Heegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformization

Massimo Bertolini^{1} Henri Darmon^{2}

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Quaternion algebras, upper half planes, and trees . . . . . . . . . . . . 4 2 The p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . 11 3 Generalities on Mumford curves . . . . . . . . . . . . . . . . . . . 17 4 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Computing the p-adic Abel-Jacobi map . . . . . . . . . . . . . . . 29 7 Proof of the main results . . . . . . . . . . . . . . . . . . . . . 34

References . . . . . . . . . . . . . . . . . . . . . . . . . 35

Acknowledgements. Work on this paper began while the second author was visiting the Università di Pavia, supported by a fellowship from the Consiglio Nazionale delle Ricerche (CNR). It was completed during a visit by the first author at the Institute for Advanced Study, funded partly by NSF grant DMS 9304580. Both authors gratefully acknowledge the financial support of the CNR, FCAR and NSERC, as well as the hospitality of the Institute for Advanced Study, Princeton University, and the Mathematical Sciences Research Institute in Berkeley.

Introduction

Let E/ℚ be a modular elliptic curve of conductor N, and let K be an imaginary quadratic field. Rankin’s method gives the analytic continuation and functional equation for the Hasse-Weil L-function L(E/K, s). When the sign of this functional equation is −1, a Heegner point α_{K} is defined on E(K) using a modular curve or a Shimura curve parametrization of E.

In the case where all the primes dividing N are split in K, the Heegner point comes from a modular curve parametrization, and the formula of Gross-Zagier [GZ] relates its Néron-Tate canonical height to the first derivative of L(E/K, s) at s = 1. Perrin-Riou [PR] later established a p-adic analogue of the Gross-Zagier formula, expressing the p-adic height of α_{K} in terms of a derivative of the 2-variable p-adic Lfunction attached to E/K. At around the same time, Mazur, Tate and Teitelbaum [MTT] formulated a p-adic Birch and Swinnerton-Dyer conjecture for the p-adic Lfunction of E associated to the cyclotomic ℤ_{p}-extension of ℚ, and discovered that this L-function acquires an extra zero when p is a prime of split multiplicative reduction for E. The article [BD1] proposed analogues of the Mazur-Tate-Teitelbaum conjectures for the p-adic L-function of E associated to the anticyclotomic ℤ_{p}- extension of K. In a significant special case, the conjectures of [BD1] predict a p-adic analytic construction of the Heegner point α_{K} from the first derivative of the anticyclotomic p-adic L-function. (Cf. conjecture 5.8 of [BD1].) The present work supplies a proof of this conjecture.

1Università di Pavia, Dipartimento di Matematica. E-mail: massimo@dragon.ian.pv.cnr.it

2McGill University, Mathematics Department. E-mail: darmon@math.mcgill.ca

We state a simple case of our main result; a more general version is given in section 7. Assume from now on that N is relatively prime to disc(K), that E is semistable at all the primes which divide N and are inert in K/ℚ, and that there is such a prime, say p. Let O_{K} be the ring of integers of K, and let u_{K} := \frac{1}{2}#O_{K}^{×}. (Thus, u_{K} = 1 unless K = ℚ(i) or ℚ(\sqrt{−3}).)

Note that the curve E/K_{p} has split multiplicative reduction, and thus is equipped with the Tate p-adic analytic uniformization

Φ_{Tate} : K_{p}^{×} ⟶ E(K_{p}),

whose kernel is the cyclic subgroup of K_{p}^{×} generated by the Tate period q ∈ pℤ_{p}.

Let H be the Hilbert class field of K, and let H_{∞} be the compositum of all the ring class fields of K of conductor a power of p. Write

G_{∞} := Gal(H_{∞}/H), G̃_{∞} := Gal(H_{∞}/K), Δ := Gal(H/K).

By class field theory, the group G_{∞} is canonically isomorphic to K_{p}^{×}/ℚ_{p}^{×}O_{K}^{×}, which can also be identified with a subgroup of the group K_{p,1}^{×} of elements of K_{p}^{×} of norm 1, by sending z to (\frac{z}{z̄})^{u_{K}}, where z̄ denotes the complex conjugate of z in K_{p}^{×}.

A construction of [BD1], sec. 2.7 and 5.3, based on ideas of Gross [Gr], and recalled in section 2, gives an element L_{p}(E/K) in the completed integral group ring ℤ[[G̃_{∞}]] which interpolates the special values of the classical L-function of E/K twisted by complex characters of G̃_{∞}. We will show (section 2) that L_{p}(E/K) belongs to the augmentation ideal Ĩ of ℤ[[G̃_{∞}]]. Let L′_{p}(E/K) denote the image of L_{p}(E/K) in Ĩ/Ĩ^{2} = G̃_{∞}. The reader should view L′_{p}(E/K) ∈ G̃_{∞} as the first derivative of L_{p}(E/K) evaluated at the central point. One shows that the element L′_{p}(E/K) actually belongs to G_{∞} ⊂ G̃_{∞}, so that it can (and will) be viewed as an element of K_{p}^{×} of norm 1.

Using the theory of Jacquet-Langlands, and the assumption that E is modular, we will define a surjective map η_{f} : J ⟶ Ẽ, where Ẽ is an elliptic curve isogenous to E over ℚ, and J is the Jacobian of a certain Shimura curve X. The precise definitions of X, J, η_{f} and Ẽ are given at the end of section 4. At the cost of possibly replacing E with an isogenous curve, we assume from now on in the introduction that E = Ẽ. (This will imply that E is the “strong Weil curve” for the Shimura curve parametrization.)

A special case of our main result is:

Theorem A The local point Φ_{Tate}(L′_{p}(E/K)) in E(K_{p}) is a global point in E(K).

When L′_{p}(E/K) is non-trivial, theorem A gives a construction of a rational point on E(K) from the first derivative of the anticyclotomic p-adic L-function of E/K, in much the same way that the derivative at s = 0 of the Dedekind zeta-function of a real quadratic field leads to a solution of Pell’s equation. A similar kind of phenomenon was discovered by Rubin [Ru] for elliptic curves with complex multiplication, with the exponential map on the formal group of E playing the role of the Tate parametrization. See also a recent result of Ulmer [U] for the universal elliptic curve over the function field of modular curves over finite fields.

We now state theorem A more precisely. In section 5, a Heegner point α_{K} ∈ E(K) is defined as the image by η_{f} of certain divisors supported on CM points of X. Let ᾱ_{K} be the complex conjugate of α_{K}.

Theorem B Let w = 1 (resp. w = −1) if E/ℚ_{p} has split (resp. non-split) multiplicative reduction. Then

Φ_{Tate}(L′_{p}(E/K)) = α_{K} − wᾱ_{K}.

Theorem B, which relates the Heegner point α_{K} to the first derivative of a p-adic L-function, can be viewed as an analogue in the p-adic setting of the theorem of Gross-Zagier, and also of the p-adic formula of Perrin-Riou [PR]. Unlike these results, it does not involve heights of Heegner points, and gives instead a p-adic analytic construction of a Heegner point.

Observe that G_{∞} is isomorphic to ℤ_{p} × ℤ/(p + 1)ℤ, so that its torsion subgroup is of order p + 1. Choosing an anticyclotomic logarithm λ mapping G_{∞} onto ℤ_{p} determines a map from ℤ[[G_{∞}]] to the formal power series ring ℤ_{p}[[T]]. Let L_{p}(E/K) be the image of L_{p}(E/K) in ℤ_{p}[[T]], and L′_{p}(E/K) the derivative of L_{p}(E/K) with respect to T evaluated at T = 0. Since Φ_{Tate} is injective on K_{p,1}^{×}, theorem B implies:

Corollary C The derivative L′_{p}(E/K) is non-zero if and only if the point α_{K} − wᾱ_{K} is of infinite order.

Corollary C gives a criterion in terms of the first derivative of a p-adic L-function for a Heegner point coming from a Shimura curve parametrization to be of infinite order. Work in progress of Keating and Kudla suggests that a similar criterion (involving the Heegner point α_{K} itself) can be formulated in terms of the first derivative of the classical L-function, in the spirit of the Gross-Zagier formula.

The work of Kolyvagin [Ko] shows that if α_{K} is of infinite order, then E(K) has rank 1 and � (E/K) is finite. By combining this with corollary C, one obtains

Corollary D If L′_{p}(E/K) is non-zero, then E(K) has rank 1 and � (E/K) is finite.

The formula of theorem B is a consequence of the more general result given in section 7, which relates certain Heegner divisors on jacobians of Shimura curves to derivatives of p-adic L-functions. The main ingredients in the proof of this theorem are (1) a construction, based on ideas of Gross, of the anticyclotomic p-adic L-function of E/K, (2) the explicit construction of [GVdP] of the p-adic Abel-Jacobi map for Mumford curves, and (3) the Cerednik-Drinfeld theory of padic uniformization of Shimura curves.

1 Quaternion algebras, upper half planes, and trees

Definite quaternion algebras Let N^{−} be a product of an odd number of distinct primes, and let B be the (unique, up to isomorphism) definite quaternion algebra of discriminant N^{−}. Fix a maximal order R ⊂ B. (There are only finitely many such maximal orders, up to conjugation by B^{×}.)

For each prime ℓ, we choose certain local orders in B_{ℓ} := B ⊗ ℚ_{ℓ}, as follows.

1. If ℓ is any prime which does not divide N^{−}, then B_{ℓ} is isomorphic to the algebra of 2 × 2 matrices M_{2}(ℚ_{ℓ}) over ℚ_{ℓ}. Any maximal order of B_{ℓ} is isomorphic to M_{2}(ℤ_{ℓ}), and all maximal orders are conjugate by B_{ℓ}^{×}. We fix the maximal order

R_{ℓ} := R ⊗ ℤ_{ℓ}.

2. If ℓ is a prime dividing N^{−}, then B_{ℓ} is the (unique, up to isomorphism) quaternion division ring over ℚ_{ℓ}. We let

R_{ℓ} := R ⊗ ℤ_{ℓ},

as before. The valuation on ℤ_{ℓ} extends uniquely to R_{ℓ}, and the residue field of R_{ℓ} is isomorphic to 𝔽_{ℓ^{2}}, the finite field with ℓ^{2} elements. We fix an orientation of R_{ℓ}, i.e., an algebra homomorphism

𝔬_{ℓ}^{−} : R_{ℓ} ⟶ 𝔽_{ℓ^{2}}.

Note that there are two possible choices of orientation for R_{ℓ}.

3. For each prime ℓ which does not divide N^{−}, and each integer n ≥ 1, we also choose certain oriented Eichler orders of level ℓ^{n}. These are Eichler orders \underline{R}_{ℓ}^{(n)} of level ℓ^{n} contained in R_{ℓ}, together with an orientation of level ℓ^{n}, i.e., an algebra homomorphism

𝔬_{ℓ}^{+} : \underline{R}_{ℓ}^{(n)} ⟶ ℤ/ℓ^{n}ℤ.

We will sometimes write \underline{R}_{ℓ} for the oriented Eichler order \underline{R}_{ℓ}^{(1)} of level ℓ.

For each integer M = ∏_{i} ℓ_{i}^{n_{i}} which is prime to N^{−}, let R(M) be the (oriented) Eichler order of level M in R associated to our choice of local Eichler orders:

R(M) := B ∩ (∏_{ℓ|̸M} R_{ℓ} ∏_{ℓ_{i}} \underline{R}_{ℓ_{i}}^{(n_{i})}).

We view R(M) as endowed with the various local orientations 𝔬_{ℓ}^{+} and 𝔬_{ℓ}^{−} for the primes ℓ which divide MN^{−}, and call such a structure an orientation on R(M). We will usually view R(M) as an oriented Eichler order, in what follows.

Let ℤ̂ = ∏_{ℓ} ℤ_{ℓ} be the profinite completion of ℤ, and let

B̂ := B ⊗ ℤ̂ = ∏_{ℓ} B_{ℓ}

be the adelization of B. Likewise, if R_{0} is any order in B (not necessarily maximal), let R̂_{0} := R_{0} ⊗ ℤ̂.

