0}(N)). The following lemma relates the various spaces of forms on T to spaces of classical modular forms.

Lemma 1.3 1. The function which to f(e, z) associates f_{0}(z) := f_{e_{∗}}(z) induces an isomorphism from S_{2}(E, Γ̃) to S_{2}(Γ_{0}(N)).

2. The function which to f(v, z) associates f_{0}(z) := f_{v_{∗}}(z) induces an isomorphism from S_{2}(V, Γ̃) to S_{2}(Γ_{0}(M)).

3. The function which to f(e, z) associates f_{0}(z) := f_{e_{∗}}(z) induces an isomorphism from S_{2}(T , Γ) to S_{2}^{new−p}(Γ_{0}(N)).

Proof: To prove part 1, note that by definition of S_{2}(E, Γ̃), the form f_{0}(z) is a cusp form on Γ_{0}(N). The fact that Γ̃ acts transitively on E(T ) implies that f ∈ S_{2}(E, Γ̃) is completely determined by its restriction to {e_{∗}} × H, so that the assignment f ↦ f_{0} is injective. To show surjectivity, note that any f in S_{2}(Γ_{0}(N)) can be extended to a form on S_{2}(E, Γ̃) by the rule

f_{e}(z)dz = f_{0}(γz)dγz  (44)

if e = γ^{−1}e_{∗} with γ ∈ Γ̃. Part 2 is proved in an identical way, after observing that the stabiliser of v_{∗} in Γ̃ is equal to Γ_{0}(M). Finally, to prove part 3, consider the following natural diagram in which the first row is taken from equation (39) and the vertical maps are those of lemma 1.3:

0 ⟶  S_{2}(T , Γ)  ⟶^{i} S_{2}(E, Γ̃) ⟶  S_{2}(V, Γ̃) ⊕ S_{2}(V, Γ̃)
↓  ↓  ↓
0 ⟶ S_{2}^{new−p}(Γ_{0}(N)) ⟶ S_{2}(Γ_{0}(N)) ⟶ S_{2}(Γ_{0}(M)) ⊕ S_{2}(Γ_{0}(M)).

This diagram commutes: the commutativity of the first square follows directly from equation (34) defining i, in view of the fact that e_{∗} is an even edge. The commutativity of the second square follows from the choice of α satisfying (36) which shows that γ_{1}e_{∗}, … γ_{p+1}e_{∗} is a complete list of edges with source v_{∗}, while γ_{1}αe_{∗}, … γ_{p+1}αe_{∗} is a complete list of edges with target v_{∗}. Part 3 of lemma 1.3 follows from the five lemma.

The Hecke operators T_{ℓ} (ℓ |̸N) act on the spaces S_{2}(E, Γ̃), S_{2}(V, Γ̃), and S_{2}(T , Γ) via the identifications of lemma 1.3. More precisely, for each prime

ℓ, write the double coset Γ ( \begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix} ) Γ as a disjoint union of left cosets:

Γ ( \begin{matrix}ℓ & 0 \\ 0 & 1 \end{matrix} ) Γ = ∪_{j}γ_{j}Γ.  (45)

Then T_{ℓ}f is given by

(T_{ℓ}f)(e, z)dz = ∑_{j} f(γ_{j}^{−1}e, γ_{j}^{−1}z)d(γ_{j}^{−1}z).  (46)

Similarly, for each prime ℓ dividing M, let W_{ℓ} denote the Atkin-Lehner involution at ℓ acting on S_{2}^{new−p}(Γ_{0}(N)) or on S_{2}(T , Γ) via the identification of lemma 1.3. Writing N = ℓ^{n}N′, choose integers x, y, z, t such that

ℓ^{n}xt − N′yz = 1, and set α_{ℓ} = ( \begin{matrix}ℓ^{n} & y \\ Nz & ℓ^{n}t \end{matrix} ) .  (47)

The actions of W_{ℓ} are given by the rules

(W_{ℓ}f_{0})(z)dz = f_{0}(α_{ℓ}z)d(α_{ℓ}z)  (48)
(W_{ℓ}f)(e, z)dz = f(α_{ℓ}e, α_{ℓ}z)d(α_{ℓ}z),  (49)

for all f_{0} ∈ S_{2}(Γ_{0}(N)) and for all f ∈ S_{2}(T , Γ).

The involution W_{p} on S_{2}(Γ_{0}(N)) defined in (40) plays a particularly important role in our discussion.

Lemma 1.4 Let f_{0} be a form in S_{2}^{new−p}(Γ_{0}(N)) and let f be the form in S_{2}(T , Γ) associated to it by the identification of lemma 1.3. Denote by W_{p}f the form attached to W_{p}f_{0} by this identification. Then

(W_{p}f)(e, z)dz := −f(αe, αz)d(αz).  (50)

(Note the minus sign appearing in this formula.) Proof: Let e be any even edge of T , so that e = γe_{∗} for some γ ∈ Γ. A direct calculation using definition (40) of W_{p} and property (36) satisfied by α shows that

W_{p}f(e, z)dz = W_{p}f_{0}(γ^{−1}z)d(γ^{−1}z) = f_{0}(αγ^{−1}z)d(αγ^{−1}z)
= f(e_{∗}, αγ^{−1}z)d(αγ^{−1}z) = −f(ē_{∗}, αγ^{−1}z)d(αγ^{−1}z)
= −f(ē, γαγ^{−1}z)d(γαγ^{−1}z)
= −f(γαγ^{−1}e, γαγ^{−1}z)d(γαγ^{−1}z) = −f(αe, αz)d(αz),

where the last equality follows from the fact that γαγ^{−1} belongs to Γ̃ − Γ. A similar reasoning works if e is odd, and the lemma follows.

Let f_{0} be a normalized newform on Γ_{0}(N) having rational Fourier coefficients, so that it corresponds to an elliptic curve E over ℚ of conductor N by the Eichler–Shimura construction. Let f be the form in S_{2}(T , Γ) which is related to f_{0} by lemma 1.3, so that f_{e_{∗}} = f_{0}. The form f_{0} is an eigenvector for W_{p} acting on on S_{2}(Γ_{0}(N)). It is known that

W_{p}f_{0} = −f_{0} if E has split multiplicative reduction at p  (51)
W_{p}f_{0} = f_{0} if E has non-split multiplicative reduction at p. (52)

Let w be the negative of the eigenvalue of W_{p} acting on f_{0}, so that w = 1 if E has split multiplicative reduction at p, and w = −1 if E has non-split multiplicative reduction at p.

Lemma 1.5 The form f satisfies the following transformation rule under the group Γ̃ ⊃ Γ:

f(γe, γz)d(γz) = w^{|γ|}f(e, z)dz, for all γ ∈ Γ̃.

Proof: A direct consequence of the transformation property of f_{0} under W_{p} and lemma 1.4.

1.2 Mixed period integrals

The discussion preceding definition 1.2 suggests that f should be viewed as a system of residues for a “form ω of weight (2, 2)” on (H_{p} × H)/Γ, even though the ω itself is not defined. It is natural in this light to seek to attach to f periods analogous to the periods of modular forms of weight (2, 2) on a Hilbert modular surface.

By assumption 2, the elliptic curve E is isomorphic to the strong Weil curve in its isogeny class. Let φ : H^{∗}/Γ_{0}(N) ⟶ E(ℂ) be the strong Weil parametrisation attached to E. Letting ω_{E} denote the Néron differential of E, one has

φ^{∗}(ω_{E}) = 2πic_{φ}f_{0}(z)dz,  (53)

where c_{φ}, the so-called Manin constant, is a rational number which is known to be equal to ±1 in many cases (cf. [Ed]).

Choose elements x, y in the extended upper half-plane H^{∗} := H ∪ ℙ_{1}(ℚ). The function κ̃_{f}{x→y} : E(T ) ⟶ ℂ defined by

κ̃_{f}{x→y}(e) := 2πic_{φ} ∫_{x}^{y} f_{e}(z)dz  (54)

is a complex-valued harmonic cocycle on T , as follows immediately from the harmonicity properties of f itself. At the same time, the elements of E(T ) correspond to basic compact open subsets of ℙ_{1}(ℚ_{p}) by setting

U(e_{∗}) := ℤ_{p} ⊂ ℙ_{1}(ℚ_{p}),

and defining, for any edge e = γe_{∗} with γ ∈ GL_{2}(ℚ_{p}),

U(e) := γU(e_{∗}) = γℤ_{p} = {x ∈ ℙ_{1}(ℚ_{p}) such that γ^{−1}x ∈ ℤ_{p}}.  (55)

Thus κ̃_{f}{x→y} gives rise to a complex-valued distribution μ̃_{f}{x→y} on the boundary ℙ_{1}(ℚ_{p}) of H_{p} by the rule

μ̃_{f}{x→y}(U(e)) = κ̃_{f}{x→y}(e).  (56)

This distribution can only be integrated against locally constant complexvalued functions on ℙ_{1}(ℚ_{p}). For the purposes of p-adic integration, it is desirable that κ̃_{f}{x→y} take on integral, or at least p-adic integral, values. Fortunately, this can be achieved, provided that x and y belong to ℙ_{1}(ℚ). For in this case, the values of κ̃_{f}{x→y}(e) can be expressed in terms of the modular symbols

λ̃_{E}(a, b) := 2πic_{φ} ∫_{∞}^{− \frac{a}{b}} f_{0}(z)dz,  (57)

defined in [MTT], §8, where they are simply called 2πic_{φ}λ_{E}(a, b). More precisely, choose γ ∈ Γ̃ such that γe = e_{∗}, and write γx = −\frac{a}{b}, γy = −\frac{c}{d}. Then

κ̃_{f}{x→y}(e) = μ̃_{f}{x→y}(U(e)) = w^{|γ|}(λ̃_{E}(c, d) − λ̃_{E}(a, b)). (58)

Let Ω denote the Néron lattice attached to the elliptic curve E. The following proposition, proved under assumption 2, plays a key role in the definition of integrals on (H_{p} × H)/Γ.

Proposition 1.6 (Drinfeld-Manin) The ℤ-module Λ ⊂ ℂ generated by the modular symbols λ̃_{E}(a, b) is contained in Ω.

Proof: From equations (57) and (53),

λ̃_{E}(a, b) := 2πic_{φ} ∫_{∞}^{− \frac{a}{b}} f_{0}(z)dz = ∫_{φ([∞,− \frac{a}{b}])} ω_{E},  (59)

where φ([∞, −\frac{a}{b}]) is the image of the path joining a to b in H^{∗} under the modular parametrisation φ. By the theorem of Drinfeld and Manin, all cusps in H^{∗} map to rational torsion points in E. Hence by assumption 2, φ([∞, −\frac{a}{b}]) is a closed path on E(ℂ) and hence λ̃_{E}(a, b) belongs to Ω.

If E(ℝ) has two components then Ω is generated by a positive real period Ω_{+} and a purely imaginary period Ω_{−}. If E(ℝ) has one connected component, then Ω is contained with index two in the lattice spanned by Ω_{+} and Ω_{−}, where Ω_{+} (resp. Ω_{−}) denotes the real (resp. imaginary) half-period attached to E. In either case, thanks to proposition 1.6, one can write

λ̃_{E}(a, b) = λ_{E}^{+}(a, b) · Ω_{+} + λ_{E}^{−}(a, b) · Ω_{−},  (60)

with λ_{E}^{±}(a, b) ∈ ℤ. Choose a sign w_{∞} = ±1 and let λ_{E}(a, b) denote λ_{E}^{+}(a, b) (resp. λ_{E}^{−}(a, b)) if w_{∞} = 1 (resp. w_{∞} = −1). Write κ_{f} (resp. μ_{f}) for the ℤ-valued harmonic cocycle on T (resp. distribution on ℙ_{1}(ℚ_{p})) attached to this modular symbol, so that, with γ, a, b, c and d as in (58),

κ_{f}{x→y}(e) = μ_{f}{x→y}(U(e)) = w^{|γ|}(λ_{E}(c, d) − λ_{E}(a, b)). (61)

It is worth recording the following lemma which will be used repeatedly in the sequel:

Lemma 1.7 For all γ ∈ Γ̃, x, y ∈ ℙ_{1}(ℚ), and e ∈ E(T ),

κ_{f}{γx→γy}(γe) = w^{|γ|}κ_{f}{x→y}(e).

