Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting

Massimo Bertolini Henri Darmon Adrian Iovita Michael Spiess

November 23, 2000

Contents

1 Preliminaries 5 1.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Harmonic cocycles . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Schneider’s distribution . . . . . . . . . . . . . . . . . . . . . . 12

2 The p-adic L-function 14 2.1 Complex L-functions . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The basic strategy . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Embeddings of K into B . . . . . . . . . . . . . . . . . . . . . 17 2.4 The p-adic distribution attached to an embedding . . . . . . . 19 2.5 Interpolation of classical special values . . . . . . . . . . . . . 26 2.6 The p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . 35

3 Proof of the main identities 38 3.1 Teitelbaum’s L-invariant . . . . . . . . . . . . . . . . . . . . . 38 3.2 The split case . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 The inert case . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Introduction

Let ϕ = ∑ a_{n}q^{n} be an eigenform of even weight k ≥ 2 on Γ_{0}(N). The classical L-function L(ϕ, s) admits an analytic continuation to the entire complex plane, and a functional equation which relates its values at s and k − s. Of special arithmetic interest for the present work is the central value L(ϕ, k/2).

For example, when k = 2, the Birch and Swinnerton-Dyer conjecture relates the behaviour of L(ϕ, s) at s = 1 to the arithmetic of the abelian variety A_{ϕ} associated to ϕ by the Eichler-Shimura construction. In [MTT], a p-adic variant of the Birch and Swinnerton-Dyer conjecture is formulated with L(ϕ, s) replaced by a p-adic analogue L_{p}(ϕ, s) attached to the cyclotomic ℤ_{p}-extension of ℚ.

When p divides N exactly and a_{p} = 1 (which implies that A_{ϕ} has split multiplicative reduction at p), the function L_{p}(ϕ, s) vanishes at s = 1. In this case the conjectures of [MTT] imply the following relationship between the first derivative L′_{p}(ϕ, 1) and the special value L(ϕ, 1):

L′_{p}(ϕ, 1) = L(ϕ) · L(ϕ, 1)/Ω,  (1)

where Ω is an appropriate real period and L(ϕ), the so-called L-invariant, is an isogeny invariant of A_{ϕ/ℚ_{p}}. It is defined by using the Tate-Morikawa theory of p-adic uniformisation of abelian varieties with multiplicative reduction at p. For example, when ϕ has rational Fourier coefficients, so that A_{ϕ} is an elliptic curve, then

L(ϕ) = \frac{log_{p}(q)}{ord_{p}(q)},

where q ∈ ℚ_{p}^{×} is the Tate period associated to A_{ϕ} over ℚ_{p}, and log_{p} is the branch of the p-adic logarithm such that log p = 0. The conjectured relation (1) was proved by Greenberg and Stevens [GS] using Hida’s theory of padic families of ordinary eigenforms and the two-variable p-adic L-functions attached to them.

Subsequently, much work has gone into formulating and proving similar conjectures for modular forms of higher (even) weight. The following generalisation of (1) was conjectured in this setting,

L′_{p}(ϕ, k/2) = L(ϕ) · L(ϕ, k/2)/Ω,  (2)

with several possible definitions for the L-invariant.

1. The first candidate for L(ϕ), denoted L_{T} (ϕ), was proposed by Teitelbaum [Tei], relying on the Jacquet-Langlands correspondence between forms on Γ_{0}(N) and on certain indefinite quaternion algebras, and on the Cerednik-Drinfeld theory of p-adic uniformisation of Shimura curves. In particular, L_{T} (ϕ) is only defined when ϕ can be associated via the Jacquet-Langlands correspondence to a modular form on such a Shimura curve.

2. A second invariant L_{C}(ϕ) was proposed by Coleman [Co], based on his theory of p-adic integration on the modular curve X_{0}(N).

3. Finally, a third L-invariant L_{F M}(ϕ) was introduced by Fontaine and Mazur [Mz1] in terms of the semistable Dieudonné module of the local padic Galois representation attached to ϕ.

Proofs of equation (2) have been announced by Stevens and by Kato, Kurihara and Tsuji, with L(ϕ) replaced by L_{C}(ϕ) and L_{F M}(ϕ) respectively.

Parallel to these developments, a p-adic conjecture of Birch and Swinnerton-Dyer type (for weight two modular forms) is formulated in [BD1], with the cyclotomic ℤ_{p}-extension of ℚ replaced by the anticyclotomic ℤ_{p}- extension of a quadratic imaginary field K. In the anticyclotomic setting, the p-adic L-function admits a construction which appeals to the p-adic analytic uniformization of Shimura curves, unlike the cyclotomic setting, where the complex uniformisation is required in the definition of modular symbols. Thus the anticyclotomic setting is more amenable to the purely p-adic approach initially proposed by [Sch] for the cyclotomic setting. Furthermore, new exceptional zero phenomena emerge which have no counterpart in the situation explored by Mazur, Tate and Teitelbaum, and which can be approached through other methods, most notably the theory of complex multiplication. (See for example the formulae of [BD2] and [BD3].)

When the prime p splits in K, the article [BD4] gives a proof of formula (1) in the anticyclotomic context. This proof differs from the one given by Greenberg and Stevens in the cyclotomic case, making no use of Hida families or two-variable p-adic L-functions, but relying on p-adic integration on Shimura curves in an essential way. Since this ingredient is precisely the one that enters both into the definition of Teitelbaum’s L-invariant and of the anticyclotomic p-adic L-function, it is natural to build on the methods of [BD4] to prove the anticyclotomic analogue of Teitelbaum’s conjecture. This is one task carried out in this paper. (Cf. theorem 3.4 of section 3.2.)

The case where p is inert in K (which for weight 2 is treated in [BD3]) is different: both the p-adic and classical L-functions vanish at the central point, and the discrepancy between them cannot be accounted for by a simple L-invariant. This case is discussed in section 3.3; theorem 3.5 expresses L′_{p}(ϕ, k/2) in terms of a p-adic Coleman integral between two CM points on the p-adic upper half plane, and can be viewed as a generalisation of one of the main results of [BD3] to weight k > 2. The arithmetic interpretation of this integral will be explored in a future work.

Note that by combining the above mentioned results of Stevens with those of Kato-Kurihara-Tsuji implies the equality of the (cyclotomic) Coleman and Fontaine-Mazur L-invariants. In general, a direct, local comparison of these L-invariants is investigated in [CI] where it is shown that they are equal when they are defined. This applies to the present work, so that the anticyclotomic analogue of L_{T} (ϕ) could be replaced by L_{C}(ϕ) or L_{F M}(ϕ) in the statement of theorem 3.4.

The present work has its roots in a graduate course taught by one of the authors (Iovita) at McGill University in the Spring of 1998, in which the connection between the p-adic L-function of [BD1] and Schneider’s rigid analytic L-transforms was emphasized. The first three authors were then led to recast the proofs of the main results of [BD3] and [BD4] in this framework, while a similar approach was being developed independently by the fourth author. What emerged were proofs of the main results of [BD3] and [BD4] which, while not fundamentally different from the originals, are more elegant and conceptual, and clarify the role of the underlying machinery of p-adic integration. In particular, the use of Schneider’s p-adic boundary distributions and Teitelbaum’s p-adic Poisson transform, implicit in [BD3] and [BD4], is made explicit. This point of view has been influential in later work, notably [Da].

Because of the more conceptual point of view taken in the present work, even the reader interested only in the weight two results of [BD3] and [BD4] may find it helpful to study this paper after setting k = 2. The authors also felt it would be worthwhile to generalize the results of [BD3] and [BD4] to higher weight, a context in which the original approach had seemed unduly burdensome to carry through.

Acknowledgements: It is a pleasure to thank Andreas Schweizer and Glenn Stevens for many inspiring discussions on quaternion algebras, p-adic integration and p-adic L-functions. Bertolini thanks the University of Strasbourg, Darmon the ETH (Zurich), and Iovita the University of Münster for their hospitality while much of this research was carried out. Two of the authors (Darmon and Iovita) were partially supported by CICMA (Centre Interuniversitaire en Calcul Mathématique Algébrique). Finally the authors thank the referee for helpful suggestions which led to significant improvements in the article.

1 Preliminaries

1.1 Review

In this section we review the main ingredients which enter our constructions and fix the notations.

The p-adic upper half plane. Let ℂ_{p} be the completion of the algebraic closure ℚ̄_{p} of ℚ_{p}, and let

H_{p} := ℙ_{1}(ℂ_{p}) − ℙ_{1}(ℚ_{p})

be Drinfeld’s p-adic upper half plane. It has a natural structure of a rigid analytic space (see [SS] where a more general situation is treated or [BD5] §3.1 for details on what is need here and where the notations are consistent with the ones in the current paper.) The group PGL_{2}(ℚ_{p}) acts on H_{p} by fractional linear transformations. Fix once and for all an embedding of ℚ̄ into ℚ̄_{p}, and hence ℂ_{p}. Write T = T_{p} for the Bruhat-Tits tree of PGL_{2}(ℚ_{p}) and denote by E^{→}(T ) the set of its oriented edges. If e ∈ E^{→}(T ) we denote by V (e) the inverse image under reduction of e. It is a wide open annulus in H_{p}.

Quaternion algebras. Let B be an indefinite quaternion algebra over ℚ, i.e., a central simple algebra of rank 4 satisfying

B ⊗_{ℚ} ℝ ≃ M_{2}(ℝ).

An order in B is a subring of B which is of rank 4 as a ℤ-module. A maximal order is an order which is contained in no larger order, and an Eichler order is the intersection of two maximal orders. (For the definition of the level of an Eichler order, see [Vi], ch. I, §4.)

Let B be a definite quaternion algebra, i.e., a quaternion algebra over ℚ satisfying

B ⊗_{ℚ} ℝ ≃ ℍ,

where ℍ = ℝ + ℝi + ℝj + ℝk is Hamilton’s skew field of real quaternions. The algebra B does not satisfy the Eichler condition, and in general contains several distinct conjugacy classes of maximal orders. (The number of such classes is called the type number of B, cf. [Vi], ch. V.)

Fix a prime p for which B splits. A ℤ[1/p]-order in B is a subring of B which is stable under multiplication by ℤ[1/p] and is of rank 4 as a ℤ[1/p]- module. A maximal ℤ[1/p]-order of B is a ℤ[1/p]-order which is contained in no larger ℤ[1/p]-order, and an Eichler ℤ[1/p]-order is the intersection of two maximal ℤ[1/p]-orders. For more details on quaternion algebras see [BD5] §3.2.

Shimura curves. Let S be a finite set of places of ℚ of odd cardinality containing the place ∞, and let N^{+} be an integer which is not divisible by any prime in S. Then a Shimura curve X over ℚ can be associated to the data (S, N^{+}) and described as follows. The presentation of the material is inspired by [Gr1], ch.IV.

Let B be the indefinite quaternion algebra ramified exactly at the places in S − {∞}, let R be an Eichler order in B of level N^{+}, and let R^{max} be a maximal order containing R. Let R_{1}^{×} be the group of elements of R of reduced norm 1, and let Γ_{∞} := ι_{∞}(R_{1}^{×}) ⊂ SL_{2}(ℝ). Then the Riemann surface X(ℂ) can be described as the quotient H/Γ_{∞}. See [BD 5] for more details.

Assume that S − {∞} is non-empty and let p ∈ S be a rational prime. Let B be the (definite) quaternion algebra ramified precisely at the places in S − {p}, and let R be an Eichler ℤ[1/p]-order in B of level N^{+}. Let us fix an isomorphism ι: B ⊗ ℚ_{p} ⟶ M_{2}(ℚ_{p}). It induces a group homomorphism, also denoted ι by abuse of notation, ι: B^{×} ⟶ PGL_{2}(ℚ_{p}). Let Γ = ι(R_{1}^{×}) ⊂ PSL_{2}(ℚ_{p}) be the image under ι of the elements of norm 1 in R. Then the rigid analytic curve X(ℂ_{p}) can be described as the quotient H_{p}/Γ. This description of the Shimura curve X follows from a theorem of Cerednik-Drinfeld, see [Ce] and [Dr]. Detailed proofs of these results can be found in [JL1] and [BC]. p-adic modular forms. For the following definition recall the quotient X(ℂ_{p}) = H_{p}/Γ from the paragraph on Shimura curves above. For all M =

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℂ_{p}), write

(f|_{k}M)(z) := \frac{det(M)^{k/2}}{(cz + d)^{k}} f(Mz).

Definition 1.1 A p-adic modular form of weight k on Γ (or on X) is a global rigid analytic function f on H_{p} satisfying

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

We denote by S_{k}^{rig}(Γ) the space of p-adic modular forms of weight k for the group Γ.

