p-adic periods, p-adic L-functions and the p-adic uniformization of Shimura curves

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

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Definite quaternion algebras and graphs . . . . . . . . . . . . . . . . 4 3 The p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . . 8 4 The theory of p-adic uniformization of Shimura curves . . . . . . . . . 14 5 p-adic Shintani cycles and special values of complex L-functions . . . . . 17 6 p-adic Shintani cycles and derivatives of p-adic L-functions . . . . . . . 19 7 Proof of theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . 22 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1 Introduction

Let E/ℚ be a modular elliptic curve of conductor N, and let p be a prime of split multiplicative reduction for E. Write ℂ_{p} for a fixed completion of an algebraic closure of ℚ_{p}. Tate’s theory of p-adic uniformization of elliptic curves yields a rigid-analytic, Gal(ℂ_{p}/ℚ_{p})-equivariant uniformization of the ℂ_{p}-points of E

(1)  0 → q^{ℤ} → ℂ_{p}^{×} ⟶^{Φ_{Tate}} E(ℂ_{p}) → 0,

where q ∈ pℤ_{p} is the p-adic period of E.

Mazur, Tate and Teitelbaum conjectured in [MTT] that the cyclotomic p-adic L-function of E/ℚ vanishes at the central point to order one greater than that of its classical counterpart. Furthermore, they proposed a formula for the leading coefficient of such a p-adic L-function. In the special case where the analytic rank of E(ℚ) is zero, they predicted that the ratio of the special value of the first derivative of the cyclotomic p-adic L-function and the algebraic part of the special value of the complex L-function of E/ℚ is equal to the quantity

\frac{log_{p}(q)}{ord_{p}(q)}

1Partially supported by GNSAGA (C.N.R.); M.U.R.S.T., progetto nazionale “Geometria algebrica”; Human Capital and Mobility Programme of the European Community, under contract ERBCH RXCT940557.

2Partially supported by grants from FCAR, NSERC and by an Alfred P. Sloan research award.

(where log_{p} is Iwasawa’s cyclotomic logarithm), which is defined purely in terms of the p-adic uniformization of E. Greenberg and Stevens [GS] gave a proof of this special case. See also the work of Boichut [Boi] in the case of analytic rank one.

The article [BD1] formulates an analogue of the conjectures of [MTT] in which the cyclotomic ℤ_{p}-extension of ℚ is replaced by the anticyclotomic ℤ_{p}-extension of an imaginary quadratic field K. When p is split in K and the sign of the functional equation of L(E/K, s) is +1, this conjecture relates the first derivative of the anticyclotomic p-adic L-function of E to the anticyclotomic logarithm of the p-adic period of E. The present paper supplies a proof of this conjecture. Our proof is based on the theory of p-adic uniformization of Shimura curves. More precisely, assume that K is an imaginary quadratic field with (disc(K), N) = 1 such that:

(i) p is split in K;

(ii) E is semistable at the rational primes which divide N and are inert in K; (iii) the number of these rational primes is odd.

The complex L-function L(E/K, s) of E over K has a functional equation and an analytic continuation to the whole complex plane. Under our assumptions, the sign of the functional equation of L(E/K, s) is +1 (cf. [GZ], p. 71), and hence L(E/K, s) vanishes to even order at s = 1.

Fix a positive integer c prime to N, and let O be the order of K of conductor c. Let H_{n} be the ring class field of K of conductor cp^{n}, with n ≥ 0, and let H_{∞} be the union of the H_{n}. By class field theory, the Galois group Gal(H_{∞}/H_{0}) is identified with O^{×}\(O_{K} ⊗ ℤ_{p})^{×}/ℤ_{p}^{×} ≃ ℤ_{p} × ℤ/((p − 1)/u)ℤ, with u := \frac{1}{2}#O^{×}. Moreover, Gal(H_{0}/K) is identified with the Picard group Pic(O). Set

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

Thus, G_{∞} is isomorphic to the product of ℤ_{p} by a finite abelian group. Choose a prime 𝔭 of K above p. Identify K_{𝔭} with ℚ_{p}, and let

rec_{p} : ℚ_{p}^{×} → G_{∞}

be the reciprocity map of local class field theory. Define the integral completed group ring of G_{∞} to be

ℤ[[G_{∞}]] := lim_{n^{←}} ℤ[G_{n}],

where the inverse limit is taken with respect to the natural projections of group rings.

In section 3, we recall the construction explained in [BD1], section 2.7 of an element

L_{p}(E/K) ∈ ℤ[[G_{∞}]]

attached to (E, H_{∞}/K), which interpolates the special values L(E/K, χ, 1) of L(E/K, s) twisted by finite order characters of G_{∞}. The construction of this p-adic L-function is based on the ideas of Gross [Gr] and a generalization due to Daghigh [Dag]. We will show that L_{p}(E/K) belongs to the augmentation ideal I of ℤ[[G_{∞}]]. Let L′_{p}(E/K) be the natural image of L_{p}(E/K) in I/I^{2} = G_{∞}. The element L′_{p}(E/K) should be viewed as the first derivative of L_{p}(E/K) at the central point.

Let f = ∑_{n≥1} a_{n}q^{n} be the newform attached to E, and let

Ω_{f} := 4π^{2} ∫∫_{H/Γ_{0}(N)} |f(τ)|^{2}dτ ∧ idτ̄

be the Petersson inner product of f with itself. We assume that E is the strong Weil curve for the Shimura curve parametrization defined in section 4. Set d := disc(O), and let n_{f} be the positive integer defined later in this introduction, and specified further in section 2. Our main result (stated in a special case: see theorem 6.4 for the general statement) is the following.

Theorem 1.1 Suppose that c = 1. The equality (up to sign)

L′_{p}(E/K) = \frac{rec_{p}(q)}{ord_{p}(q)}\sqrt{L(E/K, 1)Ω_{f}^{−1} · d^{\frac{1}{2}} u^{2}n_{f}}

holds in I/I^{2} ⊗ ℚ.

For the convenience of the reader, we now briefly sketch the strategy of the proof of theorem 1.1.

Write the conductor N of E as pN^{+}N^{−}, where N^{+}, resp. N^{−} is divisible only by primes which are split, resp. inert in K. Under our assumptions, N^{−} has an odd number of prime factors, and pN^{−} is squarefree. Denote by B the definite quaternion algebra over ℚ of discriminant N^{−}, and fix an Eichler order R of B of level N^{+}p. Let Γ be the subgroup of elements of ℚ_{p}^{×}\R[\frac{1}{p}]^{×} whose norm has even p-adic valuation, and set N := Hom(Γ, ℤ). The module N is a free abelian group, and is equipped with the action of a Hecke algebra 𝕋 attached to modular forms of level N which are new at N^{−}p. In section 2, we will also define a canonical free quotient N_{sp} of N, which is stable for the action of 𝕋 and is such that the image of 𝕋 in End(N_{sp}) corresponds to modular forms which are split multiplicative at p. Let π_{f} be the idempotent of 𝕋 ⊗ ℚ associated with f, and let n_{f} be a positive integer such that η_{f} := n_{f}π_{f} belongs to 𝕋. Denote by N^{f} the submodule of N on which 𝕋 acts via the character

ϕ_{f} : 𝕋 → ℤ, T_{n} ↦ a_{n}

defined by f. By the multiplicity-one theorem, the module N^{f} is isomorphic to ℤ. The operator η_{f} yields a map (denoted in the same way by an abuse of notation) η_{f} : N → N^{f}, which factors through N_{sp}. We will define an element L_{p}(N_{sp}/K) ∈ N_{sp} ⊗ ℤ[[G_{∞}]], such that (up to sign)

(η_{f} ⊗ id)(L_{p}(N_{sp}/K)) = c_{p} · L_{p}(E/K),

where c_{p} := ord_{p}(q). Recall that the derivative L′_{p}(E/K) of L_{p}(E/K) belongs to N^{f} ⊗ G_{∞} = G_{∞}.

On the other hand, the module N is related to the theory of p-adic uniformization of Shimura curves. Let B be the indefinite quaternion algebra of discriminant pN^{−}, and let R be an Eichler order of B of level N^{+}. Write X for the Shimura curve over ℚ associated with R (see section 4), and J for the jacobian of X. A theorem of Cerednik ([Cer]), combined with the theory of jacobians of Mumford curves ([GVdP]), yields a rigid-analytic uniformization

(2)  0 → Λ → N ⊗ ℂ_{p}^{×} ⟶^{Φ} J(ℂ_{p}) → 0,

where Λ is the lattice of p-adic periods of J. The Tate uniformization (1) is obtained from the sequence (2) by applying the operator η_{f} to the Hecke modules N ⊗ ℂ_{p}^{×} and J(ℂ_{p}) of (2). In particular, the p-adic period q of E can be viewed as an element of the module N^{f} ⊗ ℂ_{p}^{×}, and in fact one checks it belongs to N^{f} ⊗ ℚ_{p}^{×} = ℚ_{p}^{×}. An explicit calculation of p-adic periods, combined with a formula for L(E/K, 1) given in [Gr] and [Dag], will prove theorem 1.1.

A similar strategy was used in [BD2], when p is inert in K and the sign of the functional equation of L(E/K, s) is −1, to obtain a p-adic analytic construction of a Heegner point in terms of the first derivative of an anticyclotomic p-adic L-function.

It is worth observing that an analogous strategy has not (yet) proven to work in the case of the cyclotomic ℤ_{p}-extension of ℚ. The difficulty is that of relating in a natural way the construction of the cyclotomic p-adic L-function, which is defined in terms of modular symbols, to the p-adic uniformization of Shimura curves. P. Schneider [Sch] has proposed the definition of a p-adic L-function based on the notion, which stems directly from the theory of p-adic uniformization, of rigid-analytic modular symbol. C. Klingenberg [Kl] has proven an exceptional zero formula similar to theorem 1.1 for this rigid-analytic p-adic L-function. However, the relation (if any) between Schneider’s p-adic L-function and the cyclotomic p-adic L-function considered in [MTT] is at present mysterious.

The reader is also referred to Teitelbaum’s paper [T], where the theory of p-adic uniformization of Shimura curves is used to formulate analogues of the conjectures of [MTT] for cyclotomic p-adic L-functions attached to modular forms of higher weight.

The proof by Greenberg and Stevens [GS] of the cyclotomic “exceptional zero” formula of [MTT] follows a completely different strategy from the one of this paper, and is based on Hida’s theory of p-adic families of modular forms.

Finally, let us mention that Kato, Kurihara and Tsuji [KKT] have recently announced more general results on the conjectures of [MTT], which make use of an Euler System constructed by Kato from modular units in towers of modular function fields.

2 Definite quaternion algebras and graphs

Keep the notations and assumptions of the introduction. In particular, recall that K is an imaginary quadratic field, and B is a definite quaternion algebra of discriminant N^{−}. Given a rational prime ℓ, and orders O of K and S of B, set

K_{ℓ} := K ⊗ ℤ_{ℓ}, B_{ℓ} := B ⊗ ℤ_{ℓ}, O_{ℓ} := O ⊗ ℤ_{ℓ}, S_{ℓ} := S ⊗ ℤ_{ℓ}.

Denote by ℤ̂ = ∏ ℤ_{ℓ} the profinite completion of ℤ. Set

K̂ := K ⊗ ℤ̂, B̂ := B ⊗ ℤ̂, Ô := O ⊗ ℤ̂ = ∏ O_{ℓ}, Ŝ := S ⊗ ℤ̂ = ∏ S_{ℓ}.

