p-Adic Interpolation of Automorphic Periods for GL_{2}

Jeanine Van Order^{1}

Received: May 1, 2015 Revised: July 27, 2017

Communicated by Otmar Venjakob

Abstract. We give a new and representation theoretic construction of p-adic interpolation series for central values of self-dual Rankin-Selberg L-functions for GL_{2} in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of GL_{2} over a totally real field corresponding to a 𝔭-ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying GL_{2}-representation can be chosen to take values in a discrete valuation ring. A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works. Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements. To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a p-adic interpolation series for central derivative values when the root number is generically equal to −1, in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves.

2010 Mathematics Subject Classification: Primary 11F67; Secondary 11F41, 11F33, 11F70

1This research was supported in part by the ERC Advanced Grant 247049.

1. Introduction

Let F be a totally real number field, and let K be a totally imaginary quadratic extension of F. We shall write A_{K} and A_{F} to denote the respective adele rings of K and F, as well as η = η_{K/F} : A_{F}^{×}/F^{×} ⟶ {±1} ⊂ C^{×} to denote the quadratic idele class character of F associated to K/F. Let π = ⊗_{v}π_{v} be a cuspidal automorphic representation of GL_{2}(A_{F}). We shall assume for simplicity that π has trivial central character ω = ⊗_{v}ω_{v} = 1, although one could more generally allow ω = η. Fix a prime ideal 𝔭 ⊂ O_{F} with underlying rational prime p. We shall also assume the following simplifying conditions throughout:

Hypothesis 1.1. The cuspidal representation π = ⊗_{v}π_{v} is a holomorphic discrete series of parallel weight 2 at each real place of F. The prime-to-𝔭-part 𝔑′ of its conductor 𝔑 = c(π) ⊂ O_{F} is coprime to the relative discriminant of K over F, whence 𝔑 ⊂ O_{F} admits a unique factorization in O_{F} as 𝔑 = 𝔭^{δ}𝔑^{+}𝔑^{−}, where 𝔑^{+} denotes the product of primes v | 𝔑′ which split in K/F, and 𝔑^{−} the product of primes v | 𝔑′ which remain inert in K/F. Assume too that δ ∈ {0, 1}, that 𝔑^{−} is squarefree, that each prime of O_{F} dividing 𝔑^{+} splits in K, and also that 𝔭 splits in K if δ = 1.

Let Ω : A_{K}^{×}/K^{×} ⟶ C^{×} be an idele class character of K of finite order whose restriction to A_{F}^{×} is trivial (i.e. a ring class character), and let π(Ω) = ⊗_{v}π(Ω)_{v} denote the automorphic representation of GL_{2}(A_{F}) it induces. The representation π⊗π(Ω) is then self-dual, and the corresponding GL_{2}×GL_{2} Rankin-Selberg L-function L(s, π × Ω) = L(s, π × π(Ω)) has real-valued coefficients and root number. To be more precise, recall that this L-function is defined for a complex variable s ∈ C with ℜ(s) > 1 by the absolutely convergent Euler product

(1)  L(s, π × Ω) = ∏_{v∈V_{F}} L(s, π_{v} × π(Ω)_{v})

taken over the set V_{F} of all places of F. It has well-known analytic continuation by the work of Jacquet [16] and Jacquet-Langlands [18], and in this setting satisfies the symmetric functional equation

(2)  L(s, π × Ω) = ϵ(s, π × Ω)L(1 − s, π × Ω).

Additionally, the global root number

(3)  ϵ(1/2, π × Ω) = ∏_{v∈V_{F}} ϵ(1/2, π_{v} × Ω_{v}) ∈ S^{1}

p-Adic Interpolation of Automorphic Periods 1469

appearing in the ϵ-factor ϵ(s, π × Ω) = (c(π × Ω))^{s−1/2}ϵ(1/2, π × Ω) is realvalued, hence contained in the set {±1} = R ∩ S^{1} of real numbers of complex modulus 1. This root number ϵ(1/2, π × Ω) has the following simple description. Given an ideal 𝔠 ⊂ O_{F}, let O_{𝔠} = O_{F} + 𝔠O_{K} to denote the O_{F} -order of conductor 𝔠 in K, and Pic(O_{𝔠}) = A_{K}^{×}/K^{×}Ô_{𝔠}^{×}K_{∞}^{×} its class group. An idele class character of K which factors through such a class group Pic(O_{𝔠}) is said to be a ring class character of conductor 𝔠. In fact, we shall only consider ring class characters of 𝔭-power conductor, and hence those which factor through the profinite group X defined by

X = \underleftarrow{lim}_{n} X_{n} = \underleftarrow{lim}_{n} Pic(O_{𝔭^{n}}).