Hecke operators. Let N be a positive integer, and fix a prime p which divides N exactly. The space S_{k}(Γ_{0}(N)) of cusp forms of weight k on Γ_{0}(N), and the space S_{k}^{new}(Γ_{0}(N)) of newforms on this group, are endowed with an action of the commuting Hecke operators T_{n} for each n ≥ 1, defined in the standard way. (See for example [MTT], ch. I, §4.) Note that, following the conventions of [MTT], when ℓ is a prime dividing N the symbol T_{ℓ} is used to denote the Hecke operator sometimes written U_{ℓ} in other articles.

The space S_{k}(Γ_{0}(N)) is also acted on by the Atkin-Lehner involutions W_{ℓ} for each prime ℓ|N. (In [MTT], ch. I, § 5, the involution W_{ℓ} is called w_{ℓ^{a}}, where ℓ^{a} is the maximal power of ℓ dividing N.) The normalized newforms in S_{k}(Γ_{0}(N)) are also eigenvectors for these involutions. Let w = ±1 denote the negative of the eigenvalue of W_{p} acting on ϕ,

W_{p}(ϕ) = −wϕ, so that T_{p}(ϕ) = wp^{\frac{k−2}{2}} ϕ.

The form ϕ is said to be of split multiplicative type if w = 1, and of non-split multiplicative type if w = −1.

Let S be a set of places of ℚ of odd cardinality containing {∞}, and suppose that

N = N^{+} ∏_{ℓ∈S−∞} ℓ,

with N^{+} not divisible by any prime in S. Let X be the Shimura curve attached to the data (S, N^{+}). By abuse of notation, let T_{n} denote the n-th Hecke correspondence on X, defined for example as in [JL2]. When ℓ ∉ S is a prime which does not divide N^{+}, (resp. divides N^{+}), the correspondence T_{ℓ} is of bidegree ℓ+1 (resp. ℓ), just like its X_{0}(N)-counterpart. When ℓ belongs to S, the operator T_{ℓ} corresponds to an involution on X. (Cf. for example [BD1], sec. 1.5, where T_{p} is denoted W_{p}^{−}.) The involution T_{p} can be described in terms of the rigid p-adic uniformisation of X. More precisely, the group Γ arising in this uniformisation is contained in Γ̃ := ι(R^{×}) ⊂ PGL_{2}(ℚ_{p}) with index two. Choose any element γ̃ ∈ Γ̃ − Γ. Then

T_{p}(f) = f|_{k}γ̃ for all f ∈ S_{k}^{rig}(Γ).

The Jacquet-Langlands correspondence. Crucial to the constructions of this paper is the Jacquet-Langlands correspondence which allows the normalised newform ϕ = ∑ a_{n}q^{n} on Γ_{0}(N) to be replaced by a modular form on a Shimura curve associated to the appropriate quaternion algebra. Let X be the Shimura curve defined in the previous section.

Theorem 1.2 (Jacquet-Langlands) There exists a p-adic modular form f of weight k for the group Γ satisfying

T_{ℓ}(f) = a_{ℓ}f, ∀ℓ ∉ S,
T_{p}(f) = wf.

This function is unique, up to scaling by a non-zero scalar in ℂ_{p}.

Recall that w is the negative of the sign of the Atkin-Lehner involution W_{p} acting on ϕ. Thus f is fixed by the involution T_{p}, and hence is Γ̃-invariant, if and only if ϕ is of split multiplicative type at p.

1.2 Harmonic cocycles

Harmonic cocycles are treated in both [Sch] and [Tei] but as our point of view is slightly different we prefer to restate the definitions and main properties of these objects here. Let Γ be the subgroup of PSL_{2}(ℚ_{p}) defined above and let M be a ℂ_{p}[Γ]-module, where the action of Γ is written on the left.

Definition 1.3 An M-valued harmonic cocycle on T is an M-valued function on E^{→}(T ) satisfying

c(e) = −c(ē),  ∑_{source(e)=v} c(e) = 0, ∀v ∈ T .

Write C_{har}(M) for the ℂ_{p}-vector space of M-valued harmonic cocycles, and C_{har}(M)^{Γ} for the space of Γ-equivariant harmonic cocycles, i.e., harmonic cocycles c satisfying

c(γe) = γ · c(e), for all γ ∈ Γ.

Let P_{k−2} be the (k − 1)-dimensional ℂ_{p}-vector space of polynomials of degree ≤ k − 2 with coefficients in ℂ_{p}. It is endowed with a right action of GL_{2}(ℚ_{p}) (and actually of PGL_{2}(ℚ_{p})) by the rule

P (x) · β := \frac{(cx + d)^{k−2}}{(det(β))^{\frac{k−2}{2}}} P (\frac{ax + b}{cx + d}), β = ( \begin{matrix}a & b \\ c & d \end{matrix} ) , P ∈ P_{k−2}.

In this way P_{k−2}^{∨} := hom(P_{k−2}, ℂ_{p}) inherits a left PGL_{2}(ℚ_{p})-action by

β · R(P) := R(P · β), ∀ P ∈ P_{k−2}.

Let U := ad^{0}(B)⊗det be the representation of B^{×} consisting of the elements of B of reduced trace zero on which B^{×} acts on the right by the rule

u · b := (b^{−1}ub) det(b) = b̄ub.

The function which to every u ∈ U associates the polynomial P_{u}(x) ∈ P_{2} (with coefficients in ℚ_{p}) given by

P_{u}(x) = trace (ι(u) ( \begin{matrix}x & −x^{2} \\ 1 & −x \end{matrix} )) = trace (ι(u) ( \begin{matrix}x \\ 1 \end{matrix} ) ( 1 −x )) (3)

satisfies P_{u·b}(x) = P_{u}(x) · ι(b), and hence induces a linear map from U ⊗ ℂ_{p} to P_{2} which intertwines the B^{×} and PGL_{2}(ℚ_{p}) actions.

The symmetric bilinear ℚ-valued pairing on U arising from the Cartan-Killing form on ad^{0}(B),

⟨u_{1}, u_{2}⟩_{1} := \frac{1}{2}trace(u_{1}ū_{2})

satisfies ⟨u_{1} · b, u_{2}⟩_{1} = ⟨u_{1}, u_{2} · b̄⟩_{1}, so that ⟨u_{1} · b, u_{2} · b⟩_{1} = ⟨u_{1}, u_{2}⟩_{1}norm(b)^{2}. This pairing gives rise to a perfect symmetric pairing ⟨ , ⟩_{j} on Sym^{j}(U) by the rule

⟨u_{1} ⋯ u_{j}, v_{1} ⋯ v_{j}⟩_{j} = \frac{1}{j!} ∑_{σ∈S_{j}}⟨u_{1}, v_{σ1}⟩_{1} ⋯ ⟨u_{j}, v_{σj}⟩_{1}.  (4)

Thus the space Sym^{\frac{k−2}{2}} (U) is identified with its own dual via the pairing ⟨ , ⟩_{\frac{k−2}{2}} . Dualizing the natural surjection

Sym^{\frac{k−2}{2}} (U) ⊗ ℂ_{p} ⟶ P_{k−2}  (5)

induced from the map U ⟶ P_{2} of equation (3) yields an inclusion

P_{k−2}^{∨} ⟶ Sym^{\frac{k−2}{2}} (U) ⊗ ℂ_{p}.  (6)

In this way P_{k−2}^{∨} inherits a bilinear pairing arising from ⟨ , ⟩_{\frac{k−2}{2}} , denoted simply by ⟨ , ⟩.

If L is any subfield of ℂ_{p}, write P_{k−2}(L) for the natural image of the map Sym^{\frac{k−2}{2}} (U) ⊗ L ⟶ P_{k−2} arising from equation (5), and let P_{k−2}^{∨}(L) := hom(P_{k−2}(L), L). This definition yields a rational structure on P_{k−2} and P_{k−2}^{∨}. Note that in general P_{k−2}(ℚ) is not the space of polynomials with rational coefficients.

Definition 1.4 A harmonic cocycle of weight k on T is a P_{k−2}^{∨}-valued harmonic cocycle.

Set C_{har}(k) := C_{har}(P_{k−2}^{∨}), and C_{har}(k)^{Γ} := C_{har}(P_{k−2}^{∨})^{Γ}.

Following Schneider [Sch], [Tei], associate to a rigid analytic modular form f of weight k on Γ a harmonic cocycle c_{f} ∈ C_{har}(k) by the rule

c_{f}(e)(r) = res_{e}(f(z)r(z)dz), r(z) ∈ P_{k−2},  (7)

where res_{e} is the p-adic annular residue along the oriented wide open annulus V (e) in ℙ_{1}(ℂ_{p}), defined by

res_{e}(ω) := res_{V (e)}(ω|_{V (e)}).

The fact that c_{f} is harmonic follows from the p-adic residue formula. (Cf. [Sch].)

Definition 1.5 For γ ∈ PGL_{2}(ℚ_{p}) denote by |γ| :=ord_{p}(det (\overline{γ}))(mod 2), where \overline{γ} is any lift of γ in GL_{2}(ℚ_{p}).

Lemma 1.6 The cocycle c_{f} satisfies the following transformation with respect to Γ̃

c_{f}(γe) = w^{|γ|}γ · c_{f}(e), ∀γ ∈ Γ̃.

Proof: For all r ∈ P_{k−2}, and γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ),

c_{f}(γe)(r) = res_{γe}(f(z)r(z)dz) = res_{e}(f(γz)r(γz)d(γz))
= res_{e}(w^{|γ|} \frac{(cz + d)^{k}}{(detγ)^{\frac{k}{2}}} f(z)\frac{(cz + d)^{2−k}}{(detγ)^{\frac{2−k}{2}}} (r(z) · γ)\frac{detγ}{(cz + d)^{2}} dz)
= w^{|γ|}res_{e}(f(z)(r(z) · γ)dz) = w^{|γ|}c_{f}(e)(r(z) · γ)
= w^{|γ|}γ · c_{f}(e)(r).

Set

⟨c_{f}, c_{f}⟩ = ∑_{e∈ E^{→}(T )/Γ} w_{e}⟨c_{f}(e), c_{f}(e)⟩,

where the sum is taken over a set of representatives for the Γ-orbits in E^{→}(T ), the integer w_{e} is the cardinality of the stabiliser of e in Γ, and the pairing on P_{k−2}^{∨} is the one defined by equation (6) and the sentence after it.

Let now ϕ be a normalized newform of weight k ≥ 2 on Γ_{0}(N) and f ∈ S_{k}^{rig}(Γ) be the rigid analytic modular form on Γ attached to it by section 1.1.

The form f is only well-defined up to multiplication by a non-zero scalar. The following definition is introduced to remove this ambiguity.

Definition 1.7 An eigenform f ∈ S_{k}^{rig}(Γ) is said to be normalised if its associated cocycle c_{f} ∈ C_{har}(k) satisfies

⟨c_{f}, c_{f}⟩ = 1.

Note that the normalised eigenform f ∈ S_{k}^{rig}(Γ) attached to ϕ is well defined, up to a sign. Suppose from now on that f is normalised in this way.

Recall that K_{ϕ} ⊂ ℂ_{p} is the finite extension of ℚ generated by the Fourier coefficients of ϕ. The normalised eigenform f satisfies the following rationality property.

Lemma 1.8 The P_{k−2}^{∨}-valued cocycle c_{f} takes values in P_{k−2}^{∨}(K_{f}), where K_{f} is an extension of K_{ϕ} of degree ≤ 2.

Proof: The space of Γ-equivariant P_{k−2}^{∨}(ℚ)-valued cocycles gives a rational structure C_{har}(k)_{ℚ}^{Γ} on C_{har}(k)^{Γ} which is preserved by the Hecke operators, and on which the pairing ⟨ , ⟩ takes values in ℚ. Hence the one-dimensional eigenspace of C_{har}(k)^{Γ} attached to ϕ contains a K_{ϕ}-rational vector c̃_{f} ∈ C_{har}(k)_{ℚ}^{Γ} ⊗ K_{ϕ}. Since ⟨c̃_{f}, c̃_{f}⟩ belongs to K_{ϕ},the lemma follows, with K_{f} = K_{ϕ}(\sqrt{⟨c̃_{f}, c̃_{f}⟩}).

1.3 Schneider’s distribution

This section reviews a construction of Schneider which associates to a rigid analytic modular form f of weight k on Γ a “boundary distribution” μ_{f}.

An end of T is an equivalence class of sequences (e_{n})_{n=1}^{∞} of elements e_{n} ∈ E^{→}(T ) satisfying target(e_{n}) = source(e_{n+1}), and target(e_{n+1}) ≠ source(e_{n}), two such sequences (e_{n}) and (e′_{n}) being identified if there exist N and N′ with e_{N+j} = e′_{N′+j} for all j ≥ 0. Let E_{∞}(T ) be the space of ends on T . It is identified with ℙ_{1}(ℚ_{p}) by the rule

(e_{n}) ↦ lim_{n} b_{e_{n}}(∞),

where b_{e_{n}} is the coset in PGL_{2}(ℚ_{p}) associated to e_{n}. (See [BD 5] §3.1.) The space E_{∞}(T ) thus inherits a natural topology coming from the p-adic topology on ℙ_{1}(ℚ_{p}). Each edge e ∈ E^{→}(T ) corresponds to a compact open subset U(e) of E_{∞}(T ) consisting of all ends having a representative which contains e.