Fix an Eichler order R of B of level N^{+}p. Equip R with an orientation, i.e., a collection of algebra homomorphisms

𝔬_{ℓ}^{+} : R → ℤ/ℓ^{n}ℤ, ℓ^{n}∥N^{+}p,
𝔬_{ℓ}^{−} : R → 𝔽_{ℓ^{2}}, ℓ | N^{−}.

The group B̂^{×} acts transitively (on the right) on the set of Eichler orders of level N^{+}p by the rule

S ∗ b̂:= (b̂^{−1}Ŝb̂) ∩ B.

The orientation on R induces an orientation on R ∗ b̂, and the stabilizer of the oriented order R is equal to ℚ^{×}R̂^{×}. This sets up a bijection between the set of oriented Eichler orders of level N^{+}p and the coset space ℚ^{×}R̂^{×}\B̂^{×}. Likewise, there is a bijection between the set of oriented Eicher orders of level N^{+}p modulo conjugation by B^{×} and the double coset space

R̂^{×}\B̂^{×}/B^{×}.

Set Γ_{+} := ℚ_{p}^{×}\R[\frac{1}{p}]^{×} and, as in the introduction, let Γ be the image in Γ_{+} of the elements in R[\frac{1}{p}]^{×} whose reduced norm has even p-adic valuation.

Lemma 2.1 Γ has index 2 in Γ_{+}.

Proof. See [BD2], lemma 1.5.

Let T be the Bruhat-Tits tree associated with the local algebra B_{p}. The set of vertices V(T ) of T is equal to the set of maximal orders in B_{p}. The set E⃗(T ) of oriented edges of T is equal to the set of oriented Eichler orders of level p in B_{p}. Thus, E⃗(T ) can be identified with the coset space ℚ_{p}^{×}R_{p}^{×}\B_{p}^{×}, by mapping b_{p} ∈ B_{p}^{×} to R_{p} ∗ b_{p} = b_{p}^{−1}R_{p}b_{p}. Similarly, if \underline{R}_{p} is a maximal order in B_{p} containing R_{p}, we will identify V(T ) with the coset space ℚ_{p}^{×}\underline{R}_{p}^{×}\B_{p}^{×}. Define the graphs

G := T /Γ, G_{+} := T /Γ_{+}.

By strong approximation ([Vi], p. 61), there is an identification

E⃗(G_{+}) = R̂^{×}\B̂^{×}/B^{×}

of the set of oriented edges of G_{+} with the set of conjugacy classes of oriented Eichler orders of level N^{+}p.

Fixing a vertex v_{0} of T gives rise to an orientation of T in the following way. A vertex of T is called even, resp. odd if it has even, resp. odd distance from v_{0}. The direction of an edge is said to be positive if it goes from the even to the odd vertex. Since Γ sends even vertices to even ones, and odd vertices to odd ones, the orientation of T induces an orientation of G. Define a map

κ : E(G) → E⃗(G_{+})

from the set of edges of G to the set of oriented edges of G_{+}, by mapping an edge {v, v′} (mod Γ) of G, where v and v′ are vertices of T and we assume that v is even, to the oriented edge (v, v′) (mod Γ_{+}) of G_{+}.

Lemma 2.2 The map κ is a bijection.

Proof. Suppose that (v, v′) (mod Γ_{+}) = (u, u′) (mod Γ_{+}). Thus, there is γ ∈ Γ_{+} such that γv = u and γv′ = u′. If v and u are both even, γ must belong to Γ, and this proves the injectivity of κ. As for surjectivity, (v, v′) (mod Γ_{+}) is the image by κ of {v, v′} (mod Γ) if v is even, and of {wv, wv′} (mod Γ), where w is any element of Γ_{+} − Γ, if v is odd.

Given two vertices v and v′ of T , write path(v, v′) for the natural image in ℤ[E(G)] of the unique geodesic on T joining v with v′. For example, if v and v′ are even vertices joined by 4 consecutive edges e_{1}, e_{2}, e_{3}, e_{4}, by our convention for orienting the edges of T , path(v, v′) is the image in ℤ[E(G)] of e_{1} − e_{2} + e_{3} − e_{4}.

There is a coboundary map

∂^{∗} : ℤ[V(G)] → ℤ[E(G)],

which maps the image in V(G) of an odd, resp. even vertex v of T to the image in ℤ[E(G)] of the formal sum of the edges of T emanating from v, resp. the opposite of this sum. There is also a boundary map

∂_{∗} : ℤ[E(G)] → ℤ[V(G)],

which maps an edge e to the difference v′−v of its vertices, where v is the even vertex and v′ is the odd vertex of e. The integral homology, resp. the integral cohomology of the graph G is defined by H_{1}(G, ℤ) = ker(∂_{∗}), resp. H^{1}(G, ℤ) = coker(∂^{∗}).

Let

⟨ , ⟩ : ℤ[E(G)] × ℤ[E(G)] → ℤ

be the pairing on ℤ[E(G)] defined by the rule ⟨e_{i}, e_{j}⟩ := ω_{e_{i}}δ_{ij}, where the e_{i} are the elements of the standard basis of ℤ[E(G)] and ω_{e_{i}} is the order of the stabilizer in Γ of a lift of e_{i} to T . Likewise, let

⟨⟨ , ⟩⟩ : ℤ[V(G)] × ℤ[V(G)] → ℤ

be the pairing on ℤ[V(G)] defined by ⟨⟨v_{i}, v_{j}⟩⟩ := ω_{v_{i}}δ_{ij}, where the v_{i} are the elements of the standard basis of ℤ[V(G)] and ω_{v_{i}} is the order of the stabilizer in Γ of a lift of v_{i} to T .

We use the notation M to indicate the module H^{1}(G, ℤ). Let Γ̄ be the maximal torsion-free abelian quotient of Γ. As in the introduction, write N for Hom(Γ̄, ℤ). Given an element γ ∈ Γ, denote by γ̄ the natural image of γ in Γ̄.

Lemma 2.3

(i) The map from Γ̄ to H_{1}(G, ℤ) which sends γ̄ ∈ Γ̄ to the cycle path(v_{0}, γv_{0}), where v_{0} is any vertex of G and γ is any lift of γ̄ to Γ, is an isomorphism.

(ii) The map from M to N which sends m ∈ M to the homomorphism

γ̄ ↦ ⟨path(v_{0}, γv_{0}), m⟩

is injective and has finite cokernel.

Proof (sketch). Part (i) is proved in [Se]. Part (ii) follows from part (i), and from the fact that the maps ∂^{∗} and ∂_{∗} are adjoint with respect to the pairings defined above.

Write M_{sp} for the maximal torsion-free quotient of M/(w+1)M, with w ∈ Γ_{+} −Γ. By part (i) of lemma 2.3, the action of w ∈ Γ_{+} − Γ on H_{1}(G, ℤ) induces an action of w on N. Write N_{sp} for the maximal torsion-free quotient of N /(w + 1)N. We have an induced map from M_{sp} to N_{sp}, which is injective and has finite cokernel.

The module ℤ[E(G)] is equipped with the natural action of an algebra 𝕋̃ generated over ℤ by the Hecke correspondences T_{ℓ} for ℓ ∤ N and U_{ℓ} for ℓ | N, coming from its double coset description: see [BD1], sec. 1.5. The module H_{1}(G, ℤ) is stable under the action of 𝕋̃. Hence, by part (i) of lemma 2.3, the algebra 𝕋̃ also acts on the modules N and N_{sp}. Let 𝕋 and 𝕋_{sp} denote the image of 𝕋̃ in End(N) and End(N_{sp}), respectively. Thus, there are natural surjections 𝕋̃ → 𝕋 → 𝕋_{sp}. By an abuse of notation, we will denote by T_{ℓ} and U_{ℓ} also the natural images in 𝕋 and 𝕋_{sp} of T_{ℓ} and U_{ℓ} .

The next proposition clarifies the relation between the modules N and N_{sp} and the theory of modular forms.

Proposition 2.4 Let ϕ be an algebra homomorphism from 𝕋, resp. 𝕋_{sp} to ℂ, and let a_{n} := ϕ(T_{n}). Then, the a_{n} are the Fourier coefficients of a normalized eigenform of level N, which is new at N^{−}p, resp. is new at N^{−}p and is split multiplicative at p. Conversely, any such modular form arises as above from a character of 𝕋, resp. 𝕋_{sp}.

Proof. Eichler’s trace formula identifies the Hecke-module ℤ[E(G)] with a space of modular forms of level N which are new at N^{−}. Moreover, the algebra 𝕋 can also be viewed as the Hecke algebra of the module M defined above, and proposition 1.4 of [BD2] shows that M is equal to the “p-new” quotient of ℤ[E(G)]. This proves the statement of proposition 2.4 concerning characters of 𝕋. The abelian variety associated to a p-new modular form f is split multiplicative at p if and only if U_{p}f = f. Moreover, the Atkin-Lehner involution at p acts on a p-new modular form as −U_{p}, and acts on M as Γ_{+}/Γ. This concludes the proof of proposition 2.4.

Modular parametrizations, I

We now make a specific choice of the operator η_{f} (where f is the newform of level N attached to E) considered in the introduction, that will be used in formulating the results in the sequel of the paper.

As stated in lemma 2.3, Γ̄ can be identified with the homology group H_{1}(G, ℤ) ⊂ ℤ[E(G)]. Thus, when convenient, we will tacitly view elements of Γ̄ as contained in ℤ[E(G)]. The restriction of the pairing on ℤ[E(G)] defined above to Γ̄ yields the monodromy pairing (denoted in the same way by an abuse of notation)

⟨ , ⟩ : Γ̄ × Γ̄ → ℤ.

Let ℤ[E(G)]^{f}, resp. Γ̄^{f} be the submodule of ℤ[E(G)], resp. Γ̄ on which 𝕋̃, resp. 𝕋 acts via the character associated with f. Note that the quotient of ℤ[E(G)] by Γ̄ is torsion-free, and thus there is a canonical identification ℤ[E(G)]^{f} = Γ̄^{f}. Let e^{f} be a generator of Γ̄^{f} ≃ ℤ.

Define the “modular parametrizations”

π_{∗} : Γ̄ → Γ̄^{f}, π^{∗} : Γ̄^{f} → Γ̄

by π_{∗}(e) := ⟨e, e^{f}⟩e^{f} and π^{∗}(e^{f}) := e^{f}. Since

(π^{∗} ∘ π_{∗})^{2} = ⟨e^{f}, e^{f}⟩(π^{∗} ∘ π_{∗}),

we obtain that π^{∗} ∘ π_{∗} is equal to ⟨e^{f}, e^{f}⟩π_{f}, where π_{f} is the idempotent of 𝕋 ⊗ ℚ associated with f. From now on, we will assume that the operator η_{f} is defined by

η_{f} := π^{∗} ∘ π_{∗},

so that the integer n_{f} is equal to ⟨e^{f}, e^{f}⟩.

As observed in the introduction, the operator η_{f} induces a map N → ℤ, which is well-defined up to sign. Since f has split multiplicative reduction at p, this map factors through a map N_{sp} → ℤ. By an abuse of notation, we will indicate by η_{f} both the above maps.

Remark 2.5 The module Γ̄ can be identified with the character group associated with the reduction modulo p of Pic^{0}(X), where X is the Shimura curve considered in the introduction. As will be explained in section 4, the map π^{∗} ∘ π_{∗} on Γ̄ is induced by functoriality from a modular parametrization Pic^{0}(X) → E.