The multiplicative group B̂^{×} acts (on the left) on the set of all oriented Eichler orders of a given level M by the rule

b ∗ R_{0} := B ∩ (bR̂_{0}b^{−1}), b ∈ B̂^{×}, R_{0} ⊂ B.

(Note that b ∗ R_{0} inherits a natural orientation from the one on R_{0}.) This action of B̂^{×} is transitive, and the stabilizer of the oriented order R(M) is precisely R̂(M)^{×}. Hence the choice of R(M) determines a description of the set of all oriented Eichler orders of level M, as the coset space R̂(M)^{×}\B̂^{×}. Likewise, the conjugacy classes of oriented Eichler orders of level M are in bijection with the double coset space R̂(M)^{×}\B̂^{×}/B^{×}.

Let N^{+} be an integer which is prime to N^{−}, and let p be a prime which does not divide N^{+}N^{−}. We set

N = N^{+}N^{−}p.

Let Γ be the group of elements in R(N^{+})[\frac{1}{p}]^{×} of reduced norm 1. Of course, the definition of Γ depends on our choice of local orders, but:

Lemma 1.1 The group Γ depends on the choice of the R_{ℓ} and \underline{R}_{ℓ}^{(n)}, only up to conjugation in B^{×}.

Proof. This follows directly from strong approximation ([Vi], p. 61).

The p-adic upper half plane attached to B Fix an unramified quadratic extension K_{p} of ℚ_{p}. Define the p-adic upper half plane (attached to the quaternion algebra B) as follows:

H_{p} := Hom(K_{p}, B_{p}).

Remark. The group GL_{2}(ℚ_{p}) acts naturally on ℙ^{1}(K_{p}) by Möbius transformations, and the choice of an isomorphism η : B_{p} ⟶ M_{2}(ℚ_{p}) determines an identification of H_{p} with ℙ^{1}(K_{p}) − ℙ^{1}(ℚ_{p}). This identification sends ψ ∈ H_{p} to one of the two fixed points for the action of ηψ(K_{p}^{×}) on ℙ^{1}(K_{p}). More precisely, it sends ψ to the unique fixed point P ∈ ℙ^{1}(K_{p}) such that the induced action of K_{p}^{×} on the tangent line T_{P} (ℙ^{1}(K_{p})) = K_{p} is via the character z ↦ \frac{z}{z̄}. More generally, a choice of an embedding B_{p} ⟶ M_{2}(K_{p}) determines an isomorphism of H_{p} with a domain Ω in ℙ^{1}(K_{p}). In the literature, the p-adic upper half plane is usually defined to be ℙ^{1}(ℂ_{p}) − ℙ^{1}(ℚ_{p}) = ℂ_{p} − ℚ_{p}, where ℂ_{p} is the completion of (an) algebraic closure of ℚ_{p}. From this point of view, it might be more appropriate to think of H_{p} as the K_{p}-rational points of the p-adic upper half plane. But in this work, the role of the complex numbers in the p-adic context is always played, not by ℂ_{p}, but simply (and more naively) by the quadratic extension K_{p}.

We will try as much to possible to work with the more “canonical” definition of the upper half plane, which does not depend on a choice of embedding of B_{p} into M_{2}(K_{p}). The upper half plane H_{p} is endowed with the following natural structures.

1. The group B_{p}^{×} acts naturally on the left on H_{p}, by conjugation. This induces a natural action of the discrete group Γ on H_{p}.

2. An involution ψ ↦ ψ̄, defined by the formula:

ψ̄(z) := ψ(z̄),

where z ↦ z̄ is the complex conjugation on K_{p}.

The Bruhat-Tits tree attached to B. Let T be the Bruhat-Tits tree of B_{p}^{×}/ℚ_{p}^{×}. The vertices of T correspond to maximal orders in B_{p}, and two vertices are joined by an edge if the intersection of the corresponding orders is an Eichler order of level p. An edge of T is a set of two adjacent vertices on T , and an oriented edge of T is an ordered pair of adjacent vertices of T . We denote the set of edges (resp. oriented edges) of T by E(T ) (resp.

E^{→}(T )).

The edges of T correspond to Eichler orders of level p, and the oriented edges are in bijection with the oriented Eichler orders of level p.

Since T is a tree, there is a distance function defined on the vertices of T in a natural way. We define the distance between a vertex v and an edge e to be the distance between v and the furthest vertex of e.

The group B_{p}^{×} acts on T via the rule

b ∗ R_{0} := bR_{0}b^{−1}, b ∈ B_{p}^{×}, R_{0} ∈ T .

This action preserves the distance on T . In particular, the group Γ acts on T by isometries.

Fix a base vertex v_{0} of T . A vertex is said to be even (resp. odd) if its distance from v_{0} is even (resp. odd). This notion determines an orientation on the edges of T , by requiring that an edge always go from the even vertex to the odd vertex. The action of the group B_{p}^{×} does not preserve the orientation, but the subgroup of elements of norm 1 (or, more generally, of elements whose norm has even padic valuation) sends odd vertices to odd vertices, and even ones to even ones. In particular, the group Γ preserves the orientation we have defined on T .

The reduction map Let O_{p} be the ring of integers of K_{p}. Given ψ ∈ H_{p}, the image ψ(O_{p}) is contained in a unique maximal order R_{ψ} of B_{p}. In this way, any ψ ∈ H_{p} determines a vertex R_{ψ} of T . We call the map ψ ↦ R_{ψ} the reduction map from H_{p} to T , and denote it

r : H_{p} ⟶ T .

For an alternate description of the reduction map r, note that the map ψ from K_{p} to B_{p} determines an action of K_{p}^{×} on the tree T . The vertex r(ψ) is the unique vertex which is fixed under this action.

The lattice M Let G := T /Γ be the quotient graph. Since the action of Γ is orientation preserving, the graph G inherits an orientation from T . Let E(G) be the set of (unordered) edges of G, and let V(G) be its set of vertices. Write ℤ[E(G)] and ℤ[V(G)] for the modules of formal ℤ-linear combinations edges and vertices of G, respectively.

There is a natural boundary map ∂_{∗} (compatible with our orientation)

∂_{∗} : ℤ[E(G)] ⟶ ℤ[V(G)]

which sends an edge {a, b} to a − b, with the convention that a is the odd vertex and b is the even vertex in {a, b}. There is also a coboundary map

∂^{∗} : ℤ[V(G)] ⟶ ℤ[E(G)]

defined by

∂^{∗}(v) = ± ∑_{ṽ∈e} e,

where the sum is taken over the images in E(G) of the p + 1 edges of T containing an arbitrary lift ṽ of v to T . The sign in the formula for ∂^{∗} is +1 if v is odd, and −1 if v is even.

Recall the canonical pairings defined by Gross [Gr] on ℤ[E(G)] and on ℤ[V(G)]. If e is an edge (resp. v is a vertex) define w_{e} (resp. w_{v}) to be the order of the stabilizer for the action of Γ of (some) lift of e (resp. v) to T . Then

⟨e_{i}, e_{j}⟩ = w_{e_{i}}δ_{ij},
⟨⟨v_{i}, v_{j}⟩⟩ = w_{v_{i}}δ_{ij}.

Extend these pairings by linearity to the modules ℤ[(E(G)] and ℤ[V(G)].

Lemma 1.2 The maps ∂_{∗} and ∂^{∗} are adjoint with respect to the pairings ⟨ , ⟩ and ⟨⟨ , ⟩⟩, i.e.,

⟨e, ∂^{∗}v⟩ = ⟨⟨∂_{∗}e, v⟩⟩.

Proof. By direct computation.

Define the module M as the quotient

M := ℤ[E(G)]/image(∂^{∗}).

Given two vertices a and b of T , they are joined by a unique path, which may be viewed as an element of ℤ[E(G)] in the natural way. Note that because of our convention for orienting T , if a and b are even vertices (say) joined by 4 consecutive edges e_{1}, e_{2}, e_{3} and e_{4}, then the path from a to b is the formal sum

path(a, b) = e_{1} − e_{2} + e_{3} − e_{4} ∈ ℤ[E(G)].

Note that we have the following properties of the path function:

path(a, b) = −path(b, a), path(a, b) + path(b, c) = path(a, c).

Also, if a and b are Γ-equivalent, then path(a, b) belongs to H_{1}(G, ℤ) ⊂ ℤ[E(G)].

Proposition 1.3 The map from M to Hom(Γ, ℤ) which sends m ∈ M to the function

γ ↦ ⟨path(v_{0}, γv_{0}), m⟩

is injective and has finite cokernel.

Proof. The pairing ⟨ , ⟩ gives an injective map with finite cokernel

M ⟶ Hom(ker(∂_{∗}), ℤ).

But

ker(∂_{∗}) = H_{1}(G, ℤ).

Let Γ^{ab} denote the abelianization of Γ. Then the map of Γ^{ab} to H_{1}(G, ℤ) which sends γ to path(v_{0}, γv_{0}) is an isomorphism modulo torsion (cf. [Se]). The proposition follows.

Relation of M with double cosets We now give a description of M in terms of double cosets which was used in [BD1], sec. 1.4.

More precisely, let

J_{N^{+}p,N^{−}} = ℤ[R̂(N^{+}p)^{×}\B̂^{×}/B^{×}]

be the lattice defined in [BD1], sec. 1.4. (By previous remarks, the module J_{N^{+}p,N^{−}} is identified with the free ℤ-module

ℤR_{1} ⊕ ⋯ ⊕ ℤR_{t}

generated by the conjugacy classes of oriented Eichler orders of level N^{+}p in the quaternion algebra B.) Likewise, let

J_{N^{+},N^{−}} = ℤ[R̂(N^{+})^{×}\B̂^{×}/B^{×}].

In [BD1], sec. 1.7, we defined two natural degeneracy maps

J_{N^{+},N^{−}} ⟶ J_{N^{+}p,N^{−}},

and a module J_{N^{+}p,N^{−}}^{p−new} to be the quotient of J_{N^{+}p,N^{−}} by the image of J_{N^{+},N^{−}} ⊕ J_{N^{+},N^{−}} under these degeneracy maps.

Proposition 1.4 The choice of the oriented Eichler order R(N^{+}p) determines an isomorphism between M and J_{N^{+}p,N^{−}}^{p−new}.

The proof of proposition 1.4 uses the following lemma:

Lemma 1.5 There exists an element γ ∈ R(N^{+})[\frac{1}{p}]^{×} whose reduced norm is an odd power of p.

Proof. Let F be an auxiliary imaginary quadratic field of prime discriminant such that all primes dividing N^{+} are split in F and all primes dividing N^{−} are inert in F. Such an F exists, by Dirichlet’s theorem on primes in arithmetic progressions. By genus theory, F has odd class number, and hence its ring of integers O_{F} contains an element a of norm p^{k}, with k odd. Fix an embedding of O_{F} in the Eichler order R(N^{+}), and let γ be the image of a in R(N^{+})[\frac{1}{p}]^{×}.

Proof of proposition 1.4. Recall that \underline{R}_{p} ⊂ B_{p} denotes our fixed local Eichler order of level p. By strong approximation, we have

R̂(N^{+}p)^{×}\B̂^{×}/B^{×} = \underline{R}_{p}^{×}ℚ_{p}^{×}\B_{p}^{×}/R(N^{+})[\frac{1}{p}]^{×}.

The group \underline{R}_{p}^{×}ℚ_{p}^{×} is the stabilizer of an ordered edge of T . Hence \underline{R}_{p}^{×}ℚ_{p}^{×}\B_{p}^{×} is

identified with the set E^{→}(T ) of ordered edges on T , and the double coset space

R_{p}^{×}ℚ_{p}^{×}\B_{p}^{×}/R(N^{+})[\frac{1}{p}]^{×} is identified with the set of ordered edges E^{→}(G_{+}) on the quotient graph G_{+} := T /R(N^{+})[\frac{1}{p}]^{×}.