Locally analytic distributions. We would like now to recall a few facts on p-adic distributions. We will adopt the point of view and notations from [St]. Suppose that X ⊂ ℙ_{1}(ℂ_{p}) is a compact subset. Let us recall the following well known definition. Let α, β ∈ ℙ_{1}(ℂ_{p}) be represented by α = (a_{1}, a_{2}) and β = (b_{1}, b_{2}) with a_{1}, a_{2}, b_{1}, b_{2} ∈ O_{ℂ_{p}} such that g.c.d.(a_{1}, a_{2}) =g.c.d.(b_{1}, b_{2}) = 1 and set d(α, β) := p^{−ord_{p}(a_{1}b_{2}−a_{2}b_{1})}. Then d is a well defined non-archimedean metric on ℙ_{1}(ℂ_{p}). For every integer n ≥ 1 denote by

X[n] := {x ∈ ℙ_{1}(ℂ_{p}) | there is y ∈ X with d(x, y) ≤ p^{−n}}.

Then X[n] is an affinoid subdomain of ℙ_{1}(ℂ_{p}), in fact it is a finite disjoint union of affinoid balls. Let A(X[n]) be the ℚ_{p}-algebra of rigid analytic functions on X[n], and denote by ρ^{(n)} : A(X[n]) ⟶ A(X[n + 1]) the restriction map. Then for each n ≥ 1, A(X[n]) is a Banach algebra over ℚ_{p} for the spectral norm || • ||_{n} and ρ^{(n)} is a continuous homomorphism of Banach algebras. Let us denote

A(X) := lim_{→} A(X[n])

and call it the ℚ_{p}-algebra of locally analytic functions on X. We endow this algebra with the Morita topology. Let us now fix X = ℙ_{1}(ℚ_{p}) and write simply A for A(X). Let us go back to the notations at the beginning of this section. The cocycle c_{f} associated to f by equation (7) gives rise to a continuous linear functional μ_{f} on the space of locally analytic functions on ℚ_{p} with compact support, defined by the basic relation

μ_{f}(r · χ_{U(e)}) := ∫_{U(e)} r(x)dμ_{f}(x) = c_{f}(e)(r),  (8)

for all r ∈ P_{k−2} and e edge of T such that U(e) ⊂ ℚ_{p}. Here χ_{U(e)} denotes the characteristic function of U(e) in ℚ_{p}. (See [Sch]). Then μ_{f} extends uniquely to the space, denoted A_{k}, of locally analytic ℂ_{p}-valued functions on ℙ_{1}(ℚ_{p}) having a pole of order at most k − 2 at ∞. (See [Tei], proposition 9.) This linear functional will be also denoted by μ_{f}.

Lemma 1.9 If r is any polynomial of degree ≤ k − 2, then

∫_{ℙ_{1}(ℚ_{p})} r(x)dμ_{f}(x) = 0.

Proof: Let v be any vertex of T . By the finite additivity of μ_{f},

∫_{ℙ_{1}(ℚ_{p})} r(x)dμ_{f}(x) = ∑_{e,source(e)=v} ∫_{U(e)} r(x)dμ_{f}(x).

The lemma follows from (8) combined with the harmonicity of c_{f}.

Now define a weight k − 2 action of PGL_{2}(ℚ_{p}) on A_{k} by the rule

(φ ∗ γ)(x) := \frac{(cx + d)^{k−2}}{(detγ)^{\frac{k−2}{2}}} φ(γ ∗ x) for φ ∈ A_{k}, γ ∈ PGL_{2}(ℚ_{p}).

We have Lemma 1.10 The boundary distribution μ_{f} satisfies the following transformation property with respect to Γ̃

μ_{f}(φ ∗ γ) = w^{|γ|}μ_{f}(φ) for all γ ∈ Γ̃ and φ ∈ A.

Proof: It is enough to check the formula for functions of the form r · χ_{U(e)}, where r is any polynomial of degree ≤ k − 2, e any edge of T and now χ_{U(e)} is the characteristic function of U(e) in ℙ_{1}(ℚ_{p}). For this apply lemma 1.6 and the definition of μ_{f}.

Remark: Let P ∈ P_{k−2} having coefficients in ℚ_{p}. Let us consider the free A-module of rank one M_{P} := P · A ⊂ A_{k} and endow it with the topology induced from the Morita topology of A. Then the restriction of μ_{f} to M_{P} is a continuous linear functional i.e. a distribution. (See [Tei] proposition 9.)

The p-adic Poisson transform. The following result can be viewed as a p-adic analogue of the Poisson inversion formula: it allows a rigid analytic modular form to be recovered from its associated boundary distribution.

Proposition 1.11 (Teitelbaum) Let f be a rigid analytic modular form of weight k on Γ and let μ_{f} be the associated distribution on ℙ_{1}(ℚ_{p}). Then

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

Proof: See [Tei], theorem 3. Note that the integrand \frac{1}{z−t} is a bounded analytic function of t so that the integral in the theorem converges.

2 The p-adic L-function

Let K be an imaginary quadratic field. The goal of this chapter is to define the anticyclotomic p-adic L-function attached to ϕ and K. As explained in section 2.2, its construction is inspired by Schneider’s “rigid analytic” definition [Sch] of p-adic L-functions in the cyclotomic case. For simplicity, the following assumption is made throughout.

Assumption 2.1 The discriminant of K is relatively prime to the level N of ϕ.

Thus one may write

N = pN^{+}N^{−},

where N^{+} (resp. N^{−}) is divisible only by primes which are split (resp. inert) in K. The following assumption is also made for simplicity:

Assumption 2.2 The integer N^{−} is square-free.

2.1 Complex L-functions

The classical L-function attached to ϕ is defined by

L(ϕ, s) = ∏_{ℓ|N}(1 − a_{ℓ}ℓ^{−s})^{−1} ∏_{ℓ|̸N}(1 − a_{ℓ}ℓ^{−s} + ℓ^{k−1−2s})^{−1} = ∑_{n=1}^{∞} a_{n}n^{−s},

where the Euler product is taken over the rational primes. It has an analytic continuation and a functional equation relating its values at s and k − s. More precisely, the function

Λ(ϕ, s) := N^{s/2}(2π)^{−s}Γ(s)L(ϕ, s)

extends holomorphically to the entire complex plane and satisfies

Λ(ϕ, s) = (−1)^{k/2}w_{N}Λ(ϕ, k − s),

where w_{N} = ±1 is the eigenvalue of the Atkin-Lehner involution W_{N} acting on ϕ.

More germane to the present article is the L-function of ϕ over K, defined by

L(ϕ/K, s) = L(ϕ, s)L(ϕ, ε, s),

where ε is the quadratic Dirichlet character attached to K and L(ϕ, ε, s) = ∑ a_{n}ϵ(n)n^{−s} is the twisted L-function. For each prime ℓ ∤ N write (x − α_{ℓ})(x − β_{ℓ}) := x^{2} − a_{ℓ}x + ℓ^{k−1}, and set a(ℓ^{t}) = α_{ℓ}^{t} + β_{ℓ}^{t}. The function L(ϕ/K, s) factors into an Euler product,

L(ϕ/K, s) = ∏_{v|N}(1 − a_{ℕv}ℕv^{−s})^{−1} ∏_{v|̸N}(1 − a(ℕv)ℕv^{−s} + ℕv^{k−1−2s})^{−1}, (9)

the product being taken this time over all the finite places v of K. The function L(ϕ/K, s) can also be expressed as a sum of partial L-functions,

L(ϕ/K, s) = ∑_{𝔞}L(ϕ/K, 𝔞, s).

Here the 𝔞’s range over the ideal classes of K, and

L(ϕ/K, 𝔞, s) = ( ∑_{(n,N)=1^{n=1}}^{∞} \frac{ε(n)}{n^{2s−k+1}} ) (∑_{n=1}^{∞} \frac{a_{n}r_{𝔞}(n)}{n^{s}}) ,  (10)

where r_{𝔞}(n) is the number of integral ideals of norm n in the class of 𝔞.

More generally, let O_{c} be the order of K of conductor c, and let G_{c} = Pic(O_{c}) be the Picard group of rank one projective O_{c}-modules. If 𝔞 is a class in G_{c}, define L(ϕ/K, 𝔞, s) as in equation (10). The group G_{c} is identified with the Galois group of an abelian extension K_{c} of K, the so-called ring class field of conductor c. If χ : G_{c} ⟶ ℂ^{×} is any character, set

L(ϕ/K, χ, s) := ∑ χ(𝔞)L(ϕ/K, 𝔞, s),

the sum being taken over all elements of G_{c}. Like L(ϕ/K, s), and unlike the partial L-functions L(ϕ/K, 𝔞, s), the functions L(ϕ/K, χ, s) admit an Euler product decomposition analogous to (9), and a simple functional equation. Proposition 2.3 Suppose that χ is a ring class character of conductor p^{n} where n > 0. The L-function L(ϕ/K, χ, s) satisfies the functional equation

Λ(ϕ/K, χ, s) := (2π)^{−2s}Γ(s)^{2}(ND)^{s}L(ϕ/K, χ, s) = −ε(N^{−})Λ(ϕ/K, χ, k − s)

relating its values at s and k − s. For a discussion in weight 2, see [PR], sec. 5.2.

The anticyclotomic p-adic L-function studied in this work interpolates the special values of L(ϕ/K, χ, s) at the central point s = k/2, where χ ranges over ring class characters of p-power conductor. Proposition 2.3 shows that L(ϕ/K, χ, k/2) vanishes identically for all such χ, if the number of primes dividing N^{−} is even. This justifies the following basic assumption which is made throughout the paper: Assumption 2.4 The sign ϵ_{K} := −ϵ(N^{−}) in the functional equation of proposition 2.3 is equal to 1, i.e., in light of assumptions 2.1 and 2.2, N^{−} is a product of an odd number of primes.

2.2 The basic strategy

The construction of the p-adic L-function attached to ϕ and K proceeds as follows.

1. Write N^{−} = ℓ_{1} ⋯ ℓ_{r} for the prime factorisation of N^{−}, and set S = {p, ∞, ℓ_{1}, …, ℓ_{r}}. Note that S is a set of places of odd cardinality, by assumption 2.4. Let X be the Shimura curve associated to the data (S, N^{+}) as in section 1.1, and let ω be the modular form of weight k on X attached to ϕ by the Jacquet-Langlands theorem (theorem 1.2).

2. Applying the Cerednik-Drinfeld theorem, see 1.1 let f ∈ S_{k}^{rig}(Γ) be the rigid analytic modular form on H_{p} associated to ω. It satisfies invariance properties under a p-adic discrete group Γ arising from the definite quaternion algebra B ramified at the primes in S − {p}. More precisely, Γ = ι(R_{1}^{×}), where R is an Eichler ℤ[1/p]-order of level N^{+} in B and R_{1}^{×} is the group of elements of reduced norm 1 in R^{×}. Assume that f is normalised as in definition 1.7.

3. The p-adic L-function L_{p}(f, K, s) is defined as a p-adic Mellin transform of Schneider’s distribution μ_{f} attached to f as in section 1.3, along the p-adic points of a (suitably chosen) maximal torus in B^{×}/ℚ^{×}, isomorphic to K^{×}/ℚ^{×}.

It is the last step of this construction which remains to be explained. In sections 2.3 and 2.4, a distribution on (K⊗ℚ_{p})^{×}/ℚ_{p}^{×} is defined using Schneider’s distribution. Section 2.5 explains how this distribution interpolates special values of classical L-functions of ϕ over K. The construction of the p-adic L-function is then carried out in section 2.6

2.3 Embeddings of K into B

Let O_{K} be the ring of integers of K, and O = O_{K}[1/p] its ring of p-integers. An orientation of the Eichler order R is a surjective ring homomorphism

𝔬 : R ⟶ (ℤ/N^{+}ℤ) × ∏_{ℓ|N^{−}} 𝔽_{ℓ^{2}}.

The pair (R, 𝔬) is called an oriented Eichler order. Likewise, an orientation on O is a surjective homomorphism O ⟶ (ℤ/N^{+}ℤ) × ∏_{ℓ|N^{−}} 𝔽_{ℓ^{2}}. This amounts to choosing, for each prime ℓ dividing N^{+}, a prime ideal of K above ℓ, and, for each prime ℓ dividing N^{−}, an identification of the residue field of K at ℓ with 𝔽_{ℓ^{2}}. Fix orientations on R and O once and for all.

An embedding Ψ : K ⟶ B is called an oriented optimal embedding of O into R if

1. Ψ(K) ∩ R = Ψ(O), so that Ψ induces an embedding of O into R.