3 The p-adic L-function

Let O_{n} denote the order of K of conductor cp^{n}, n ≥ 0. (We will usually write O instead of O_{0}.) Equip the orders O_{n} with compatible orientations, i.e., with compatible algebra homomorphisms

𝔡_{ℓ}^{+} : O_{n} → ℤ/ℓ^{m}ℤ, ℓ^{m}∥N^{+}p,
𝔡_{ℓ}^{−} : O_{n} → 𝔽_{ℓ^{2}}, ℓ | N^{−}.

An algebra homomorphism of O_{n} into an oriented Eichler order S of level N^{+}p is called an oriented optimal embedding if it respects the orientation on O_{n} and on S, and does not extend to an embedding of a larger order into S. Consider pairs (R_{ξ}, ξ), where R_{ξ} is an oriented Eichler order of level N^{+}p and ξ is an element of Hom(K, B) which restricts to an oriented optimal embedding of O_{n} into R_{ξ}. A Gross point of conductor cp^{n} (n ≥ 0) is a pair as above, taken modulo the action of B^{×}.

By our previous remarks, a Gross point can be viewed naturally as an element of the double coset space

W := (R̂^{×}\B̂^{×} × Hom(K, B))/B^{×}.

(See [Gr], sec. 3 for more details.) Strong approximation gives the identification

W = (E⃗(T ) × Hom(K, B))/Γ_{+}.

By lemma 2.2, there is a natural map of ℤ-modules ℤ[W] → ℤ[E(G)], where ℤ[W] is the module of finite formal ℤ-linear combinations of elements of W. The Hecke algebra 𝕋̃ of ℤ[E(G)] acts naturally also on ℤ[W] (see [BD1], sec. 1.5), in such a way that the above map is 𝕋̃-equivariant.

The group G_{n} = Pic(O_{n}) = Ô_{n}^{×}\K̂^{×}/K^{×} acts simply transitively on the Gross points of conductor cp^{n} by the rule

σ(R_{ξ}, ξ) := (R_{ξ} ∗ ξ̂(σ)^{−1}, ξ),

where ξ̂ denotes the extension of ξ to a map from K̂ to B̂.

Now, fix a Gross point P_{0} = (R_{0}, ξ_{0}) (mod B^{×}) of conductor c. By the above identification, P_{0} corresponds to a pair (e⃗_{0}, ξ_{0}) ∈ E⃗(T ) × Hom(K, B), modulo the action of Γ_{+}. As above, the origin v_{0} of e⃗_{0} determines an orientation of T . Let e⃗ be one of the p oriented edges of T originating from e⃗_{0}. All the Gross points corresponding to pairs (e⃗, ξ_{0}) as above have conductor cp, except for one, which has conductor c. Fix an end

(e⃗_{0}, e⃗_{1}, …, e⃗_{n}, …),

such that (e⃗_{1}, ξ_{0}) defines a Gross point of conductor cp. Then, (e⃗_{n}, ξ_{0}) defines a Gross point P_{n} of conductor cp^{n}, for all n ≥ 0.

Denote by Norm_{H_{n+1}/H_{n}} the norm operator ∑_{g∈Gal(H_{n+1}/H_{n})} g.

Lemma 3.1

1) Let u = \frac{1}{2}#O^{×}. The equality

U_{p}P_{0} = uNorm_{H_{1}/H_{0}}P_{1} + σ_{𝔭}P_{0}

holds in ℤ[W] for a prime 𝔭 above p, where σ_{𝔭} ∈ Gal(H_{0}/K) denotes the image of 𝔭 by the Artin map.

2) For n ≥ 1,

U_{p}P_{n} = Norm_{H_{n+1}/H_{n}}P_{n+1}.

Proof. It follows from the definition of the operator U_{p} (see [BD1], sec. 1.5) and the action of Pic(O_{n}) on the Gross points.

The picture below, drawn in the case p = 2, illustrates geometrically the relation between the Galois action and the action of the Hecke correspondence U_{p}.

By lemma 2.3, the natural map from ℤ[W] to ℤ[E(G)] induces maps from ℤ[W] to the modules N and N_{sp}. These maps are Hecke-equivariant.

The Gross points P_{n} give rise to a p-adic distribution on G_{∞} with values in the module N_{sp} as follows. Given g ∈ G_{n}, denote by e_{n}^{g} the natural image of P_{n}^{g} in N_{sp}. For n ≥ 0, define the truncated p-adic L-function

L_{p,n}(N_{sp}/K) := ∑_{g∈G_{n}} e_{n}^{g} · g^{−1} ∈ N_{sp} ⊗ ℤ[G_{n}].

Note that L_{p,n}(N_{sp}/K) is well-defined up to multiplication by elements of G_{n}. For n ≥ 1, let ν_{n} : ℤ[G_{n}] → ℤ[G_{n−1}] be the natural projection of groups rings.

Lemma 3.2

1) The equality

ν_{1}(L_{p,1}(N_{sp}/K)) = u^{−1}(1 − σ_{𝔭})L_{p,0}(N_{sp}/K)

holds in N_{sp} ⊗ ℤ[G_{0}].

2) For n ≥ 2, the equality

ν_{n}(L_{p,n}(N_{sp}/K)) = L_{p,n−1}(N_{sp}/K)

holds in N_{sp} ⊗ ℤ[G_{n−1}].

Proof. By proposition 2.4, the operator U_{p} acts as +1 on N_{sp}. The claim follows from lemma 3.1 and the fact that N_{sp} is torsion-free.

Define the p-adic L-function attached to N_{sp} to be

L_{p}(N_{sp}/K) := lim_{n^{←}} L_{p,n}(N_{sp}/K) ∈ N_{sp} ⊗ ℤ[[G_{∞}]].

We now define the p-adic L-function attached to E. Observe that the maximal quotient Γ̄_{f} of Γ̄ on which 𝕋 acts via the character associated with f is isomorphic to ℤ. Let e_{f} be a generator of Γ̄_{f}. The monodromy pairing on Γ̄ induces a ℤ-valued pairing on Γ̄^{f} × Γ̄_{f}. Write ĉ_{p} for the positive integer |⟨e^{f}, e_{f}⟩|.

Lemma 3.3 The element (η_{f} ⊗ id)(L_{p}(N_{sp}/K)) ∈ ℤ[[G_{∞}]] is divisible by ĉ_{p}.

Proof. Consider the maps

π̃_{∗} : ℤ[E(G)] → ℤ[E(G)]^{f}, π̃^{∗} : ℤ[E(G)]^{f} → ℤ[E(G)]

defined by π̃_{∗}(e) := ⟨e, e^{f}⟩e^{f} and π̃^{∗}(e^{f}) := e^{f}. (The modular parametrizations π_{∗} and π^{∗} introduced in section 2 are obtained from these maps by restriction.) Hence, η̃_{f} := π̃^{∗} ∘ π̃_{∗} is an element of 𝕋̃, equal to ⟨e^{f}, e^{f}⟩π̃_{f}, where π̃_{f} is the idempotent in 𝕋̃ ⊗ ℚ associated with f. We have a commutative diagram

ℤ[E(G)] ⟶ N
η̃_{f} ↓  ^{η_{f}} ↓
ℤ[E(G)]^{f} ⟶ N^{f},

where the upper horizontal map is defined in lemma 2.3, and the lower horizontal map is the restriction of the upper one. Note that N^{f} is equal to Hom(Γ̄_{f}, ℤ), and therefore is generated by the homomorphism e_{f} ↦ 1. With our choices of generators for ℤ[E(G)]^{f} and N^{f}, the lower map of the above diagram is described as multiplication by the integer ⟨e^{f}, e_{f}⟩. The proof of lemma 3.2 also shows that mapping the Gross points of conductor cp^{n} to ℤ[E(G)]^{f} by the map η̃_{f} yields a p-adic distribution in ℤ[E(G)]^{f} ⊗ ℤ[[G_{∞}]]. By the above diagram, the image of this distribution in N^{f} ⊗ ℤ[[G_{∞}]] is equal to (η_{f} ⊗ id)(L_{p}(N_{sp}/K)). This proves the lemma.

Remark 3.4 In section 4, we will show that the integers ĉ_{p} and c_{p} are equal.

Define the p-adic L-function attached to E to be

L_{p}(E/K) = ĉ_{p}^{−1}(η_{f} ⊗ id)(L_{p}(N_{sp}/K)) ∈ ℤ[[G_{∞}]].

Observe that L_{p}(N_{sp}/K) and L_{p}(E/K) are well-defined up to multiplication by elements of G_{∞}.

Recall the quantities Ω_{f} and d defined in the introduction.

Theorem 3.5 Let χ : G_{∞} → ℂ^{×} be a finite order character of conductor cp^{n}, with n ≥ 1. Then the equality

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

holds.

Proof. See [Gr], [Dag], and [BD1], section 2.10.

Remark 3.6

1) Theorem 3.5 suggests that L_{p}(E/K) should really be viewed as the square root of a p-adic L-function, and hence we should define the anticyclotomic p-adic L-function of E to be L_{p}(E/K) ⊗ L_{p}(E/K)^{∗}, where ∗ denotes the involution of ℤ[[G_{∞}]] given on group-like elements by g ↦ g^{−1}. See section 2.7 of [BD1] for more details.

2) More generally, the p-adic L-function L_{p}(N_{sp}/K) interpolates special values of the complex L-series attached to the modular forms on 𝕋_{sp} (described in proposition 2.4).

Let σ_{𝔭} be as in lemma 3.1. Denote by H the subextension of H_{0} which is fixed by σ_{𝔭}, and set

G_{n} := Gal(H_{n}/H), G_{∞} := Gal(H_{∞}/H),
Σ := Gal(H_{0}/H) = G_{0}, Δ := Gal(H/K).

Note the exact sequences of Galois groups

0 → G_{n} → G_{n} → Δ → 0,
0 → G_{∞} → G_{∞} → Δ → 0.

The group Δ is naturally identified with the Picard group Pic(O[\frac{1}{p}]), and G_{∞} is equal to the image of the reciprocity map rec_{p} : ℚ_{p}^{×} → G_{∞} (where we have identified ℚ_{p}^{×} with K_{𝔭}^{×}). Let I be the kernel of the augmentation map ℤ[[G_{∞}]] → ℤ, and let I_{Δ} be the kernel of the augmentation map ℤ[[G_{∞}]] → ℤ[Δ].

Lemma 3.7

i) L_{p}(N_{sp}/K) belongs to N_{sp} ⊗ I_{Δ}. ii) L_{p}(E/K) belongs to I_{Δ}.

Proof. There are canonical isomorphisms

ℤ[[G_{∞}]]/I_{Δ} = ℤ[G_{n}]/I_{Δ,n} = ℤ[Δ],

where I_{Δ,n} is the natural image of I_{Δ} in ℤ[G_{n}]. By lemma 3.2, the image of L_{p}(N_{sp}/K) in N_{sp} ⊗ (ℤ[[G_{∞}]]/I_{Δ}) is equal to the image of L_{p,1}(N_{sp}/K) in N_{sp} ⊗ (ℤ[G_{1}]/I_{Δ,1}) = N_{sp} ⊗ ℤ[Δ]. The first part of the lemma now follows from lemma 3.2, 1). The second part follows directly from the first.