But the map which sends {x, y} ∈ E(G) to (x, y) ∈ E^{→}(G_{+}) if x is even, and to

(y, x) if x is odd, is a bijection between E(G) and E^{→}(G_{+}). For, if {x, y} and {x′, y′}

have the same image in E^{→}(G_{+}), then there is an element of R(N^{+})[\frac{1}{p}]^{×} which sends the odd vertex in {x, y} to the odd vertex in {x′, y′} and the even vertex in {x, y} to the even vertex in {x′, y′}. This element is necessarily in Γ, since it sends an odd vertex to an odd vertex. Hence the edges {x, y} and {x′, y′} are Γ-equivalent, and our map is one-one. To check surjectivity, let γ be the element of R(N^{+})[\frac{1}{p}]^{×}

given by lemma 1.5. Then the element (x, y) of E^{→}(G_{+}) is the image of {x, y} if x is even and y is odd, and is the image of {γx, γy} if x is odd and y is even. To sum up, we have shown that the choice of the Eichler order R(N^{+}p) determines a canonical bijection between J_{N^{+}p,N^{−}} and ℤ[E(G)]. Likewise, one shows that the Eichler order R(N^{+}) determines a canonical bijection between J_{N^{+},N^{−}} and the set of vertices V(G_{+}), and between J_{N^{+},N^{−}} ⊕ J_{N^{+},N^{−}} and ℤ[V(G)]. (The resulting map from ℤ[V(G_{+})] ⊕ ℤ[V(G_{+})] to ℤ[V(G)] sends a pair (v, w) to v_{+} − w_{−}, where where v_{+} and w_{−} are lifts of v and w to vertices of G, which are even and odd respectively.) Finally, from the definition of the degeneracy maps given in [BD1] one checks that the following diagram commutes up to sign:

\begin{matrix}J_{N^{+},N^{−}} ⊕ J_{N^{+},N^{−}} & ⟶ & ℤ[V(G)] \\ ↓ & & ↓ ∂^{∗} \\ J_{N^{+}p,N^{−}} & ⟶ & ℤ[E(G)] \end{matrix} ,

where the horizontal maps are the identifications we have just established, and the left vertical arrow is the difference of the two degeneracy maps. (Which is only well-defined up to sign). From this, it follows that M = ℤ[E(G)]/image(∂^{∗}) is identified with the module

J_{N^{+}p,N^{−}}^{p−new} = J_{N^{+}p,N^{−}}/image(J_{N^{+},N^{−}} ⊕ J_{N^{+},N^{−}})

of [BD1].

Hecke operators The lattice M is equipped with a natural Hecke action, coming from its description in terms of double cosets. (Cf. [BD1], sec. 1.5.) Let 𝕋 be the Hecke algebra acting on M. Recall that N = N^{+}N^{−}p. The following is a consequence of the Eichler trace formula, and is a manifestation of the Jacquet-Langlands correspondence between automorphic forms on GL_{2} and quaternion algebras.

Proposition 1.6 If ϕ : 𝕋 ⟶ ℂ is any algebra homomorphism, and a_{n} = ϕ(T_{n}) (for all n with gcd(n, N^{−}p) = 1), then the a_{n} are the Fourier coefficients of a normalized eigenform of weight 2 for Γ_{0}(N). Conversely, every normalized eigenform of weight 2 on Γ_{0}(N) which is new at p and at the primes dividing N^{−} corresponds in this way to a character ϕ.

Given a normalized eigenform f on X_{0}(N), denote by O_{f} the order generated by the Fourier coefficients of f and by K_{f} the fraction field of O_{f}. Assuming that f is new at p and at N^{−}, let π_{f} ∈ 𝕋 ⊗ K_{f} be the idempotent associated to f by proposition 1.6. Let n_{f} ∈ O_{f} be such that η_{f} := n_{f}π_{f} belongs to 𝕋 ⊗ O_{f}.

Let M^{f} ⊂ M⊗O_{f} be the sublattice on which 𝕋 acts via the character associated to f. The endomorphism η_{f} induces a map, still denoted η_{f} by an abuse of notation,

η_{f} : M → M^{f}.

In particular, if f has integer Fourier coefficients, then M^{f} is isomorphic to ℤ. Fixing such an isomorphism (i.e., choosing a generator of M^{f}), we obtain a map

η_{f} : M → ℤ,

which is well-defined up to sign.

2 The p-adic L-function

We recall the notations and assumptions of the introduction: E is a modular elliptic curve of conductor N, associated to an eigenform f on Γ_{0}(N); K is a quadratic imaginary field of discriminant D relatively prime to N. Furthermore:

1. the curve E has good or multiplicative reduction at all primes which are inert in K/ℚ;

2. there is at least one prime, p, which is inert in K and for which E has multiplicative reduction;

3. the sign in the functional equation for L(E/K, s) is −1.

Write

N = N^{+}N^{−}p,

where N^{+}, resp. N^{−} is divisible only by primes which are split, resp. inert in K. Note that by our assumptions, N^{−} is square-free and not divisible by p.

Lemma 2.1 Under our assumptions, N^{−} is a product of an odd number of primes.

Proof. By page 71 of [GZ], the sign in the functional equation of the complex L-function L(E/K, s) is (−1)^{#{ℓ|N^{−}p}+1}. The result follows.

Let c be an integer prime to N. We modify slightly the notations of the introduction, letting H denote now the ring class field of K of conductor c, and H_{n} the ring class field of conductor cp^{n}. We write H_{∞} = ⋃ H_{n}, and set

G_{n} := Gal(H_{n}/H), G̃_{n} := Gal(H_{n}/K),
G_{∞} := Gal(H_{∞}/H), G̃_{∞} := Gal(H_{∞}/K), Δ := Gal(H/K).

(Thus, the situation considered in the introduction corresponds to the special case where c = 1.) There is an exact sequence of Galois groups

0 ⟶ G_{∞} ⟶ G̃_{∞} ⟶ Δ ⟶ 0,

and, by class field theory, G_{∞} is canonically isomorphic to K_{p}^{×}/ℚ_{p}^{×}O_{K}^{×}.

The completed integral group rings ℤ[[G_{∞}]] and ℤ[[G̃_{∞}]] are defined as the inverse limits of the integral group rings ℤ[G_{n}] and ℤ[G̃_{n}] under the natural projection maps. We set

M[G_{n}] := M ⊗ ℤ[G_{n}],

M[[G_{∞}]] := lim_{n^{←}} M[G_{n}] = M ⊗ ℤ[[G_{∞}]],

and likewise for G_{n} and G_{∞} replaced by G̃_{n} and G̃_{∞}. The groups G_{∞} and G̃_{∞} act naturally on M[[G_{∞}]] and M[[G̃_{∞}]] by multiplication on the right.

In this section, we review the construction of a p-adic L-function L_{p}(M/K), in a form adapted to the calculations we will perform later. A slightly modified version of this construction is given in section 2.7 of [BD1]. It is based on results of Gross [Gr] on special values of the complex L-functions attached to E/K, and on their generalization by Daghigh [Dag].

Let

Ω_{f} := 4π^{2} ∫∫_{H_{∞}/Γ} |f(τ)|^{2}dτ ∧ idτ̄

be the complex period associated to the cusp form f. Write d for the discriminant of the order O of conductor c, u for one half the order of the group of units of O and n_{f} for the integer defined at the end of section 1 by the relation η_{f} = n_{f}π_{f}.

Theorem 2.2 There is an element L_{p}(M/K) ∈ M[[G̃_{∞}]], well-defined up to right multiplication by G̃_{∞}, with the property that

|χ(η_{f}(L_{p}(M/K)))|^{2} = \frac{L(f/K, χ, 1)}{Ω_{f}} \sqrt{d} · (n_{f}u)^{2},

for all finite order complex characters χ of G̃_{∞} and all modular forms f associated to 𝕋 as in proposition 1.6.

Proof. See [Gr], [Dag] and [BD1], sec. 2.7.

Corollary 2.3 Setting

L_{p}(E/K) := η_{f}(L_{p}(M/K)) ∈ ℤ[[G̃_{∞}]],

where f is the modular form associated to E, one has

|χ(L_{p}(E/K))|^{2} = \frac{L(E/K, χ, 1)}{Ω_{f}} \sqrt{d} · (n_{f}u)^{2},

for all finite order characters χ of G̃_{∞}.

Remark. One sees that the interpolation property of corollary 2.3 determines L_{p}(E/K) uniquely, up to right multiplication by elements in G̃_{∞}, if it exists. The existence amounts to a statement of rationality and integrality for the special values L(E/K, χ, 1). The construction of L_{p}(M/K) (and hence, of L_{p}(E/K)) is based on the notion of Gross points of conductor c and cp^{n}.

Gross points of conductor c Recall that O is the order of conductor c in the maximal order O_{K}, where we assume that c is prime to N. We equip O with an orientation of level N^{+}N^{−}, i.e., for each ℓ^{n}||N^{+}, an algebra homomorphism

𝔬_{ℓ}^{+} : O ⟶ ℤ/ℓ^{n}ℤ,

and for each ℓ|N^{−}, an algebra homomorphism

𝔬_{ℓ}^{−} : O ⟶ 𝔽_{ℓ^{2}}.

An embedding ξ : O ⟶ R_{ξ} of O into an oriented Eichler order R_{ξ} of level dividing N^{+} is called an oriented embedding if it respects the orientations on O and on R_{ξ}, i.e., if the diagrams

O  ⟶^{ξ} R_{ξ}  O ⟶^{ξ} R_{ξ}
↓ 𝔬_{ℓ}^{+}  ↓ 𝔬_{ℓ}^{+}  ↓ 𝔬_{ℓ}^{−}  ↓ 𝔬_{ℓ}^{−}
ℤ/ℓ^{n}ℤ = ℤ/ℓ^{n}ℤ  𝔽_{ℓ^{2}} = 𝔽_{ℓ^{2}}

commute, for all ℓ which divide N^{+}N^{−}.

The embedding ξ is called optimal if it does not extend to an embedding of any larger order into R_{ξ}. The group B^{×} acts naturally on the set of oriented optimal embeddings of conductor c, by conjugation:

b(R_{ξ}, ξ) := (bR_{ξ}b^{−1}, bξb^{−1}).

Definition 2.4 A Gross point of conductor c and level N^{+}N^{−} is a pair (R_{ξ}, ξ) where R_{ξ} is an oriented Eichler order of level N^{+} in B, and ξ is an oriented optimal embedding of O into R_{ξ}, taken modulo conjugation by B^{×}.

We denote by Gr(c) the set of all Gross points of conductor c and level N^{+}N^{−}.

Given ξ ∈ Hom(K, B), we denote by ξ̂ ∈ Hom(K̂, B̂) the natural extension of scalars.

The group

Δ = Pic(O) = Ô^{×}\K̂^{×}/K^{×}

acts on the Gross points, by the rule

σ(R_{ξ}, ξ) := (ξ̂(σ) ∗ R_{ξ}, ξ).

Lemma 2.5 The group Δ acts simply transitively on the Gross points of conductor c.

Proof. See [Gr], sec. 3.

One says that (R_{ξ}, ξ) is in normal form if

R_{ξ} ⊗ ℤ_{ℓ} = R_{ℓ} for all ℓ |̸Np,
R_{ξ} ⊗ ℤ_{ℓ} = \underline{R}_{ℓ}^{(n)} as oriented Eichler orders, for all ℓ^{n}||N^{+},
R_{ξ} ⊗ ℤ_{ℓ} = R_{ℓ} as oriented orders, for all ℓ|N^{−}.

(Note in particular that we have imposed no condition on R_{ξ} ⊗ ℤ_{p} in this definition.) Choose representatives (R_{1}, ψ_{1}), (R_{2}, ψ_{2}), …, (R_{h}, ψ_{h}) for the Gross points of conductor c, written in normal form. (This can always be done, by strong approximation.) Note that

R_{i}[\frac{1}{p}] = R[\frac{1}{p}] as oriented Eichler orders,

and that the orders R_{i} are completely determined by the local order R_{i} ⊗ ℤ_{p}. Let v_{1}, …, v_{h} be the vertices on T associated to the maximal orders R_{1} ⊗ ℤ_{p}, …, R_{h} ⊗ ℤ_{p}. The vertex v_{i} is equal to r(ψ_{i}), i.e., it is the image of ψ_{i} (viewed as a point on H_{p} in the natural way) by the reduction map to T .

Gross points of conductor cp^{n} Let n ≥ 1, and let O_{n} denote the order of K of conductor cp^{n}.