Proof: By definition,

κ̃_{f}{γx→γy}(γe) = ∫_{γx}^{γy} f_{γe}(z)dz = w^{|γ|} ∫_{γx}^{γy} f_{e}(γ^{−1}z)d(γ^{−1}z)
= w^{|γ|} ∫_{x}^{y} f_{e}(z)dz = w^{|γ|}κ̃_{f}{x→y}(e).

The same result with κ̃_{f} replaced by κ_{f} follows at once.

The next lemma complements lemma 1.7 by describing the transformation behaviour of κ_{f} under the entire group R^{×} which contains Γ̃ with index two.

Lemma 1.8 If α_{∞} ∈ R^{×} is any element of determinant −1, then

κ_{f}{α_{∞}x→α_{∞}y}(α_{∞}e) = w_{∞}κ_{f}{x→y}(e).

Proof: It is enough to show this for a single such α_{∞}, say the matrix

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). In this case it follows directly from the relation

λ_{E}(−a, b) = w_{∞}λ_{E}(a, b)  (62)

satisfied by the modular symbol λ_{E} attached to the choice of sign w_{∞}. (Cf. [MTT].)

For any γ ∈ R^{×}, set sgn(γ) = 0 if det(γ) > 0, and sgn(γ) = 1 if det(γ) < 0. Lemmas 1.7 and 1.8 can be combined into the following transformation formula satisfied by κ_{f} under the action of R^{×}:

κ_{f}{γx→γy}(γe) = w^{|γ|}w_{∞}^{sgn(γ)}κ_{f}{x→y}(e), for all γ ∈ R^{×}.  (63)

Because the values μ_{f}{x→y}(U(e)) are integral and hence p-adically bounded as e ∈ E(T ), the distribution μ_{f}{x→y} defines a p-adic measure on ℙ_{1}(ℚ_{p}). In particular, if h is any bounded locally analytic ℂ_{p}-valued function on ℙ_{1}(ℚ_{p}), the integral

∫_{ℙ_{1}(ℚ_{p})} h(t)dμ_{f}{x→y}(t) ∈ ℂ_{p}  (64)

can be defined in the usual way.

Digression on p-adic modular forms, continued. To motivate definition 1.9 below of the periods attached to ω, it is useful to return to the digression about p-adic modular forms and the setting where f is a rigid analytic modular form on H_{p}/Γ, with Γ a discrete arithmetic subgroup of SL_{2}(ℚ_{p}). The function κ_{f} : E(T ) ⟶ ℂ_{p} encoding the residues of f gives rise to a p-adic distribution μ_{f} on ℙ_{1}(ℚ_{p}) which is p-adically bounded, since T /Γ is a finite graph. In [Te], it is proved that the weight two modular form f can be recovered from its boundary distribution μ_{f} by the elegant Poisson inversion formula

f(z) = ∫_{ℙ_{1}(ℚ_{p})} \frac{1}{z − t}dμ_{f}(t).  (65)

To the modular form f and a choice of two endpoints z_{1}, z_{2} ∈ H_{p} is attached the Coleman p-adic line integral ∫_{z_{1}}^{z_{2}} f(z)dz which depends on a choice of padic logarithm log : ℂ_{p}^{×} ⟶ ℂ_{p}. Taking this Coleman integral on both sides of equation (65) and formally interchanging the order of integration on the right suggests the identity

∫_{z_{1}}^{z_{2}} f(z)dz = ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − z_{2}}{t − z_{1}} ) dμ_{f}(t),  (66)

which can be justified rigorously as in [BDIS] or alternately can be adopted as a definition of the Coleman line integral in this setting.

Returning to the original setting, but guided by formula (66), the following definition, depending similarly on a choice of log, imposes itself naturally.

Definition 1.9 Let z_{1} and z_{2} be elements of H_{p}, and let x, y ∈ ℙ_{1}(ℚ).

∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω := ∫_{ℙ_{1}(ℚ_{p})} log (\frac{t − z_{2}}{t − z_{1}} ) dμ_{f}\{x→y\}(t) ∈ ℂ_{p}.  (67)

The following lemma shows that this definition is well-behaved:

Lemma 1.10 The double integrals of definition 1.9 satisfy the following properties:

∫_{z_{1}}^{z_{3}}∫_{x}^{y}ω = ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω + ∫_{z_{2}}^{z_{3}}∫_{x}^{y}ω;  (68)
∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{3}}ω = ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{2}}ω + ∫_{zi_{1}}^{z_{2}}∫_{x_{2}}^{x_{3}}ω;  (69)
∫_{γz_{1}}^{γz_{2}}∫_{γx}^{γy}ω = w^{|γ|}w_{∞}^{sgn(γ)} ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω, for all γ ∈ R^{×}.  (70)

Proof: The first and second identity are a direct consequence of definition 1.9, while the third follows from equation (63).

Caveat: Once again, the reader should not be misled by this notation into thinking that ω is defined by itself; only its system of p-adic residues, described by f, is defined, but this is enough to make sense of definition 1.9. Of course, the notation is meant to be suggestive, and the reader should view the left hand side of definition 1.9 as a period for a form of weight (2, 2) on (H_{p} × H)/Γ, with the complex period Ω_{+} or Ω_{−} “factored out”.

To obtain stronger formulae, it is preferable to avoid choosing a p-adic logarithm, exploiting the fact that κ_{f}{x→y} is ℤ-valued to define

× ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω := × ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − z_{2}}{t − z_{1}} ) dμ_{f}\{x→y\}(t) ∈ ℂ_{p}^{×}.  (71)

Here × ∫ denotes the multiplicative integral, in which limits of products replace the usual limits of Riemann sums. More precisely,

× ∫_{ℙ_{1}(ℚ_{p})} g(t)dμ(t) := lim_{C} ∏_{U_{α}∈C} g(t_{α})^{μ(U_{α})},  (72)

where the limit is taken over increasingly fine covers C = {U_{α}}_{α} of ℙ_{1}(ℚ_{p}) by disjoint compact open subsets, with t_{α} ∈ U_{α}. This limit exists if log g is locally analytic and g takes values in a compact subset of ℂ_{p}^{×}, as is the case for the integrand appearing in (71).

The multiplicative integral has the advantage that it does not rely on a choice of p-adic logarithm. It is related to its additive counterpart by the formula

∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω = log (× ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω) .  (73)

Since any p-adic logarithm vanishes on the torsion in ℂ_{p}^{×}, the multiplicative integral carries more information than the additive one. Note also that it is ℂ_{p}^{×}, and not ℂ_{p}, which arises most naturally in Tate’s p-adic uniformisation theory of elliptic curves with multiplicative reduction.

Properties analogous to those of lemma 1.10, with addition replaced by multiplication, hold for the multiplicative integral:

Lemma 1.11 The double multiplicative integral of definition 1.9 satisfies the following properties:

× ∫_{z_{1}}^{z_{3}}∫_{x}^{y}ω = × ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω × × ∫_{z_{2}}^{z_{3}}∫_{x}^{y}ω;  (74)
× ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{3}}ω = × ∫_{z_{1}}^{z_{2}}∫_{x_{1}}^{x_{2}}ω × × ∫_{z_{1}}^{z_{2}}∫_{x_{2}}^{x_{3}}ω;  (75)

× ∫_{γz_{1}}^{γz_{2}}∫_{γx}^{γy}ω = (× ∫_{z_{1}}^{z_{2}}∫_{x}^{y}ω)^{w^{|γ|}w_{∞}^{sgn(γ)}} , for all γ ∈ R^{×}.  (76)

Proof: The proof is identical to that of lemma 1.10.

2 Periods attached to split tori

2.1 Preliminary calculations

Let K = ℚ × ℚ, let Ψ : K ⟶ M_{2}(ℚ) be a ℚ-algebra embedding, and let c be a positve integer which is relatively prime to N. One says that Ψ is an optimal embedding of conductor c if the subring O := Ψ^{−1}(R) is the ℤ[1/p]-order in K of conductor c, so that

O = {(u, v) ∈ ℤ[1/p] × ℤ[1/p] such that u ≡ v (mod c)}.  (77)

An orientation on O is a ring homomorphism

𝔬 : O ⟶ ℤ/Mℤ.  (78)

If M = M_{1}M_{2} is a factorisation of M into a product of two relatively prime integers, the homomorphism 𝔬_{M_{1},M_{2}} defined by

𝔬_{M_{1},M_{2}}((a, b)) = (a mod M_{1}, b mod M_{2}) ∈ ℤ/M_{1}ℤ × ℤ/M_{2}ℤ = ℤ/Mℤ (79)

is an orientation, and all orientations are of this form.

An optimal embedding Ψ of conductor c gives rise to an orientation 𝔬_{Ψ} on O, sending x ∈ O to the residue class modulo M of the upper left hand entry of Ψ(x). The embedding Ψ is said to be oriented if

𝔬_{Ψ} = 𝔬_{M,1}.  (80)

Lemma 2.1 Let M = M_{1}ℓ^{n}M_{2} be a factorization of M into three relatively prime integers M_{1}, ℓ^{n} and M_{2}. If 𝔬_{Ψ} = 𝔬_{M_{1},ℓ^{n}M_{2}}, and α_{ℓ} is the matrix of (47) used to define the Atkin-Lehner involution W_{ℓ}, then

𝔬_{α_{ℓ}Ψα_{ℓ}^{−1}} = 𝔬_{M_{1}ℓ^{n},M_{2}}.

Proof: This follows by a direct calculation with matrices using the definition of α_{ℓ} given in (47).

For each integer ν satisfying gcd(ν, c) = 1, define the embedding Ψ_{ν} of K into M_{2}(ℚ) by the rule

Ψ_{ν}(a, a) = ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) , Ψ_{ν}(c, 0) = ( \begin{matrix}c & ν \\ 0 & 0 \end{matrix} ) .  (81)

Note that Ψ_{ν} is an oriented optimal embedding of conductor c. Conversely:

Lemma 2.2 If Ψ is any oriented optimal embedding of K of conductor c, it is Γ-conjugate to the embedding Ψ_{ν} for some ν with gcd(ν, c) = 1. The image of ν in (ℤ/cℤ)^{×}/⟨p^{2}⟩ is uniquely determined by Ψ.

Proof: The embedding Ψ maps (c, 0) to a matrix in M_{2}(ℤ[1/p]) of determinant 0 and trace c, so that

Ψ(c, 0) = ( \begin{matrix}r & s \\ t & u \end{matrix} ) , r ≡ c (mod M), ru − ts = 0, r + u = c.

Let x and y be relatively prime integers satisfying the equation tx − ry = 0. Note that M divides y, since M divides t and r is a unit modulo M. Hence one can choose γ ∈ Γ of the form

γ = ( \begin{matrix}x & x′ \\ y & y′ \end{matrix} ) .  (82)

A direct calculation now shows that

γ^{−1}Ψ(c, 0)γ = ( \begin{matrix}c & ν \\ 0 & 0 \end{matrix} ) ,  (83)

where ν ∈ ℤ[1/p] is relatively prime to c. Conjugating by the matrices

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}p & 0 \\ 0 & p^{−1} \end{matrix} ) shows that Ψ_{ν} and Ψ_{ν′} are conjugate to each

other in Γ if and only if the natural images of ν and ν′ in (ℤ/Mℤ)^{×}/⟨p^{2}⟩ are the same.

Recall from the introduction that the torus Ψ(K^{×}) acting on H^{∗} has exactly two fixed points x_{Ψ} and y_{Ψ}, and that Ψ̄(K^{×}) ∩ Γ is free of rank one, generated by an element γ_{Ψ} ∈ Γ. The period I_{Ψ} is then defined by choosing z ∈ H_{p} and setting

I_{Ψ} = × ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω.  (84)

Lemma 2.3 The period I_{Ψ} does not depend on the choice of z ∈ H_{p} that was made to define it. Furthermore, it depends only on the Γ-conjugacy class of Ψ.