Since I_{Δ} is contained in I, the element L_{p}(N_{sp}/K) belongs to N_{sp}⊗I and L_{p}(E/K) belongs to I. Denote by

L′_{p}(N_{sp}/K), L′_{p}(N_{sp}/H)

the natural image of L_{p}(N_{sp}/K) in N_{sp} ⊗ I/I^{2} = N_{sp} ⊗ G_{∞} and N_{sp} ⊗ I_{Δ}/I_{Δ}^{2} = N_{sp} ⊗ ℤ[Δ] ⊗ G_{∞}, respectively. Likewise, let

L′_{p}(E/K), L′_{p}(E/H)

be the natural image of L_{p}(E/K) in I/I^{2} = G_{∞} and I_{Δ}/I_{Δ}^{2} = ℤ[Δ] ⊗ G_{∞}, respectively. The above elements should be viewed as derivatives of p-adic L-functions at the central point.

In order to carry out the calculations of the next sections, it is useful to observe that the derivatives L′_{p}(N_{sp}/K) and L′_{p}(N_{sp}/H) can be expressed in terms of the derivatives of certain partial p-adic L-functions. Set h := #(Δ). Fix Gross points of conductor c

P_{0} = P_{0}^{1}, …, P_{0}^{h},

corresponding to pairs (R_{0}^{i}, ξ_{0}^{i}), i = 1, …, h, which are representatives for the Σorbits of the Gross points of conductor c. Writing [P_{0}^{i}] for the Σ-orbit of P_{0}^{i}, let δ_{i} be the element of Δ such that

[δ_{i}P_{0}^{1}] = [P_{0}^{i}].

Suppose that P_{0}^{i} corresponds to a pair (e⃗_{0}(i), ξ_{0}^{i}) ∈ E⃗(T ) × Hom(K, B), modulo the action of Γ_{+}. Fix ends

(e⃗_{0}(i), e⃗_{1}(i), …, e⃗_{n}(i), …)

such that (e⃗_{1}(i), ξ_{0}^{i}) defines a Gross point of conductor cp. Thus, (e⃗_{n}(i), ξ_{0}^{i}) defines a Gross point P_{n}^{i} of conductor cp^{n}, for all n ≥ 0. For g ∈ G_{n}, let e_{n}(i)^{g} denote the natural image of (P_{n}^{i})^{g} in N_{sp}. Let

L_{p,n}(N_{sp}/H, P_{0}^{i}) := ∑_{g∈G_{n}} e_{n}(i)^{g} · g^{−1} ∈ N_{sp} ⊗ ℤ[G_{n}].

The proof of lemma 3.2 also shows that the elements L_{p,n}(N_{sp}/H, P_{0}^{i}) are compatible under the maps induced by the natural projections of group rings. Thus, we may define the partial p-adic L-function attached to N_{sp} and P_{0}^{i} to be

L_{p}(N_{sp}/H, P_{0}^{i}) := lim_{n^{←}} L_{p,n}(N_{sp}/H, P_{0}^{i}) ∈ N_{sp} ⊗ ℤ[[G_{∞}]].

Observe that L_{p}(N_{sp}/H, P_{0}^{i}) depends only on the Σ-orbit of P_{0}^{i}, up to multiplication by elements of G_{∞}.

Let I_{H} be the kernel of the augmentation map ℤ[[G_{∞}]] → ℤ. Like in the proof of lemma 3.7, one checks that L_{p}(N_{sp}/H, P_{0}^{i}) belongs to I_{H}. Write L′_{p}(N_{sp}/H, P_{0}^{i}) for the natural image of L_{p}(N_{sp}/H, P_{0}^{i}) in N_{sp} ⊗ I_{H}/I_{H}^{2} = N_{sp} ⊗ G_{∞}. Thus,

L′_{p}(N_{sp}/H, P_{0}^{i}) = lim_{n^{←}} L′_{p,n}(N_{sp}/H, P_{0}^{i}),

where

(3)  L′_{p,n}(N_{sp}/H, P_{0}^{i}) = ∑_{g∈G_{n}} e_{n}(i)^{g} ⊗ g^{−1}.

We obtain directly:

Lemma 3.8 i)

L′_{p}(N_{sp}/K) = ∑_{i=1}^{h} L′_{p}(N_{sp}/H, P_{0}^{i}).

ii)

L′_{p}(N_{sp}/H) = ∑_{i=1}^{h} L′_{p}(N_{sp}/H, P_{0}^{i}) · δ_{i}^{−1}.

4 The theory of p-adic uniformization of Shimura curves

For more details on the results stated in this section, the reader is referred to [BC], [Cer], [Dr], [GVdP] and [BD2].

Let B be the indefinite quaternion algebra over ℚ of discriminant N^{−}p, and let R be an Eichler order of B of level N^{+}. Denote by X the Shimura curve over ℚ associated with the order R. We refer the reader to [BC] and [BD2], section 4 for the definition of X via moduli. Here we content ourselves with recalling Cerednik’s theorem, which describes a rigid-analytic uniformization of X. Write

H_{p} := ℂ_{p} − ℚ_{p}

for the p-adic upper half plane. The group GL_{2}(ℚ_{p}) acts (on the left) on H_{p} by linear fractional transformations. Thus, fixing an isomorphism

ψ : B_{p} → M_{2}(ℚ_{p})

induces an action of Γ on H_{p}. This action is discontinuous, and the rigid-analytic quotient Γ\H_{p} defines the ℂ_{p}-points of a non-singular curve X over ℚ_{p}. The curves X and X are equipped with the action of Hecke algebras 𝕋_{X} and 𝕋_{X}, respectively ([BC], [BD1]).

By lemma 2.1, the action of Γ_{+}/Γ induces an involution W of X. Let ℚ_{p^{2}} be the unique unramified quadratic extension of ℚ_{p} contained in ℂ_{p}, and let τ be the generator of Gal(ℚ_{p^{2}}/ℚ_{p}). Denote by ♮ ∈ H^{1}(⟨τ⟩, Aut(X)) the class of the cocycle mapping τ to W, and write X^{♮} for the curve over ℚ_{p} obtained by twisting X by ♮.

Theorem 4.1 (Cerednik) There is a Hecke-equivariant isomorphism X ≃ X^{♮} of curves over ℚ_{p}. In particular, X and X are isomorphic over ℚ_{p^{2}}.

Proof. See [Cer], [Dr], [BC].

Building on theorem 4.1, the results in [GVdP] yield a rigid-analytic description of the jacobian of X. If D = P_{1} + ⋯ + P_{r} − Q_{1} − ⋯ − Q_{r} ∈ Div^{0}(H_{p}) is a divisor of degree zero on H_{p}, define the theta function

ϑ(z; D) = ∏_{ϵ∈Γ} \frac{(z − ϵP_{1}) ⋯ (z − ϵP_{r})}{(z − ϵQ_{1}) ⋯ (z − ϵQ_{r})} .

Write δ̄ for the natural image in Γ̄ of an element δ of Γ. For all δ in Γ, the above theta function satisfies the functional equation

ϑ(δz; D) = ϕ_{D}(δ̄)ϑ(z; D),

where ϕ_{D} is an element of Hom(Γ̄, ℂ_{p}^{×}) = N ⊗ ℂ_{p}^{×} which does not depend on z. For γ ∈ Γ, the number ϕ_{(γz)−(z)}(δ̄) does not depend on the choice of z ∈ H_{p}, and depends only on the image of γ in Γ̄. This gives rise to a pairing

[ , ] : Γ̄ × Γ̄ → ℚ_{p}^{×}.

The pairing [ , ] is bilinear and symmetric. The next proposition explains the relation between [ , ] and the monodromy pairing ⟨ , ⟩ : Γ̄ × Γ̄ → ℤ defined in section 2.

Proposition 4.2 The pairings ⟨ , ⟩ and ord_{p} ∘ [ , ] are equal.

Proof. See [M], th. 7.6.

It follows that ord_{p} ∘ [ , ] is positive definite, so that themap

j : Γ̄ → N ⊗ ℚ_{p}^{×}

induced by [ , ] is injective and has discrete image. Set Λ := j(Γ̄). Given a divisor D of degree zero on X (ℂ_{p}) = Γ\H_{p}, let D̃ denote an arbitrary lift to a degree zero divisor on H_{p}. The automorphy factor ϕD̃ depends on the choice of the lift D̃, but its image in (N ⊗ ℂ_{p}^{×})/Λ depends only on D. Thus, the assignment D ↦ ϕD̃ gives a well-defined map from Div^{0}(X (ℂ_{p})) to (N ⊗ ℂ_{p}^{×})/Λ.

Proposition 4.3 The map Div^{0}(X (ℂ_{p})) → (N ⊗ ℂ_{p}^{×})/Λ defined above is trivial on the group of principal divisors, and induces a Hecke-equivariant isomorphism from the ℂ_{p}-points of the jacobian J of X to (N ⊗ ℂ_{p}^{×})/Λ.

Proof. See [GVdP], VI.2 and VIII.4, and also [BC], ch. III.

Let

Φ : N ⊗ ℂ_{p}^{×} → J (ℂ_{p})

stand for the map induced by (the inverse of) the isomorphism defined in proposition 4.3.

Modular parametrizations, II The map η_{f} : N → ℤ defined in section 2 induces a map

η_{f} ⊗ id : N ⊗ ℂ_{p}^{×} → ℂ_{p}^{×}.

The Jacquet-Langlands correspondence [JL] implies that the quotient abelian variety η_{f}J is an elliptic curve ℚ-isogenous to E. From now on, we assume that E = η_{f}J is the strong Weil curve for the parametrization by the Shimura curve X. By an abuse of notation, we denote by η_{f} also the surjective map

J(ℂ_{p}) → E(ℂ_{p})

induced by η_{f}.

Let Λ^{f} be the submodule of Λ on which 𝕋 acts via the character ϕ_{f}.

Proposition 4.4 The kernel q^{ℤ} of Φ_{Tate} is canonically equal to the module Λ^{f}, and the following diagram

0 ⟶ Λ ⟶ N ⊗ ℂ_{p}^{×} ⟶^{Φ} J (ℂ_{p}) ⟶ 0
η_{f} ↓  ^{η_{f} ⊗id}↓  ^{η_{f}} ↓
0 ⟶ Λ^{f} ⟶ ℂ_{p}^{×}  ⟶^{Φ_{Tate}} E(ℂ_{p}) ⟶ 0

is Hecke-equivariant and commutes up to sign.

Proof. The right-most square in the above diagram is a consequence of proposition 4.3, combined with theorem 4.1 and the fact that f is split-multiplicative at p. In order to obtain the left-most square, it is enough to prove that the kernel of Φ_{Tate} is equal to Λ^{f}. Note that the target ℂ_{p}^{×} = N^{f} ⊗ ℂ_{p}^{×} of the map η_{f} ⊗ id is naturally a submodule of N ⊗ℂ_{p}^{×}, since the quotient of N by N^{f} is torsion-free. By definition, E(ℂ_{p}) may similarly be viewed as an abelian subvariety of J (ℂ_{p}). It follows that Φ_{Tate} can be described as the restriction of Φ to ℂ_{p}^{×}. In particular, ker(Φ_{Tate}) is equal to Λ ∩ ℂ_{p}^{×}. In turn, this last module is equal to Λ^{f}.

Corollary 4.5 The integer ĉ_{p} = |⟨e^{f}, e_{f}⟩| (introduced in lemma 3.3) is equal to c_{p}.

Proof. Working through the definition of the maps in the diagram of proposition 4.4 shows that [e^{f}, e_{f}] is equal to q^{±1}. The claim follows from proposition 4.2.