Definition 2.6 A Gross point of conductor cp^{n} and level N is a pair (R_{ξ}, ξ) where R_{ξ} is an oriented Eichler order of level N^{+}p in B, and ξ is an oriented optimal embedding of O_{n} into R_{ξ}, taken modulo conjugation by B^{×}.

To make definition 2.6 complete, we need to clarify what we mean by an orientation at p of the optimal embedding ξ. (For the primes which divide N^{+}N^{−}, the meaning is exactly the same as before.) The oriented Eichler order R_{ξ} ⊗ ℤ_{p} corresponds to an ordered edge on T , whose source and target correspond to maximal orders R_{1} and R_{2} respectively. We require that ξ still be an optimal embedding of O_{n} into R_{2}. (It then necessarily extends to an optimal embedding of O_{n−1} into R_{1}.)

We let Gr(cp^{n}) be the set of Gross points of level cp^{n}, and we set

Gr(cp^{∞}) := ⋃_{n=1}^{∞} Gr(cp^{n}).

The group G̃_{n} = Ô_{n}^{×}\K̂^{×}/K^{×} acts on Gr(cp^{n}) by the rule

σ(R_{ξ}, ξ) := (ξ̂(σ) ∗ R_{ξ}, ξ).

Lemma 2.7 The group G̃_{n} acts simply transitively on Gr(cp^{n}).

Proof. See [Gr], sec. 3.

In particular, the group G̃_{∞} acts transitively on Gr(cp^{∞}). As before, we say that a Gross point (R_{ξ}, ξ) of conductor cp^{n} is in normal form if

R_{ξ} ⊗ ℤ_{ℓ} = R_{ℓ} for all ℓ |̸Np,
R_{ξ} ⊗ ℤ_{ℓ} = \underline{R}_{ℓ}^{(n)} as oriented Eichler orders, for all ℓ^{n}||N^{+},
R_{ξ} ⊗ ℤ_{ℓ} = R_{ℓ} as oriented orders, for all ℓ|N^{−}.

Recall the representatives (R_{1}, ψ_{1}), …,(R_{h}, ψ_{h}) for the Gross points of conductor c that were chosen in the previous paragraph.

Lemma 2.8 Every point in Gr(cp^{∞}) is equivalent to an element in normal form, and can be written as (R_{0}, ψ_{i}), where ψ_{i} ∈ {ψ_{1}, …, ψ_{h}}, and R_{0} ⊗ ℤ_{p} is an oriented Eichler order of level p. A point in Gr(cp^{∞}) described by a pair (R_{0}, ψ_{i}) is of level cp^{n}, where n is the distance between the edge associated to R_{0} on T , and the vertex associated to R_{i}.

Proof. The first statement follows from strong approximation, and the second from a direct calculation.

By lemma 2.8, the set Gr(cp^{∞}) can be described by the system of representatives

E(T ) × {ψ_{1}, …, ψ_{h}}.

The action of G_{∞} = K_{p}^{×}/ℚ_{p}^{×} on Gr(cp^{∞}) in this description is simply

σ(R_{0}, ψ_{i}) := (ψ̂_{i}(σ) ∗ R_{0}, ψ_{i}).

Construction of L_{p}(M/K)

Choose one of the representatives of Gr(c), say, (v_{1}, ψ_{1}). Choose an end of T originating from v_{1}, i.e., a sequence e_{1}, e_{2}, …, e_{n}, … of consecutive edges originating from v_{1}. By lemma 2.8, the Gross points (e_{n}, ψ_{1}) are a sequence of Gross points of conductor cp^{n}. Consider the formal expression

(−1)^{n} ∑_{σ∈G̃_{n}} σ(e_{n}, ψ_{1}) · σ^{−1}

and let L_{p,n}(M/K) denote its natural image in M[G̃_{n}].

Lemma 2.9 The elements L_{p,n}(M/K) (n ≥ 1) are compatible under the natural projection maps M[G̃_{n+1}] ⟶ M[G̃_{n}].

Proof. This follows directly from the definiton of the action of G̃_{n} on Gr(cp^{n}) given above, and from the definition of the coboundary map ∂^{∗}. They yield that the formal expression Norm_{K_{n+1}/K_{n}}(e_{n+1}, ψ_{1}) + (e_{n}, ψ_{1}) is in the image of the coboundary map ∂^{∗}, and hence is zero in M. The lemma follows.

Lemma 2.9 implies that we can define an element

L_{p}(M/K) ∈ M[[G̃_{∞}]]

by taking inverse limit of the L_{p,n}(M/K) via the projections M[G_{n+1}] → M[G_{n}].

The element L_{p}(M/K) satisfies the conclusions of theorem 2.2. It should be thought of as a p-adic L-function (or rather, the square root of a p-adic L-function) over K, associated to modular forms for 𝕋. If f is any such modular form, then the element η_{f}L_{p}(M/K) is equal to the element θ_{N_{∗}^{+},N_{∗}^{−}} defined in [BD1], sec. 5.3 (in the special case when f has rational coefficients).

Note that L_{p}(M/K) depends on the choice of the initial point (v_{1}, ψ_{1}), and on the end e_{1}, …, e_{n}, … of T originating from v_{1}, but only up to multiplication (on the right) by an element of G̃_{∞}.

Recall the augmentation ideal I of ℤ[[G̃_{∞}]] described in the introduction. More generally, let I_{Δ} be the kernel of the augmentation map ℤ[[G̃_{∞}]] ⟶ ℤ[Δ].

Lemma 2.10 L_{p}(M/K) belongs to M ⊗ I. In fact, L_{p}(M/K) belongs to M ⊗ I_{Δ}.

Proof. Since Δ acts simply transitively on (v_{1}, ψ_{1}), …, (v_{h}, ψ_{h}), let σ_{i} be the element such that

σ_{i}v_{1} = v_{i}.

Let I_{Δ} denote, by abuse of notation, the image of I_{Δ} in ℤ[G̃_{n}]. Note that we have the canonical isomorphisms

ℤ[[G̃_{∞}]]/I_{Δ} = ℤ[G̃_{n}]/I_{Δ} = ℤ[Δ].

By the compatibility lemma 2.9, the image of L_{p}(M/K) in M[[G̃_{∞}]]/I_{Δ} is equal to the image of L_{p,1}(M/K) in M[G_{1}]/I_{Δ} = M[Δ], which is equal to:

∑_{i=1}^{h}(∑_{v_{i}∈e} e) · σ_{i}^{−1}.

But each of the terms in the inner sum belongs to the image of ∂^{∗}, and hence is 0 in M. Thus, L_{p}(M/K) belongs to M ⊗ I_{Δ}, and also to M ⊗ I, since I_{Δ} ⊂ I.

Remark. If χ is any character of Δ and f is any modular form attached to 𝕋, then the functional equation of L(f/K, χ, s) has sign −1, and hence L(f/K, χ, 1) = 0 for all such characters. The interpolation formula of theorem 2.2 implies then that L_{p}(M/K) belongs to I_{Δ}. The point of the proof of lemma 2.10 is that the construction of L_{p}(M/K) also implies this directly, without using the relation with L-function values.

Let

L′_{p}(M/K) ∈ M ⊗ (I/I^{2}) = M ⊗ G̃_{∞}

and

L′_{p}(M/H) ∈ M ⊗ (I_{Δ}/I_{Δ}^{2}) = M[Δ] ⊗ G_{∞} = M[Δ] ⊗ (K_{p,1}^{×})

be the natural images of the element L_{p}(M/K). Since L_{p}(M/K) is well-defined up to right multiplication by G̃_{∞}, the element L′_{p}(M/K) is canonical, and does not depend on the choice of (v_{1}, ψ_{1}) or on the choice of the end of T originating from v_{1}. The element L′_{p}(M/H) is well defined, up to right multiplication by an element of Δ.

We now give an explicit description of L′_{p}(M/K) and L′_{p}(M/H) in Hom(Γ, K_{p,1}^{×}) which will be used in the calculations of section 6 and 7. Let ψ be any point in H_{p}, corresponding to a local embedding of K_{p} into B_{p}. The embedding ψ gives rise to an action of K_{p}^{×}/ℚ_{p}^{×} on the tree T by multiplication on the right, fixing the vertex v_{0} := r(ψ). Choose a sequence of ends e_{1}, …, e_{n}, … originating from v_{0}, and let

L′_{p,n}(ψ) = (−1)^{n} ∑_{σ∈G_{n}} ψ(σ)(e_{n}) ⊗ σ^{−1}

be the element of M⊗G_{n} (here we denote by e_{n} the element in M associated to the edge e_{n}). The elements L′_{p,n}(ψ) are compatible under the obvious projection maps M ⊗ G_{n+1} ⟶ M ⊗ G_{n}, and hence the element L′_{p}(ψ) ∈ M ⊗ G_{∞} can be defined as the inverse limit of the L′_{p,n}(ψ) under the natural projections. By proposition 1.3, we may view L′_{p}(ψ) as an element of Hom(Γ, K_{p,1}^{×}), given by

L′_{p}(ψ)(δ) = lim_{n^{←}}⟨path(v_{0}, δv_{0}), L′_{p,n}(ψ)⟩ ∈ G_{∞} = K_{p,1}^{×}, ∀δ ∈ Γ.

In this notation, we have

L′_{p}(M/K) = ∑_{i=1}^{h} L′_{p}(ψ_{i}),

L′_{p}(M/H) = ∑_{σ∈Δ} L′_{p}(ψ_{1}^{σ})σ^{−1}.

3 Generalities on Mumford curves

Following [Jo-Li], we call a smooth complete curve X over K_{p} an admissible curve over K_{p} if it admits a model X over the ring of integers O_{p} of K_{p}, such that:

(i) the scheme X is proper and flat over O_{p};

(ii) the irreducible components of the special fiber X_{(p)} are rational and defined over O_{p}/(p) ≃ 𝔽_{p^{2}}, and the singularities of X_{(p)} are ordinary double points defined over O_{p}/(p);

(iii) if x ∈ X_{(p)} is a singular point, then the completion Ô_{X ,x} of the local ring O_{X ,x} is O-isomorphic to the completion of the local ring O[[X, Y]]/(XY − p^{m}) for a positive integer m.

Let Γ be a finitely generated subgroup of PGL_{2}(K_{p}), acting on ℙ^{1}(ℂ_{p}) by Möbius transformations. A point z ∈ ℙ^{1}(ℂ_{p}) is said to be a limit point for the action of Γ if it is of the form z = lim g_{n}(z_{0}) for a sequence of distinct elements g_{n} of Γ. Let I ⊂ ℙ^{1}(ℂ_{p}) denote its set of limit points and let Ω_{p} = ℙ^{1}(K_{p}) − I. The group Γ is said to act discontinuously, or to be a discontinuous group, if Ω_{p} ≠ ∅. A fundamental result of Mumford, extended by Kurihara, establishes a 1-1 correspondence between conjugacy classes of discontinuous groups and admissible curves.

Theorem 3.1 Given an admissible curve X over K_{p}, there exists a discontinuous group Γ ⊂ PGL_{2}(K_{p}), unique up to conjugation, such that X(K_{p}) is isomorphic to Ω_{p}/Γ. Conversely, any such quotient is an admissible curve over K_{p}.

Proof. See [Mu] and [Ku].

If D = P_{1} + ⋯ + P_{r} − Q_{1} − ⋯ − Q_{r} ∈ Div^{0}(Ω_{p}) is a divisor of degree zero on Ω_{p}, define the theta function

θ(z; D) = ∏_{γ∈Γ} \frac{(z − γP_{1}) ⋯ (z − γP_{r})}{(z − γQ_{1}) ⋯ (z − γQ_{r})},

with the convention that z − ∞ = 1.

Let Γ_{ab} := Γ/[Γ, Γ] be the abelianization of Γ, and let Γ̄ := Γ_{ab}/(Γ_{ab})_{tor} be its maximal torsion-free quotient.

Lemma 3.2 There exists ϕ_{D} ∈ Hom(Γ, K_{p}^{×}) such that θ(δz; D) = ϕ_{D}(δ)θ(z; D), for all δ in Γ. Furthermore, the map ϕ_{D} factors through Γ̄, so that ϕ_{D} can be viewed as an element of Hom(Γ̄, K_{p}^{×}).