Proof: The integral I_{Ψ} is independent of z, since

× ∫_{z_{1}}^{γ_{Ψ}z_{1}}∫_{x_{Ψ}}^{y_{Ψ}}ω ÷ × ∫_{z_{2}}^{γ_{Ψ}z_{2}}∫_{x_{Ψ}}^{y_{Ψ}}ω = × ∫_{z_{1}}^{z_{2}}∫_{x_{Ψ}}^{y_{Ψ}}ω ÷ × ∫_{γ_{Ψ}z_{1}}^{γ_{Ψ}z_{2}}∫_{x_{Ψ}}^{y_{Ψ}}ω,

by property (74) of the multiplicative integral stated in lemma 1.11. By property (76),

× ∫_{z_{1}}^{z_{2}}∫_{x_{Ψ}}^{y_{Ψ}}ω = × ∫_{γ_{Ψ}z_{1}}^{γ_{Ψ}z_{2}}∫_{x_{Ψ}}^{y_{Ψ}}ω,

and it follows that I_{Ψ} does not depend on z. Replacing Ψ by αΨα^{−1} with α ∈ Γ̃, the period attached to αΨα^{−1} becomes

× ∫_{z}^{αγ_{Ψ}α^{−1}z}∫_{αx_{Ψ}}^{αy_{Ψ}}ω = (× ∫_{α^{−1}z}^{γ_{Ψ}α^{−1}z}∫_{x_{Ψ}}^{y_{Ψ}}ω)^{w^{|α|}} = (× ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω)^{w^{|α|}} ,  (85)

where the first equality follows from property (76) of the integral, and the second equality follows from the independence of I_{Ψ} on z.

The goal of section 2 is to prove theorem 1 of the introduction. We begin by disposing of it in the following trivial special case.

Lemma 2.4 Suppose that E has non-split multiplicative reduction at p and that p has odd order in (ℤ/cℤ)^{×}. Then I_{Ψ} = ±1.

Proof: Let t be the order of p in (ℤ/cℤ)^{×} and let α = Ψ(p^{t}, 1) ∈ Γ̃ − Γ. Then Ψ = αΨα^{−1}. On the other hand, by equation (85), I_{αΨα^{−1}} = I_{Ψ}^{−1}.

Because of lemma 2.4, the following assumption will be made in the rest of section 2.

Assumption 2.5 When w = −1, the order of p in (ℤ/cℤ)^{×} is even.

By interchanging x_{Ψ} and y_{Ψ} if necessary, assume that y_{Ψ} is an attractive fixed point for γ_{Ψ} and that x_{Ψ} is a repulsive fixed point, so that, for all t ∈ H_{p},

γ_{Ψ}^{n}(t) → y_{Ψ} as n → ∞,  γ_{Ψ}^{n}(t) → x_{Ψ} as n → −∞.  (86)

Let M_{Ψ}(t) be a Möbius transformation with coefficients in ℚ_{p} sending y_{Ψ} to 0 and x_{Ψ} to ∞. For example, if x_{Ψ} and y_{Ψ} are not equal to ∞, one may take

M_{Ψ}(t) = \frac{t − y_{Ψ}}{t − x_{Ψ}} .

Note that M_{Ψ}(t) is well-defined up to multiplication by a scalar in ℚ_{p}^{×}.

The element γ_{Ψ} acts discretely on the complement ℙ_{1}(ℚ_{p}) − FP_{Ψ} of the fixed-point set FP_{Ψ} := {x_{Ψ}, y_{Ψ}}. In fact, Lemma 2.6

M_{Ψ}(γ_{Ψ}t) = p^{s}M_{Ψ}(t),

where

s = 2 × the order of p^{2} in (ℤ/cℤ)^{×}.

(Note that by assumption 2.5, s is equal to the order of p in (ℤ/cℤ)^{×} when w = −1.) Proof: Note that the Möbius tranformations M_{Ψ}(γ_{Ψ}t) and M_{Ψ}(t) both send y_{Ψ} to 0 and x_{Ψ} to ∞, so they must differ by multiplication by a scalar in ℚ_{p}^{×}. A direct calculation shows that this scalar is equal to p^{s}, using the fact that

γ_{Ψ} = Ψ̄ (( \begin{matrix}p^{s/2} & 0 \\ 0 & p^{−s/2} \end{matrix} ))

is the image in PSL_{2}(ℚ_{p}) of a matrix having eigenvalues p^{s/2} and p^{−s/2}.

Lemma 2.6 allows the calculation of a fundamental region F_{Ψ} for the action of the element γ_{Ψ} on ℙ_{1}(ℚ_{p}) − FP_{Ψ}:

F_{Ψ} = {t ∈ ℙ_{1}(ℚ_{p}) − FP_{Ψ} such that 0 ≤ ord_{p}(M_{Ψ}(t)) < s} . (87)

Points of the boundary ℙ_{1}(ℚ_{p}) of the p-adic upper half-plane H_{p} correspond to ends of T ; let path(x_{Ψ}, y_{Ψ}) be the infinite path on T joining the ends associated to x_{Ψ} and y_{Ψ}. For each vertex v on this path, let

U(v) ⊂ ℙ_{1}(ℚ_{p}) − FP_{Ψ}

be the compact open subset corresponding to all points associated to ends originating from v and not passing through any edge in path(x_{Ψ}, y_{Ψ}). The vertices of path(x_{Ψ}, y_{Ψ}) can be indexed consecutively by subscripts j ∈ ℤ so that

U(v_{j}) = {t ∈ ℙ_{1}(ℚ_{p}) − FP_{Ψ} such that ord_{p}(M_{Ψ}(t)) = j} ,  (88)

so that

F_{Ψ} = U(v_{0}) ∪ U(v_{1}) ∪ ⋯ ∪ U(v_{s−1}).  (89)

Let e_{j} be the edge on path(x_{Ψ}, y_{Ψ}) joining v_{j−1} to v_{j}. Note that

γ_{Ψ}(v_{j}) = v_{j+s},  γ_{Ψ}(e_{j}) = e_{j+s}.  (90)

Set

m_{Ψ} = κ_{f}{x_{Ψ}→y_{Ψ}}(e_{0}) = κ_{f}{x_{Ψ}→y_{Ψ}}(e_{s}).  (91)

The remainder of section 2.1 is devoted to proving the following proposition, a first step in the evaluation of I_{Ψ}.

Proposition 2.7 For all z ∈ H_{p},

I_{Ψ} = × ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω = (× ∫_{F_{Ψ}} M_{Ψ}(t)dμ_{f}{x_{Ψ}→y_{Ψ}}(t)) × p^{sm_{Ψ}}.

In particular, I_{Ψ} belongs to ℚ_{p}^{×}.

Proof: By definition,

× ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω = × ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − γ_{Ψ}z}{t − z} ) dμ_{f}\{x_{Ψ}→y_{Ψ}\}(t).  (92)

For conciseness, write μ_{f} for μ_{f}{x_{Ψ}→y_{Ψ}} and κ_{f} for κ_{f}{x_{Ψ}→y_{Ψ}} in the remainder of section 2.

Lemma 2.8 The measure μ_{f} is invariant under multiplication by γ_{Ψ}, i.e,. μ_{f}(γ_{Ψ}U) = μ_{f}(U) for all compact open subsets U of ℙ_{1}(ℚ_{p}).

Proof: This follows directly from lemma 1.7.

Lemma 2.9 If c ∈ ℂ_{p}^{×} is any non-zero constant, then

× ∫_{F_{Ψ}} cdμ_{f}(t) = 1.

Proof: This is simply because μ_{f}(F_{Ψ}) = κ_{f}(e_{0}) − κ_{f}(e_{s}) = 0.

Proof of proposition 2.7: To evaluate the integral appearing in the right hand side of equation (92), break up the region of integration ℙ_{1}(ℚ_{p}) as follows:

ℙ_{1}(ℚ_{p}) = U_{−}(n) ∪ U_{+}(n) ∪ ⋃_{j=−n}^{n} γ_{Ψ}^{j}F_{Ψ},  (93)

where

U_{−}(n) = {t ∈ ℙ_{1}(ℚ_{p}) − FP_{Ψ} s.t. ord_{p}(M_{Ψ}(t)) < −ns} ∪ {x_{Ψ}},
U_{+}(n) = {t ∈ ℙ_{1}(ℚ_{p}) − FP_{Ψ} s.t. ord_{p}(M_{Ψ}(t)) ≥ (n + 1)s} ∪ {y_{Ψ}}.

Invoking lemma 2.8, define

I(n) := ∏_{j=−n}^{n} × ∫_{γ_{Ψ}^{j}F_{Ψ}} (\frac{t − γ_{Ψ}z}{t − z} ) dμ_{f}(t) = ∏_{j=−n}^{n} × ∫_{F_{Ψ}} (\frac{γ_{Ψ}^{j}t − γ_{Ψ}z}{γ_{Ψ}^{j}t − z} ) dμ_{f}(t). (94)

Observe that

μ_{f}(U_{−}(n)) = −κ_{f}(e_{0}) = −m_{Ψ}, μ_{f}(U_{+}(n)) = κ_{f}(e_{s}) = m_{Ψ}. (95)

Hence by definition of the multiplicative integral,

× ∫_{ℙ_{1}(ℚ_{p})} (\frac{t − γ_{Ψ}z}{t − z} ) dμ_{f}(t) = (\frac{x_{Ψ} − γ_{Ψ}z}{x_{Ψ} − z} )^{−m_{Ψ}} (\frac{y_{Ψ} − γ_{Ψ}z}{y_{Ψ} − z} )^{m_{Ψ}} lim_{n→∞} I(n)

= (\frac{M_{Ψ}(γ_{Ψ}z)}{M_{Ψ}(z)})^{m_{Ψ}} lim_{n⟶∞} I(n) = p^{sm_{Ψ}} lim_{n→∞} I(n), by lemma 2.6. (96)

Note that the integrand \frac{γ_{Ψ}^{j}t−γ_{Ψ}z}{γ_{Ψ}^{j}t−z} appearing in the definition of I(n) and the function \frac{t−γ_{Ψ}^{−j+1}z}{t−γ_{Ψ}^{−j}z} differ by a non-zero scalar multiple (depending on z but not on t), so that by lemma 2.9

× ∫_{F_{Ψ}} (\frac{γ_{Ψ}^{j}t − γ_{Ψ}z}{γ_{Ψ}^{j}t − z} ) dμ_{f}(t) = × ∫_{F_{Ψ}} (\frac{t − γ_{Ψ}^{−j+1}z}{t − γ_{Ψ}^{−j}z} ) dμ_{f}(t).  (97)

Interchanging the order of summation and integration in the definition (94) of I(n),

I(n) = × ∫_{F_{Ψ}} ( ∏_{j=−n}^{n} \frac{t − γ_{Ψ}^{−j+1}z}{t − γ_{Ψ}^{−j}z} ) dμ_{f}(t)  (98)
= × ∫_{F_{Ψ}} \frac{(t − γ_{Ψ}^{n+1}z)}{(t − γ_{Ψ}^{−n}z)}dμ_{f}(t)  (99)
= × ∫_{F_{Ψ}} M(γ_{Ψ}^{n+1}z; γ_{Ψ}^{−n}z; t)dμ_{f}(t),  (100)

where for a, b ∈ H, M(a; b; t) is any Möbius transformation sending a to 0 and b to ∞.

By (86), there exists a sequence of scalars λ_{n} ∈ ℂ_{p}^{×} such that the ℂ_{p}^{×}- valued functions λ_{n}M(γ_{Ψ}^{n+1}z; γ_{Ψ}^{−n}z; t) converge to M_{Ψ}(t) uniformly on compact subsets of ℙ_{1}(ℚ_{p}) − {x_{Ψ}, y_{Ψ}} as n → ∞, so that

lim_{n→∞} I(n) = × ∫_{F_{Ψ}} M_{Ψ}(t)dμ_{f}(t).  (101)

Combining this with equations (92) and (96),

× ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω = (× ∫_{F_{Ψ}} M_{Ψ}(t)dμ_{f}{x_{Ψ}→y_{Ψ}}(t)) × p^{sm_{Ψ}},  (102)

as was to be shown.