5 p-adic Shintani cycles and special values of complex Lfunctions

Let P_{0} = (R_{0}, ξ_{0}) (mod B^{×}) be a Gross point of conductor c. The point P_{0} determines a p-adic cycle 𝔠(P_{0}) ∈ Γ̄ in the following way. By strong approximation, we may assume that the representative (R_{0}, ξ_{0}) for P_{0} is such that the oriented orders R_{0}[\frac{1}{p}] and R[\frac{1}{p}] are equal. Thus, ξ_{0} induces an embedding of O[\frac{1}{p}] into R[\frac{1}{p}], which we still denote by ξ_{0}. The image by ξ_{0} of a fundamental p-unit in O[\frac{1}{p}] having norm of even p-adic valuation determines an element γ = γ(P_{0}) of Γ. This element is well-defined up to conjugation and up to inversion, and up to multiplication by the image of torsion elements of O^{×}.

More explicitly, write k for the order of σ_{𝔭} in Pic(O) (where σ_{𝔭} is as in lemma 3.1), and set 𝔭^{k} = (v) with v ∈ O. Let ι be 1, resp. 2 if k is even, resp. odd. Then γ is the image of ξ_{0}(v)^{ι} in Γ.

Definition. The p-adic Shintani cycle 𝔠 = 𝔠(P_{0}) attached to P_{0} is the natural image of γ in Γ̄.

This terminology is justified in the remark 5.4 below. Observe that 𝔠 is well-defined up to sign.

Denote by ℤ[E(G)]_{sp} the maximal torsion-free quotient of ℤ[E(G)]/(w+1)ℤ[E(G)], where w is any element of Γ_{+} − Γ. Recall the element η̃_{f} ∈ 𝕋̃ defined in the proof of lemma 3.3, mapping to η_{f} by the natural projection 𝕋̃ → 𝕋. The next lemma relates the p-adic cycle 𝔠 to the image in N_{sp} of the Gross point P_{0}.

Lemma 5.1 The natural images in ℤ[E(G)]_{sp} of 𝔠 and ∑_{σ∈Σ} ιP_{0}^{σ} are equal. In particular, η_{f}𝔠 is equal to the image of ∑_{σ∈Σ} ι(η̃_{f}P_{0}^{σ}) in ℤ[E(G)].

Proof. (In order to visualize the geometric content of this proof, the reader may find it helpful to refer to the picture in section 3.) Set P_{i} := σ_{𝔭}^{i}P_{0}, for i = 0, …, k − 1. By part 1 of lemma 3.1 and the definition of the action of U_{p} on the Bruhat-Tits tree, we can fix representatives (e⃗_{i}, ξ_{0}) for the Gross points P_{i} so that the e⃗_{i} are consecutive oriented edges of T . With notations as at the beginning of this section, let γ_{+} ∈ Γ_{+} be the image of ξ_{0}(v). Thus, γ = γ_{+}^{ι}. Call v_{0} the origin of e⃗_{0}. If ι = 1, the even vertex of the edge e⃗_{k−1} is equal to γv_{0}. If ι = 2, i.e., γ_{+} belongs to Γ_{+} −Γ, then

e⃗_{0}, …, e⃗_{k−1}, γ_{+}e⃗_{0}, …, γ_{+}e⃗_{k−1}

is a sequence of consecutive oriented edges, and the even vertex of γ_{+}v⃗_{k−1} is equal to γv_{0}. Note that ∑_{σ∈Σ} ιP_{0}^{σ} is equal in ℤ[E⃗(G_{+})] to e⃗_{0} + e⃗_{1} + … + e⃗_{k−1} if ι = 1, and to

e⃗_{0} + e⃗_{1} + … + e⃗_{k−1} + γ_{+}e⃗_{0} + γ_{+}e⃗_{1} + … + γ_{+}e⃗_{k−1}

if ι = 2. Denote by e_{i} the unoriented edge of T corresponding to e⃗_{i}, and let w be any element of Γ_{+} − Γ. By definition of the bijection κ of lemma 2.2, the following equalities hold in ℤ[E(G)]:

κ^{−1}(e⃗_{0} + … + e⃗_{k−1}) = e_{0} + we_{1} + … + e_{k−2} + we_{k−1} if ι = 1,

κ^{−1}(e⃗_{0} + … + e⃗_{k−1} + γ_{+}e⃗_{0} + γ_{+}e⃗_{1} + … + γ_{+}e⃗_{k−1})) = e_{0} + we_{1} + … + e_{k−1} + w(γ_{+}e_{0}) + (γ_{+}e_{1}) + … + w(γ_{+}e_{k−1}) if ι = 2.

Projecting the right hand sides of the above equalities to ℤ[E(G)]_{sp}, and keeping into account that w acts as −1 on this module, gives in both cases path(v_{0}, γv_{0}).

The next proposition elucidates the relation between the p-adic Shintani cycle defined above and the special values of the complex L-function of E/K. Following the notations of section 3, fix Gross points P_{0} = P_{0}^{1}, …, P_{0}^{h} which are representatives for the Σ-orbits of the Gross points of conductor c, and list the elements of Δ so that [δ_{i}P_{0}^{1}] = [P_{0}^{i}], where [P_{0}^{i}] denotes the Σ-orbit of P_{0}^{i}. As above, the Gross point P_{0}^{i} determines a p-adic Shintani cycle 𝔠_{i} ∈ Γ̄, with 𝔠_{1} = 𝔠. Given a complex character χ : Δ → ℂ^{×} of Δ, set

𝔠_{H} := ∑_{i=1}^{h} 𝔠_{i} ⊗ δ_{i}^{−1} ∈ Γ̄ ⊗ ℤ[Δ],

𝔠_{K,χ} := χ(𝔠_{H}) = ∑_{i=1}^{h} 𝔠_{i} ⊗ χ(δ_{i})^{−1} ∈ Γ̄ ⊗ ℤ[χ].

If χ is the trivial character, we will also write 𝔠_{K} as a shorthand for 𝔠_{K,χ}. Extend the pairing ⟨ , ⟩ on Γ̄ to a hermitian pairing on Γ̄ ⊗ ℤ[χ].

Proposition 5.2 Suppose that χ is primitive. The following equality holds:

⟨η_{f}𝔠_{K,χ}, 𝔠_{K,χ}⟩ = \frac{L(E/K, χ, 1)}{Ω_{f}} \sqrt{d} · (ιu)^{2} · n_{f}.

Proof. In view of lemma 5.1, this is simply a restatement of the results of [Gr] and [Dag].

Recall the maps j : Γ̄ → N ⊗ ℚ_{p}^{×} and η_{f} ⊗ id : N ⊗ ℂ_{p}^{×} → ℂ_{p}^{×} defined in section 4. By an abuse of notation, we denote in the same way the maps obtained by extending scalars to ℤ[χ].

Corollary 5.3 The equality

(η_{f} ⊗ id)(j(𝔠_{K,χ})) = q ⊗ ρ

holds in ℚ_{p}^{×} ⊗ ℤ[χ], where ρ ∈ ℤ[χ] satisfies

|ρ|^{2} = \frac{L(E/K, χ, 1)}{Ω_{f}} \sqrt{d} · (ιu)^{2} · n_{f}.

Proof. By proposition 4.4 combined with the definition of η_{f} given in section 2, ρ is equal to ⟨𝔠_{K,χ}, e^{f}⟩ ∈ ℤ[χ]. Hence

|ρ|^{2} = ⟨𝔠_{K,χ}, e^{f}⟩⟨e^{f}, 𝔠_{K,χ}⟩
= ⟨η_{f}𝔠_{K,χ}, 𝔠_{K,χ}⟩.

The claim follows from proposition 5.2.

Remark 5.4 Let F be a real quadratic field and let ψ : F → M_{2}(ℚ) be an embedding. Assume that ψ maps the ring of integers O_{F} to the Eichler order M_{0}(N) of integral matrices with lower left entry divisible by N. Since the homology group H_{1}(X_{0}(N), ℤ) can be identified with the maximal torsion-free abelian quotient of Γ_{0}(N), the image by ψ of a fundamental unit in O_{F} of norm 1 determines an integral homology cycle 𝔰 ∈ H_{1}(X_{0}(N), ℤ). Shintani [Sh] proved that the cycle 𝔰 encodes the special values of the classical L-series over F attached to newforms on X_{0}(N). In light of proposition 5.2, the element 𝔠 can be viewed as a p-adic analogue of the cycle 𝔰.

6 p-adic Shintani cycles and derivatives of p-adic L-functions

Let P_{0} be a Gross point of conductor c. In section 5, we attached to P_{0} a p-adic cycle 𝔠 ∈ Γ̄, and proved in proposition 5.2 that 𝔠 is related to the special values of the complex L-function of E/K. Our main result (theorem 6.1 below) shows that 𝔠 is also related to the first derivative of the p-adic L-function defined in section 3. By combining these results we will obtain theorem 1.1.

Write \underline{j} for the composite map

Γ̄ ⟶^{j} N ⊗ ℚ_{p}^{×} → N_{sp} ⊗ ℚ_{p}^{×} → N_{sp} ⊗ G_{∞},

where the second map is induced by the natural projection of N onto N_{sp}, and the third map is induced by rec_{p} : ℚ_{p}^{×} → G_{∞}. Our main result is the following.

Theorem 6.1 The following equality holds up to sign in N_{sp} ⊗ G_{∞}:

L′_{p}(N_{sp}/H, P_{0})^{ι} = \underline{j}(𝔠).

Recall the definition of the elements 𝔠_{H} and 𝔠_{K} given in section 5. By lemma 3.8, we obtain directly:

Corollary 6.2

(i) The following equality holds up to sign in N_{sp} ⊗ ℤ[Δ] ⊗ G_{∞}:

L′_{p}(N_{sp}/H)^{ι} = \underline{j}(𝔠_{H}).

(ii) The following equality holds up to sign in N_{sp} ⊗ G_{∞}:

L′_{p}(N_{sp}/K)^{ι} = \underline{j}(𝔠_{K}).

By applying the operator η_{f} to both sides of the equalities of corollary 6.2, and using corollary 4.5 and the definitions of the p-adic L-functions attached to N_{sp} and E , we find:

Corollary 6.3

(i) The following equality holds up to sign in ℤ[Δ] ⊗ G_{∞}:

c_{p}L′_{p}(E/H)^{ι} = \underline{j}(η_{f}𝔠_{H}).

(ii) The following equality holds up to sign in G_{∞}:

c_{p}L′_{p}(E/K)^{ι} = \underline{j}(η_{f}𝔠_{K}).

Proof of theorem 1.1 Combine corollary 6.3 with corollary 5.3.

By combining corollary 6.3 with corollary 5.3, we also obtain the following generalization of theorem 1.1. Let L′_{p}(E/K, χ) stand for the element χ(L′_{p}(E/H)) of G_{∞} ⊗ ℤ[χ].

Theorem 6.4 Suppose that χ is primitive. The following equalities hold up to sign:

c_{p}L′_{p}(E/K, χ) = rec_{p}(q) ⊗ ρ in G_{∞} ⊗ ℤ[\frac{1}{2}][χ]

and

L′_{p}(E/K, χ) = \frac{rec_{p}(q)}{ord_{p}(q)} ⊗ ρ in G_{∞} ⊗ ℚ[χ],

where

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

Corollary 6.5 The derivative L′_{p}(E/K, χ) is non-zero in G_{∞} ⊗ ℚ[χ] if and only if the classical special value L(E/K, χ, 1) is non-zero.