2. Ψ is compatible with the chosen orientations on O and R in the sense that the following diagram commutes

\begin{matrix}O & ⟶^{Ψ} & R \\ ↘ & & 𝔬 ↙ \end{matrix} ; (ℤ/N^{+}ℤ) × ∏_{ℓ|N^{−}} 𝔽_{ℓ^{2}}.

The group R^{×} acts naturally by conjugation on the set of oriented optimal embeddings. Write emb(O, R) for the set of all oriented optimal embeddings of O into R, taken modulo conjugation by R_{1}^{×}.

Denote by Δ = Pic(O) the Picard group of projective modules of rank one over O. (In classical language, it is the quotient of the class group of K, by the subgroup generated by the classes of the prime ideals above p.) The group Δ acts naturally on emb(O, R) as follows (see also [Gr2], sec. 1 and 3). Let 𝔞 ⊂ O be a projective O-module representing a class α ∈ Δ, chosen in such a way that 𝔞 ⊗ ℤ_{ℓ} = O ⊗ ℤ_{ℓ} for all ℓ|N^{+}N^{−}, and let Ψ : K ⟶ B be an oriented optimal embedding of O into R. The right order of the left R-ideal RΨ(𝔞), denoted R_{𝔞}, is an Eichler order of level N^{+} in B, which inherits an orientation from the one on R, since R_{𝔞} ⊗ ℤ_{ℓ} = R_{ℓ} for all ℓ|N^{+}N^{−}. The right action of Ψ(O) on RΨ(𝔞) yields an embedding Ψ̃_{𝔞} : O ⟶ R_{𝔞}, which is compatible with the orientations on O and R_{𝔞}. Since all Eichler ℤ[1/p]-orders of level N^{+} in B are conjugate, there exists a ∈ B^{×} such that

ord_{p}(aā) ≡ 0 (mod 2),  a ∈ R_{ℓ}^{×} for all ℓ|N^{+}N^{−},  R = aR_{𝔞}a^{−1}.

(11)

Set

α · Ψ := aΨ̃_{𝔞}a^{−1}.  (12)

It can be checked that α · Ψ is an oriented optimal embedding of K into B, and that its image in emb(O, R) depends only on the image of 𝔞 in Pic(O) and of Ψ in emb(O, R), so that (12) gives a well-defined action of Δ on emb(O, R). The following result is proved as in [Gr2], sec. 3.

Proposition 2.5 The group Δ = Pic(O) acts freely on emb(O, R). The set emb(O, R)is the disjoint union of two Δ-orbits under this action.

In particular, emb(O, R) is finite and has cardinality 2h, where h = #Δ.

2.4 The p-adic distribution attached to an embedding

Let Ψ : K ⟶ B be an oriented optimal embedding of O into R. It induces an embedding

Ψ : K_{p} ⟶ B_{p}, where K_{p} := K ⊗ ℚ_{p}.

Let us recall that we have fixed an isomorphism ι : B_{p} ⟶ M_{2}(ℚ_{p}). The map ιΨ : K_{p} ⟶ M_{2}(ℚ_{p}) induces an embedding of K_{p}^{×}/ℚ_{p}^{×} into PGL_{2}(ℚ_{p}), which will also be denoted by ιΨ by abuse of notation. Thus Ψ gives rise to an action of K_{p}^{×}/ℚ_{p}^{×} on the boundary ℙ_{1}(ℚ_{p}) of H_{p} by the rule

α∗x := ιΨ(α)(x),  α ∈ K_{p}^{×}/ℚ_{p}^{×}, x ∈ ℙ_{1}(ℚ_{p}).  (13)

Choose a base point c ∈ ℙ_{1}(ℚ_{p}). The element b_{Ψ} := Ψ(\sqrt{−D}) is an element of B of reduced trace 0. Let P_{Ψ} be the polynomial P_{b_{Ψ}} in P_{2} defined as in equation (3) of section 1.2. Note that

P_{Ψ} · (ιΨ(α)) = P_{Ψ},  (14)

for all α ∈ K_{p}^{×}.

Case 1: p is inert in K. The (compact) torus ιΨ(K_{p}^{×}/ℚ_{p}^{×}) ⊂ PGL_{2}(ℚ_{p}) acts simply transitively on the boundary ℙ_{1}(ℚ_{p}). The choice of c determines an isomorphism η_{Ψ,c} : K_{p}^{×}/ℚ_{p}^{×} ⟶ ℙ_{1}(ℚ_{p}). The torus ιΨ(K_{p}^{×}) has two fixed points in H_{p}, denoted a and ā, which belong to K_{p} and are interchanged by Gal(K_{p}/ℚ_{p}). Let us now fix the base point c = ∞. We have a natural homeomorphism K_{p}^{×}/ℚ_{p}^{×} ≅ K_{p,1}^{×} sending x (mod ℚ_{p}^{×}) to x/\overline{x} ∈ K_{p,1}^{×}. Here K_{p,1}^{×} denotes the compact subgroup of K_{p}^{×} of elements of norm 1. Let us denote by η_{Ψ} the composition

η_{Ψ} : G := K_{p,1}^{×} ⟶ K_{p}^{×}/ℚ_{p}^{×} ⟶ ℙ_{1}(ℚ_{p}).

We then have the following formulas

η_{Ψ}(α) = \frac{(aα − ā)}{α − 1},  η_{Ψ}^{−1}(x) = \frac{x − ā}{x − a}.  (15)

Pullback by η_{Ψ} and η_{Ψ}^{−1} on functions preserve local analyticity and so we get a natural, continuous isomorphism

η_{Ψ∗} = (η_{Ψ}^{−1})^{∗}: A(G) ⟶ A,

where the notations of section 1.3 are used. Recall that we have defined a natural, continuous linear functional μ_{f} on the free A-module of rank one

PΨ^{\frac{k−2}{2}} · A ⊂ A_{k} in section 1.3. We will use it to define the locally analytic distribution μ_{f,Ψ,c} on G = K_{p,1}^{×}, which will be simply denoted by μ_{Ψ} whenever f and the basepoint c are fixed, by

μ_{f,Ψ,c}(φ) = μ_{Ψ}(φ) := μ_{f}(PΨ^{\frac{k−2}{2}} × (η_{Ψ}^{−1})^{∗}(φ)), for φ ∈ A(G).

Case 2: p is split. The torus ιΨ(K_{p}^{×}/ℚ_{p}^{×}) ⊂ PGL_{2}(ℚ_{p}) has precisely two fixed points a, b ∈ ℙ_{1}(ℚ_{p}), and it acts simply transitively on the complement ℙ_{1}(ℚ_{p}) − {a, b}. The choice of a base point c in this complement determines an identification

η_{Ψ,c} : K_{p}^{×}/ℚ_{p}^{×} ⟶ ℙ_{1}(ℚ_{p}) − {a, b}

by the rule

η_{Ψ,c}(α) := α∗c.  (16)

Choose c so that the Möbius transformation sending (a, b, c) to (0, 1, ∞) has the property that its determinant has even p-adic valuation. Base points satisfying this property will be called Ψ-normalised. Let w(α) be the locally constant function of α ∈ ℚ_{p}^{×} defined by

w(α) := w^{ord_{p}(α)},  (17)

where w = ±1 is the sign defined in section 1.1.

Now choose a prime 𝔭 of K above p. This choice defines identifications

K_{p}^{×}/ℚ_{p}^{×} = (K ⊗ ℚ_{p})^{×}/ℚ_{p}^{×} = (K_{𝔭}^{×} × K× ; 𝔭̄)/ℚ_{p}^{×} and a group isomorphism

ϵ: K_{p}^{×}/ℚ_{p}^{×} ⟶ K_{𝔭}^{×} = ℚ_{p}^{×},

which has the basic property that if x ∈ K^{×}, then

ϵ([x]) = \frac{x}{x̄} ∈ K_{𝔭}^{×} = ℚ_{p}^{×}.

Let u_{0} be a generator of the rank one group O^{×}/ℤ[1/p]^{×} and let u ∈ ℚ_{p}^{×} be the image of u_{0}/ū_{0} in K_{𝔭} = ℚ_{p}, normalised so that ord_{p}(u) > 0. In other words u = ϵ([u_{0}]). Let us now fix a Ψ-normalized base point c ∈ ℙ_{1}(ℚ_{p}) − {a, b}, and denote by η_{Ψ} the following composition

η_{Ψ} : ℚ_{p}^{×} ⟶^{ϵ^{−1}} K_{p}^{×}/ℚ_{p}^{×} ⟶^{η_{Ψ,c}} ℙ_{1}(ℚ_{p}) − {a, b}.

It extends to a fractional linear transformation η_{Ψ} : ℙ_{1}(ℚ_{p}) ⟶ ℙ_{1}(ℚ_{p}) sending 0 to a, ∞ to b and 1 to c. Obviously, η_{Ψ} extends uniquely to a fractional linear transformation denoted by the same symbol ℙ_{1}(ℂ_{p}) ⟶ ℙ_{1}(ℂ_{p}).

Distributions on ℚ_{p}^{×} with compact support. Let us recall the notations of section 1.3, where from now on X := ℙ_{1}(ℚ_{p}). Fix a parameter z on ℙ_{1}(ℂ_{p}), which fixes by restriction a parameter on ℙ_{1}(ℚ_{p}). This determines a natural inclusion ℚ_{p}^{×} ⊂ ℙ_{1}(ℚ_{p}). For every integer n ≥ 1 denote by

X_{c}[n] := X[n] − (B[0, p^{−n}] ∪ B[∞, p^{−n}]).

Then for all n ≥ 1 X_{c}[n] is a ℚ_{p}-affinoid subdomain of ℂ_{p}^{×} ⊂ ℙ_{1}(ℂ_{p}) (in fact it is a finite, disjoint union of closed balls in ℂ_{p}^{×}). For every n ≥ 1 let us also denote by A[n] := A(X[n]) and by A_{c}[n] the ring of rigid analytic functions on X_{c}[n] extended by zero to X[n]−{0, ∞}. A[n] and A_{c}[n] are naturally Banach algebras over ℚ_{p}. We have natural, continuous maps ρ^{(n)} : A[n] ⟶ A[n + 1] which is restriction, ψ^{(n)} : A[n] ⟶ A_{c}[n] which is restriction from X[n] to X_{c}[n] followed by extension by zero to X[n]−{0, ∞} and ψ_{!}^{(n)} : A_{c}[n] ⟶ A[n] which is extension by zero. We denote by ρ_{c}^{(n)} : A_{c}[n] ⟶ A_{c}[n + 1] the composition ψ^{(n+1)} ∘ ρ^{(n)} ∘ ψ_{!}^{(n)}. Then Lemma 2.6 For every n ≥ 1 the following diagram commutes:

A_{c}[n] ⟶^{ρ_{c}^{(n)}} A_{c}[n + 1]
ψ_{!}^{(n)} ↓  ↓ ψ_{!}^{(n+1)}
A[n] ⟶^{ρ^{(n)}} A[n + 1].

Proof: The proof of the lemma is a calculation which is omitted.

We now define

A_{c} := lim_{→} A_{c}[n],

where the connecting homomorphisms are the ρ_{c}^{(n)}’s. We’ll refer to the ring A_{c} as the ring of locally analytic functions on ℚ_{p}^{×} with compact support. We endow this ring with the Morita topology. Lemma 2.6 implies that we have a natural, continuous ring homomorphism ψ_{!} : A_{c} ⟶ A induced by the family of maps ψ_{!}^{(n)}. We’ll refer to this map as the “extension by zero” map.

Let us recall the map η_{Ψ} from the beginning of this paragraph. It has the basic property that if x, y ∈ ℚ_{p}^{×} then

η_{Ψ}(xy) = Ψ(ϵ^{−1}(x)) ∗ η_{Ψ}(y).

Notice that both η_{Ψ} and η_{Ψ}^{−1} preserve local analyticity. Let us recall from section 1.3 that we have defined a natural, continuous linear functional μ_{f}

on the A-module PΨ^{\frac{k−2}{2}} · A ⊂ A_{k}. We use μ_{f} to define a continuous linear functional μ_{f,Ψ,c} on A_{c}, which will be simply denoted by μ_{Ψ} for the rest of the section and whenever f and c are fixed. Let φ ∈ A_{c} then

μ_{f,Ψ,c}(φ) = μ_{Ψ}(φ) := μ_{f}(PΨ^{\frac{k−2}{2}} × (η_{Ψ}^{−1})^{∗}(ψ_{!}(φ × w))),

where w is the function defined in equation (17).

Let u ∈ ℚ_{p}^{×} with ord_{p}(u) > 0 defined at the beginning of the paragraph and let us consider the group generated by u, i.e. let σ := ⟨u⟩ ⊂ ℚ_{p}^{×}. Our next goal is to define a locally analytic distribution on G := ℚ_{p}^{×}/σ. Let us first denote by F a fundamental domain in ℚ_{p}^{×} for the action of σ, for example:

F := {x ∈ ℚ_{p}^{×} | p|u| ≤ |x| ≤ 1}.