2.2 Evaluation of ord_{p}(I_{Ψ})

Define the so-called Winding element attached to Ψ by choosing a vertex of T and setting

W_{Ψ} = ∑_{v→γ_{Ψ}v} κ_{f}{x_{Ψ}→y_{Ψ}}(e),

the sum being taken over all edges in the path joining v to γ_{Ψ}v.

Lemma 2.10 The winding element W_{Ψ} does not depend on the choice of v ∈ V(T ) that was made to define it, and it depends only on the Γ-conjugacy class of Ψ.

Replacing v by v′ changes W_{Ψ} by the quantity

∑_{v′→γ_{Ψ}v′} κ_{f}{x_{Ψ}→y_{Ψ}}(e) − ∑_{v→γ_{Ψ}v} κ_{f}{x_{Ψ}→y_{Ψ}}(e)

= ∑_{v′→v} κ_{f}{x_{Ψ}→y_{Ψ}}(e) − ∑_{γ_{Ψ}v′→γ_{Ψ}v} κ_{f}{x_{Ψ}→y_{Ψ}}(e) = 0,

where the first equality follows from the harmonicity of κ_{f}{x_{Ψ}→y_{Ψ}} and the fact that T is simply connected, while the second equality follows from the Γ-invariance of κ_{f} (lemma 1.7).

Proposition 2.11

ord_{p}(I_{Ψ}) = W_{Ψ}.

Proof: By proposition 2.7,

ord_{p} (× ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω) = ord_{p} (× ∫_{F_{Ψ}} M_{Ψ}(t)dμ_{f}(t) × p^{sm_{Ψ}})  (103)
= (∫_{F_{Ψ}} ord_{p}(M_{Ψ}(t))dμ_{f}(t)) + sm_{Ψ}. (104)

But the function g(t) := ord_{p}(M_{Ψ}(t)) is locally constant on F_{Ψ}, and satisfies

g(t) = i, for all t ∈ U(v_{i}).  (105)

Hence

ord_{p} (× ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω) = (∑_{i=0}^{s−1} i(κ_{f}(e_{i}) − κ_{f}(e_{i+1}))) + sκ_{f}(e_{s}) (106)

= ∑_{i=1}^{s} κ_{f}(e_{i}) = ∑_{v_{0}→γ_{Ψ}v_{0}} κ_{f}(e) = W_{Ψ},  (107)

and the result follows.

Lemma 2.1 implies that any embedding Ψ of conductor c is conjugate, under the action of the group generated by the matrices α_{ℓ}, to an oriented optimal embedding of conductor c. In particular, since the newform f_{0} is an eigenvector for all the Atkin-Lehner involutions, one has

Lemma 2.12 Given any embedding Ψ : K ⟶ M_{2}(ℚ) of conductor c, there exists an oriented embedding Ψ′ of conductor c for which

I_{Ψ′} = I_{Ψ}^{±1}.

Hence, to show conjecture 3 or theorem 1 it is enough to show it for all oriented optimal embeddings. Lemma 2.2 allows us to focus exclusively on the embeddings of the special form described in (81). When Ψ = Ψ_{ν}, the following proposition evaluates the winding number W_{Ψ} in terms of the modular symbols λ_{E}(a, b) attached to E in equation (57) of section 1.

Proposition 2.13 Let J be the coset of (ℤ/cℤ)^{×} consisting of a such that a/ν ≡ p^{j} (mod c) for some j = j(a). Then

ord_{p}(I_{Ψ_{ν}}) = W_{Ψ_{ν}} = ∑_{a∈J} w^{j(a)}λ_{E}(a, c).  (108)

Proof: Keeping the notations of the discussion preceding the statement of proposition 2.7, note that

W_{Ψ} = κ_{f}{x_{Ψ}→y_{Ψ}}(e_{0}) + ⋯ + κ_{f}{x_{Ψ}→y_{Ψ}}(e_{s−1}).  (109)

A direct calculation reveals that the two fixed points for the action of Ψ(K^{×}) on ℙ_{1}(ℚ) are

x_{Ψ} = ∞ and y_{Ψ} = −\frac{ν}{c} ,  (110)

and that a fundamental region for the action of γ_{Ψ} on ℙ_{1}(ℚ_{p}) − {x_{Ψ}, y_{Ψ}} is therefore given by

F_{Ψ} = \{t ∈ ℙ_{1}(ℚ_{p}) such that 0 ≤ ord_{p} (t + \frac{ν}{c} ) < s\} ,  (111)

where s is twice the order of p^{2} in (ℤ/cℤ)^{×}. By proposition 2.7, and since M_{Ψ}(t) can be chosen to be equal to t + \frac{ν}{c},

U(e_{j}) = \{t ∈ ℙ_{1}(ℚ_{p}) such that ord_{p} (t + \frac{ν}{c} ) ≥ j\} ,  (112)
= \{t ∈ ℙ_{1}(ℚ_{p}) such that p^{−j} (t + \frac{ν}{c} ) ∈ ℤ_{p}\} .  (113)

Hence U(e_{j}) = U(γ^{−1}e_{∗}), where γ = ( \begin{matrix}1 & −ν′ \\ 0 & p^{j} \end{matrix} ). Noting that

γx_{Ψ} = ∞, γ(y_{Ψ}) = \frac{(−ν − cν′)/p^{j}}{c} , |γ| = j,  (114)

we find

μ_{f}(U(e_{j})) = κ_{f}{x_{Ψ}→y_{Ψ}}(e_{j}) = w^{j}λ_{E}(a, c),  (115)

where a is an integer which is defined modulo c by the congruence a ≡ νp^{−j}. The result follows.

Let χ be a Dirichlet character of conductor c prime to N.

Lemma 2.14 If χ(p) = w, then

∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)W_{Ψ_{ν}} = s ∑_{a∈(ℤ/cℤ)^{×}} χ(a)λ_{E}(a, c).

Proof: This follows from proposition 2.13 by a direct calculation.

Recall the choice of sign w_{∞} that was made in defining the modular symbol λ_{E}(a, c).

Lemma 2.15 If χ(p) = w and χ(−1) = −w_{∞}, then

∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)W_{Ψ_{ν}} = 0.

Proof: This follows directly from lemma 2.14 and from the relation

λ_{E}(−a, b) = w_{∞}λ_{E}(a, b)  (116)

satisfied by the modular symbols attached to the choice of sign w_{∞}.

Let

L(f_{0}, χ, s) = L(E/ℚ, χ, s) = ∑_{n=1}^{∞} a_{n}χ(n)n^{−s}  (117)

be the L-series of E/ℚ twisted by χ. Write

τ(χ) := ∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)e^{2πiν/c}  (118)

for the Gauss sum attached to χ.

Proposition 2.16 If χ(p) = w and χ(−1) = w_{∞}, then

∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)W_{Ψ_{ν}} = \frac{sc}{τ(χ)} \frac{L(E/ℚ, χ, 1)}{c_{φ}Ω_{w_{∞}}}.

Proof: This follows by combining lemma 2.14 with equation (8.6) of §I.8 of [MTT]. (The discrepancy involving the factors of 2πi and c_{φ} are accounted for by the different normalisations used in the definition of λ_{E}(a, c) in [MTT] and in this article.)

Lemma 2.17 There exist infinitely many Dirichlet characters χ of conductor prime to N satisfying

χ(p) = w, χ(−1) = w_{∞}, ∑_{ν∈(ℤ/cℤ)^{×}} χ(ν)W_{Ψ_{ν}} ≠ 0.

Proof: In view of proposition 2.16, it is enough to show that there exist infinitely many Dirichlet characters of conductor prime to N satisfying

χ(p) = w, χ(−1) = w_{∞}, L(E/ℚ, χ, 1) ≠ 0.  (119)

This can be proved by considering averages of L(E/ℚ, χ, 1) as χ ranges over characters of conductor c satisfying χ(p) = w and χ(−1) = w_{∞}, and showing that such averages are non-zero as c becomes large. In fact, when M ≠ 1 or w_{∞} ≠ −1, it is enough to consider averages over quadratic characters χ. In the exceptional situation where N = p and ww_{∞} = −1, the sign in the functional equation of L(E/ℚ, χ, s) is always −1. It then becomes necessary to allow χ to be a non-quadratic character, so that this sign does not force the vanishing of L(E/ℚ, χ, s) at s = 1. See for example [MM] where nonvanishing results for twists of L-series, and the averaging techniques used to obtain them, are explained in detail.

Remarks:

1. The non-vanishing of the expression ∑_{a} χ(a)λ_{E}(a, c), for some Dirichlet character χ, is an elementary consequence of the fact that the paths between elements of ℙ_{1}(ℚ) generate the rational homology of the curve H^{∗}/Γ_{0}(N) so that the modular symbols λ_{E}(a, c) cannot vanish identically. It seems more difficult to exploit this property of the modular symbols to establish the corresponding non-vanishing as χ ranges over the smaller subset of characters satisfying χ(p) = w and χ(−1) = w_{∞}, without exploiting the connection between modular symbols and special values of twisted L-series and invoking analytic arguments to establish non-vanishing theorems for such twists.

2. The fact that E has split (resp. non-split) multiplicative reduction at p when w = 1 (resp. when w = −1) combined with the fact that χ(p) = w implies that the factor ord_{p}(q) appears as one of the fudge factors (attached to the prime p) in the algebraic part of the special value L(E/ℚ, χ, 1) predicted by the Birch and Swinnerton-Dyer conjecture. Hence for all such χ, one expects that

ord_{p}(q) divides \frac{L(E/ℚ, χ, 1)}{Ω_{w_{∞}}}.  (120)

This can be used to deduce the corresponding divisibility of ord_{p}(I_{Ψ}) by ord_{p}(q), at least away from the primes dividing s, c and c_{φ}, and lends some support for conjecture 3. (Cf the remarks immediately following the statement of theorem 1 in the introduction.)

The divisibility of ord_{p}(I_{Ψ}) by ord_{p}(q) will be established in many cases, independently of any conjectures, in [BD7].

2.3 Evaluation of log(I_{Ψ})

Set Ψ = Ψ_{ν} as before. The following proposition evaluates log(I_{Ψ}) explicitly in terms of modular symbols.

Proposition 2.18 Let J_{n} be the coset in (ℤ/p^{n}cℤ)^{×} consisting of a such that a/ν ≡ p^{j} (mod c) for some j = j(a). Then

log(I_{Ψ}) = ∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω = lim_{n→∞} w^{n} ∑_{a∈J_{n}} w^{j(a)} log(a)λ_{E}(a, p^{n}c).

Note that the expression w^{j(a)} which appears in the right-hand limit is well defined, since assumption 2.5 makes the parity of j well-defined in the case where w = −1.