Proof. By theorem 6.4, one is reduced to showing that rec_{p}(q) is a non-torsion element of G_{∞}, i.e., q^{p−1} does not belong to the kernel of the reciprocity map. But elements in this kernel are algebraic over ℚ, and q is known to be transcendental by a result of Barré-Sirieix, Diaz, Gramain and Philibert [BSDGP].

Remark 6.6 Theorem 1.1 was conjectured in [BD1], section 5.1 in a slightly different form. We conclude this section by studying the compatibility of theorem 1.1 (and its generalization theorem 6.4) with the conjectures of [BD1]. For simplicity, assume throughout this remark that the elliptic curve E is semistable, so that N is squarefree, and that E is isolated in its isogeny class, so that the action of Gal(ℚ̄/ℚ) on the ℓ-torsion points of E is irreducible for all primes ℓ.

Let p_{1} ⋯ p_{n}q_{1} ⋯ q_{n} be a prime factorization of the squarefree integer pN^{−}, with p_{1} = p. Denote by X_{1} the Shimura curve X, and by X_{n+1} the classical modular curve X_{0}(N). For i = 2, …, n, denote by X_{i} the Shimura curve associated with an Eichler order of level N^{+}p_{1} ⋯ p_{i−1}q_{1} ⋯ q_{i−1} in the indefinite quaternion algebra of discriminant p_{i} ⋯ p_{n}q_{i} ⋯ q_{n}. Since E is modular, the Jacquet-Langlands correspondence [JL] implies that E is parametrized by the jacobian J_{i} of the curve X_{i}, i = 1, …, n + 1. Let

ϕ_{i} : J_{i} → E

be the strong Weil parametrization of E by J_{i}. Thus, the morphism ϕ_{i} has connected kernel, and its dual ϕ_{i}^{∨} : E → J_{i} is injective. The endomorphism ϕ_{i} ∘ϕ_{i}^{∨} of E is multiplication by an integer d_{X_{i}}, called the degree of the modular parametrization of E by the Shimura curve X_{i}.

If ℓ | N, denote by c_{ℓ} the order of the group of connected components of E at ℓ.

Theorem 6.7 (Ribet-Takahashi) Under our assumptions: i)

\frac{d_{X_{0}(N)}}{d_{X}} = c_{p_{1}} ⋯ c_{p_{n}}c_{q_{1}} ⋯ c_{q_{n}};

ii)

⟨e^{f}, e^{f}⟩ = d_{X}c_{p} .

Proof. Part i) follows from theorem 1 of [RT]. Part ii) follows from section 2 of [RT]. The results of [RT] exclude the case where N^{+} is prime, but a forthcoming paper of S. Takahashi will deal with this case as well.

By combining theorem 6.7 with the relation Ω_{f} = d_{X_{0}(N)} · Ω_{E}, where Ω_{E} is the complex period of E, we find that the formula of theorem 1.1 (and likewise for theorem 6.4) becomes

L′_{p}(E/K) = \frac{rec_{p}(q)}{ord_{p}(q)}\sqrt{L(E/K, 1)Ω_{E}^{−1} · d^{\frac{1}{2}} u^{2} ∏_{ℓ|N^{−}} c_{ℓ}^{−1}},

which is the same as conjecture 5.3 of [BD1].

7 Proof of theorem 6.1

First, we give an explicit description of certain group actions on the p-adic upper half plane and on the Bruhat-Tits tree depending on our choice of a Gross point P_{0} of conductor c. Then, we compute the value \underline{j}(𝔠), for 𝔠 as in sections 5 and 6. I Group actions on H_{p} and T Let K_{p} := K ⊗ ℚ_{p}. Our choice of a prime 𝔭 above p determines an identification of K_{p} = K_{𝔭} × K𝔭̄ with ℚ_{p} × ℚ_{p}.

As in section 5, choose a representative (R_{0}, ξ_{0}) for the Gross point P_{0} such that R_{0}[\frac{1}{p}] and R[\frac{1}{p}] are equal. Let (e⃗_{0}, ξ_{0}) be a pair corresponding to P_{0}, and denote by v_{0} the origin of e⃗_{0}. Set R_{0,p} := R_{0} ⊗ ℤ_{p}, and let \underline{R}_{0,p} be the maximal order of B_{p} corresponding to v_{0}. Recall the isomorphism

ψ : B_{p} → M_{2}(ℚ_{p})

fixed in section 4. We may, and will from now on, choose ψ so that:

i) ψ maps \underline{R}_{0,p} onto M_{2}(ℤ_{p});

ii) ψ ∘ ξ_{0} maps (x, y) ∈ K_{p} = ℚ_{p} × ℚ_{p} to the diagonal matrix ( \begin{matrix}x & 0 \\ 0 & y \end{matrix} ).

Condition i) allows us to identify T = ℚ_{p}^{×}\underline{R}_{0,p}^{×}\B_{p}^{×} with PGL_{2}(ℤ_{p})\PGL_{2}(ℚ_{p}). Viewing K_{p}^{×} as a subgroup of GL_{2}(ℚ_{p}) thanks to the embedding ψ ∘ ξ_{0} yields actions of K_{p}^{×} on H_{p} and on T = PGL_{2}(ℤ_{p})\PGL_{2}(ℚ_{p}), factoring through K_{p}^{×}/ℚ_{p}^{×}. Identify this last group with ℚ_{p}^{×} by mapping a pair (x, y) modulo ℚ_{p}^{×} to xy^{−1}. Under this identification, an element x of ℚ_{p}^{×} acts on H_{p} as multiplication by x,

and on T as conjugation by the matrix ( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} ).

Recall the element v ∈ O ⊂ K_{p}^{×} defined in section 5 by 𝔭^{k} = (v). Identify as above v with an element \underline{w} of ℚ_{p}^{×}. Note that \underline{w} is equal to p^{k} times a p-adic unit. Set G̃_{∞} := ℚ_{p}^{×} = p^{ℤ} × ℤ_{p}^{×}. Define the quotients of G̃_{∞}

Σ̃ := ℚ_{p}^{×}/ℤ_{p}^{×} = p^{ℤ},  G̃_{n} := p^{ℤ} × (ℤ_{p}/p^{n}ℤ_{p})^{×}, n ≥ 1.

To simplify slightly the computation, assume from now on that O^{×} = {±1}. (If O^{×} ≠ {±1}, then K has discriminant −3 or −4, and the exact sequences below have to be modified to account for the non-trivial units of O. The computations in this case follow closely those presented in the paper.) Class field theory yields the exact sequence

0 → ⟨\underline{w}⟩ → G̃_{∞} ⟶^{rec_{p}} G_{∞} → 0,

and the induced sequences

0 → ⟨\underline{w}⟩ → Σ̃ → Σ → 0, 0 → ⟨\underline{w}⟩ → G̃_{n} → G_{n} → 0.

For n ≥ 0, denote by ℤ_{p}^{(n)} ⊂ G̃_{∞} the subgroup of elements of ℤ_{p}^{×} which are congruent to 1 modulo p^{n}.

Definition. We say that a vertex v of T has level n, and write ℓ(v) = n, if the stabilizer of v for the action of G̃_{∞} is equal to ℤ_{p}^{(n)}. Likewise, we say that an edge e of T has level n, and write ℓ(e) = n, if the stabilizer of e for the action of G̃_{∞} is ℤ_{p}^{(n)}.

Note that the group G̃_{n} (Σ̃ if n = 0) acts simply transitively on the vertices and edges of level n. By definition of the action of G̃_{∞} on T , v_{0} is a vertex of level 0. Thus, the set of vertices of level 0 is equal to the Σ̃-orbit of v_{0}. More generally, the set of vertices of level n can be described as the G̃_{n}-orbit of a vertex v_{n}, whose distance from v_{0} is n and whose distance from all the other vertices in the orbit Δ̃v_{0} is > n.

By using the standard coordinate, identify ℙ^{1}(ℂ_{p}) with ℂ_{p} ∪ {∞} and H_{p} with ℙ^{1}(ℂ_{p}) − ℙ^{1}(ℚ_{p}). In particular, view 0 and ∞ as elements of ℙ^{1}(ℚ_{p}). Recall the element γ = γ(P_{0}) of Γ defined in section 5. Since the reduced norm of γ has positive valuation, our choice of the isomorphism ψ yields

(4)  lim_{n→+∞} γ^{n}z = 0,  lim_{n→−∞} γ^{n}z = ∞

for all z ∈ H_{p}. Note also that 0 and ∞ are the fixed points for the action of G̃_{∞} on ℙ^{1}(ℂ_{p}).

Let H_{p}(ℚ_{p^{2}}) = ℚ_{p^{2}} −ℚ_{p} be the ℚ_{p^{2}}-points of the p-adic upper half plane. Define the reduction map

r : H_{p}(ℚ_{p^{2}}) → V(T )

as follows. Given z ∈ H_{p}(ℚ_{p^{2}}), let Q_{z} denote the stabilizer of z in GL_{2}(ℚ_{p}), together with the zero matrix. Then Q_{z} is a field isomorphic to ℚ_{p^{2}}, and this gives rise to an embedding of ℚ_{p^{2}} in M_{2}(ℚ_{p}) (well-defined up to an isomorphism of ℚ_{p^{2}}). Write ℤ_{p^{2}} for the ring of integers of ℚ_{p^{2}}, and let S be the unique maximal order of M_{2}(ℚ_{p}) containing the image of ℤ_{p^{2}} by the above embedding. We have r(z) = S. (See also [BD2], section 1.)

Lemma 7.1

1) The reduction map r is GL_{2}(ℚ_{p})-equivariant. In particular, r is equivariant for the group actions defined above.

2) Write ℤ_{p^{2}} = ℤ_{p}α + ℤ_{p}. We have r^{−1}(v_{0}) = ℤ_{p}^{×}α + ℤ_{p}.

3) If z_{1} and z_{2} are mapped by r to adjacent vertices of respective levels n and n+1, then z_{1}z_{2}^{−1} ≡ 1 (mod p^{n}).

Proof.

1) Let z be an element of H_{p}(ℚ_{p^{2}}), and let B be a matrix in GL_{2}(ℚ_{p}). If f : ℚ_{p^{2}} → M_{2}(ℚ_{p}) is an embedding fixing z, then BfB^{−1} is an embedding fixing Bz. Suppose that S is the maximal ideal containing f(ℤ_{p^{2}}). Then BSB^{−1} = S ∗ B^{−1} is the maximal ideal containing the image of ℤ_{p^{2}} by BfB^{−1}. Thus, r(Bz) = S ∗ B^{−1}, as was to be shown.

2) Suppose to fix ideas that p > 2. Then, we may assume that α = \sqrt{ν}, where the integer ν is not a squaremodulo p. (The case p = 2 can be dealt with in a similar way, for instance by taking α = (1 + \sqrt{−3})/2.) A direct computation shows that

Q\sqrt{ν} = \{( \begin{matrix}b & aν \\ a & b \end{matrix} ) : a, b ∈ ℚ_{p}\}.

Mapping the above matrix to a\sqrt{ν} + b yields an isomorphism of Q\sqrt{ν} onto ℚ_{p^{2}}. Thus, r(\sqrt{ν}) is equal to v_{0} = M_{2}(ℤ_{p}). Given z = a\sqrt{ν} + b ∈ H_{p}(ℚ_{p^{2}}), we have

z = B\sqrt{ν}, where B is the matrix ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ). By part 1,

r(z) = BM_{2}(ℤ_{p})B^{−1}.