Then F is a compact open subset of ℚ_{p}^{×} and ℚ_{p}^{×} = ∪_{i∈ℤ}u^{i}F. To give a function φ: G ⟶ ℂ_{p} is equivalent to giving a function φ: ℚ_{p}^{×} ⟶ ℂ_{p} such that φ(ux) = φ(x) for all x ∈ ℚ_{p}^{×}. Such a function is determined by its restriction to F. We say that a function φ on G is locally analytic if φχ_{F} is the restriction to ℚ_{p}^{×} of an element of A_{c}, where χ_{F} is the characteristic function of the set F ⊂ ℚ_{p}^{×}.

Definition 2.7 Let A(G) denote the ℚ_{p}-algebra of locally analytic functions on G.

Let us remark that the map “restriction to F” defines an isomorphism between A(F) and A(G). We will identify these two algebras whenever convenient and endow A(G) with the natural topology on A(F).

We define an action of the discrete subgroup σ ⊂ ℚ_{p}^{×} on A_{c} by (φ × u)(x) := φ(ux) for all x ∈ ℚ_{p}^{×} and φ ∈ A_{c}. Obviously this action is continuous and it defines an action of σ on linear functionals on A_{c}.

Proposition 2.8 u × μ_{Ψ} = μ_{Ψ}, i.e. μ_{Ψ} is σ-invariant.

Proof: First of all, we know that Ψ(ϵ^{−1}(u)) = Ψ(u_{0}) ∈ Γ̃ and w × u = w^{ord_{p}(u)}w. Let φ ∈ A_{c} and let us write η instead of η_{Ψ} for this proof. Then

(u × μ_{Ψ})(φ) = μ_{Ψ}(φ × u) = μ_{f}(PΨ^{\frac{k−2}{2}} × (η^{−1})^{∗}(ψ_{!}((φ × u) × w))).

Let us denote g := ψ_{!}(φ × w). Then

PΨ^{\frac{k−2}{2}} × (η^{−1})^{∗}(ψ_{!}((φ × u) × w))(x) = w^{ord_{p}(u)}(PΨ^{\frac{k−2}{2}} × (η^{−1})^{∗}(g × u))(x)

= w^{ord_{p}(u)}PΨ^{\frac{k−2}{2}} (x)g(uη^{−1}(x)) = w^{ord_{p}(u)}PΨ^{\frac{k−2}{2}} (x)g(η^{−1}(Ψ(u_{0}) ∗ x))

= w^{ord_{p}(u)}PΨ^{\frac{k−2}{2}} (x)((η^{−1})^{∗}(g))(Ψ(u_{0}) ∗ x)

= w^{ord_{p}(u)}[(PΨ^{\frac{k−2}{2}} (η^{−1})^{∗}(g)) ∗ Ψ(u_{0})](x).

Therefore PΨ^{\frac{k−2}{2}} × (η^{−1})^{∗}(g × u) = w^{ord_{p}(u)}(PΨ^{\frac{k−2}{2}} × (η^{−1})^{∗}(g)) ∗ Ψ(u_{0}). But

ord_{p}(u) + |Ψ(u_{0})| = 0 (mod 2) and the result follows upon applying lemma 1.10.

Lemma 2.9 Let V be a compact open subset of ℚ_{p}^{×} and φ ∈ A_{c}. Then

∫_{u^{i}V} φdμ_{Ψ} = ∫_{V} (φ × u^{i})dμ_{Ψ},

for all i ∈ ℤ.

Proof: We have

∫_{u^{i}V} φdμ_{Ψ} = μ_{Ψ}(φχ_{u^{i}V}) = μ_{Ψ}(φ(χ_{V} × u^{−i})) =

= μ_{Ψ}(((φ × u^{i})χ_{V}) × u^{−i}) = μ_{Ψ}((φ × u^{i})χ_{V}) = ∫_{V} (φ × u^{i})dμ_{Ψ}.

Corollary 2.10 If φ ∈ A(G) then

∫_{u^{i}F} φdμ_{Ψ} = ∫_{F} φdμ_{Ψ},

for all i ∈ ℤ.

Definition 2.11 We define a locally analytic distribution μ_{f,Ψ,c} (denoted μ_{Ψ} whenever f and c are fixed) on G i.e. a continuous linear functional on A(G) by

μ_{f,Ψ,c}(φ) = ∫_{G} φdμ_{f,Ψ,c} := ∫_{F} φdμ_{f,Ψ,c}, for all φ ∈ A(G).

For the rest of this section, the assumption that w = 1 will be made. First of all

Remark: If φ ∈ A then PΨ^{\frac{k−2}{2}} × (η_{Ψ}^{−1})^{∗}(φ) is an element of the free A-

module of rank one PΨ^{\frac{k−2}{2}} · A. Therefore μ_{f,Ψ,c} extends to a continuous linear functional on A denoted by the same symbol and given by the formula:

μ_{f,Ψ,c}(φ) = μ_{Ψ}(φ) := μ_{f}(PΨ^{\frac{k−2}{2}} × (η_{Ψ}^{−1})^{∗}(φ)).

We end this section with a few properties of the above defined distribution on G, which will be used in chapter 3.

Lemma 2.12 Let φ ∈ A[n_{0}] for some n_{0} ∈ ℕ and suppose that i_{0} ∈ ℕ is such that ord_{p}(u^{i_{0}}) ≥ n_{0}. Moreover suppose that φ(0) = φ(∞) = 0. Then there is a constant C_{φ} such that for all i ≥ i_{0}, if V denotes the closed ball around 0 or around ∞ of radius |u^{i}| then

| ∫_{V} φdμ_{Ψ}| ≤ C_{φ}|u^{i}|.

Proof: It is enough to treat the case V = B[0, |u^{i}|]. Let D := B[0, |u^{i_{0}}|], then V = u^{i−i_{0}}D. Using the results above,

∫_{u^{i−i_{0}}D} φdμ_{Ψ} = ∫_{D}(φ × u^{i−i_{0}})dμ_{Ψ}.

Because φ(0) = 0, it follows that φ(z) = ∑_{j≥1} a_{j}z^{j} for z ∈ D, and

(φ × u^{i−i_{0}})(z) = u^{i−i_{0}}(∑_{j≥1} a_{j}u^{(j−1)(i−i_{0})}z^{j}).

Therefore

| ∫_{V} φdμ_{Ψ}| ≤ |u^{i}|(|u^{−i_{0}}| sup_{j≥1}(|a_{j}u^{(j−1)(i−i_{0})} ∫_{D} z^{j}dμ_{Ψ}(z)|) ≤ |u^{i}|C_{φ},

where

C_{φ} := |u^{−i_{0}}| sup_{j≥1} |a_{j} ∫_{D} z^{j}dμ_{Ψ}(z)|.

Note that the supremum is finite by the continuity of μ_{Ψ}.

Finally we have the following

Proposition 2.13 Let φ ∈ A be such that φ(0) = φ(∞) = 0. Then

a) ∑_{i∈ℤ}(φ|_{ℚ_{p}^{×}} × u^{i}) ∈ A(G).

b) We have

∫_{ℙ_{1}(ℚ_{p})} φdμ_{Ψ} := μ_{Ψ}(φ) = ∫_{G}(∑_{i∈ℤ}φ|_{ℚ_{p}^{×}} × u^{i})dμ_{Ψ}.

Proof: a) Let us denote by ψ(x) := ∑_{i∈ℤ} φ(u^{i}x), for all x ∈ ℚ_{p}^{×}. Then the convergence of the series and the invariance of ψ with respect to the action of σ is clear. Moreover if denote by ψ_{N} := ∑_{i=−N}^{N}(φ|_{ℚ_{p}^{×}} × u^{i}) for every N ∈ ℕ, then one sees that ψ = lim_{N→∞} ψ_{N} where the limit is uniform on compact open subsets of ℚ_{p}^{×}. We leave it to reader to show that ψ|_{F}, ψ_{N}|_{F} ∈ A(G) for all N and that the limit above is a limit in A(G). In the application of this proposition to the proof of theorem 3.3 these facts will be obvious.

b) We have

∫_{ℙ_{1}(ℚ_{p})} φdμ_{Ψ} − ∫_{F} ψ_{N}dμ_{Ψ} = ∫_{ℙ(ℚ_{p})} φdμ_{Ψ} − ∫_{∪_{i=−N}^{N} u^{i}F} φdμ_{Ψ} =

= ∫_{B[0,|u^{N+1}|]} φdμ_{Ψ} + ∫_{B[∞,|u^{N} |]} φdμ_{Ψ}.

For N big enough, φ ∈ A[ord_{p}(u^{N})] and from lemma 1.6 we have

| ∫_{ℙ(ℚ_{p})} φdμ_{Ψ} − ∫_{F} ψ_{N}dμ_{Ψ}| ≤ |u^{N}|C′_{φ} ⟶^{N→∞} 0.

The fact that μ_{Ψ} is a continuous linear functional on A_{c} imply that

lim_{N→∞} ∫_{F} ψ_{N}dμ_{Ψ} = ∫_{F} ψdμ_{Ψ} = ∫_{G} ψdμ_{Ψ}.

2.5 Interpolation of classical special values

This section describes how the distributions μ_{f,Ψ} interpolate special values of the classical L-function L(ϕ/K, s) at s = k/2, justifying for the definition of the p-adic L-function that is given in section 2.6, definitions 2.20 and 2.21. The interpolation properties of μ_{f,Ψ} are summarized in formula (18) and proposition 2.19. Strictly speaking, these are not required for the construction of the p-adic L-function given in section 2.6, but merely provide motivation and justification for it. Therefore, in a first reading, the reader may wish to skip directly to section 2.6 after looking at (18) and the statement of proposition 2.19, postponing the proofs for later.

Galois-theoretic preliminaries. An abelian extension L/K is called anticyclotomic if it is Galois over ℚ and if the involution in Gal(K/ℚ) acts as −1 on Gal(L/K) by conjugation. Let K_{∞} denote the maximal anticyclotomic extension of K which is unramified outside p. Let H denote the Hilbert class field of K, and let H_{p} be the field fixed by a Frobenius element at p in Gal(H/K) (i.e., the maximal unramified abelian extension of K in which p splits completely). Let K_{n} denote the ring class field of K of conductor p^{n}, so that K_{0} = H. The fields K_{n} form a tower of extensions containing the anticyclotomic ℤ_{p}-extension:

ℚ ⊂ K ⊂ H_{p} ⊂ H ⊂ K_{1} ⊂ ⋯ ⊂ K_{n} ⊂ · · · .

Class field theory provides a Galois theoretic interpretation of the p-adic group G defined in section 1.3. More precisely,

G = Gal(K_{∞}/H_{p})

if O_{K}^{×} = {±1}. For simplicity assume from now on that O_{K}^{×} = {±1}, a condition which is satisfied as soon as Disc(O_{K}) < −4. We denote

G^{0} = Gal(K_{∞}/H), G^{n} = Gal(K_{∞}/K_{n}), Δ = Gal(H_{p}/K),

(where Δ is the group defined in section 2.3). Write

G̃ := Gal(K_{∞}/K).

This group fits into the exact sequence

1 ⟶ G ⟶ G̃ ⟶ Δ ⟶ 1,

and it can be identified with the Picard group of rank one projective Omodules 𝔞 equipped with a trivialisation at p, i.e., a fixed isomorphism K_{p} ⟶ 𝔞 ⊗ ℚ_{p}. More precisely, let K̂ denotes the finite adèles of K, and write

Ô = ∏_{ℓ} O_{ℓ}, Ô_{#} = ∏_{ℓ≠p} O_{ℓ}.

Class field theory identifies G̃ with K̂^{×}/K^{×}ℚ̂^{×}Ô_{#}^{×}. An element in this coset space with representative (g_{ℓ}) corresponds to the projective module 𝔞 = K ∩ ∏_{ℓ≠p} O_{ℓ}g_{ℓ}, with trivialisation at p sending 1 to g_{p}. Let Λ be the set of pairs (Ψ, c) taken modulo conjugation by R_{1}^{×}, where Ψ is an oriented optimal embedding of O into R, and c is a Ψ-normalised base point (if p is inert in K, every base point is defined to be Ψ-normalised.)

Lemma 2.14 The action of Δ on emb(O, R) lifts to a free action of G̃ on Λ, which is the union of two orbits under this action.

The description of this action proceeds along the same lines as in section 2.3. Let 𝔞 ⊂ O be a representative for a class α ∈ G̃, i.e., a projective rank one O-module together with a K_{p}-generator t for 𝔞⊗ℚ_{p}. Let R_{𝔞} denote the right order of the left R-ideal RΨ(𝔞) of section 2.3, and let g = Ψ(t) ∈ R_{𝔞} ⊗ ℚ_{p}. Choosing an element a ∈ B^{×} satisfying the conditions of equation (11) of section 2.3, and letting Ψ̃_{𝔞} be the embedding defined there, set

α · (Ψ, c) = (Ψ′, c′), where Ψ′ = aΨ̃_{a}a^{−1}, c′ = ι(aga^{−1})(c).