Proof of proposition 2.18: By proposition 2.7, and since M_{Ψ}(t) can be chosen to be equal to t + \frac{ν}{c},

∫_{z}^{γ_{Ψ}z}∫_{x_{Ψ}}^{y_{Ψ}}ω = log(I_{Ψ}) = ∫_{F_{Ψ}} log (t + \frac{ν}{c} ) dμ_{f}\{x_{Ψ}→y_{Ψ}\}(t).  (121)

As in the discussion preceding the proof of proposition 2.7, write

F_{Ψ} = U(v_{0}) ∪ ⋯ ∪ U(v_{s−1}),  (122)

where

U(v_{j}) = \{t ∈ ℙ_{1}(ℚ_{p}) such that ord_{p} (t + \frac{ν}{c} ) = j\} .  (123)

For each positive integer n, the compact open subset U(v_{j}) can be further decomposed as

U(v_{j}) = ∪_{a∈(ℤ/p^{n}ℤ)^{×}}U_{j,a},  (124)

where

U_{j,a} = \{t ∈ U(v_{j}) such that p^{−j} (t + \frac{ν}{c} ) ≡ a (mod p^{n})\} .  (125)

Choose an integer ν′ such that

ν′ ≡ −\frac{ν}{c} (mod p^{n+s}),  (126)

and observe that

U_{j,a} = U(γ^{−1}e_{∗}), where γ = ( \begin{matrix}1 & −ν′ − p^{j}a \\ 0 & p^{n+j} \end{matrix} ) ∈ R^{×}.  (127)

Noting that

γx_{Ψ} = ∞, γ(y_{Ψ}) = \frac{(−ν − cν′)/p^{j} − ac}{cp^{n}}, |γ| = n + j,  (128)

we find by equation (61) of section 1 that

μ_{f}{x_{Ψ}→y_{Ψ}}(U_{j,a}) = w^{n+j}λ_{E}(a_{j}, cp^{n}),  (129)

where a_{j} is an integer which is defined modulo cp^{n} by the congruences

a_{j} ≡ ac (mod p^{n}), a_{j} ≡ νp^{−j} (mod c).  (130)

It follows that

∫_{F_{Ψ}} log (t + \frac{ν}{c} ) dμ_{f}\{x_{Ψ}→y_{Ψ}\}(t) = lim_{n→∞} w^{n} ∑_{a} w^{j(a)} log(a)λ_{E}(a, p^{n}c),

(131)

where the sum ranges over a ∈ J_{n}. The proposition follows.

We now turn to the proof of theorem 1 of the introduction.

Proof of theorem 1. By combining propositions 2.18 and 2.13, one is reduced to showing that for each ν in (ℤ/cℤ)^{×}/⟨p^{2}⟩,

lim_{n→∞} w^{n} ∑_{a∈J_{n}} w^{j(a)} log(a)λ_{E}(a, p^{n}c) = \frac{log(q)}{ord_{p}(q)} ∑_{a∈J} w^{j(a)}λ_{E}(a, c).  (132)

Let χ : (ℤ/cℤ)^{×} ⟶ ℂ_{p}^{×} be a Dirichlet character with the property that χ(p) = w, so that in particular χ factors through (ℤ/cℤ)^{×}/⟨p^{2}⟩. Recall the sign w_{∞} that was used to define λ_{E}(a, b). If χ(−1) = −w_{∞}, then a direct calculation using the relation λ_{E}(−a, b) = w_{∞}λ_{E}(a, b) shows that

lim_{n→∞} w^{n} ∑_{a∈(ℤ/p^{n}cℤ)^{×}} χ(a) log(a)λ_{E}(a, p^{n}c) = ∑_{a∈(ℤ/cℤ)^{×}} χ(a)λ_{E}(a, c) = 0. (133)

If χ(−1) = w_{∞}, then the exceptional zero conjecture of Mazur, Tate and Teitelbaum ([MTT], §13, conjecture 1) proved by Greenberg and Stevens [GS] states that for all such χ,

lim_{n→∞} w^{n} ∑_{a∈(ℤ/p^{n}cℤ)^{×}} χ(a) log(a)λ_{E}(a, p^{n}c) = \frac{log(q)}{ord_{p}(q)} ∑_{a∈(ℤ/cℤ)^{×}} χ(a)λ_{E}(a, c),

(134)

where the sums now are taken over all congruence classes a in (ℤ/p^{n}cℤ)^{×} and (ℤ/cℤ)^{×} respectively. Hence relation (134) holds for all Dirichlet characters χ such that χ(p) = w. But the relation expressed in equation (132) can be written as a ℂ_{p}-linear combination of the relations expressed in equation (134). Theorem 1 follows.

3 The cohomology of Γ

3.1 The cohomology of M-symbols

Recall from the introduction that an M-symbol with values in an abelian group C is a function

m{ → } : ℙ_{1}(ℚ) × ℙ_{1}(ℚ) ⟶ C  (135)

satisfying

m{x→y} + m{y→z} = m{x→z},  m{x→y} = −m{y→x}, (136)

for all x, y, z ∈ ℙ_{1}(ℚ). The group of ℂ_{p}-valued M-symbols is denoted M, and more generally the group of C-valued M-symbols is denoted by M(C). Recall also that Γ acts on M(C) by the rule

(γm){x→y} := m{γ^{−1}x→γ^{−1}y}.  (137)

The cohomology groups H^{i}(Γ, M) play the crucial role in this section. Recall that Γ acts on T with e_{∗} as fundamental region. The stabiliser of e_{∗} in Γ is equal to Γ_{0}(N) and the stabiliser of s(e_{∗}) (resp of t(e_{∗})) is equal to Γ_{0}(M) (resp Γ′_{0}(M) := α_{p}Γ_{0}(M)α_{p}^{−1}). Proposition 13 of sec. II.2.8 of [Se1], applied to the case M = M and G = Γ acting on T , yields a natural exact sequence of ℂ_{p}-vector spaces

\begin{matrix}M^{Γ_{0}(M)} ⊕ M^{Γ′_{0}(M)} ⟶ M^{Γ_{0}(N)} ⟶^{θ} H^{1}(Γ, M) \\ ⟶ H^{1}(Γ_{0}(M), M) ⊕ H^{1}(Γ′_{0}(M), M). \end{matrix}  (138)

To describe the map θ explicitly, note that a Γ_{0}(N)-invariant M-symbol m{x→y} gives rise to a unique system of M-symbols m_{e} indexed by the edges of T , satisfying m_{e_{∗}}{x→y} = m{x→y} as well as

m_{γe}{γx→γy} = m_{e}{x→y}, mē{x→y} = −m_{e}{x→y},  (139)

for all x, y ∈ ℙ_{1}(ℚ) and γ ∈ Γ. One can then write

(θm)(γ){x→y} = ∑_{v_{∗}→γv_{∗}} m_{e}{x→y}.  (140)

All the cohomology groups appearing in the exact sequence (138) are endowed with a natural action of the Hecke operators T_{ℓ} with ℓ |̸N, defined as in [Sh], §8.3. Furthermore these groups are equipped with the “Atkin-Lehner

involution W_{∞} at ∞”, defined using the matrix α_{∞} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) of deter-

minant −1 which belongs to the normalisers of the groups Γ_{0}(N), Γ_{0}(M), Γ′_{0}(M), and Γ in R^{×}. On M-symbols W_{∞} is defined by the rule

(W_{∞}m){x→y} = m{α_{∞}x→α_{∞}y} = m{−x→ − y},  (141)

and on H^{1}(Γ, M) by the rule

(W_{∞}c)(γ){x→y} = c(γ^{α_{∞}}){−x→ − y}.  (142)

A direct calculation exploiting the explicit description of θ given by (140) shows that the maps arising in (138) are compatible with the action of the Hecke operators T_{ℓ} as well as with the involution W_{∞}. For example, to check this last compatibility note that

θ(W_{∞}m)(γ){x→y} = ∑_{v_{a}st→γv_{∗}}(W_{∞}m)_{e}{x→y} = ∑_{v_{∗}→γv_{∗}} m_{α_{∞}e}{−x→ − y}. (143)

As e ranges over the edges in the path joining v to γv, the edges α_{∞}e range over the edges in the path joining α_{∞}v_{∗} = v_{∗} to α_{∞}γv_{∗} = γ^{α_{∞}}v_{∗}. Hence the right hand side of (143) is equal to

∑_{v_{∗}→γ^{α∞}v_{∗}} m_{e}{−x→ − y} = W_{∞}(θ(m))(γ){x→y}.  (144)

Given the newform in S_{2}(T , Γ) satisfying

T_{ℓ}(f) = a_{ℓ}(f)f  (145)

for all ℓ |̸N, and a ℂ_{p}-vector space H equipped with the commuting actions of the Hecke operators T_{ℓ} and the involution W_{∞}, denote by H^{f} the f-isotypic component of H, defined as the set of classes c ∈ H for which

T_{ℓ}(c) = a_{ℓ}(f)c for all ℓ |̸N.  (146)

Denote also by H^{w_{∞}} the space on which the involution W_{∞} acts with the eigenvalue w_{∞}, and by H^{f,w_{∞}} the intersection of H^{f} and H^{w_{∞}}.

Proposition 3.1 The space (M^{Γ_{0}(N)})^{f,w_{∞}} is a one-dimensional ℂ_{p}-vector space.

Proof: Let F be the space of ℂ_{p}-valued functions on ℙ_{1}(ℚ), endowed with the natural action of Γ. The assignment d : F ⟶ M defined by

(df){x→y} := f(y) − f(x)  (147)

defines a surjective homomorphism of Γ-modules, with kernel the space of constant functions, identified with ℂ_{p}. Taking the Γ_{0}(N)-cohomology of the exact sequence

0 ⟶ ℂ_{p} ⟶ F ⟶^{d} M ⟶ 0  (148)

yields the exact sequence of cohomology groups

F^{Γ_{0}(N)} ⟶ M^{Γ_{0}(N)} ⟶ H^{1}(Γ_{0}(N), ℂ_{p})  (149)

which is compatible under the natural action of the Hecke operators and the involution W_{∞}. On the other hand, the space F^{Γ_{0}(N)} is Eisenstein, i.e.,

If x ∈ F^{Γ_{0}(N)}, then T_{ℓ}x = (ℓ + 1)x for all ℓ |̸N.  (150)

Hence (F^{Γ_{0}(N)})^{f} = 0. By the Eichler-Shimura period isomorphism (cf. [Sh]. §8.2), the space H^{1}(Γ_{0}(N), ℂ_{p})^{f} is a two-dimensional ℂ_{p}-vector space, on which the involution W_{∞} acts with eigenvalues 1 and −1. Proposition 3.1 follows at once from this.

Proposition 3.2 The map θ induces an isomorphism from (M^{Γ_{0}(N)})^{f,w_{∞}} to H^{1}(Γ, M)^{f,w_{∞}}.

Proof: The arguments used in the proof of proposition 3.1 show that the action of the Hecke operators on H^{0}(Γ_{0}(M), M) factors through the natural image of the algebra of Hecke operators in the endomorphism ring of the space of modular forms of weight 2 on Γ_{0}(M). Since the Hecke eigenvalues attached to f are those of a newform of level N, it follows that

H^{0}(Γ_{0}(M), M)^{f} = 0.  (151)

Pursuing the long exact sequence in cohomology attached to the Γ_{0}(M)- cohomology of the sequence (148) yields the sequence

H^{1}(Γ_{0}(M), F) ⟶ H^{1}(Γ_{0}(M), M) ⟶ H^{2}(Γ_{0}(M), ℂ_{p}).  (152)

As a Γ_{0}(M)-module, F can be written as a direct sum of induced modules:

F = ⊕_{x}Ind_{Γ_{x}}^{Γ_{0}(M)}ℂ_{p},  (153)

where the direct sum is taken over a system of representatives for the Γ_{0}(M)- orbits in ℙ_{1}(ℚ), and Γ_{x} denotes the stabiliser of x in Γ_{0}(M). By Shapiro’s lemma,

H^{1}(Γ_{0}(M), F) = ⊕_{x}H^{1}(Γ_{x}, ℂ_{p}),  (154)

and the action of the Hecke algebra on this module is Eisenstein, so that H^{1}(Γ_{0}(M), F)^{f} = 0. Furthermore it is well-known that H^{2}(Γ_{0}(M), ℂ_{p}) = 0. (For example, if Γ_{0}(M) acts on H without fixed points, then it is a free group.) Hence it follows that

H^{1}(Γ_{0}(M), M)^{f} = 0.  (155)

The result now follows from (151) and (155) (and the corresponding statements with Γ_{0}(M) replaced by the conjugate subgroup Γ′_{0}(M)) combined with the exact sequence (138).

Corollary 3.3 The vector space H^{1}(Γ, M)^{f,w_{∞}} is one-dimensional over ℂ_{p}.

Proof: This follows from propositions 3.1 and 3.2.