Proof. See [GVdP], p. 47, (2.3.1), and ch. VIII, prop. (2.3).

Let

Φ_{AJ} : Div^{0}(Ω_{p}) ⟶ Hom(Γ̄, K_{p}^{×})

be the map which associates to the degree zero divisor D the automorphy factor ϕ_{D}. The reader should think of this map as a p-adic Abel-Jacobi map.

Given δ ∈ Γ, the number ϕ_{(z)−(δz)}(β) does not depend on the choice of z ∈ Ω_{p}, and depends only on the image of α and β in Γ̄. Hence it gives rise to a well-defined pairing

( , ) : Γ̄ × Γ̄ → K_{p}^{×}.

Lemma 3.3 The pairing ( , ) is bilinear, symmetric, and positive definite (i.e., ord_{p} ∘ ( , ) is positive definite). Hence, the induced map

j : Γ̄ → Hom(Γ̄, K_{p}^{×})

is injective and has discrete image.

Proof. See [GVdP], VI.2. and VIII.3.

Given a divisor D of degree zero on X(K_{p}) = Ω_{p}/Γ, let D̃ denote an arbitrary lift to a degree zero divisor on Ω_{p}. Let Λ := j(Γ̄). The automorphy factor ϕD̃ depends on the choice of D̃, but its image in Hom(Γ̄, K_{p}^{×})/Λ depends only on D. Thus Φ_{AJ} induces a map Div^{0}(X(K_{p})) ⟶ Hom(Γ̄, K_{p}^{×})/Λ, which we also call Φ_{AJ} by abuse of notation.

Proposition 3.4 The map Div^{0}(X(K_{p})) ⟶ Hom(Γ̄, K_{p}^{×})/Λ defined above is trivial on the group of principal divisors, and induces an identification of the K_{p}-rational points of the jacobian J of X over K_{p} with Hom(Γ̄, K_{p}^{×})/Λ.

Proof. See [GVdP], VI.2. and VIII.4.

To sum up, we have:

Corollary 3.5 The diagram

Div^{0}(Ω_{p}) ⟶^{Φ_{AJ}} Hom(Γ̄, K_{p}^{×})
↓  ↓
Div^{0}(X(K_{p})) ⟶^{Φ_{AJ}}  J(K_{p})

commutes.

4 Shimura curves

Let B be the indefinite quaternion algebra of discriminant N^{−}p, and let R be an (oriented) maximal order in B (which is unique up to conjugation). Likewise, for each M prime to N^{−}p, choose an oriented Eichler order R(M) of level M contained in R.

Let X be the Shimura curve associated to the Eichler order R(N^{+}), as in [BD1], sec. 1.3.

I Moduli description of X The curve X/ℚ is a moduli space for abelian surfaces with quaternionic multiplication and N^{+}-level structure. More precisely, the curve X/ℚ coarsely represents the functor F_{ℚ} which associates to every scheme S over ℚ the set of isomorphism classes of triples (A, i, C), where

1. A is an abelian scheme over S of relative dimension 2;

2. i : R → End_{S}(A) is an inclusion defining an action of R on A;

3. C is an N^{+}-level structure, i.e., a subgroup scheme of A which is locally isomorphic to ℤ/N^{+}ℤ and is stable and locally cyclic under the action of R(N^{+}). See [BC], ch. III and [Rob] for more details.

Remarks

1. The datum of the level N^{+} structure is equivalent to the data, for each ℓ^{n}||N^{+}, of a subgroup C_{ℓ} which is locally isomorphic to ℤ/ℓ^{n}ℤ and is locally cyclic for the action of R(N^{+}).

2. For each ℓ dividing N^{−}p, let I ⊂ R_{ℓ} be the maximal ideal of R_{ℓ}. The subgroup scheme A_{I} of points in A killed by I is a free R_{ℓ}/I ≃ 𝔽_{ℓ^{2}}-module of rank one, and the orientation 𝔬_{ℓ}^{−} : R_{ℓ} ⟶ 𝔽_{ℓ^{2}} allows us to view A_{I} canonically as a one-dimensional 𝔽_{ℓ^{2}}-vector space.

II Complex analytic description of X Let

B_{∞} := B ⊗ ℝ ≃ M_{2}(ℝ).

Define the complex upper half plane associated to B to be

H_{∞} := Hom(ℂ, B_{∞}).

Note that a choice of isomorphism η : B_{∞} ⟶ M_{2}(ℝ) determines an isomorphism of H_{∞} with the union ℂ − ℝ of the “usual” complex upper half plane

{z ∈ ℂ : Imz > 0}

with the complex lower half plane, by sending ψ ∈ Hom(ℂ, B_{∞}) to the unique fixed point P of ηψ(ℂ^{×}) such that the induced action of ℂ^{×} on the complex tangent line T_{P} (ℂ − ℝ) = ℂ is by the character z ↦ \frac{z}{z̄}.

Let Γ_{∞} = R(N^{+})^{×} be the group of invertible elements in R(N^{+}) (i.e., having reduced norm equal to ±1). This group acts naturally on H_{∞} via the action of B_{∞}^{×} by conjugation.

Proposition 4.1 The Shimura curve X over ℂ is isomorphic to the quotient of the complex upper half plane H_{∞} attached to B_{∞} by the action of Γ_{∞}, i.e.,

X(ℂ) = H_{∞}/Γ_{∞}.

Proof. See [BC], ch. III, and [Rob].

In particular, an abelian surface A over ℂ with quaternionic multiplications by R and level N^{+} structure determines a point ψ ∈ H_{∞} = Hom(ℂ, B_{∞}) which is welldefined modulo the natural action of Γ_{∞}. We will now give a description of the assignment A ↦ ψ. Although not used in the sequel, this somewhat non-standard description of the complex uniformization is included to motivate the description of the p-adic uniformization of X which follows from the work of Cerednik and Drinfeld.

The complex upper half plane as a moduli space. We first give a “moduli” description of the complex upper half plane H_{∞} := Hom(ℂ, B_{∞}) as classifying complex vector spaces with quaternionic action and a certain “rigidification”.

Definition 4.2 A quaternionic space (attached to B_{∞}) is a two-dimensional complex vector space V equipped with a (left) action of B_{∞}, i.e., an injective homomorphism i : B_{∞} ⟶ End_{ℂ}(V).

Let V_{ℝ} be the 4-dimensional real vector space underlying V.

Lemma 4.3 The algebra End_{B_{∞}}(V_{ℝ}) is isomorphic (non-canonically) to B_{∞}.

Proof. The natural map

B_{∞} ⊗ End_{B_{∞}}(V_{ℝ}) ⟶ End_{ℝ}(V_{ℝ}) ≃ M_{4}(ℝ)

is an isomorphism, and hence End_{B_{∞}}(V_{ℝ}) is abstractly isomorphic to the algebra B_{∞}.

Definition 4.4 A rigidification of the quaternionic space V is an isomorphism

ρ : B_{∞} ⟶ End_{B_{∞}}(V_{ℝ}).

A pair (V, ρ) consisting of a quaternionic space V and a rigidification ρ is called a rigidified quaternionic space.

There is a natural notion of isomorphism between rigidified quaternionic spaces.

Proposition 4.5 There is a canonical bijection betwen H_{∞} and the set of isomorphism classes of rigidified quaternionic spaces. Proof. Given ψ ∈ H_{∞} = Hom(ℂ, B_{∞}), we define a rigidified quaternionic space as follows. Let V = B_{∞}, viewed as a two-dimensional complex vector space by the rule

λv := vψ(λ), v ∈ V, λ ∈ ℂ.

The left multiplication by B_{∞} on V endows V with the structure of quaternionic space. The right multiplication of B_{∞} on V is then used to define the rigidification B_{∞} ⟶ End_{B_{∞}}(V_{ℝ}).

Conversely, given a rigidified quaternionic space (V, ρ), one recovers the point ψ in H_{∞} by letting ψ(λ) be ρ^{−1}(m_{λ}), where m_{λ} is the endomorphism in End_{B_{∞}}(V_{ℝ}) induced by multiplication by the complex number λ.

One checks that these two assignments are bijections between H_{∞} and the set of isomorphism classes of rigidified quaternionic spaces, and that they are inverses of each other.

We now describe the isomorphism X(ℂ) = H_{∞}/Γ_{∞} given in proposition 4.1. Let A be an abelian surface over ℂ with quaternionic multiplication by R and level N^{+} structure. Then the Lie algebra V = Lie(A) is a quaternionic space in a natural way. (The quaternionic action of B_{∞} is induced by the action of R on the tangent space, by extension of scalars from ℤ to ℝ.) Moreover, V is equipped with an R-stable sublattice Λ which is the kernel of the exponential map V ⟶ A.

Lemma 4.6

1. The endomorphism ring End_{R}(Λ) is isomorphic (non-canonically) to R.

2. The set of endomorphisms in End_{R}(Λ) which preserve the level N^{+}-structure on Λ is isomorphic (non-canonically) to the Eichler order R(N^{+}).

Proof. The natural map

B ⊗ (End_{R}(Λ) ⊗ ℚ) ⟶ End_{ℚ}(Λ ⊗ ℚ) ≃ M_{4}(ℚ)

is an isomorphism, and hence End_{R}(Λ) ⊗ ℚ is abstractly isomorphic to the quaternion algebra B. Furthermore, the natural map

End_{R}(Λ) ⟶ End_{ℤ}(Λ)

has torsion-free cokernel, and hence End_{R}(Λ) is a maximal order in B. Likewise, one sees that the subalgebra of End_{R}(Λ) preserving the level N^{+} structure (viewed as a submodule of \frac{1}{N^{+}} Λ/Λ) is an Eichler order of level N^{+}.

Fix an isomorphism

ρ_{0} : R ⟶ End_{R}(Λ),

having the following properties.

1. For each ℓ^{n}||N^{+}, ρ_{0}(R(N^{+})) ⊗ ℤ_{ℓ} preserves the subgroup C_{ℓ} (viewed as a subgroup of \frac{1}{ℓ^{n}} Λ/Λ). By the remark 1 above, R(N^{+}) operates on C_{ℓ} via a homomorphism R(N^{+}) ⟶ ℤ/ℓ^{n}ℤ. In addition, we require that this homomorphism be equal to the orientation 𝔬_{ℓ}^{+}.

2. For all ℓ|N^{−}p, the algebra R_{ℓ} acts on \frac{1}{ℓ}Λ/Λ, and stabilizes the subspace V corresponding to A_{I} (where I is the maximal ideal of R_{ℓ}). By the remark 2 above, V is equipped with a canonical 𝔽_{ℓ^{2}}-vector space structure, and ρ_{0}(R_{ℓ}) acts 𝔽_{ℓ^{2}}- linearly on it. We require that the resulting homomorphism R_{ℓ} ⟶ 𝔽_{ℓ^{2}} be equal to the orientation 𝔬_{ℓ}^{−}.

With these conventions, the homomorphism ρ_{0} is well-defined, up to conjugation by elements in Γ_{∞}. Let ρ : B_{∞} ⟶ End_{B_{∞}}(V_{ℝ}) be the map induced from ρ_{0} by extension of scalars from ℤ to ℝ. The pair (V, ρ) is a rigidified quaternionic space, which depends only on the isomorphism class of A, up to the action of Γ_{∞} on ρ by conjugation. The pair (V, ρ) thus gives a well-defined point on H_{∞}/Γ_{∞} associated to A.

It is a worthwhile exercise for the reader to check that this complex analytic description of the moduli of abelian varieties with quaternionic multiplications corresponds to the usual description of the moduli space of elliptic curves as H_{∞}/SL_{2}(ℤ), in the case where the quaternion algebra B is M_{2}(ℚ).

III p-adic analytic description of X The fundamental theorem of Cerednik and Drinfeld states that X is an admissible curve over ℚ_{p} and gives an explicit description of the discrete subgroup attached to X by theorem 3.1. More precisely, let B, R, and Γ ⊂ R(N^{+})[\frac{1}{p}]^{×} be as in section 1. (So that B is the definite quaternion algebra obtained from B by the Cerednik “interchange of invariants” at p.) Then we have:

Theorem 4.7 (Cerednik-Drinfeld) The set of K_{p}-rational points of the Shimura curve X is isomorphic to the quotient of the p-adic upper half plane H_{p} attached to B by the natural action of Γ, i.e.,

X(K_{p}) = H_{p}/Γ.