By fixing a representative Ψ for an element of emb(O, R), and a Ψ-normalised base point c, a locally analytic distribution μ_{f,K} on G̃ will be defined as follows. First, for all δ ∈ Δ choose and fix a lift α_{δ} ∈ G̃. This allows us to view G̃ as the disjoint union of the orbits α_{δ}G for δ ∈ Δ. Let φ: G̃ ⟶ ℂ_{p} be a function. We say that φ is locally analytic if

(φ|_{α_{δ}G}) × α_{δ} ∈ A(G) for all δ ∈ Δ,

where let us recall (φ × α_{δ})(x) := φ(α_{δ}x) for all x ∈ G. The set of locally analytic functions on G̃ will be denoted A(G̃). To define μ_{f,K}, let φ ∈ A(G̃). Then set

μ_{f,K}(φ) = ∫G̃ φdμ_{f,K} := ∑_{δ∈Δ} μ_{f,Ψ_{δ},c_{δ}}(φ|_{α_{δ}G} × α_{δ}),

where, if δ ∈ Δ then (Ψ_{δ}, c_{δ}) = α_{δ}(Ψ, c).

For further applications, we want to calculate the values of μ_{f,K} on locally constant functions on G̃.

Lemma 2.15 For all α ∈ G̃,

μ_{f,K}(αG^{n}) = μ_{f,Ψ′,c′}(G^{n}), where (Ψ′, c′) = α · (Ψ, c).

Proof: Let δ be the projection of α on Δ. Then, by the definition above,

μ_{f,K}(αG^{n}) := μ_{f,K}(χ_{αG^{n}}) = μ_{f,Ψ_{δ},c_{δ}}(aG^{n}) := μ_{f,Ψ_{δ},c_{δ}}(χ_{aG^{n}}),

where a = α_{δ}^{−1}α ∈ G. We have

(Ψ′, c′) = α(Ψ, c) = a(α_{δ}(Ψ, c)) = a(Ψ_{δ}, c_{δ}) = (Ψ_{δ}, Ψ_{δ}(a) ∗ c_{δ}).

A calculation, using the fact that both c′ and c_{δ} are Ψ′ = Ψ_{δ} normalised based points, shows that

μ_{f,Ψ_{δ},c_{δ}}(aG^{n}) = μ_{f,Ψ_{δ},Ψ_{δ}(a)∗c_{δ}}(G^{n}) = μ_{f,Ψ′,c′}(G^{n}).

The lemma shows that the restriction of μ_{f,K} to locally constant functions is independent of the choices of the α_{δ}’s. Moreover

Lemma 2.16 The distribution μ_{f,K} depends on the choice of (Ψ, c), only up to translation by an element of G̃, and up to multiplication by w. Its restriction to G is equal to μ_{f,Ψ,c}.

Proof: If (Ψ, c) is replaced by (γΨγ^{−1}, γc) with γ ∈ R_{1}^{×}, the associated distribution is unchanged, by lemma 1.10. If (Ψ, c) and (Ψ′, c′) = α · (Ψ, c) are in the same G̃-orbit, the associated distributions differ by translation by α. Finally, if (Ψ, c) and (Ψ′, c′) belong to different G̃-orbits, the associated distributions differ by translation by an element of G̃, and multiplication by w.

Choose a complex embedding of the field K_{f} introduced in lemma 1.8. Since the cocycle c_{f} takes values in P_{k−2}^{∨}(K_{f}) (lemma 1.8), the p-adic distribution μ_{f,K} can also be viewed as a complex-valued distribution, against which locally constant ℂ-valued functions on G̃ can be integrated.

In particular, let χ : G̃ ⟶ ℂ^{×} be a continuous character of finite order. By lemma 2.16, the integral ∫G̃ χ(α)dμ_{f,K}(α) depends on the choice of (Ψ, c) only up to multiplication by a root of unity, so that its complex norm is well defined.

Define the multiplier M(χ) by

M(χ) = \{ \begin{matrix}1 if χ is ramified; \\ (1 − wχ(frob_{𝔭}))(1 − wχ̄(frob_{𝔭})) otherwise. \end{matrix}

Write u_{K} = #O_{K}^{×}/2, let D_{K} be the discriminant of K, and denote by (ϕ, ϕ) the Petersson scalar product of ϕ with itself.

The distribution μ_{f,K} is expected to satisfy the following interpolation property with respect to special values of the L-function of ϕ over K.

|∫G̃ χ(x)dμ_{f,K}(x)|^{2} = M(χ)u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, χ, k/2)}{(ϕ, ϕ)}.  (18)

The remainder of this section elucidates the relation between (18) and the calculations explained in [Gr2] for weight k = 2 and in [Ha] for general even weight. These articles treat the case—too restrictive for the application to (18)—of a modular form ϕ of prime level N, and an unramified character χ. Specializing temporarily to the setting considered by [Gr2] and [Ha], take N^{−} = N to be prime and set N^{+} = 1. Let B denote the definite quaternion algebra ramified at N^{−} and choose a maximal ℤ-order \underline{R} of B. Let B_{0} be the space of elements of B of trace 0, and set

ˆ\underline{R} := ∏_{ℓ}(\underline{R} ⊗ ℤ_{ℓ}),  B̂ := ˆ \underline{R} ⊗ ℚ.

The article [Ha] introduces a vector bundle V by the rule

V = ( ˆ\underline{R}^{×}\B̂^{×} × Sym^{\frac{k−2}{2}} (B_{0}))/B^{×},

where the action of B^{×} on Sym^{\frac{k−2}{2}} (B_{0}) is as described in page 543 of [Ha]. Thanks to equation (3) of section 1.2, the Picard group of V maps to the space of P_{k−2}-valued functions on ˆ \underline{R}^{×}\B̂^{×} which are invariant under the action of B^{×}. Let P (V) denote this image, i.e.,

P (V) := \{c : ˆ \underline{R}^{×}\B̂^{×} ⟶ P_{k−2} | c(xb) = c(x) · ι(b), for all b ∈ B^{×}\}.

It is a finite-dimensional ℂ_{p}-vector space since an element c is entirely determined by its values on a system of representatives for the finite double coset space ˆ \underline{R}^{×}\B̂^{×}/B^{×} (cf. [Vi], ch. V). The space P (V) is also endowed with a linear action of the Hecke operators T_{n} defined as in [Ha], sec. 4, and is isomorphic, as a Hecke module, to the space of modular forms of weight k on Γ_{0}(N^{−}) with ℂ_{p}-coefficients.

In the paragraph before ch. 4 of [Ha], the author defines Heegner elements in V. These elements are described by pairs (Ψ, \underline{R}_{Ψ}) modulo conjugation by B^{×}, where 1. \underline{R}_{Ψ} is a maximal ℤ-order of B. (By strong approximation there are

only finitely many such orders, up to conjugation in B^{×}.)

2. Ψ : K ⟶ B is an oriented embedding which is optimal relative to O_{K} and \underline{R}_{Ψ}, in the sense that Ψ(K) ∩ \underline{R}_{Ψ} = Ψ(O_{K}).

Such a pair yields an element in V represented by ((b_{ℓ})_{ℓ}; Ψ(\sqrt{−D})^{\frac{k−2}{2}} ), where (b_{ℓ}) represents the coset describing \underline{R}_{Ψ} as in [Ha]. The group Pic(O_{K}) acts transitively on the set of Heegner elements in a manner similar to proposition 2.5. (Cf. also [Ha].)

Fix a Heegner element (Ψ, \underline{R}_{Ψ}), and, for 𝔞 ∈ Pic(O_{K}) let

v_{𝔞} = 𝔞 · (Ψ, \underline{R}_{Ψ}),

viewed as an element of P (V). If χ is any character of Pic(O_{K}), write v_{χ} = ∑_{𝔞} χ(𝔞)v_{𝔞}, and let v_{f,χ} ∈ P (V) ⊗ ℂ be the projection of this vector to the f-isotypic component for the action of the Hecke algebra. Recall the natural pairing ⟨ , ⟩ of equation (6) of section 1.2. It naturally induces a symmetric pairing on the spaces C_{har}(k) and C_{har}(P_{k−2})^{Γ} of Γ-equivariant harmonic cocycles, as well as on the space P (V). Use the same symbol ⟨ , ⟩ to denote these pairings, by abuse of notation. The following formula is proved by Gross in weight 2 [Gr2], and by Hatcher for even weight k ≥ 2 (cf. proposition 8.2. of [Ha]):

Proposition 2.17 (Gross, Hatcher)

⟨v_{f,χ}, v_{f,χ̄}⟩ = u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, χ, k/2)}{(ϕ, ϕ)}.

Remark: Note that the factor (\frac{k−2}{2})!^{2} which is present in prop. 8.2 of [Ha] does not appear in proposition 2.17, because of the different normalisation that is used for the inner product on P_{k−2}^{∨}. (Compare equation (4) of section 1.2 with equation (3.1) of [Ha].)

To establish equation (18), it is first necessary to capture the modular forms on Γ_{0}(N^{+}N^{−}p). In order to do this, introduce an auxiliary Γ_{0}(N^{+}p)- structure and replace the maximal order \underline{R} by an Eichler order of level N^{+}p in the definite quaternion algebra of discriminant N^{−}, to be denoted by the same letter, \underline{R}, from now on. By strong approximation, the double coset space ˆ \underline{R}^{×}\B̂^{×}/B^{×} is identified with \underline{R}_{p}^{×}\B_{p}^{×}/R^{×}, where R := \underline{R}[1/p]. Note that R is an Eichler ℤ[1/p]-order of level N^{+} in B, as in the previous sections. Under ι the space \underline{R}_{p}^{×}\B_{p}^{×} is identified with the space E^{→}(T ) of edges of T .

Thus P (V) is identified with the space of functions c̃ : E^{→}(T ) ⟶ P_{k−2} satisfying

c̃(γ^{−1}e) = c̃(e) · γ, for all γ ∈ Γ̃ = ι(R^{×}).

Suppose that c̃ is an eigenfunction for the Hecke correspondences satisfying T_{n}c̃ = a_{n}c̃ for all n. The fact that ϕ is new at p implies that

∑_{source(e)=v} c̃(e) = 0, ∑_{target(e)=v} c̃(e) = 0,  for all v ∈ V(T ).

(Cf. [BD3], prop. 1.4.) However, c̃ does not satisfy the rule c̃(ē) = −c̃(e) in general, but merely the formula

c̃(ē) = −wc̃(e),

where −w is the eigenvalue of the Atkin-Lehner involution at p acting on ϕ, which is equal to −1 when ϕ is of split multiplicative type, and 1 otherwise.

To turn c̃ into a harmonic cocycle when ϕ is of non-split multiplicative type, recall the base vertex v^{∘} on T and call a vertex even (resp. odd) if its distance from v^{∘} is even (resp. odd). An edge is then said to be positively oriented if its source is even and its target is odd, and negatively oriented otherwise. Note that this orientation is reversed by elements in Γ̃ − Γ, but is preserved by the index two subgroup Γ. Now given c̃, define similarly as in the proof of prop. 1.4 of [BD3],

c(e) = \{ \begin{matrix}c̃(e) if e is positively oriented, \\ −c̃(ē) if e is negatively oriented. \end{matrix}

The assignment c̃ ↦ c defines a Hecke equivariant isomorphism between the space of p-new vectors in P (V) and the space C_{har}(P_{k−2})^{Γ} of Γ-equivariant harmonic cocycles on T with values in P_{k−2}.

For the purposes of equation (18), it is also necessary to generalize the notion of Heegner elements. A Heegner element of level p^{n} is now described by a pair (Ψ, \underline{R}_{Ψ}), taken modulo conjugation by B^{×}, satisfying 1. \underline{R}_{Ψ} is an Eichler order of level N^{+}p in B; by strong approximation it

can be assumed without loss of generality that \underline{R}_{Ψ}[1/p] = R, and this is done from now on.

2. Ψ is an optimal embedding of O_{n}, the order in O_{K} of conductor p^{n}, i.e., Ψ(K)∩\underline{R}_{Ψ} = Ψ(O_{n}). Note that when n ≥ 1, such optimal embeddings always exist, whereas an optimal embedding of O_{K} into an Eichler order of level N^{+}p only exists if p is split in K/ℚ.

To give a pair (Ψ, \underline{R}_{Ψ}) as above is equivalent to giving a pair (Ψ, e) satisfying

1. Ψ(K) ∩ R = Ψ(O),

2. e is an edge of T which is at distance n from the vertices fixed by ιΨ(G^{0}).