3.2 Proof of theorem 4

Given τ ∈ H_{p}, a 1-cocycle c̃_{f,τ} ∈ Z^{1}(Γ, M(ℂ_{p}^{×})) is defined by setting

c̃_{f,τ}(γ){x→y} = × ∫_{τ}^{γτ}∫_{x}^{y}ω.  (156)

The image of c̃_{f,τ} in H^{1}(Γ, M(ℂ_{p}^{×})), denoted c_{f}, does not depend on τ. Let ord_{p}(c_{f}) ∈ H^{1}(Γ, M) be the cohomology class obtained by applying the function ord_{p} : ℂ_{p}^{×} ⟶ ℚ ⊂ ℂ_{p} to c_{f}, and let log(c_{f}) be the image of c_{f} in H^{1}(Γ, M) under the logarithm map.

Lemma 3.4 The class ord_{p}(c_{f}) is non-zero. (And hence generates the onedimensional ℂ_{p}-vector space H^{1}(Γ, M)^{f,w_{∞}}.)

Proof: Suppose that ord_{p}(c_{f}) = 0. Then the one-cocycle ord_{p}(c̃_{f,τ}) is the coboundary of an M-symbol η ∈ M:

ord_{p}(c̃_{f,τ})(γ){x→y} = η{γ^{−1}x→γ^{−1}y} − η{x→y}.  (157)

Let Ψ be an oriented embedding of ℚ×ℚ of conductor c prime to M in M_{2}(ℚ). Letting γ = γ_{Ψ} and (x, y) = (x_{Ψ}, y_{Ψ}), the right-hand term of equation (157) disappears and one finds

W_{Ψ} = 0 for all such Ψ.  (158)

This contradicts lemma 2.17.

We are now ready to prove theorem 4 of the introduction:

Proof of theorem 4: Since ord_{p}(c_{f}) and log(c_{f}) each belong to the onedimensional vector space H^{1}(Γ, M)^{f,w_{∞}}, and since ord_{p}(c_{f}) ≠ 0, there exists a constant L_{f} ∈ ℂ_{p} such that log(c_{f}) = L_{f}ord_{p}(c_{f}). Hence there is an M-symbol η ∈ M such that

log(c̃_{f,τ}(γ){x→y}) = L_{f}ord_{p}(c̃_{f,τ}(γ){x→y}) + (η{γ^{−1}x→γ^{−1}y} − η{x→y}),

for all γ ∈ Γ and x, y ∈ ℙ_{1}(ℚ). Choosing an embedding Ψ of conductor prime to N and letting (x, y) = (x_{Ψ}, y_{Ψ}) and γ = γ_{Ψ} as in the proof of lemma 3.4 shows that

log(I_{Ψ}) = L_{f}ord_{p}(I_{Ψ}).  (159)

By proposition 2.17 it is possible to choose a Ψ for which ord_{p}(I_{Ψ}) = W_{Ψ} is non-zero. It now follows from theorem 1 that

L_{f} = \frac{log(q)}{ord_{p}(q)},  (160)

as was to be shown.

Remark: While the evaluation of the constants L_{f} requires the full strength of the Greenberg-Stevens theorem, the equality log(c_{f}) = L_{f}ord_{p}(c_{f}) for some L_{f} ∈ ℂ_{p} is sufficient to imply that log(I_{Ψ}) = L_{f}ord_{p}(I_{Ψ}) for all Ψ, with L_{f} not depending on Ψ. This in turn implies that the extra zero occurring in the exceptional zero conjecture of Mazur, Tate and Teitelbaum is of “local type” in the sense of [MTT], ch I, §19. In other words, the factor L_{f}, which describes the discrepancy between the first derivative of the p-adic L-function attached to E in [MTT] and the special value of the classical L-function attached to E, is invariant under twists by Dirichlet characters χ for which χ(p) = 1. Note that our proof of this fact is based on little more than purely formal arguments involving the cohomology of Γ.

3.3 Indefinite integrals

Theorem 4 justifies the slightly stronger multiplicative refinement of it that is formulated in conjecture 5 of the introduction. Assume this conjecture. Let τ be an element of H_{p} and let L be the field generated by τ over ℚ_{p}. The conjecture guarantees the existence of an M-symbol η_{f,τ} ∈ M(L^{×}/q^{ℤ}) satisfying

× ∫_{τ}^{γτ}∫_{x}^{y}ω = η_{f,τ}{γ^{−1}x→γ^{−1}y} ÷ η_{f,τ}{x→y} (mod q^{ℤ})  (161)

for all γ ∈ Γ and x, y ∈ ℙ_{1}(ℚ). The M-symbol η_{f,τ} is defined uniquely by this property modulo elements of H^{0}(Γ, M(L^{×}/q^{ℤ})).

Let Γ′ ⊂ Γ denote the smallest normal subgroup of Γ generated by the commutators and the rational elements in Γ. (An element of Γ is said to be rational if its fixed points belong to ℙ_{1}(ℚ).)

Lemma 3.5 The group Γ′ is of finite index in Γ. It is equal to Γ if M = 1.

Proof: The first part of this lemma follows for example from theorem 2 of [Me] (see also Théorème 3 of [Se2]). The second part follows from a direct calculation, using the fact that SL_{2}(ℤ) is equal to its commutator subgroup.

Remark: The main result of [Me], stating that the group Γ has the congruence subgroup property, makes it possible to calculate Γ′, and bound its index, in any specific situation. See [BD7] for a more complete discussion.

Let e_{Γ} denote the exponent of the finite abelian group Γ/Γ′.

Proposition 3.6 Let C be a Γ-module and let t be the exponent of the e_{Γ}- torsion module C[e_{Γ}]. If x and y are in the same Γ-orbit in ℙ_{1}(ℚ), and m belongs to M(C)^{Γ}, then

t · m{x→y} = 0.

Proof: It follows directly from the Γ-invariance of m that the function which to γ associates m{x→γx} is a homomorphism from Γ to C. Hence its kernel contains the commutator subgroup of Γ. It also contains the rational elements, since if γy = y for some y ∈ ℙ_{1}(ℚ),

m{x→γx} = m{x→y} + m{y→γx} = m{x→y} + m{y→x} = 0.

Proposition 3.6 follows.

Let Q_{0} = (L^{×}/q^{ℤ})[e_{Γ}] be the e_{Γ}-torsion subgroup of L^{×}/q^{ℤ}, and let Q ⊂ L^{×} denote the preimage of Q_{0} under the natural projection. It follows from proposition 3.6 that the image of η_{f,τ}{x→y} in L^{×}/Q does not depend on the choice of M-symbol η_{f,τ} satisfying (161), if x and y belong to the same Γ-orbit in ℙ_{1}(ℚ). This makes it possible to define the indefinite integral attached to τ by the rule

× ∫ ^{τ}∫_{x}^{y}ω := the natural image of η_{f,τ}{x→y} in L^{×}/Q,  (162)

for all such x, y. Note that when M = 1, lemma 3.5 implies that Q = q^{ℤ}.

Lemma 3.7 The indefinite multiplicative integral of (23) satisfies the following properties:

× ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω = × ∫ ^{τ_{2}}∫_{x}^{y}ω ÷ × ∫ ^{τ_{1}}∫_{x}^{y}ω, (mod Q)  (163)
× ∫ ^{τ}∫_{x_{1}}^{x_{3}}ω = × ∫ ^{τ}∫_{x_{1}}^{x_{2}}ω × × ∫ ^{τ}∫_{x_{2}}^{x_{3}}ω, (mod Q)  (164)

× ∫ ^{γτ}∫_{γx}^{γy}ω = (× ∫ ^{τ}∫_{x}^{y}ω)^{w^{|γ|}w_{∞}^{sgn(γ)}} , (mod Q)  (165)

for all γ ∈ R^{×}.

Proof: The first identity is proved by showing that the coboundary of the Msymbol η_{f,τ_{2}} ÷ η_{f,τ_{1}} (viewed as a 0-cochain with values in M(L^{×}/q^{ℤ})) agrees with the coboundary of the M-symbol

m_{τ_{1},τ_{2}}{x→y} := × ∫_{τ_{1}}^{τ_{2}}∫_{x}^{y}ω.

The second relation follows directly from the definitions. The third is proved by showing that the L^{×}/q^{ℤ}-valued M-symbol γη_{τ} satisfies the defining property of the M-symbol w^{|γ|}w_{∞}^{sgn(γ)}η_{γτ}, so that these two M-symbols are equal up to elements in M(L^{×}/q^{ℤ})^{Γ}.

4 Periods attached to real quadratic fields

Suppose in this section that the prime p splits in the real quadratic field K, and let Ψ denote an algebra embedding of K into M_{2}(ℚ).

Choose a base point τ ∈ H_{p} and x ∈ ℙ_{1}(ℚ), and define a ℂ_{p}^{×}-valued function on Γ × Γ by the rule

⟨α, β⟩_{Ψ} := × ∫_{τ}^{α^{−1}τ}∫_{x}^{βx}ω ÷ × ∫_{τ}^{β^{−1}τ}∫_{x}^{αx}ω  (166)

Lemma 4.1 If α and β commute, the expression ⟨α, β⟩_{Ψ} does not depend on the choices of τ ∈ H_{p} and x ∈ ℙ_{1}(ℚ) that were made to define it.

Proof: The element ⟨α, β⟩_{Ψ} can be expressed in terms of the ℂ_{p}^{×}-valued 2cocycle d̃_{τ,x} ∈ Z^{2}(Γ, ℂ_{p}^{×}) defined by setting

d̃_{τ,x}(α, β) := c̃_{f,τ}(α^{−1}){x→βx} = × ∫_{τ}^{α^{−1}τ}∫_{x}^{βx}ω.  (167)

More precisely, one has

⟨α, β⟩_{Ψ} = d̃_{τ,x}(α, β) ÷ d̃_{τ,x}(β, α).  (168)

A direct calculation shows that the natural image d of d̃_{τ,x} in H^{2}(Γ, ℂ_{p}^{×}) does not depend on the choices of τ and x, so that a different choice would have the effect of multiplying d by a coboundary of the form h(α, β) = g(α) × g(β) ÷ g(αβ). The corresponding change in ⟨α, β⟩_{Ψ} is then given by h(α, β)÷h(β, α). Since α and β commute, this ratio of coboundaries is equal to 1 and the result follows.

The Dirichlet S-unit theorem implies that the group O_{1}^{×} is free of rank two, so that Ψ̄(K) ∩ Γ = Ψ̄(O_{1}^{×}) is free of rank two as well.

Lemma 4.2 The restriction of ⟨ , ⟩_{Ψ} to Ψ̄(O_{1}^{×}) is a bilinear alternating ℂ_{p}^{×}-valued pairing.

Proof: Note that the restriction of ⟨ , ⟩_{Ψ} to Ψ̄(O_{1}^{×}) does not depend on the choice of τ and x, by lemma 4.1. A direct calculation based on the definitions shows that ⟨ , ⟩_{Ψ} is bilinear and alternating.

Let γ_{1} and γ_{2} ∈ Γ be ℤ-module generators for Ψ(K) ∩ Γ. The period I_{Ψ} ∈ ℂ_{p}^{×} attached to Ψ is defined by setting

I_{Ψ} := ⟨γ_{1}, γ_{2}⟩_{Ψ}.  (169)

Note that {I_{Ψ}, I_{Ψ}^{−1}} is independent of the choice of basis (γ_{1}, γ_{2}).

Proof of theorem 6: Let ord_{p}(d_{f}) and log(d_{f}) be the classes in H^{2}(Γ, ℂ_{p}) obtained from d_{f} by applying ord_{p} and log respectively. Theorem 4 implies that

log(d_{f}) = \frac{log(q)}{ord_{p}(q)}ord_{p}(d_{f}).  (170)

The first part of theorem 6 follows immediately from (168) and (170). It also follows from these arguments that the period I_{Ψ} belongs to q^{ℤ} assuming conjecture 5.