Under this identification, the involution ψ ↦ ψ̄ of H_{p} corresponds to the involution τw_{p} of X(K_{p}), where τ is the complex conjugation in Gal(K_{p}/ℚ_{p}), and w_{p} is the Atkin-Lehner involution of X at p.

Proof. See [C], [Dr] and [BC].

In particular, an abelian surface A over K_{p} with quaternionic multiplications by R and level N^{+} structure determines a point ψ ∈ H_{p} = Hom(K_{p}, B_{p}) which is well-defined modulo the natural action of Γ. We will now give a precise description of the assignment A ↦ ψ. Crucial to this description is Drinfeld’s theorem that the p-adic upper half plane H_{p} parametrizes isomorphism classes of certain formal groups with a quaternionic action, and a suitable “rigidification”.

The p-adic upper half plane as a moduli space. We review Drinfeld’s moduli interpretation of the (K_{p}-rational points of the) p-adic upper half plane H_{p}. Roughly speaking, H_{p} classifies formal groups of dimension 2 and height 4 over O_{p}, equipped with an action of our fixed local order R_{p} and with a “rigidification” of their reduction modulo p.

In order to make this precise, we begin with a few definitions. Let as usual k be O_{p}/(p)(≃ 𝔽_{p^{2}}).

Definition 4.8 A 2-dimensional commutative formal group V over O_{p} is a formal R_{p}-module (for brevity, a FR-module) if it has height 4 and there is an embedding

i : R_{p} → End(V).

The F R-modules play the role of the quaternionic spaces of the previous section. Let V̄ be the formal group over k deduced from V by extension of scalars from O_{p} to k. It is equipped with the natural action of R_{p} given by reduction of endomorphisms. Let End^{0}(V̄) := End(V̄) ⊗ ℚ_{p} be the algebra of quasi-endomorphisms of V̄, and let End_{B_{p}}^{0}(V̄) be the subalgebra of quasi-endomorphisms which commute with the action of B_{p}.

Lemma 4.9

1. The algebra End^{0}(V̄) is isomorphic (non-canonically) to M_{2}(B_{p}).

2. The algebra End_{B_{p}}^{0}(V̄) is isomorphic (non-canonically) to the matrix algebra B_{p} over ℚ_{p}. Proof. The formal group V̄ is isogenous to the formal group of a product of two supersingular elliptic curves in characteristic p. Part 1 follows. Part 2 can then be seen by noting that the natural map

B_{p} ⊗ End_{B_{p}}^{0}(V̄) ⟶ End^{0}(V̄) ≃ M_{2}(B_{p})

is an isomorphism, so that End_{B_{p}}^{0}(V̄) is abstractly isomorphic to the matrix algebra B_{p}.

Denote by B_{p,u}^{×} the subgroup of elements of B_{p}^{×} whose reduced norm is a p-adic unit.

Definition 4.10

1. A rigidification of the F R-module V is an isomorphism

ρ : B_{p} ⟶ End_{B_{p}}^{0}(V̄),

subject to the condition of being “positively oriented at p”, i.e., that the two maximal orders R_{p} and ρ^{−1}(End_{R_{p}}(V̄)) of B_{p} are conjugated by an element of B_{p,u}^{×}.

2. A pair (V, ρ) consisting of an F R-module V and a rigidification ρ is called a rigidified F R-module.

3. Two rigidified modules (V, ρ) and (V′, ρ′) are said to be isomorphic if there is an isomorphism ϕ : V → V′ of formal groups over O_{p}, such that the induced isomorphism

ϕ^{∗} : End_{B_{p}}^{0}(V̄) → End_{B_{p}}^{0}(V̄′)

satisfies the relation ϕ^{∗} ∘ ρ = ρ′.

Remark. In [Dr] and [BC], a rigidification of a F R-module V is defined to be a quasi-isogeny of height zero from a fixed F R-module Φ̄ to the reduction V̄ modulo p of V. This definition is equivalent to the one we have given, once one has fixed an isomorphism between B_{p} and End_{B_{p}}^{0}(Φ̄). The definition given above is in a sense “base-point free”.

Recall that B_{p,u}^{×} acts (on the left) on H_{p} via the natural action of B_{p}^{×} on H_{p} by conjugation. Note that B_{p,u}^{×} acts on the left on (the isomorphism classes of) rigidified F R-modules, by

b(V, ρ) := (V, ρ^{b}) for b in B_{p,u}^{×},

where ρ^{b}(x) is equal to ρ(b^{−1}xb) for x in B_{p}.

Theorem 4.11 (Drinfeld)

1. The p-adic upper half plane H_{p} is a moduli space for the isomorphism classes of rigidified F R-modules over O_{p}. In particular, there is a bijective map

Ψ : {(V, ρ) : (V, ρ) a rigidified F R−module} /(isomorphisms) ⟶^{∼} Hom(K_{p}, B_{p}).

2. The map Ψ is B_{p,u}^{×}-equivariant.

Proof. See [Dr] and [BC], chapters I and II. For part 2, see in particular [BC], ch. II, sec. 9.

Corollary 4.12 All F R-modules have formal multiplication by O_{p}.

Proof. If V is a F R-module, equip V with a rigidification ρ. By theorem 4.11, the pair (V, ρ) determines a point P_{(V,ρ)} of the p-adic upper half plane H_{p}. Note that the stabilizer of P_{(V,ρ)} for the action of B_{p,u}^{×} is isomorphic to O_{p}^{×}. The claim now follows from part 2 of theorem 4.11.

Remark. As we will explain in the next paragraph, if V is an F R-module, there exists an abelian surface A over O_{p} with quaternionic multiplication by R, whose formal group Â (with the induced action of R_{p}) is isomorphic to V. Of course, quite often one has End_{R}(A) ≃ ℤ, even though End_{R}(V) contains O_{p} by corollary 4.12. In fact, combining Drinfeld’s theory with the theory of complex multiplication shows the existence of an uncountable number of such abelian surfaces such that (i) End_{R}(A) = ℤ; (ii) End_{R_{p}}(Â) ≃ O_{p}. (A similar phenomenon for elliptic curves has been observed by Lubin and Tate [LT].)

We give a description of the bijection Ψ, which follows directly from Drinfeld’s theorem. By lemma 4.12, identify End_{R_{p}}(V) with O_{p}. Let ψ : K_{p} ⟶ B_{p} be the map induced by the composition

O_{p} ⟶ End_{B_{p}}^{0}(V̄) ⟶ B_{p},

where the first map is given by the reduction modulo p of endomorphisms, and the second map is just ρ^{−1}. Then Ψ(V, ρ) = ψ.

We now use Drinfeld’s theorem to describe the p-adic uniformization of the K_{p}- rational points of the Shimura curve X, i.e., the isomorphism

X(K_{p}) = H_{p}/Γ.

The curve X has a model X over ℤ_{p}. Given a point in X(K_{p}), we may extend it to a point in X (O_{p}). In other words, given a pair (A, i, C), where A is an abelian surface over K_{p} with quaternionic action by i, and C is a level N^{+}-structure, we may extend it to a similar pair (\underline{A}, \underline{i}, \underline{C}) of objects over O_{p}. We write (Ā, ī, C̄) for the reduction modulo p of (\underline{A}, \underline{i}, \underline{C}). A p-quasi endomorphism of Ā is an element in End(Ā)⊗ℤ[\frac{1}{p}]. The algebra of all p-quasi endomorphisms is denoted by End^{(p)}(Ā).

Likewise, we denote by End_{R}^{(p)}(Ā) the algebra of p-quasi-endomorphisms which commute with the action of R. Let B_{p∞} be the quaternion algebra over ℚ ramified at p and ∞, and let R_{p∞} be a maximal order of B_{p∞}.

Lemma 4.13

1. The algebra End^{(p)}(Ā) is isomorphic to M_{2}(R_{p∞}[\frac{1}{p}]).

2. The algebra End_{R}^{(p)}(Ā) is isomorphic to R[\frac{1}{p}].

3. The subalgebra of endomorphisms preserving the level N^{+}-structure C̄ on Ā is isomorphic to the Eichler order R(N^{+})[\frac{1}{p}].

Proof.

1. The abelian variety Ā is p-isogenous to a product of a supersingular elliptic curve in characteristic p with itself. Part 1 follows. To see part 2, observe that the natural map

R[\frac{1}{p}] ⊗ End_{R}^{(p)}(Ā) ⟶ End^{(p)}(Ā) ≃ M_{2}(R_{p∞}[\frac{1}{p}])

is an isomorphism, and hence End_{R}^{(p)}(Ā)⊗ℚ is abstractly isomorphic to the quaternion algebra B. Furthermore, the natural map

End_{R}^{(p)}(Ā) ⟶ End^{(p)}(Ā)

has torsion-free cokernel, and hence End_{R}^{(p)}(Ā) is a maximal ℤ[\frac{1}{p}]-order in B. Like-

wise, one sees that the subalgebra of End_{R}^{(p)}(Ā) preserving the level N^{+} structure C̄ is abstractly isomorphic to the Eichler order R(N^{+})[\frac{1}{p}].

Fix an isomorphism

ρ_{0} : R[\frac{1}{p}] ⟶ End_{R}^{(p)}(Ā),

having the following properties.

1. For each ℓ^{n}||N^{+}, we require that ρ_{0}(R(N^{+})) ⊗ ℤ_{ℓ} preserves the subgroup C_{ℓ}, so that it operates on it via a homomorphism R(N^{+}) ⟶ ℤ/ℓ^{n}ℤ. We impose, in addition, that this homomorphism be equal to the orientation 𝔬_{ℓ}^{+}.

2. For all ℓ|N^{−}, the algebra R_{ℓ} acts on Ā_{ℓ} via ρ_{0}, and stabilizes the subspace corresponding to A_{I} (where I is the maximal ideal of R_{ℓ}.) By remark 2 in part I of this section, A_{I} is equipped with a canonical 𝔽_{ℓ^{2}}-vector space structure, and ρ_{0}(R_{ℓ}) acts 𝔽_{ℓ^{2}}-linearly on it. We require that the resulting homomorphism R_{ℓ} ⟶ 𝔽_{ℓ^{2}} be equal to the orientation 𝔬_{ℓ}^{−}.

3. Let V̄ be the formal group of Ā, and let ρ : B_{p} ⟶ End_{B_{p}}^{0}(V̄) be the map induced by ρ_{0} by extension of scalars from ℤ[\frac{1}{p}] to ℚ_{p}. We require that ρ^{−1}(End_{R_{p}}(V̄)) be conjugate to R_{p} by an element of B_{p,u}^{×}.

With these conventions, the homomorphism ρ_{0} is well-defined, up to conjugation by elements in Γ. The pair (V, ρ) is a rigidified F R-module, which is completely determined by the isomorphism class of A, up to the action of Γ on ρ by conjugation. Thus, (V, ρ) gives a well-defined point on H_{p}/Γ associated to A.

IV Shimura curve parametrizations We denote by π_{CD} the Cerednik-Drinfeld p-adic analytic uniformization

π_{CD} : H_{p} ⟶ X(K_{p}),

which induces a map Div^{0}(H_{p}) ⟶ Div^{0}(X(K_{p})), also denoted π_{CD} by abuse of notation. The Jacobian J of X_{/K_{p}} is therefore uniformized by a p-adic torus, and by proposition 3.4 and corollary 3.5, we have:

Corollary 4.14 The map π_{CD} induces a p-adic uniformization

Φ_{CD} : Hom(Γ̄, K_{p}^{×}) ⟶ J(K_{p}),

such that the following diagram commutes:

Div^{0}(H_{p}) ⟶^{Φ_{AJ}} Hom(Γ̄, K_{p}^{×})
π_{CD} ↓  ↓ Φ_{CD}
Div^{0}(X(K_{p})) ⟶^{Φ_{AJ}}  J(K_{p}).