For example, the Eichler order \underline{R}_{Ψ} is equal to R ∩ ι^{−1}(M_{e}), where M_{e} ⊂ M_{2}(ℤ_{p}) is the local Eichler order of level p attached to the edge e. The group G̃_{n} = Pic(O_{n}) acts naturally on the set of Heegner elements of level p^{n}, and the action of G = K_{p}^{×}/ℚ_{p}^{×} is given by

α · (Ψ, e) = (Ψ, ιΨ(α)e).

Fix a Heegner element (Ψ, \underline{R}_{Ψ}) of level p^{n} and write as before v_{𝔞} := 𝔞·(Ψ, \underline{R}_{Ψ}), with 𝔞 ∈ G̃_{n}. Since it is only the images of the elements v_{𝔞} in P (V) which matter in the calculations, the v_{𝔞} are viewed as elements of P (V) from now on. The projection of v_{𝔞} onto the f-isotypic component of C_{har}(P_{k−2})^{Γ}, denoted v_{𝔞,f}, can be viewed as elements of C_{har}(P_{k−2})^{Γ} by the identification of the previous paragraph, and can be written as

v_{𝔞,f} = ⟨v_{𝔞}, c_{f}⟩c_{f}^{∨},

where c_{f}^{∨} is the vector attached to f in the basis for C_{har}(P_{k−2})^{Γ} dual to the basis of normalised eigenforms in C_{har}(k)^{Γ}. It follows from the definition of μ_{f,K} that, after a suitable choice of base point (Ψ, c) of Λ, compatible with the choice (Ψ, \underline{R}_{Ψ}) made to define v_{𝔞}, that

⟨v_{𝔞,f}, c_{f}⟩ = μ_{f,K}(𝔞̃ · G^{n}),

where 𝔞̃ is any lift of 𝔞 ∈ G̃_{n} to G̃. Hence if χ factors through G̃_{n},

⟨v_{f,χ}, c_{f}⟩ = ± ∫G̃ χ(x)dμ_{f,K}(x), where v_{f,χ} = ∑ χ(𝔞)v_{𝔞,f}.  (19)

The following generalization of the proposition 2.17 is expected to hold

Conjecture 2.18 : In the notations above,

⟨v_{f,χ}, v_{f,χ̄}⟩ = M(χ)u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, χ, k/2)}{(ϕ, ϕ)}, if n = 0;  (20)
⟨v_{f,χ}, v_{f,χ̄}⟩ = u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, χ, k/2)}{(ϕ, ϕ)} if n > 0.  (21)

Remark The identities of conjecture 2.18 should follow from a direct generalization of the calculations carried on in [Gr2] and [Ha], but these calculations are not present in the literature. See also [Dag] and [Va] for a discussion of related topics. Granting conjecture 2.18 and combining it with formula (19) yields

|∫G̃ χ(x)dμ_{f,K}(x)|^{2} = M(χ)u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, χ, k/2)}{(ϕ, ϕ)},

and (18) follows.

Suppose now that p is split and that Ψ_{j} (1 ≤ j ≤ h) is one of the oriented optimal embeddings of O into R. Choose an even vertex v_{j} of T whose stabiliser under the action of K_{p}^{×} induced by Ψ_{j} is (O_{K} ⊗ ℤ_{p})^{×}. Let us now recall the notations of section 1.3 i.e. u_{0} be a generator of the rank one group O^{×}/ℤ[1/p]^{×} such that the image u of \frac{u_{0}}{ū_{0}} in ℚ_{p}^{×} has ord_{p}(u) > 0. Denote by δ_{j} = ιΨ_{j}(u_{0}). Given an ordered edge e, let sgn(e) be equal to 0 if e is positively oriented, and 1 if it is negatively oriented.

Proposition 2.19 Assume conjecture 2.18. Then

(∑_{j=1}^{h} ∑_{v_{j}→δ_{j}v_{j}} w^{sgn(e)}c_{f}(e)(PΨ_{j}^{\frac{k−2}{2}} ))^{2} = u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, k/2)}{(ϕ, ϕ)},

where the inner sum on the left is taken over all edges in the path joining v_{j} to δ_{j}v_{j}.

Proof: We will apply formula (18) to the trivial character χ_{triv}. Let us fix a j as above and calculate

∫_{G} χ_{triv}dμ_{f,Ψ_{j},c_{j}} = ∫_{F} w(α)dμ_{f,Ψ_{j},c_{j}}(α),

where c_{j} is the corresponding Ψ_{j}-normalised base point, and let us recall from section 1.3 that F is a fundamental domain for the action of u on ℚ_{p}^{×}. Let now E_{j} denote the set of oriented edges of T with same source as the edges on the path from v_{j} to δ_{j}v_{j}, but not containing these edges. The geometry of the action of δ_{j} on T , as explained in [BD4] §5, implies that the set

F′ := ∪_{e∈E_{j}}U(e)

is a fundamental domain for the action of δ_{j} on ℙ_{1}(ℚ_{p}) − {a_{j}, b_{j}}. Therefore, η_{Ψ_{j}}^{−1}(F′) is a fundamental domain for the action of u on ℚ_{p}^{×}. Using the fact that the base point c_{j} is Ψ_{j}-normalised we have

∫_{F} w(α)dμ_{f,Ψ_{j},c_{j}}(α) = ∫_{F′} w(α)PΨ_{j}^{\frac{k−2}{2}} dμ_{f}(α) =

= ∑_{e∈E_{j}} ∫_{U(e)} w(α)PΨ_{j}^{\frac{k−2}{2}} dμ_{f}(α) = ∑_{e∈E_{j}} w^{sgn(e)}c_{f}(e)(PΨ_{j}^{\frac{k−2}{2}} ).

Proposition 2.19 now follows using the harmonicity of c_{f} and the definition of E_{j}.

Remark: The path on T joining v_{j} to δ_{j}v_{j} is a p-adic analogue of the geodesic paths on the classical upper half plane associated by Shintani to narrow ideal classes in real quadratic fields. The integrals of classical modular forms against such paths yield special values of their L-series over the associated real quadratic field. For this reason, the path in H_{p} from z_{0} to δ_{j} ∗ z_{0}, or the path in T from v_{j} to δ_{j}v_{j}, are sometimes called p-adic Shintani cycles attached to the optimal embeddings Ψ_{j} (1 ≤ j ≤ h). See for example [BD4].

2.6 The p-adic L-function

If p is split in K, let log : ℂ_{p}^{×} ⟶ ℂ_{p} be a branch of the p-adic logarithm, normalised so that log(u) = 0. If p is inert, let log be the usual branch satisfying log(p) = 0 (although any other choice would do equally well). The logarithm gives a homomorphism log : K_{𝔭}^{×} ⟶ K_{𝔭} which is 0 on O_{1}^{×}, and hence, by passing to the quotient, a homomorphism from G to K_{𝔭} which extends uniquely to a homomorphism

log : G̃ ⟶ K_{𝔭} ⊂ ℂ_{p}.

For α ∈ G̃ and s ∈ ℤ_{p}, define

α^{s} := exp(s log(x)),

where exp is the usual p-adic exponential. Note that s ↦ α^{s} is an analytic function of s ∈ ℤ_{p}.

Definition 2.20 The p-adic L-function attached to the distribution μ_{f,K} is the function of the p-adic variable s ∈ ℤ_{p} defined by

L_{p}(f, K, s) = ∫G̃ α^{s−k/2}dμ_{f,K}(α).

Definition 2.21 Let Ψ be a representative for a class in emb(O, R) and let c be a base point. The partial p-adic L-function attached to the datum (Ψ, c) is the function of the p-adic variable s ∈ ℤ_{p} defined by

L_{p}(f, Ψ, c, s) = ∫_{G} α^{s−k/2}dμ_{f,Ψ,c}(α).

Remark: If c′ = ιΨ(α_{0})c, then

L_{p}(f, Ψ, c′, s) = (α_{0})^{s− \frac{k}{2}} L_{p}(f, Ψ, c, s).

In particular,

Lemma 2.22 The order of vanishing of L_{p}(f, Ψ, c, s) at s = k/2, and the value of the first non-vanishing derivative of L_{p}(f, Ψ, c, s), do not depend on the choice of c.

Because of this lemma, it is customary to drop the c from the notations and write L_{p}(f, Ψ, s) instead of L_{p}(f, Ψ, c, s).

If Ψ_{1} …, Ψ_{h} are representatives for the distinct classes of oriented optimal embeddings in emb(O, R), note that

L_{p}(f, K, k/2) = L_{p}(f, Ψ_{1}, k/2) + ⋯ + L_{p}(f, Ψ_{h}, k/2).  (22)

The following proposition describes the value of L_{p}(f, K, s) and L_{p}(f, Ψ, s) at the central critical point s = k/2.

Proposition 2.23 Suppose p is split in K.

1. If w = −1, then

L_{p}(f, K, k/2)^{2} = 4u_{K}^{2}D^{\frac{k−1}{2}} \frac{L(ϕ/K, k/2)}{(ϕ, ϕ)}.

2. If w = 1, then L_{p}(f, K, k/2) = 0, and in fact L_{p}(f, Ψ, k/2) = 0 for all Ψ ∈ emb(O, R).

Proof: Choose a vertex v of T which is even and fixed under the action of ιΨ(O_{K} ⊗ ℤ_{p})^{×}, and let δ = ιΨ(u). A direct evaluation shows that

L_{p}(f, Ψ, k/2) = ±(1 − w) ∑_{v→δv} w^{sgn(e)}c_{f}(e)(PΨ^{\frac{k−2}{2}} ).

The result follows directly, using (22) and proposition 2.19.

Proposition 2.24 Suppose p is inert in K. Then L_{p}(f, K, k/2) = 0, and in fact L_{p}(f, Ψ, k/2) = 0 for all Ψ ∈emb(O, R).

Proof: This is apparent from the harmonicity of c_{f} and the geometry of the action of K_{p}^{×}/ℚ_{p}^{×} on T induced by Ψ.

Let Ψ_{1}, …, Ψ_{h} be representatives for the distinct classes in emb(O, R). In the cases where p is inert in K or where p is split and w = 1, so that L_{p}(f, Ψ_{i}, k/2) = 0 for all i, the first derivative of the p-adic L-function L_{p}(f, K, s) at s = k/2 is given by:

L′_{p}(f, K, k/2) = ∫G̃ log(α)dμ_{f,K}(α) = ∑_{i=1}^{h} L′_{p}(f, Ψ_{i}, k/2),  (23)

and

L′_{p}(f, Ψ_{i}, k/2) = ∫_{G} log(α)dμ_{f,Ψ_{i}}(α).  (24)

The goal of the next chapter is to derive a formula for these first derivatives.

3 Proof of the main identities

Sections 3.1 and 3.2 contain the proofs of the main identities in the case where p is split in K. In this situation an exceptional zero arises only if w = 1, which will be assumed from now on for those sections. Section 3.3 discusses the case when p is inert in K.

3.1 Teitelbaum’s L-invariant

The definition of Teitelbaum’s L-invariant attached to f is recalled in this section. It depends on the choice of p-adic logarithm that was made in section 2.6.

First let us point out that the modular form f is actually a modular form for the group Γ̃, as a consequence of the assumption w = 1. Choose a point z_{0} ∈ H_{p}, and a vertex v ∈ T .

Proposition 3.1 (Teitelbaum) There exists L_{T} (f) ∈ ℂ_{p} and R ∈ P_{k−2}^{∨} (the latter, depending on the choice of z_{0} and v) such that, for all α ∈ Γ̃ and r ∈ P_{k−2},

∫_{z_{0}}^{α∗z_{0}} f(z)r(z)dz = L_{T} (f) · ∑_{v→αv} c_{f}(e)(r) + (α · R(r) − R(r)) .

Here the integral on the left is Coleman’s integral associated to the choice of log, and the sum on the right is taken over the edges in the path joining v to αv.

The proof is given in [Tei], sec. 1.

Definition 3.2 The scalar L_{T} (f) (which depends only on the homothety class of f) is called the Teitelbaum L-invariant associated to f.

The coboundary term (αR(r) − R(r)), which vanishes when k = 2, presents an extra difficulty in computing L_{T} (f) in the higher weight case. To avoid

this extra term, let δ be any element of Γ̃, let ψ = δ − \frac{1}{2}trace(δ) ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) ∈

M_{2}(ℚ_{p}), and write P_{ψ} for the polynomial P_{ι^{−1}(ψ)} ∈ P_{2} defined by formula (3). Note that P_{ψ} · δ = P_{ψ}, and the same invariance property holds for Pψ^{\frac{k−2}{2}} ∈ P_{k−2}. Replacing α by δ and r by Pψ^{\frac{k−2}{2}} yields a formula which does not involve any coboundary term and determines L_{T} (f) when the sum on the right is non-zero:

∫_{z_{0}}^{δ∗z_{0}} f(z)Pψ^{\frac{k−2}{2}} (z)dz = L_{T} (f) ∑_{v→δv} c_{f}(e)(Pψ^{\frac{k−2}{2}} ).  (25)

Note that both the integral and the sum in the above equation are independent of the choice of base points z_{0} and v respectively.