But BM_{2}(ℤ_{p})B^{−1} = M_{2}(ℤ_{p}) if and only if B belongs to GL_{2}(ℤ_{p}), i.e., a belongs to ℤ_{p}^{×}.

3) Set r(z_{1}) = v_{1} and r(z_{2}) = v_{2}. The edge joining v_{1} to v_{2} has level n + 1. Since G̃_{∞} = ℚ_{p}^{×} acts transitively on the edges of level n + 1, there is g ∈ ℚ_{p}^{×} such that gv_{1} and gv_{2} have distance from v_{0} equal to n and n + 1, respectively. With notations as in the proof of part 2 of this proposition, write gz_{i} = a_{i}\sqrt{ν}+b_{i}, i = 1, 2, where a_{i}, b_{i} ∈ ℤ_{p}, gcd(a_{i}, b_{i}) = 1, and p^{n} ∥ a_{1}, p^{n+1} ∥ a_{2}. Thus, the vertex gv_{i} is represented by the matrix

A_{i} = ( \begin{matrix}a_{i} & b_{i} \\ 0 & 1 \end{matrix} ) .

Our assumption on gv_{1} and gv_{2} implies that the column ( \begin{matrix}b_{2} \\ 1 \end{matrix} ) of A_{2} is a ℤ_{p}-linear

combination of the columns of A_{1}. It follows that b_{1} ≡ b_{2} (mod p^{n}), and hence

z_{1}z_{2}^{−1} = gz_{1}(gz_{2})^{−1} ≡ 1 (mod p^{n}).

II The calculation Given δ ∈ Γ, write as usual δ̄ for the natural image of δ in Γ̄. We now compute explicitly the value of j(𝔠)(δ̄) = [𝔠, δ̄], for δ ∈ Γ. We begin with the following lemma.

Lemma 7.2 Given δ ∈ Γ, we have

j(𝔠)(δ̄) = ∏_{ϵ∈S} \frac{ϵδz_{0}}{ϵz_{0}} ,

where z_{0} is any element in H_{p}, and S is any set of representatives for ⟨γ⟩\Γ.

Proof. (Cf. [M], theorem 2.8.) Let S′ be any set of representatives for Γ/⟨γ⟩. In view of the formulae (4), for any z_{0} and a in H_{p} we have the chain of equalities

j(𝔠)(δ̄) = ∏_{ϵ∈Γ} \frac{z_{0} − ϵa}{z_{0} − ϵγa} · \frac{δz_{0} − ϵγa}{δz_{0} − ϵa}

= ∏_{ϵ∈S′} ∏_{n=−∞}^{+∞} \frac{z_{0} − ϵγ^{n}a}{z_{0} − ϵγ^{n+1}a} · \frac{δz_{0} − ϵγ^{n+1}a}{δz_{0} − ϵγ^{n}a}

= ∏_{ϵ∈S′} lim_{N→+∞} \frac{z_{0} − ϵγ^{−N}a}{z_{0} − ϵγ^{N+1}a} · \frac{δz_{0} − ϵγ^{N+1}a}{δz_{0} − ϵγ^{−N}a}

= ∏_{ϵ∈S′} \frac{z_{0} − ϵ∞}{z_{0} − ϵ0} · \frac{δz_{0} − ϵ0}{δz_{0} − ϵ∞}

= ∏_{ϵ∈S′} \frac{ϵ^{−1}δz_{0}}{ϵ^{−1}z_{0}} .

Note that (S′)^{−1} is a set of representatives for ⟨γ⟩\Γ, and any set of representatives for ⟨γ⟩\Γ can be obtained in this way. The claim follows.

Lemma 7.3 Let d be an edge of T , let n be a positive integer, and let S be a set of representatives for ⟨γ⟩\Γ. Then the set {ϵ ∈ S : ℓ(ϵd) ≤ n} is finite.

Proof. If {ϵ_{i}} is a sequence of distinct elements of S such that ℓ(ϵ_{i}d) ≤ n, we can find integers k_{i} such that γ^{k_{i}}ϵ_{i}d describes only finitely many edges. This contradicts the discreteness of Γ.

We say that two elements of Γ̄ are linearly independent if they generate a rank two free abelian subgroup of Γ̄.

Proposition 7.4

1) Suppose that 𝔠 and δ̄ are linearly independent in Γ̄. There exists a set S of representatives for ⟨γ⟩\Γ such that if ϵ belongs to S, then all the elements of the coset ϵ⟨δ⟩ belong to S.

2) There exists a set S = S_{0} ∐ S_{1} of representatives for ⟨γ⟩\Γ such that: (i) the set S_{0} contains a finite number of elements which are mapped by the

isomorphism ψ to diagonal matrices of PGL_{2}(ℚ_{p}); (ii) if ϵ belongs to S_{1}, then all the elements of the coset ϵ⟨γ⟩ belong to S_{1}.

Proof. (Cf. [M], lemma 2.7)

1) Consider a decomposition of Γ as disjoint union of double cosets

Γ = ∐_{ϵ̄∈S̄}⟨γ⟩ϵ̄⟨δ⟩.

We claim that we may take S to be {ϵ̄δ^{m} : ϵ̄ ∈ S̄, m ∈ ℤ}. For, if ϵ̄δ^{m} = γ^{r}ϵ̄δ^{n}, we find δ^{m−n} = ϵ̄^{−1}γ^{r}ϵ̄. Projecting this relation to Γ̄ gives m = n.

2) Consider a decomposition of Γ as disjoint union of double cosets

Γ = ∐_{ϵ̄∈S̄}⟨γ⟩ϵ̄⟨γ⟩.

Define S_{1} to be the set of elements of Γ of the form ϵ̄γ^{m}, m ∈ ℤ, where ϵ̄ ∈ S̄ is such that ⟨γ⟩ϵ̄γ^{n} ≠ ⟨γ⟩ϵ̄γ^{m} whenever m ≠ n. As for S_{0}, we claim that it can be taken to be the set of elements ϵ̄ ∈ S̄ which do not satisfy the above condition. In such a case, there is a relation γ^{r}ϵ̄γ^{n} = ϵ̄γ^{m} for integers r and m ≠ n. Then, γ^{r} = ϵ̄γ^{m−n}ϵ̄^{−1}. By projecting this equality to Γ̄, we see that m−n = r, and hence

ϵ̄ and γ^{r} commute. Since γ^{r} is mapped by ψ to the diagonal matrix ( \begin{matrix}\underline{w}^{ιr} & 0 \\ 0 & 1 \end{matrix} ),

where ord_{p}(\underline{w}) = k > 0, a direct computation shows that ϵ̄ is also diagonal (and thus commutes with γ). Now consider the group of all the diagonal matrices in ψ(Γ). Since Γ is discrete, this group is the product of a finite group by a cyclic group containing the group generated by γ. In conclusion, the set S_{0} is finite, and

∐_{ϵ̄∈S_{0}}⟨γ⟩ϵ̄⟨γ⟩ = ∐_{ϵ̄∈S_{0}}⟨γ⟩ϵ̄.

The claim follows.

In the computation of \underline{j}(𝔠)(δ̄), we can assume that either

(I) 𝔠 and δ̄ are linearly independent, or

(II) δ̄ = 𝔠. (In fact, if the rank of Γ̄ is > 1, it is enough to consider elements as in the first case, since the linear map j(𝔠) is completely determined by the values j(𝔠)(δ̄), for 𝔠 and δ̄ linearly independent.) In the case (I), we use the notation S_{1} := S, and the symbol S_{1} will always refer to a choice of representatives for ⟨γ⟩\Γ as in part 1 of proposition 7.4. In the case (II), the symbol S = S_{0} ∐ S_{1} will stand for a choice of representatives as in part 2 of proposition 7.4.

Lemma 7.5 Let δ ∈ Γ be as in case (I) or (II) above. Then, the images in G_{∞} by the reciprocity map of j(𝔠)(δ̄) and ∏_{ϵ∈S_{1}} ϵδz_{0}/ϵz_{0} are equal.

Proof. In the case (I) there is nothing to prove. In the case (II), proposition 7.4 combined with a direct computation shows that

∏_{ϵ∈S_{0}} \frac{ϵγz_{0}}{ϵz_{0}} = \underline{w}^{ι#(S_{0})}.

Since \underline{w} is in the kernel of the reciprocity map, the claim follows.

By lemma 7.5, we are now reduced to compute the product ∏_{ϵ∈S_{1}} ϵδz_{0}/ϵz_{0}, with δ as in case (I) or (II).

We begin with some preliminary remarks. Fix an edge e of level equal to an odd integer n, having v as its vertex of level n. Moreover, assume that the distance of v from v_{0} is also equal to n. Note that the image in M of e is equal to the image in M of a Gross point of conductor cp^{n}.

Given σ̃ ∈ G̃_{n}, define μσ̃ to be equal to 1, resp. −1 if σ̃v has odd, resp. even distance from v_{0}. If ι = 1, observe that μσ̃ depends only on the image σ̄ of σ̃ in Σ under the projection induced by the reciprocity map; in this case, we write μσ̄ instead of μσ̃. If ι = 2, μσ̃ is constant on the elements σ̃ which have the same image in Σ and p-adic valuation of the same parity; moreover, the values of μσ̃ corresponding to different parities are opposite. In this case, if σ̃ projects in Σ to σ̄ and ord_{p}(σ̃) is even, we let μσ̄ stand for μσ̃.

Given an edge d of T , and σ̃ ∈ G̃_{n}, write σ̃e ≡ d if the edge σ̃e is S_{1}-equivalent to d, and σe ≈ d if the element σe of M is Γ-equivalent to d. If ι = 1, the relation σ̃e ≡ d implies that σe ≈ d. If ι = 2, σ̃e ≡ d yields σe ≈ d when ord_{p}(σ̃) is even, and σe ≈ wd, with w ∈ Γ_{+} − Γ, when ord_{p}(σ̃) is odd.

Recall that ω_{d} denotes the order of the stabilizer in Γ of d.

Lemma 7.6

1) Suppose that ι = 1. If the odd integer n is sufficiently large, the projection G̃_{n} → G_{n} induces a ω_{d}-to-1 map

{σ̃ ∈ G̃_{n} : σ̃e ≡ d} → {σ ∈ G_{n} : σe ≈ d}.

2) Suppose that ι = 2. If the odd integer n is sufficiently large, the projection G̃_{n} → G_{n} induces ω_{d}-to-1 maps

{σ̃ ∈ G̃_{n} : σ̃e ≡ d, ord_{p}(σ̃) even} → {σ ∈ G_{n} : σe ≈ d}

and

{σ̃ ∈ G̃_{n} : σ̃e ≡ d, ord_{p}(σ̃) odd} → {σ ∈ G_{n} : σe ≈ wd}.

Proof.