Combining this corollary with the canonical inclusion of M into Hom(Γ, ℤ) given by proposition 1.3, yields a natural p-adic uniformization

Φ_{CD} : M ⊗ K_{p}^{×} ⟶ J(K_{p}),

which will also be denoted Φ_{CD} by abuse of notation.

The Shimura curve X is equipped with natural Hecke correspondences (cf. [BD1], sec. 1.5), and the Hecke algebra acting on J is isomorphic to the Hecke algebra 𝕋 acting on M, in such a way that the actions of 𝕋 on M and on J are compatible with the inclusion of M into the dual of the character group of J over k. (See [BC], ch. III, sec. 5.)

Recall the endomorphism η_{f} ∈ 𝕋 attached to f which was used to define the map M ⟶ ℤ. This endomorphism also acts on Pic(X), and induces a (generically) surjective map

η_{f} : Pic(X) → Ẽ,

where Ẽ is a subabelian variety of J isogenous to E. From now on we will assume that E = Ẽ.

Proposition 4.15 The p-adic uniformizations Φ_{Tate} and Φ_{CD} of Tate and Cerednik-Drinfeld are related by the following diagram which commutes up to sign.

M ⊗ K_{p}^{×} ⟶^{Φ_{CD}} J(K_{p})
η_{f} ⊗ id ↓  η_{f} ↓
K_{p}^{×}  ⟶^{Φ_{Tate}} E(K_{p})

(Note that both of the maps η_{f} that appear in this diagram are only well-defined up to sign.)

5 Heegner points

I Moduli description We give first a moduli definition of Heegner points. Let c be as before an integer prime to N, and let O be the order of K of conductor c.

Given an abelian surface A with quaternionic multplication and level N^{+} structure, we write \underline{End}(A) to denote the algebra of endomorphisms of A (over an algebraic closure of ℚ) which commute with the quaternionic multiplications and respect the level N^{+} structure.

Definition 5.1 A Heegner point of conductor c on X (attached to K) is a point on X corresponding to an abelian surface A with quaternionic multiplication and level N^{+} structure, such that

\underline{End}(A) ≃ O.

It follows from the theory of complex multiplication that the Heegner points on X of conductor c are all defined over the ring class field of K of conductor c. (Cf. [ST].)

II Complex analytic description For the convenience of the reader we recall now how to define Heegner points using the complex analytic uniformization. (This material will not be used in our proofs, but is quite parallel to the p-adic theory, which we do use extensively.)

Given an embedding ψ of K into B, let ψ denote also, by abuse of notation, its natural image by extension of scalars in H_{∞} = Hom(ℂ, B_{∞}). An embedding ψ : K ⟶ B is said to be an optimal embedding of conductor c (relative to the Eichler order R(N^{+}) if it maps O to R(N^{+}) and does not extend to an embedding of any larger order into R(N^{+}).

Let P ∈ X(H) be a Heegner point of conductor c, corresponding to a quaternionic surface A over H. By choosing a complex embedding H ⟶ ℂ, the point P gives rise to a point P_{ℂ} in X(ℂ), which corresponds to the abelian surface A_{ℂ} obtained from A by extension of scalars from H to ℂ, via our chosen complex embedding. Let P̃ be a lift of P_{ℂ} to H_{∞} by the complex analytic uniformization of proposition 4.1.

Theorem 5.2 The Heegner point P̃ ∈ H_{∞} corresponds to an optimal embedding ψ : K ⟶ B of conductor c. Proof. Let V = Lie(A), and let the isomorphism ρ_{0} : R ⟶ End_{R}(Λ) be chosen as in the discussion following lemma 4.6. The action of ℂ by multiplication on V arises by extension of scalars from the action of the order O of conductor c on A, and hence the point ψ necessarily comes (by extension of scalars) from a global embedding of O to R(N^{+}) which is optimal.

III p-adic analytic description Let H be the ring class field of conductor c, and let P = (A, i, C) be a Heegner point of conductor c. By fixing an embedding H → K_{p}, we may view P as a point of X(K_{p}). We want to describe the Heegner points of conductor c as elements of the quotient H_{p}/Γ. Recall the Gross points of conductor c represented by the oriented optimal embeddings

ψ_{i} : O → R[\frac{1}{p}], i = 1, …, h

fixed in section 2. By lemma 2.5, the group Δ acts simply transitively on these points. The embeddings ψ_{i} determine local embeddings (which we denote in the same way by an abuse of notation)

ψ_{i} : K_{p} → B_{p}.

Theorem 5.3 The classes modulo Γ of the local embeddings ψ_{i} correspond via the Cerednik-Drinfeld uniformization to distinct Heegner points on X of conductor c, in such a way that the natural Galois action of Δ on these Heegner points is compatible with the action of Δ on the Gross points represented by the ψ_{i}.

Proof. If P ∈ X(K_{p}) is a Heegner point of conductor c, let P̄ ∈ X(k) denote the reduction modulo p of P. By our description of the p-adic uniformization, the point P corresponds to the class modulo Γ of a local embedding ψ : K_{p} → B_{p} defined in the following way. Let

ψ_{0} : O = End(P) → End(P̄)

be the map obtained by reduction modulo p of endomorphisms. Identify End(P̄)[\frac{1}{p}] with R[\frac{1}{p}] by using the conventions of section 4, so that ψ_{0} gives rise to a map from O to R[\frac{1}{p}]. Then ψ is obtained from ψ_{0} by extension of scalars from ℤ to ℚ_{p}.

By proposition 7.3 of [GZ], ψ_{0} is an optimal embedding. Moreover, ψ_{0} is Γconjugate to one of the ψ_{i}. Finally, the proof of the compatibility under the action of the group Δ is similar to that of proposition 4.2 of [BD2].

6 Computing the p-adic Abel-Jacobi map

Let ψ ∈ H_{p} = Hom(K_{p}, B_{p}) be a point on the p-adic upper half plane, and let ψ̄ be its conjugate, defined by

ψ̄(z) = ψ(z̄).

The divisor (ψ) − (ψ̄) is a divisor of degree 0 on H_{p}.

Recall the canonical element L′_{p}(ψ) ∈ M ⊗ K_{p,1}^{×} associated to ψ in section 2, using the action of K_{p}^{×} induced by ψ on the Bruhat Tits tree T . When needed, we will identify L′_{p}(ψ) with its natural image in Hom(Γ, K_{p}^{×}), by an abuse of notation.

Recall also the p-adic Abel Jacobi map

Φ_{AJ} : Div^{0}(H_{p}) ⟶ Hom(Γ, K_{p}^{×})

defined in sections 3 and 4 by considering automorphy factors of p-adic thetafunctions.

The main result of this section is:

Theorem 6.1

Φ_{AJ}((ψ) − (ψ̄)) = L′_{p}(ψ).

The rest of this section is devoted to the proof of theorem 6.1. We begin by giving explicit descriptions, and elucidating certain extra structures, which the fixing of the point ψ ∈ H_{p} gives rise to.

The algebra B_{p} We give an explicit description of the algebra B_{p}, which depends on the embedding ψ. Identify K_{p} with its image in B_{p} by ψ, and choose an element u ∈ B_{p} so that B_{p} = K_{p} ⊕ K_{p}u and u anticommutes with the elements of K_{p}, i.e., uz = z̄u for all z ∈ K_{p}. Note that u^{2} belongs to ℚ_{p}, and is a norm from K_{p} to ℚ_{p}, since the quaternion algebra B is split at p. Moreover, the element u^{2} is well-defined up to multiplication by norms from K_{p} to ℚ_{p}. We may and will fix u so that u^{2} = 1. From now on, write elements of B_{p} as a + bu, with a and b in K_{p}. The conjugate of a + bu under the canonical anti-involution of B_{p} is ā − bu. The reduced trace and norm are given by the formulae

T r(a + bu) = T r_{K/ℚ}(a), N(a + bu) = N_{K/ℚ}(a) − N_{K/ℚ}(b).

The embedding ψ allows us to view B_{p} as a two-dimensional vector space over K_{p}, on which B_{p} acts by multiplication on the right. This yields a local embeding B_{p} ⟶ M_{2}(K_{p}), defined by:

a + bu ↦ ( \begin{matrix}a & b \\ b̄ & ā \end{matrix} ) .

This embedding allows us to define an action of B_{p}^{×} on the projective line ℙ^{1}(K_{p}) (or ℙ^{1}(ℂ_{p})) by fractional linear transformations, by setting

γ(z) := \frac{az + b}{b̄z + ā}, if γ = a + bu ∈ B_{p}^{×}, z ∈ ℙ^{1}(K_{p}).

This induces an action of the group Γ on ℙ^{1}(K_{p}).

The domain Ω_{p} Let

S^{1} = {z ∈ K_{p} | zz̄ = 1}

be the p-adic “circle” of radius 1, and let Ω_{p} = ℙ^{1}(K_{p}) − S^{1}.

Lemma 6.2 The limit set of Γ acting on ℙ^{1}(K_{p}) is equal to S^{1}. In particular, the group Γ acts discontinuously on ℙ^{1}(K_{p}).

Proof. To compute the limit set of Γ, observe that if γ_{n} is a sequence of distinct elements of Γ, then one can write

γ_{n} = \frac{a_{n} + b_{n}u}{p^{e_{n}}},

with a_{n}, b_{n} ∈ O_{p}^{×}, and lim_{n⟶∞} e_{n} = ∞. Hence

N_{K_{p}/ℚ_{p}}(\frac{a_{n}}{b̄_{n}}) ≡ 1 (mod p^{2e_{n}}),

so that the limit lim γ_{n}z_{0}, if it exists, must belong to S^{1}. Conversely, let z be an element of S^{1}, and let b_{n} be a sequence of elements in B_{p}^{×} satisfying

lim_{n⟶∞}(b_{n}^{−1}∞) = z.

By the finiteness of the double coset space R_{p}^{×}\B_{p}^{×}/Γ, which follows from strong approximation, there is an element b ∈ B_{p}^{×} such that, for infinitely many n

b_{n} = r_{n}bγ_{n},

where r_{n} belongs to R_{p}^{×} and γ_{n} belongs to Γ. Assume without loss of generality (by extracting an appropriate subsequence) that this equation holds for all n. Then we have

z = lim(γ_{n}^{−1}b^{−1}r_{n}^{−1}∞).

But the sequence b^{−1}r_{n}^{−1}∞ is contained in a compact set, and hence has a convergent subsequence b^{−1}r_{k_{n}}^{−1}∞ which tends to some z_{0} ∈ ℙ^{1}(K_{p}). Hence z = lim γ_{k_{n}}^{−1}z_{0} is a limit point for Γ. Lemma 6.2 follows.

Using the embedding ψ, the “abstract” upper half plane H_{p} now becomes identified with the domain Ω_{p}.

The tree T Let v_{0} = r(ψ) be the vertex on T which is fixed by ψ(K_{p}). This vertex corresponds to the maximal order

R_{p} = O_{p} ⊕ O_{p}u,

where O_{p} is the ring of integers of K_{p}. The vertices of T are in bijection with the coset space R_{p}^{×}ℚ_{p}^{×}\B_{p}^{×}, by assigning to b ∈ B_{p}^{×} the vertex b^{−1} ∗ v_{0}.

We say that a vertex v of T has level n, and write ℓ(v) = n, if its distance from v_{0} is equal to n. A vertex is of level n if and only if it can be represented by an element of the form a + bu, where a and b belong to O_{p} and at least one of a or b is in O_{p}^{×}, and n = ord_{p}(N(a + bu)) = ord_{p}(N_{K_{p}/ℚ_{p}}(a) − N_{K_{p}/ℚ_{p}}(b)).

Likewise, we say that an edge e of T has level n, and we write ℓ(e) = n, if the distance of its furthest vertex from v_{0} is equal to n.

The reduction map We use our identification of H_{p} with Ω_{p} to obtain a reduction map

r : Ω_{p} ⟶ T

from Ω_{p} to the tree of B_{p}.

Lemma 6.3 The divisor (ψ) − (ψ̄) on H_{p} corresponds to the divisor (0) − (∞) on Ω_{p} under our identification of Ω_{p} with H_{p}.