3.2 The split case

In this section suppose that p is split in K, and let 𝔭 be a prime of K above p. Let us recall the notations of section 1.3, i.e. let u_{0} denote a generator of the rank one group O^{×}/ℤ[1/p]^{×} such that the image u of \frac{u_{0}}{ū_{0}} in ℚ_{p}^{×} has the property ord_{p}(u) > 0. If Ψ is a representative for an element of emb(O, R), write δ := ιΨ(u_{0}) ∈ Γ̃. (This gives rise to the p-adic Shintani cycle attached to Ψ, as in the discussion following proposition 2.19.) Let v be an even vertex of T which is fixed by ιΨ(O_{K} ⊗ ℤ_{p})^{×}.

Theorem 3.3

L′_{p}(f, Ψ, k/2) = L_{T} (f) ∑_{v→δv} c_{f}(e)(PΨ^{\frac{k−2}{2}} ),

where the sum on the right is taken over all edges e in the path joining v to δv.

Proof: Consider the left-hand side of (25) with this choice of δ, and note that the polynomial denoted there by P_{ψ} is a constant multiple of P_{Ψ}.

By Teitelbaum’s p-adic Poisson inversion formula (proposition 1.11),

I := ∫_{z_{0}}^{δ∗z_{0}} f(z)PΨ^{\frac{k−2}{2}} (z)dz = ∫_{z_{0}}^{δ∗z_{0}} (∫_{ℙ_{1}(ℚ_{p})} \frac{1}{z − x}dμ_{f}(x)) PΨ^{\frac{k−2}{2}} (z)dz.(26)

Observe that

\frac{PΨ^{\frac{k−2}{2}} (z) − PΨ^{\frac{k−2}{2}} (x)}{z − x} = polynomial in x of degree ≤ k − 2.

Hence, by lemma 1.9,

∫_{ℙ_{1}(ℚ_{p})} \frac{PΨ^{\frac{k−2}{2}} (z)}{z − x}dμ_{f}(x) = ∫_{ℙ_{1}(ℚ_{p})} \frac{PΨ^{\frac{k−2}{2}} (x)}{z − x}dμ_{f}(x).

Therefore the polynomial PΨ^{\frac{k−2}{2}} (z) can be replaced by PΨ^{\frac{k−2}{2}} (x) in equation (26). Applying the reasoning in the proof of theorem 4 of [Tei], reverse the order of integration to obtain

I = ∫_{ℙ_{1}(ℚ_{p})} (∫_{z_{0}}^{δ∗z_{0}} \frac{dz}{z − x}) PΨ^{\frac{k−2}{2}} (x)dμ_{f}(x)  (27)
= ∫_{ℙ_{1}(ℚ_{p})} log (\frac{δ ∗ z_{0} − x}{z_{0} − x}) PΨ^{\frac{k−2}{2}} (x)dμ_{f}(x),  (28)

where the last equality follows from the definition of the Coleman integral associated to this choice of log. Recall the Möbius transformation η_{Ψ} : ℙ_{1}(ℂ_{p}) ⟶ ℙ_{1}(ℂ_{p}) of section 2.4. Define β_{0} ∈ ℙ_{1}(ℂ_{p}) by η_{Ψ}(β_{0}) = z_{0}. Performing the change of variables x = η_{Ψ}(α) and applying the definition of μ_{Ψ} = μ_{f,Ψ,c}

I = ∫_{ℙ_{1}(ℚ_{p})} log (\frac{uβ_{0} − α}{β_{0} − α}) dμ_{Ψ}(α).  (29)

Note that if α = 0 or ∞, the integrand vanishes, because of the choice of logarithm that was made in the definition of the Coleman integral. Since log(u) = 0, observe that

∑_{n=−∞}^{∞} log (\frac{uβ_{0} − u^{n}α}{β_{0} − u^{n}α}) = lim_{N→∞} ∑_{n=−N}^{N} log (\frac{β_{0} − u^{n−1}α}{β_{0} − u^{n}α})

= lim_{N→∞} log (\frac{β_{0} − u^{−N−1}α}{β_{0} − u^{N}α}) = lim_{N→∞} log (\frac{u^{N+1}β_{0} − α}{β_{0} − u^{N}α}) = log α − log β_{0}.

Applying the proposition 2.13 it follows that

I = ∫_{G} ( ∑_{n=−∞}^{∞} log \frac{uβ_{0} − u^{n}α}{β_{0} − u^{n}α}) dμ_{Ψ}(α) = ∫_{G}(log α − log β_{0})dμ_{Ψ}(α). (30)

Note that

∫_{G} log(β_{0})dμ_{Ψ}(α) = log(β_{0})L_{p}(f, Ψ, k/2) = 0,

by proposition 2.23, (2). It follows that

I = ∫_{G} log(α)dμ_{Ψ}(α) = L′_{p}(f, Ψ, k/2),  (31)

by (24). On the other hand, formula (25) for Teitelbaum’s L-invariant shows that

I = L_{T} (f) ∑_{v→δv} c_{f}(e)(PΨ^{\frac{k−2}{2}} ).  (32)

Theorem 3.3 follows from (31) and (32).

The following theorem can be viewed as the analogue of Teitelbaum’s conjecture in the anticyclotomic setting.

Theorem 3.4 Assume conjecture 2.18. Then

L′_{p}(f, K, k/2)^{2} = L_{T} (f)^{2}u_{K}^{2}DK^{\frac{k−1}{2}} \frac{L(ϕ/K, k/2)}{(ϕ, ϕ)}.

Proof: Let Ψ_{1}, …, Ψ_{h} be distinct representatives for the oriented optimal embeddings in emb(O, R). For 1 ≤ j ≤ h, choose even vertices v_{j} of T which are fixed by ιΨ_{j}((O_{K} ⊗ℤ_{p})^{×}) and write δ_{j} = ιΨ_{j}(u_{0}). By formula (23) combined with theorem 3.3,

L′_{p}(f, K, k/2) = L_{T} (f) ∑_{j=1}^{h} ∑_{v_{j}→δ_{j}v_{j}} c_{f}(e)(PΨ_{j}^{\frac{k−2}{2}} ).

Theorem 3.4 now follows from proposition 2.19.

Remark: In [Kl] Klingenberg claims to prove an “exceptional zero conjecture”, analogous to theorem 3.3, for Schneider’s rigid analytic p-adic Lfunction. Klingenberg normalizes the Schneider L-function by choosing a special isomorphism B_{p} → M_{2}(ℚ_{p}); this depends on the choice of a normalization datum ([Kl], definition 6.3.1) which need not exist in general. Indeed, assume such a datum to be given. Then there is a maximal ℤ-order O′ in B and an element δ_{0} ∈ O′[1/p]^{×} such that for the zeros α, β ∈ ℚ_{p} of the minimal polynomial of δ_{0}/ℚ we have ord(α/β) = 1. After multiplying δ_{0} by a suitable power of p one can assume that δ_{0} belongs to O′ and that its reduced norm is p. Set K = ℚ(δ_{0}). It is an imaginary quadratic field and the minimal polynomial of δ_{0} over ℚ is of the form

f(X) = X^{2} + aX + p with a^{2} < 4p.

There are only finitely many such fields and hence there exist quaternion algebras B which do not contain any of them and thus do not admit a normalization datum. (For example, choose B such that its discriminant has a prime factor q which splits in all the fields K as above.)

In addition, theorem 4.6.1 and 4.6.2 of [Kl] are not correct as stated. In fact the proofs of these theorems actually yield the following (adopting the notation from [Kl]):

Let γ, δ ∈ Γ be such that [γ] and [δ] ∈ Γ^{ab} are non-zero and assume that L_{p}(c_{γ}, 1) = 0. Assume moreover that δ is hyperbolic. Then for every homomorphism χ ∈ Hom(ℚ_{p}^{×}, ℚ_{p}) the following holds

∫_{ℤ_{p}[δ]} χ(x)dμ_{γ}(x) = χ(⟨γ, δ⟩_{Γ}mult(δ)^{m})  (33)

Here m is the unique integer such that ⟨γ, δ⟩_{Γ}mult(δ)^{m} ∈ ℤ_{p}[δ].

Consequently, theorems 6.4.2 and corollary 6.4.1 of [Kl] do not hold as stated. (A summand log_{p}(mult(δ)^{m}) is missing on the right hand side of the equations there and it is usually non-zero.)

While it seems unlikely that there is any simple relation between Schneider’s p-adic L-function (with respect to a suitable normalization) and the cyclotomic p-adic L-function of Mazur, Tate and Teitelbaum as is speculated in ([Kl], p.313), the arguments of [Kl] can be adapted to give a proof of theorem 3.3 in the case of weight 2. Together with the result on the interpolation of classical special values of section 2.5, this yields a proof of the exceptional zero conjecture for the anticyclotomic p-adic L-function when k = 2.

We sketch briefly now how one can deduce theorem 3.3 in the weight two case from (33). For simplicity assume that K has class number one. Then ord_{𝔭}(u) = 1 and (33) gives

L′_{p}(f, K, 1) = ∫_{ℤ_{p}^{×}} log(x)dμ_{f}(x) = log(⟨δ, c_{f}⟩),

since mult(δ) = u^{2} and thus log(mult(δ)) = 0. Now ([Kl], 5.3 and proposition 3.3.1) imply that

L′_{p}(f, K, 1) = log(⟨δ, c_{f}⟩) ∑_{v→δv} c_{f}(e).

3.3 The inert case

Assume now that p is inert in K.

Theorem 3.5 Let z_{0} and z̄_{0} ∈ H_{p} be the two fixed points for ιΨ(K_{p}^{×}) acting on H_{p}. Then

L′_{p}(f, Ψ, k/2) = ± ∫ z_{0} ; z̄_{0} f(z)PΨ^{\frac{k−2}{2}} (z)dz.

Proof: Beginning with the right hand side of theorem 3.5,

I := ∫ z_{0} ; z̄_{0} f(z)PΨ^{\frac{k−2}{2}} (z)dz = ∫ z_{0} ; z̄_{0} (∫_{ℙ_{1}(ℚ_{p})} \frac{1}{z − x}dμ_{f}(x)) PΨ^{\frac{k−2}{2}} (z)dz, (34)

by Teitelbaum’s p-adic Poisson inversion formula (proposition 1.11). Now observe that

\frac{PΨ^{\frac{k−2}{2}} (z) − PΨ^{\frac{k−2}{2}} (x)}{z − x} = polynomial in x of degree ≤ k − 2.

Hence, by lemma 1.9,

∫_{ℙ_{1}(ℚ_{p})} \frac{PΨ^{\frac{k−2}{2}} (z)}{z − x}dμ_{f}(x) = ∫_{ℙ_{1}(ℚ_{p})} \frac{PΨ^{\frac{k−2}{2}} (x)}{z − x}dμ_{f}(x).

Therefore the polynomial PΨ^{\frac{k−2}{2}} (z) can be replaced by PΨ^{\frac{k−2}{2}} (x) in equation (34). Reversing the order of integration as in (27),

I = ∫_{ℙ_{1}(ℚ_{p})} (∫ z_{0} ; z̄_{0} \frac{dz}{z − x}) PΨ^{\frac{k−2}{2}} (x)dμ_{f}(x)  (35)
= ∫_{ℙ_{1}(ℚ_{p})} log (\frac{x − z_{0}}{x − z̄_{0}} ) PΨ^{\frac{k−2}{2}} (x)dμ_{f}(x)  (36)
= ∫_{G} log(α)dμ_{Ψ}(α),  (37)

where the last identity follows by making the change of variables x = η_{Ψ}(α) (cf. equation (15)). Hence

I = L′_{p}(f, Ψ, k/2),

by (24). The result follows.

Remark: By the Cerednik-Drinfeld theorem, the rigid analytic curve H_{p}/Γ has a model over ℚ given by a Shimura curve X classifying abelian surfaces with quaternionic multiplication and auxiliary level structure. (Cf. [BD3], §4.) Using Drinfeld’s moduli interpretation of H_{p}, it is shown in section 5 of [BD3] that the points z_{0} and z̄_{0} correspond to CM points on X defined over the Hilbert class field H of K. Assuming k = 2, the modular form f corresponds to a quotient A_{f} defined over ℚ of the Jacobian J of X; in this situation, the integral appearing on the right hand side of the identity of theorem 3.5 is related to the natural image of the divisor (z_{0})−(z̄_{0}) in A_{f}(H) via the p-adic Abel-Jacobi map. One recovers the main result of [BD3], which gives a construction, in terms the first derivative of the anticyclotomic p-adic L-function, of a Heegner point on A_{f}(K) and implies that this point is of infinite order when L′_{p}(f, K, 1) ≠ 0.

For general k, the integral appearing in theorem 3.5 can be interpreted as the image by a higher p-adic Abel-Jacobi map of certain CM cycles in the Chow groups of the Kuga-Sato variety attached to forms of weight k. This interpretation will be explained in a future work.

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