Remark:

1. Choose a prime 𝔭 of K above p, and let Ō^{×} be the p-adic closure of the group O^{×} in K_{𝔭}^{×} = ℚ_{p}^{×}. In [BD6], it is explained how the period I_{Ψ}, taken modulo this finite index subgroup, can be interpreted as the leading term in a θ-element which interpolates the special values of L(E/K, 1) twisted by certain finite order characters of Gal(K̄/K). On the other hand, ord_{p}(I_{Ψ}) can be expressed in terms of algebraic parts of classical special values of certain partial L-functions attached to E/K. Thus, theorem 6 can be viewed as giving an exceptional zero result for the leading terms of the theta-elements attached to elliptic curves over a real quadratic K, expressing his leading term in terms of Tate’s period attached to E.

2. Note that the identities of conjectures 3 and theorem 6 provide a natural lifting of the exceptional zero conjectures (which are formulated as identities in compact quotients of K_{p}^{×} admitting a Galois-theoretic interpretation) to the group K_{p}^{×} itself; this suggests that the exceptional zero conjectures might arise as consequences of more basic identities involving double integrals of ω, identities which can be expressed without appealing to the notion of p-adic L-functions or θ-elements. It is this point of view that provided some of the inspiration for the conjectures of the next section.

5 Heegner points attached to real quadratic fields

5.1 The main conjecture

Assume now that the prime p is inert in the real quadratic field K. For x ∈ K, let x̄ denote the Galois conjugate of x. Fix embeddings of K into ℝ and ℂ_{p}, so that K can be viewed simultaneously as a subfield of these two fields. As before, let

Ψ : K ⟶ M_{2}(ℚ)  (171)

be an algebra embedding. As in section 2, the conductor of Ψ is defined to be the conductor of the ℤ[1/p]-order O = Ψ^{−1}(R) of K. Let c denote this conductor, and make the following simplifying assumption from now on:

Assumption 5.1 The discriminant c^{2}Disc(K) of O is relatively prime to N.

The torus Ψ̄(K^{×}) acting on H_{p} by Möbius transformations has two fixed points in ℙ_{1}(K_{p})−ℙ_{1}(ℚ_{p}) ⊂ H_{p} which are interchanged by Gal(K_{p}/ℚ_{p}). Let z_{Ψ} be the unique fixed point with the property that Ψ(λ) acts on the tangent space of H_{p} at z_{Ψ} by multiplication by λ/λ̄ (i.e., such that the column vector (z_{Ψ}, 1) is an eigenvector for Ψ(λ) with eigenvalue λ), and let z̄_{Ψ} be the other fixed point. The group Ψ̄(K^{×}) ∩ Γ is free of rank one, generated by Ψ̄(u) where u is a generator for the group of units of norm one in O^{×}. Normalise γ_{Ψ} by the requirement that u be greater than 1 (relative to the chosen real embedding of K).

A period I_{Ψ} ∈ ℂ_{p}^{×} is naturally attached to Ψ by choosing a base point x ∈ ℙ_{1}(ℚ) and setting

I_{Ψ} := × ∫ z_{Ψ} ; z̄_{Ψ}∫_{x}^{γ_{Ψ}x}ω ∈ K_{p}^{×}.  (172)

Lemma 5.2 The period I_{Ψ} does not depend on the choice of x ∈ ℙ_{1}(ℚ) that was made to define it. Furthermore, it depends only on the Γ-conjugacy class of Ψ.

The proof is identical to the proof of lemma 2.3. (But observe how the roles of the places p and ∞ are interchanged in these two proofs.)

Let H^{+} denote the narrow ring class field of K of conductor c. (The definition of this class field is recalled in section 5.2.) The Galois groups Gal(H^{+}/ℚ) is a generalized dihedral group, in which any element mapping to a generator of Gal(K/ℚ) is necessarily of order two. Hence, since the prime p is inert in K/ℚ, it splits completely in H^{+}/K. Fix an embedding of H^{+} into ℂ_{p}. (This is tantamount to choosing a prime ideal of H^{+} above p.)

Conjecture 5.3 The local point

P_{Ψ}^{−} := Φ_{Tate}(I_{Ψ}) ∈ E(K_{p})

is a global point in E(H^{+}).

Conjecture 5.3 makes it clear that the period I_{Ψ} is a more subtle arithmetic invariant than in the previous situations that were treated. Indeed, one disposes of no modular construction of global points on E over ring class fields of real quadratic fields, such as is provided by the theory of complex multiplication when K is a quadratic imaginary field.

As will be seen in section 5.2, the points P_{Ψ}^{−} are not the “right” generalisation of Heegner points in this setting, because they fail to obey an analogue of the classical Shimura reciprocity law (cf. the discussion following the statement of corollary 5.12).

To obtain the appropriate generalisation, choose a Γ-orbit in ℙ_{1}(ℚ), say Γ∞ to fix ideas. Select any x in this orbit and use the indefinite multiplicative integral of (162) to define

J_{Ψ} := × ∫ ^{z_{Ψ}}∫_{x}^{γ_{Ψ}x}ω ∈ K_{p}^{×}/Q.

Lemma 5.4 The period J_{Ψ} does not depend on the choice of x ∈ Γ∞. Furthermore, it depends only on the Γ-conjugacy class of Ψ.

Proof: This follows directly from the properties of the indefinite multiplicative integral given in lemma 3.7, by manipulations identical to those presented in the proof of lemma 2.3.

Thus, J_{Ψ} is a canonical element in K_{p}^{×}/Q attached to Ψ. It is related to the period I_{Ψ} as follows:

Lemma 5.5 For all embeddings Ψ of K into M_{2}(ℚ),

J_{Ψ}/J̄_{Ψ} = × ∫ z_{Ψ} ; z̄_{Ψ}∫_{x}^{γ_{Ψ}x}ω = I_{Ψ} (mod Q).

Proof: This follows directly from property (163) of lemma 3.7 of the indefinite multiplicative integral.

Let t denote the exponent of the group Q_{0} introduced before, so that

t divides gcd(e_{Γ}, (p^{2} − 1)ord_{p}(q)).  (173)

Since raising to the power t maps Q to q^{ℤ}, the element J_{Ψ}^{t} is well-defined element in K_{p}^{×}/q^{ℤ}. In view of lemma 5.5, the following modification of conjecture 5.3 is natural:

Conjecture 5.6 The local point

P_{Ψ} := Φ_{Tate}(J_{Ψ}^{t}) ∈ E(K_{p})

is a global point in E(H^{+}).

Assuming this conjecture, one can (and will) view P_{Ψ} as global points in E(H^{+}).

5.2 A Shimura Reciprocity Law

The goal of the Shimura Reciprocity Law is to give an explicit (conjectural) description of the action of Gal(H^{+}/K) on the global points P_{Ψ}. An indispensable ingredient in formulating such a law is the concrete description of Gal(H^{+}/K) provided by class field theory.

Let I ⊂ K be a free ℤ[1/p]-submodule of rank two. The order associated to I is the set of λ ∈ K satisfying λI ⊂ I. A fractional O-ideal of K is a free ℤ[1/p]-module of rank two in K whose associated order is equal to O. Two such fractional O-ideals I_{1} and I_{2} are said to be equivalent if there exists α ∈ K^{×} such that I_{1} = αI_{2}, and are said to be strictly equivalent if α can be chosen to be of positive norm. Let Pic(O) (resp. Pic^{+}(O)) denote the group of equivalence (resp. strict equivalence) classes of fractional O-ideals in K, where the group operation is the usual multiplication of fractional ideals. Let h and h^{+} denote the cardinalities of Pic(O) and Pic^{+}(O) respectively. Note that if O^{×} contains an element of negative norm, then strict equivalence is no stronger than equivalence and h^{+} = h. Otherwise, we have h^{+} = 2h.

The reciprocity law of class field theory identifies Pic^{+}(O) and Pic(O) with the Galois groups of certain abelian extensions of K, denoted H^{+} and H respectively. The field H^{+} is the strict ring class field of K of conductor c introduced in section 5.1, and H is simply called the ring class field of K of conductor c. The extension H is the maximal totally real subfield of H^{+}. Denote by rec the isomorphism given by the reciprocity law of class field theory:

rec : Pic(O) ⟶^{≃} Gal(H/K), rec : Pic^{+}(O) ⟶^{≃} Gal(H^{+}/K). (174)

Lemma 5.7 An embedding Ψ of conductor c exists if and only if there is a ring homomorphism 𝔬 from O = Ψ^{−1}(R) to ℤ/Mℤ.

Proof: If such a Ψ exists, the algebra homomorphism 𝔬_{Ψ} : O ⟶ ℤ/Mℤ which to x ∈ O associates the upper left-hand entry of the matrix Ψ(x) (mod M) is the desired homomorphism. Conversely, given such an 𝔬, choose a ℤ[1/p]-module basis (e_{1}, e_{2}) for O with the property that the image of e_{1} in O/MO is annihilated by ker 𝔬. The action of α ∈ O on this basis is expressed by a matrix m_{α} ∈ R and the assignment α ↦ M_{α} gives the desired Ψ.

The homomorphism 𝔬_{Ψ} is called the orientation at M attached to the embedding Ψ. Conjugation by Γ (in fact, by R^{×}) preserves the orientation of Ψ, so that 𝔬_{Ψ} is an invariant of the Γ-conjugacy class of Ψ.

Fix an orientation 𝔬 : O ⟶ (ℤ/Mℤ). Let Emb(O, R) denote the set of optimal embeddings Ψ : K ⟶ M_{2}(ℚ) of conductor c satisfying:

1. 𝔬_{Ψ} = 𝔬;

2. The vertex of T fixed by Ψ(K^{×}) is an even vertex.

Since conjugation by the Atkin-Lehner matrices α_{ℓ} transitively permutes the possible orientations of Ψ, and conjugation by α_{p} ∈ Γ̃ − Γ interchanges the odd and even vertices of T , it follows that Emb(O, R) is non-empty if and only if an embedding of conductor c exists. Let Emb(O, R)/Γ denote the set of orbits of Emb(O, R) under conjugation by Γ.

Proposition 5.8 The sets Emb(O, R)/Γ and Pic^{+}(O) are in natural bijection with each other. In particular, Emb(O, R) is finite of cardinality h^{+}.

Proof: A basis (e_{1}, e_{2}) for K over ℚ is said to be positive if

det ( \begin{matrix}e_{1} & ē_{1} \\ e_{2} & ē_{2} \end{matrix} ) > 0, and ord_{p} det ( \begin{matrix}e_{1} & ē_{1} \\ e_{2} & ē_{2} \end{matrix} ) = 0 (mod 2).

Choose an element λ^{−} of K^{×} of negative norm. Given an embedding Ψ ∈ Emb(O, R), recall the distinguished fixed point z_{Ψ} of Ψ̄(K^{×}) associated to it in section 5.1 and let 𝔞_{Ψ} denote the fractional O-ideal defined by

𝔞_{Ψ} = \{ \begin{matrix}Oz_{Ψ} + O & if (z_{Ψ}, 1) is a positive basis of K over ℚ \\ λ^{−}(Oz_{Ψ} + O) & if (z_{Ψ}, 1) is a negative basis of K over ℚ. \end{matrix} (175)

Conjugating Ψ by an element α ∈ Γ has the effect of replacing 𝔞_{Ψ} by an ideal

which is equivalent to it, in the strict sense. This is because if α = ( \begin{matrix}r & s \\ t & u \end{matrix} ), then the bases (z_{Ψ}, 1) and (αz_{Ψ}, 1) have the same sign of orientation if the element (tz_{Ψ}+u) is of positive norm, and have opposite orientation otherwise. Hence the function which to Ψ associates the narrow ideal class 𝔠_{Ψ} of 𝔞_{Ψ} gives a well-defined map

η_{1} : Emb(O, R)/Γ ⟶ Pic^{+}(O).