Proof. The group ψ(K_{p}^{×}) acting on Ω_{p} by Möbius transformations fixes the points 0 and ∞, and acts on the tangent line at 0 by the character z ↦ \frac{z}{z̄}.

In general, if z is a point of Ω_{p} and b ∈ B_{p}^{×} is such that b^{−1}0 = z, then r(z) is equal to b. This implies directly part 1 and 2 of the next lemma.

Lemma 6.4

1. We have r(∞) = r(0) = v_{0}. More generally, if z ∈ Ω_{p} ⊂ K_{p} ∪ ∞ does not belong to O_{p}^{×}, then r(z) = v_{0}.

2. If z belongs to O_{p}^{×}, then the level of the vertex r(z) is equal to ord_{p}(zz̄ − 1).

3. If z_{1} and z_{2} ∈ Ω_{p} map under the reduction map to adjacent vertices on T of level n and n + 1, then

z_{1} ≡ z_{2} (mod p^{n}).

Proof. We prove part 3. Choose representatives b_{1} and b_{2} in B_{p}^{×} for r(z_{1}) and r(z_{2}), with the properties

b_{i} = x_{i} + y_{i}u, with x_{i}, y_{i} ∈ O_{p} and gcd(x_{i}, y_{i}) ∈ O_{p}^{×}.

Since the vertices corresponding to b_{1} and b_{2} are adjacent, it follows that b_{2}b_{1}^{−1} = b_{2}b̄_{1}/p^{n} has norm p and level 1. Since

b_{2}b̄_{1} = (x_{2} + y_{2}u)(x̄_{1} − y_{1}u) = (x_{2}x̄_{1} − y_{2}ȳ_{1}) − (y_{1}x_{2} − x_{1}y_{2})u,

it follows that

\frac{x_{1}}{y_{1}} ≡ \frac{x_{2}}{y_{2}} (mod p^{n}),

so that b_{1}^{−1}0 ≡ b_{2}^{−1}0 (mod p^{n}). This proves the lemma.

Let ϕ_{(0)−(∞)} ∈ Hom(Γ, K_{p}^{×}) be the automorphy factor of the p-adic theta-function associated to the divisor (0) − (∞) as in section 3. By the results of section 3, we have

Φ_{AJ}((ψ) − (ψ̄)) = ϕ_{(0)−(∞)}.

By definition, for δ ∈ Γ one has

ϕ_{(0)−(∞)}(δ) = ∏_{γ∈Γ} \frac{γδ(z_{0})}{γ(z_{0})} ,

where z_{0} is any element in the domain Ω_{p}. Suppose that r(z_{0}) = v_{0}. Let

path(v_{0}, δv_{0}) = e_{1} − e_{2} + ⋯ + e_{s−1} − e_{s}.

(Note that s is even, since δ belongs to Γ.) Write e_{j} = {v_{j}^{e}, v_{j}^{o}}, where v_{j}^{e} is the even vertex of e_{j}, and v_{j}^{o} is the odd vertex of e_{j}. Note that we have

v_{j}^{o} = v_{j+1}^{o} for j = 1, 3, …, s − 1,
v_{j}^{e} = v_{j+1}^{e} for j = 2, 4, …, s − 2,
Γv_{s}^{e} = Γv_{1}^{e}.

Thus we may choose elements z_{j}^{o} and z_{j}^{e} in Ω_{p}(K_{p}) such that r(z_{j}^{o}) = v_{j}^{o}, r(z_{j}^{e}) = v_{j}^{e}, and

z_{j}^{o} = z_{j+1}^{o} for j = 1, 3, …, s − 1,
z_{j}^{e} = z_{j+1}^{e} for j = 2, 4, …, s − 2,
z_{1}^{e} = z_{0}, z_{s}^{e} = δz_{0}.

Hence

(γz_{1}^{o})(γz_{2}^{o})^{−1} ⋯ (γz_{s−1}^{o})(γz_{s}^{o})^{−1} = 1, (γz_{2}^{e})(γz_{3}^{e})^{−1} ⋯ (γz_{s−2}^{e})(γz_{s−1}^{e})^{−1} = 1,

so that

ϕ_{(0)−(∞)}(δ) = ∏_{γ∈Γ} (\frac{γz_{1}^{o}}{γz_{1}^{e}} ) (\frac{γz_{2}^{o}}{γz_{2}^{e}} )^{−1} ⋯ (\frac{γz_{s−1}^{o}}{γz_{s−1}^{e}} ) (\frac{γz_{s}^{o}}{γz_{s}^{e}} )^{−1}

= ∏_{γ∈Γ} (\frac{γz_{1}^{o}}{γz_{1}^{e}} ) ∏_{γ∈Γ} (\frac{γz_{2}^{o}}{γz_{2}^{e}} )^{−1} ⋯ ∏_{γ∈Γ} (\frac{γz_{s−1}^{o}}{γz_{s−1}^{e}} ) ∏_{γ∈Γ} (\frac{γz_{s}^{o}}{γz_{s}^{e}} )^{−1} ,

where the last equality follows from part 3 of lemma 6.4. Fix a large odd integer n. For each 1 ≤ j ≤ s, let Γ(j) be the set of elements γ in Γ such that the set γe_{j} has level ≤ n. By lemma 6.4, we have

(†) ϕ_{(0)−(∞)}(δ) ≡ ∏_{γ∈Γ(1)} (\frac{γz_{1}^{o}}{γz_{1}^{e}} ) ∏_{γ∈Γ(2)} (\frac{γz_{2}^{o}}{γz_{2}^{e}} )^{−1} ⋯ ∏_{γ∈Γ(s)} (\frac{γz_{s}^{o}}{γz_{s}^{e}} )^{−1} (mod p^{n}).

Each of the factors in the right hand side of equation (†) can be broken up into three contributions:

∏_{Γ(j)} \frac{γz_{j}^{o}}{γz_{j}^{e}} = ∏_{ℓ(γv_{j}^{o})<n} γz_{j}^{o} · ∏_{ℓ(γv_{j}^{e})<n} γ(z_{j}^{e})^{−1} · ∏_{ℓ(γe_{j})=n} γz_{j}^{o}.

The first two factors in this last expression cancel out in the formula (†) for ϕ_{(0)−(∞)}(δ). Hence we obtain

ϕ_{(0)−(∞)}(δ) ≡ ∏_{ℓ(γe_{1})=n} γz_{1}^{o} · ∏_{ℓ(γe_{2})=n} γ(z_{2}^{o})^{−1} ⋯ ∏_{ℓ(γe_{s})=n} γ(z_{s}^{o})^{−1} (mod p^{n}).

Now, fix an edge e of level n, having v as its vertex of level n, and choose any z ∈ Ω_{p} with r(z) = v. If σ is a variable running over G_{n} (which we view as belonging to (O_{p}/p^{n}O_{p})^{×}/(ℤ/p^{n}ℤ)^{×}), write σe ≡ e_{j} if the edge σe is Γ-equivalent to e_{j}. Using the fact that G_{n} acts transitively on the set of edges of level n, we have

ϕ_{(0)−(∞)}(δ) ≡ ∏_{σe≡e_{1}}(\frac{σ}{σ̄} z)^{w_{e_{1}}} · ∏_{σe≡e_{2}}(\frac{σ}{σ̄} z)^{−w_{e_{2}}} ⋯ ∏_{σe≡e_{s}}(\frac{σ}{σ̄} z)^{−w_{es}} (mod p^{n})

= ∏_{σe≡e_{1}}(\frac{σ}{σ̄})^{w_{e_{1}}} · ∏_{σe≡e_{2}}(\frac{σ}{σ̄})^{−w_{e_{2}}} ⋯ ∏_{σe≡e_{s}}(\frac{σ}{σ̄})^{−w_{es}} · (z^{M}),

where M = ⟨path(v_{0}, δv_{0}), ∑_{σ∈G_{n}} σe⟩. Since the element ∑_{σ∈G_{n}} σe belongs to the image of ∂^{∗}, and path(v_{0}, δv_{0}) is in the kernel of ∂_{∗}, it follows that M = 0 so that:

ϕ_{(0)−(∞)}(δ) ≡ ∏_{σe≡e_{1}}(\frac{σ}{σ̄})^{w_{e_{1}}} · ∏_{σe≡e_{2}}(\frac{σ}{σ̄})^{−w_{e_{2}}} ⋯ ∏_{σe≡e_{s}}(\frac{σ}{σ̄})^{−w_{es}} .

The reader will notice that this last expression is equal to

⟨path(v_{0}, δv_{0}), L′_{p,n}(ψ)⟩.

Hence

L′_{p}(ψ) = ϕ_{(0)−(∞)} = Φ_{AJ}((ψ) − (ψ̄)),

and theorem 6.1 follows.

7 Proof of the main results

We now combine the results of the previous sections to give a proof of our main results. First, we introduce some notations. Having fixed an embedding H → K_{p}, let P_{1}, …, P_{h} in X(K_{p}) be the h distinct Heegner points of conductor c, corresponding via theorem 5.3 to our fixed optimal embeddings ψ_{1}, …, ψ_{h}. Let σ_{1}, …, σ_{h} ∈ Δ be the elements of Δ, labeled in such a way that σ_{i}(P_{1}) = P_{i}. By theorem 5.3, the Gross point corresponding to ψ_{1} is sent by σ_{i} to the Gross point corresponding to ψ_{i}. Write P_{K} ∈ Pic(X(K_{p})) for the class of the divisor P_{1} + … + P_{h}. Note that P_{K} depends on the choice of the embedding of H into K_{p}, only up to conjugation in Gal(K_{p}/ℚ_{p}). We denote by P̄_{i} the complex conjugate of P_{i}, and likewise for P̄_{K}. (No confusion should arise with the use of the notation P̄ in section 5 to indicate the reduction modulo p of the point P.) Let w_{p} stand for the Atkin-Lehner involution at p.

Theorem 7.1

1. Φ_{CD}(L′_{p}(M/K)) = Φ_{AJ}((P_{K}) − (w_{p}P̄_{K})).

2. Φ_{CD}(L′_{p}(M/H)) = ∑_{i=1}^{h} Φ_{AJ}((P_{i}) − (w_{p}P̄_{i})) · σ_{i}^{−1}.

Proof. By the formula at the end of section 2,

L′_{p}(M/K) = ∑_{i=1}^{h} L′_{p}(ψ_{i}),

where ψ_{1}, …, ψ_{h} are as above. Hence,

Φ_{CD}(L′_{p}(M/K)) = ∑_{i=1}^{h} Φ_{CD}(L′_{p}(ψ_{i})) = ∑_{i=1}^{h} Φ_{CD}(Φ_{AJ}((ψ_{i}) − (ψ̄_{i}))),

where the last equality follows from theorem 6.1. By theorems 5.3 and 4.7, and by the commutative diagram of proposition 4.14, this last expression is equal to

∑_{i=1}^{h} Φ_{AJ}(π_{CD}((ψ_{i}) − (ψ̄_{i}))) = ∑_{i=1}^{h} Φ_{AJ}((P_{i}) − (w_{p}P̄_{i})) = Φ_{AJ}((P_{K}) − (w_{p}P̄_{K})).

Part 1 follows. Part 2 is proved in a similar way.

Recall our running assumption that E = Ẽ is the subabelian variety of the Jacobian J of the Shimura curve X, and that η_{f} maps J to Ẽ. Let α_{i} = η_{f}(P_{i}) ∈ E(K_{p}), and let α_{K} = α_{1} + ⋯ + α_{h} = trace_{H/K}(α_{1}). Theorem 7.1 gives the following corollary, whose first part is the statement of theorem B of the introduction.

Corollary 7.2 Let w = 1 (resp. w = −1) if E/ℚ_{p} has split (resp. non-split) multiplicative reduction. Then the following equalities hold up to sign:

Φ_{Tate}(L′_{p}(E/K)) = α_{K} − wᾱ_{K},

Φ_{Tate}(L′_{p}(E/H)) = ∑_{i=1}^{h}(α_{i} − wᾱ_{i}) · σ_{i}^{−1}.

Proof. Apply η_{f} to the equations of theorem 7.1, using the commutative diagram of proposition 4.15.

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