1) Suppose that σ̃_{1}e ≡ d and σ̃_{2}e ≡ d, i.e., σ̃_{1}e = ϵ_{1}d and σ̃_{2}e = ϵ_{2}d, for ϵ_{1} and ϵ_{2} in S_{1}. If σ̃_{1} and σ̃_{2} have the same image in G_{n}, then σ̃_{1} = \underline{w}^{r}σ̃_{2} for r ∈ ℤ, and hence γ^{r}ϵ_{2}d = ϵ_{1}d. If r ≠ 0, i.e., σ̃_{1} ≠ σ̃_{2} and ϵ_{1} ≠ ϵ_{2}, then γ^{r}ϵ_{2}ϵ_{1}^{−1} is a non-trivial element of the stabilizer in Γ of ϵ_{1}d, which is a group of cardinality ω_{d}. Conversely, if σ̃_{1}e = ϵ_{1}d for ϵ_{1} ∈ S_{1} and if β is a non-trivial element of the stabilizer of ϵ_{1}d, we have σ̃_{1}e = βϵ_{1}d. Write βϵ_{1} = γ^{r}ϵ_{2}, r ∈ ℤ, ϵ_{2} ∈ S. Then ϵ_{1} ≠ ϵ_{2}. Note that if n is large, then ϵ_{2} belongs to S_{1}. We obtain \underline{w}^{−r}σ̃_{1}e = ϵ_{2}d. This concludes the proof of part 1.

2) The proof is exactly the same as that of part 1.

Let

path(v_{0}, δv_{0}) = d_{1} − d_{2} + ⋯ + d_{s−1} − d_{s} ∈ ℤ[E(T )].

(Note that s is even, since δ belongs to Γ.) Write d_{j} = {v_{j}^{e}, v_{j}^{o}}, where v_{j}^{e} is the even vertex of d_{j}, and v_{j}^{o} is the odd vertex of d_{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,
v_{s}^{e} = δv_{1}^{e}.

Fix z_{0} ∈ H_{p}(ℚ_{p^{2}}) such that r(z_{0}) = v_{0}. We may choose elements z_{j}^{o} and z_{j}^{e} in H_{p}(ℚ_{p^{2}}) 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

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

so that

∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} = ∏_{ϵ∈S_{1}} (\frac{ϵz_{1}^{o}}{ϵz_{1}^{e}} ) (\frac{ϵz_{2}^{o}}{ϵz_{2}^{e}} )^{−1} ⋯ (\frac{ϵz_{s}^{o}}{ϵz_{s}^{e}} )^{−1} .

Fix a large odd integer n. For each 1 ≤ j ≤ s, let S(j) be the set of elements ϵ in S_{1} such that ϵd_{j} has level ≤ n. Lemma 7.3 shows that the sets S(j) are finite. By lemma 7.1, we have the congruence

(5) ∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ≡ ∏_{ϵ∈S(1)} (\frac{ϵz_{1}^{o}}{ϵz_{1}^{e}} ) ∏_{ϵ∈S(2)} (\frac{ϵz_{2}^{o}}{ϵz_{2}^{e}} )^{−1} ⋯ ∏_{ϵ∈S(s)} (\frac{ϵz_{s}^{o}}{ϵz_{s}^{e}} )^{−1} (mod p^{n}).

Each of the factors in the right hand side of equation (5) can be broken up into three contributions:

∏_{S(j)} \frac{ϵz_{j}^{o}}{ϵz_{j}^{e}} = ∏_{ℓ(ϵv_{j}^{o})<n} ϵz_{j}^{o} · ∏_{ℓ(ϵv_{j}^{e})<n}(ϵz_{j}^{e})^{−1} · ∏_{ℓ(ϵd_{j})=n}(ϵz_{j}^{π_{j}})^{μ_{j}},

where π_{j} = o, resp. π_{j} = e if the distance of the furthest vertex of ϵd_{j} from v_{0} is odd, resp. even, and where we set μ_{j} = 1 in the first case and μ_{j} = −1 in the second case. By our choice of the set S_{1} as in proposition 7.4, the first two factors in this last expression cancel out in the formula (5). Hence we obtain

∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ≡ ∏_{ℓ(ϵd_{1})=n}(ϵz_{1}^{π_{1}})^{μ_{1}} · ∏_{ℓ(ϵd_{2})=n}(ϵz_{2}^{π_{2}})^{−μ_{2}} ⋯ ∏_{ℓ(ϵd_{s})=n}(ϵz_{s}^{π_{s}})^{−μ_{s}} (mod p^{n}).

As in the remarks before lemma 7.6, let e be an edge of level n, such that its vertex v of level n has distance from v_{0} also equal to n. Choose any z ∈ H_{p}(ℚ_{p^{2}}) with r(z) = v. Since G̃_{n} acts simply transitively on the set of edges of level n, lemma 7.1 gives

∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ≡ ∏_{σ̃e≡d_{1}}(σ̃z)^{μσ̃} · ∏_{σ̃e≡d_{2}}(σ̃z)^{−μσ̃} ⋯ ∏_{σ̃e≡d_{s}}(σ̃z)^{−μσ̃} (mod p^{n}).

By lemma 7.6, we obtain

∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ≡ ∏_{σ̃e≡d_{1}} σ̃^{μσ̃} · ∏_{σ̃e≡d_{2}} σ̃^{−μσ̃} ⋯ ∏_{σ̃e≡d_{s}} σ̃^{−μσ̃} · (z^{M}) (mod p^{n}),

where

M = \{ \begin{matrix}⟨path(v_{0}, δv_{0}), ∑_{σ∈G_{n}} μσ̄σe⟩ & if ι = 1 \\ ⟨path(v_{0}, δv_{0}), ∑_{σ∈G_{n}}(μσ̄ − μσ̄w)σe⟩ & if ι = 2. \end{matrix}

By lemma 2.3, the duality ⟨ , ⟩ induces a pairing on H_{1}(G, ℤ) × M. In the case ι = 1, one sees directly that ∑_{σ∈G_{n}} μσ̄σe has trivial image in M, so that M is zero. Consider now the case ι = 2. Since we are interested in computing \underline{j}(𝔠)(δ̄), we need only consider the image of the homomorphism j(𝔠) in N_{sp} ⊗ ℚ_{p}^{×}. Thus, we may view the above pairing as being defined on H_{1}(G, ℤ)^{−} × M_{sp}, where H_{1}(G, ℤ)^{−} indicates the “minus” eigenspace for the action of w on H_{1}(G, ℤ), and we may assume from now on that path(v_{0}, δv_{0}) belongs to H_{1}(G, ℤ)^{−}. One checks that the image ι ∑_{σ∈G_{n}} μσ̄σe in M_{sp} of the element ∑_{σ∈G_{n}}(μσ̄ − wμσ̄)σe is trivial, so that also in this case M is zero. Hence, in all cases

∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ≡ ∏_{σ̃e≡d_{1}} σ̃^{μσ̃} · ∏_{σ̃e≡d_{2}} σ̃^{−μσ̃} ⋯ ∏_{σ̃e≡d_{s}} σ̃^{−μσ̃} (mod p^{n}).

Let rec_{p,n} : G̃_{∞} → G_{n} be the composite of the reciprocity map with the natural projection of G_{∞} onto G_{n}. Suppose that ι = 1. By lemma 7.6, the above relation yields the equality in G_{n}:

rec_{p,n}( ∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ) = ∏_{σe≈d_{1}} σ^{ω_{d_{1}} μσ̄} · ∏_{σe≈d_{2}} σ^{−ω_{d_{2}} μσ̄} ⋯ ∏_{σe≈d_{s}} σ^{−ω_{ds} μσ̄}.

Recall the derivative L′_{p,n}(N_{sp}/H, P_{0}) ∈ N_{sp} ⊗ G_{n} defined in the formula (3) at the end of section 3. By the definition of the bijection κ of lemma 2.2, the right hand side of the above equality can be written as

L′_{p,n}(N_{sp}/H, P_{0})(δ̄) = ⟨path(v_{0}, δv_{0}), ∑_{g∈G_{n}} e_{n}(i)^{g} ⊗ g^{−1}⟩,

where, by an abuse of notation, ∑_{g∈G_{n}} e_{n}(i)^{g} ⊗ g^{−1} is viewed as an element of M_{sp} ⊗ G_{n}. When ι = 2, a similar computation shows that

ιL′_{p,n}(N_{sp}/H, P_{0})(δ̄) = rec_{p,n}( ∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ).

By passing to the limit, one obtains in all cases

ιL′_{p}(N_{sp}/H, P_{0})(δ̄) = rec_{p}( ∏_{ϵ∈S_{1}} \frac{ϵδz_{0}}{ϵz_{0}} ).

In other words, by definition of the map \underline{j},

L′_{p}(N_{sp}/H, P_{0})^{ι} = \underline{j}(𝔠),

as was to be shown.

References

[BC] J-F. Boutot, H. Carayol, Uniformization p-adique des courbes de Shimura: les théorèmes de Cerednik et de Drinfeld, Astérisque 196-197 (1991) 45-158.

[BD1] M. Bertolini, H. Darmon, Heegner points on Mumford-Tate curves, Inv. Math. 126, Fasc. 3 (1996) 413-453.

[BD2] M. Bertolini, H. Darmon, Heegner points, p-adic L-functions and the Cerednik-Drinfeld uniformization, Inv. Math., to appear.

[Boi] X. Boichut, On the Mazur-Tate-Teitelbaum p-adic conjecture for elliptic curves with bad reduction at p, in preparation.

[BSDGP] K. Barré-Sirieix, G. Diaz, F. Gramain, G. Philibert, Une preuve de la conjecture de Mahler-Manin, Inv. Math. 124 (1996) 1-9.

[Cer] I.V. Cerednik, Uniformization of algebraic curves by discrete arithmetic subgroups of PGL_{2}(k_{w}) with compact quotient, (in Russian) Math. Sbornik 100 (1976) 59-88. Transl. in Math. USSR. Sb. 29 (1976) 55-78.

[Dag] H. Daghigh, McGill University PhD thesis, in progress.

[Dr] V.G. Drinfeld, Coverings of p-adic symmetric regions, Funct. Anal. Appl. 10 (1976) 29-40.

[Gr] B.H. Gross, Heights and special values of L-series, CMS Conference Proceedings, H. Kisilevsky, J. Labute, Eds., 7 (1987).

[GS] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms, Inv. Math. 111 (1993) 407-447.

[GVdP] L. Gerritzen, M. Van der Put, Schottky groups and Mumford curves, LNM 817, Springer, 1980.

[GZ] B.H. Gross, D. Zagier, Heegner points and derivatives of L-series, Inv. Math. 84 (1986) 225-320.

[Kl] C. Klingenberg, On p-adic L-functions on Mumford curves, In: p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture, B. Mazur and G. Stevens, eds., Contemporary Math. 165 (1991) 277-315.

[KKT] K. Kato, M. Kurihara, Tsuji, Forthcoming work.

[JL] H. Jacquet, R.P. Langlands, Automorphic forms on GL_{2}, LNM 114, Springer.

[M] Y.I. Manin, p-adic automorphic functions, J. Sov. Math. 5 (1976) 279-333.

[MTT] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inv. Math. 84 (1986) 1-48.

[RT] K. Ribet, S. Takahashi, Parametrization of elliptic curves by Shimura curves and by classical modular curves, Proc. Nat. Acad. Sc. 94 (1997) 11110-11114.

[Sch] P. Schneider, Rigid-analytic L-transforms, In: Number Theory, Noordwijkerhout, LNM 1068, Springer, 1984.

[Sh] T. Shintani, On the construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975) 83-126.

[Se] J-P. Serre, Arbres, amalgames, SL_{2}, Astérisque 46, Soc. Math. de France, 1977.

[T] J. Teitelbaum, Values of p-adic L-functions and a p-adic Poisson kernel, Inv. Math. 101 (1990) 395-410.

[Vi] M-F. Vigneras, Arithmétique des algèbres des quaternions, LNM 800, Springer.