In the opposite direction, given an ideal class 𝔠 in Pic^{+}(O), choose a representative ideal 𝔞, and let (e_{1}, e_{2}) be a ℤ[1/p]-module basis for 𝔞 chosen so that:

1. The image of e_{1} in 𝔞/M𝔞 is annihilated by ker 𝔬.

2. The pair (e_{1}, e_{2}) is a positive basis for K/ℚ.

Given λ ∈ O, the matrix m_{λ} expressing the action of multiplication by λ on 𝔞, relative to the basis (e_{1}, e_{2}), is an element of R, and the assignment λ ↦ m_{λ} is an algebra embedding of O into R which gives rise to an optimal embedding Ψ_{𝔠} of conductor c. The class of Ψ_{𝔠} in Emb(O, R)/Γ does not depend on the choice of basis (e_{1}, e_{2}) satisfying properties 1 and 2 above, and is unaffected by replacing the ideal 𝔞 by λ𝔞 for any element λ of K of positive norm. Hence it give rise to a well-defined function

η_{2} : Pic^{+}(O) ⟶ Emb(O, R)/Γ.

The reader will check that η_{1} and η_{2} are inverse to each other and hence define bijections between the two sets.

Thanks to proposition 5.8, the set Emb(O, R)/Γ is equipped with a natural simply transitive action by Pic^{+}(O), denoted

(𝔠, Ψ) ↦ 𝔠 ∗ Ψ.

The following conjecture is an analogue of the classical Shimura Reciprocity Law.

Conjecture 5.9 The global points P_{Ψ} ∈ E(H^{+}) attached to the embeddings Ψ via conjecture 5.6 satisfy

P_{𝔠∗Ψ} = rec(𝔠)^{−1}(P_{Ψ}), for all 𝔠 ∈ Pic^{+}(O).

We now proceed to deduce properties of the points P_{Ψ} under the assumption that they satisfy conjecture 5.9.

The Action of Gal(K_{p}/ℚ_{p}). Let τ_{p} denote the generator of Gal(K_{p}/ℚ_{p}) = Gal(K/ℚ), so that τ_{p}(z) = z̄. Thanks to the chosen embedding of H^{+} into K_{p}, the involution τ_{p} can be viewed as an element of Gal(H^{+}/K). Recall the Atkin-Lehner involution W_{N} acting on S_{2}(Γ_{0}(N)) and let w_{N} denote the eigenvalue of W_{N} attached to the eigenform f_{0}.

Proposition 5.10 Assume the Shimura reciprocity law of conjecture 5.9. Then there exists σ ∈ Gal(H^{+}/K) (depending on P_{Ψ} and on the chosen embedding of H^{+} into K_{p}) such that

P̄_{Ψ} := τ_{p}(P_{Ψ}) = w_{N}σP_{Ψ}.

Proof: First note that

τ_{p}J_{Ψ} = × ∫ ^{z̄_{Ψ}}∫_{x}^{γ_{Ψ}x}ω = J_{Ψ′}^{−1},  (176)

where Ψ′ = Ψ ∘ τ_{p} is the embedding obtained from Ψ by composing with τ_{p}. Note that Ψ′ is an optimal embedding of conductor c, and that Ψ′(K^{×}) fixes an even vertex of T , since Ψ′ has the same image as Ψ. But Ψ′ does not have the same orientation as Ψ; more precisely, 𝔬_{Ψ′} = 𝔬_{Ψ} ∘ τ_{p}. By the analogue of lemma 2.1 for real quadratic embeddings, α_{M}Ψ′α_{M}^{−1} is an oriented optimal embedding. Therefore, by proposition 5.8 there exists 𝔠 ∈ Pic^{+}(O) such that

Ψ′ = α_{M}(𝔠 ∗ Ψ)α_{M}^{−1}  (177)

in Emb(O, R)/Γ. It follows from (176) and (177) and the fact that f_{0} is an eigenform for W_{M} with eigenvalue w_{M} that

τ_{p}J_{Ψ} = J_{𝔠∗Ψ}^{−w_{M}}.  (178)

Raise both sides of this equality to the t-th power, and apply Φ_{Tate}, remembering that

Φ_{Tate} ∘ τ_{p} = wτ_{p} ∘ Φ_{Tate}.  (179)

One thus finds, after setting σ = rec(𝔠)^{−1} and invoking conjecture 5.9:

wτ_{p}P_{Ψ} = −w_{M}P_{𝔠∗Ψ} = −w_{M}σP_{Ψ}.  (180)

Proposition 5.10 now follows from the fact that w is the negative of the Atkin-Lehner involution at p acting on f_{0}, so that −w_{M}/w = w_{N}.

Remark: The Heegner points arising in the classical theory of complex multiplication satisfy a relation similar to the one of proposition 5.10, but with τ_{p} replaced by complex conjugation. This is hardly surprising, since the prime p plays much the same role in our theory as the infinite place in the classical theory.

Corollary 5.11 Assume conjecture 5.9. If h^{+} = 1 then P_{Ψ} belongs to E(K)^{w_{N}}, the w_{N}-eigenspace for the action of τ_{p} on E(K).

Corollary 5.12 Assume conjecture 5.9. There exists σ ∈ Gal(H^{+}/K), depending on the embedding Ψ and on the chosen embedding of H^{+} into ℂ_{p}, such that

tP_{Ψ}^{−} = P_{Ψ} − wP̄_{Ψ} = P_{Ψ} + w_{M}σP_{Ψ}.

Remark: It follows from this corollary that if w_{M} = 1 and h^{+} = 1, then tP_{Ψ}^{−} = 2P_{Ψ}. This remark is sometimes useful because it is somewhat easier to compute P_{Ψ}^{−} than P_{Ψ}. Note however that the points P_{Ψ}^{−} are not permuted by the action of Gal(H^{+}/K), since τ_{p} does not commute with the elements of this Galois group. Hence, it is essential to work with the better-behaved points P_{Ψ} if one wishes to calculate Mordell-Weil groups of E over larger ring class fields of K.

The action of complex conjugation.

Let τ_{∞} be a complex conjugation in Gal(H^{+}/ℚ). Note that since K/ℚ is real and complex conjugation is of order 2, it belongs to the center of Gal(H^{+}/K); in particular, it does not depend on the choice of complex embedding of H^{+} used to define τ_{∞}.

Proposition 5.13 Assume conjecture 5.9. Then

τ_{∞}P_{Ψ} = w_{∞}P_{Ψ}.

Proof: By the Shimura reciprocity law,

τ_{∞}P_{Ψ} = P_{α_{∞}Ψα_{∞}^{−1}},

where α_{∞} ∈ R^{×} is a matrix of determinant −1. On the other hand, by lemma 3.7,

J_{α_{∞}Ψα_{∞}^{−1}} = J_{Ψ}^{w_{∞}}.

The result now follows by raising both sides to the power t and applying Φ_{Tate}.

Corollary 5.14 Assume conjecture 5.9. If w_{∞} = 1 then P_{Ψ} belongs to E(H).

Remark: It follows from the above corollary that if w_{∞} = 1, and the order O has class number 1, then the point P_{Ψ} belongs to E(K). If furthermore w_{N} = 1 then P_{Ψ} belongs to E(ℚ).

5.3 A Gross-Zagier conjecture

Choose a complex character

χ : Pic^{+}(O) ⟶ ℂ^{×} such that χ(τ_{∞}) = w_{∞}.  (181)

Conjecture 5.9 predicts that the ℂ-linear combination of points

P_{χ} = ∑_{𝔠∈Pic^{+}(O)} χ̄(𝔠)P_{𝔠∗Ψ}  (182)

belongs to (E(H^{+}) ⊗ ℂ)^{χ}, the χ-eigenspace for the action of Gal(H^{+}/K) on E(H^{+})⊗ℂ. In particular, when χ is the trivial character, the point P_{K} := P_{χ} is a global point in E(K).

Inspired by the formula of Gross and Zagier, one could surmise that there is a simple formula expressing the height of P_{χ} (after extending this height to a Hermitian pairing on E(H^{+}) ⊗ ℂ)) as a multiple of L′(E/K, χ, 1) by an explicit non-zero fudge factor.

Conjecture 5.15 The vector P_{χ} is non-zero if and only if

L′(E/K, χ, 1) ≠ 0.

In particular, the point P_{K} is of infinite order if and only if L′(E/K, 1) ≠ 0.

A proof of conjecture 5.15 is hard to conceive of in the absence of some machinery for tackling conjectures 7 and 5.9. A precise conjectural Gross-Zagier formula relating the height of P_{K} to L′(E/K, 1), and some numerical evidence for it, is given in [DG].

5.4 Numerical evidence

1. Let E be the elliptic curve X_{0}(11) with minimal Weierstrass equation

y^{2} + y = x^{3} − x^{2} − 10x − 20.

It has split multiplicative reduction at 11 so that w = 1 and w_{11} = −1. The real quadratic field of smallest discriminant in which 11 is inert is K = ℚ(\sqrt{2}). The sign in the functional equation for L(E/K, s) is −1, and, as predicted by the Birch and Swinnerton-Dyer conjecture, E(K) contains a point of infinite order

P = (9/2, \frac{−2 + 7\sqrt{2}}{4}) ,

which in fact generates E(K) up to torsion. Since the field K has narrow class number one, there is a unique oriented optimal embedding Ψ of K of conductor 1, up to conjugation in Γ. We have checked, to an 11-adic accuracy of 11^{−8}, that

P_{Ψ} = Φ_{Tate}(J_{Ψ}) = P.

It is instructive to compare this calculation with the significantly more cumbersome ones that were already carried out in [Da2], section 5.1. Note that the curve E is not unique in its ℚ-isogeny class. The above calculation indicates that assumption 2 is unduly restrictive and ought to be relaxed, perhaps at the cost of some minor extra complications. This question will be discussed more fully in [DG].

2. Let E be the elliptic curve of conductor 43 with minimal Weierstrass equation given by

E : y^{2} + y = x^{3} + x^{2}.

The quadratic field K = ℚ(\sqrt{37}) has narrow class number 1, and the order in K of conductor 2 has narrow class number 3. Let Ψ_{j} (j = 1, 2, 3) denote representatives of the three distinct SL_{2}(ℤ[1/43])-conjugacy classes of oriented optimal embeddings of K into M_{2}(ℚ) of conductor 2. Let τ_{j} be the fixed point for Ψ_{j}(K^{×}) acting on H_{43}, chosen as in section 5.2. A direct calculation shows that one may take τ_{1}, τ_{2} and τ_{3} to be

\frac{−3 + \sqrt{37}}{4}, \frac{−3 + \sqrt{37}}{7}, and − 6 + \sqrt{37}.  (183)

Choose w_{∞} = 1 and set

P_{j} = P_{Ψ_{j}} = Φ_{Tate}(× ∫ ^{τ_{j}}∫_{x}^{γ_{Ψ_{j}} x}ω).

After computing P_{j} = (x_{j}, y_{j}) numerically to 4 significant 43-adic digits (i.e., modulo 43^{4}) one finds:

P_{1} = (1953822 + 3156001\sqrt{37}, 1647778 + 1133177\sqrt{37}) (mod 43^{4})
P_{2} = (1953822 + 262800\sqrt{37}, 1647778 + 2285624\sqrt{37}) (mod 43^{4})
P_{3} = (2929963, 123259).  (mod 43^{4})

Further calculation reveals that:

∏_{j}(t − x_{j}) = t^{3} − 5t^{2} − 5t − 1 (mod 43^{4}),  (184)
∏_{j}(t − y_{j}) = t^{3} + 17t^{2} + 17t − 1 (mod 43^{4}).  (185)

Let f_{x}(t) and f_{y}(t) denote the polynomials in ℚ[t] appearing on the right hand sides of (184) and (185) respectively. It is natural to guess (although the author is far from being able to prove this!) that the 43-adic numbers x_{j} and y_{j} are roots of these polynomials, a guess which is buttressed by the fact that

1. The splitting field over ℚ of both f_{x}(t) and f_{y}(t) is the ring class field H of ℚ(\sqrt{37}) of conductor 2;

2. If x is a root of f_{x}(t) and y is the (unique) root of f_{y}(t) defined over ℚ(x), the pair (x, y) is a point on E(H).

Such a 43-adic calculation, leading to the discovery of global points on E defined over a cyclic cubic extension of a real quadratic field, can be viewed as providing strong evidence for conjectures 7 and 5.9.

The author is indebted to Peter Green for producing this example. More extensive numerical evidence for conjectures 7, 5.9 and 5.15 will be presented in [DG].

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