The corresponding root numbers in this setting can be described in terms of the decomposition of the conductor 𝔑 = c(π) in K, and are generically independent of the choice of Ω (see e.g. [5, Lemma 1.1]). More precisely, suppose that Ω is a ring class character factoring through X of conductor c(Ω) = 𝔭^{n} for some integer n ≥ 0. If either (i) 𝔭 does not divide 𝔑 (so that δ = 0), (ii) 𝔭 splits in K, or else (iii) the exponent n is sufficiently large, then we have the formula

(4)  ϵ(1/2, π × Ω) = (−1)^{[F:Q]} · η_{K/F} (𝔑′).

Hence, this formula (4) holds for all ring class characters Ω of X if either (i), (ii), or (iii) is true; otherwise, it holds for all but finitely many Ω factoring through X. Thus, we can define k ∈ {0, 1} to be the integer determined by the condition

(5) ϵ(1/2, π × Ω) = (−1)^{k} for all but finitely many Ω factoring through X.

The aim of this article is to give a general and completely local construction of padic L-functions in either case on k ∈ {0, 1} in this setup, i.e. a general and local construction of a measure on X which interpolates suitably-normalized central values L(1/2, π × Ω) if k = 0, or else central derivative values L′(1/2, π × Ω) if k = 1. Let us first consider the case of generic root number +1 corresponding to k = 0. In this situation, we know that π has a Jacquet-Langlands transfer to the totally definite quaternion algebra B over F whose ramification set Ram(B) consists of prime the divisors of the inert level 𝔑^{−} (the number of which is ≡ [F : Q] mod 2), together with the real places of F. Let us write π′ = ⊗_{v}π′_{v} to denote this transfer of π to an automorphic representation of B^{×}(A_{F}), so that the correspondence of Jacquet-Langands gives the relation π = JL(π′). Let Ω be a ring class character factoring through X. Let us also fix an embedding of K into B, as we can since each place of Ram(B) is inert in K. The central values L(1/2, π × Ω) can be related to the functional P_{Ω}^{B} ∈ Hom_{A_{K}^{×}} (π′, Ω) defined by the rule sending a vector φ ∈ π′ to the automorphic period

P_{Ω}^{B}(φ) = ∫_{A_{K}^{×}/K^{×}A_{F}^{×}} φ(t)Ω(t)dt.

Here, the measure dt is defined by putting on B^{×}(A_{F}) the product of the local Tamagawa measures times ζ_{F}^{S}(2), where S is some fixed finite set of places of F containing the real places and the places where π, Ω, and the fixed choice of additive character of A_{F} are unramified (see [7, §7]). Various results involving calculations with the theta correspondence, starting with the landmark theorem of Waldspurger [33], relate these integrals P_{Ω}^{B}(φ) to the central values L(1/2, π × Ω). Namely, the existence of a vector φ ∈ π′ for which P_{Ω}^{B}(φ) ≠ 0 is shown by Waldspurger to be equivalent to the nonvanishing of the central value L(1/2, π × Ω). A more precise relationship can be established in many cases by making a careful choice of test vector φ ∈ π′, i.e. a choice of decomposable vector φ = ⊗_{v}φ_{v} such that for each place v of F and nonzero local functional l_{v} ∈ Hom_{K_{v}^{×}}(π′_{v}, Ω_{v}), the value l_{v}(φ_{v}) does not vanish (cf. [10], [20], [7]). Using such a choice of vector φ ∈ π′, along with the relative trace formula (cf. [17]), the works of [20] and File-Martin-Pitale [7] establish the following relationship. Let us again write V_{F} to denote the set of all places of F. Let Δ_{F} denote the absolute discriminant of F, and Δ_{K} that of K. Let c(Ω) denote the absolute norm of the conductor of Ω (a power of p), and c(Ω_{v}) the exponent of the conductor of the local character Ω_{v} for each v ∈ V_{F}. Let S_{i} denote the set of places v ∈ V_{F} which remain inert in K. Let S(π) denote the set of finite places v ∈ V_{F} which divide the level c(π) = 𝔑, and let S(Ω) = {𝔭} the set of finite places v ∈ V_{F} where the ring class character Ω is ramified. Writing c(π_{v}) to denote the exponent of the conductor of the local representation π_{v} of GL_{2}(F_{v}), let S_{1}(π) denote the set of places v ∈ V_{F} where c(π_{v}) = 1, and S_{2}(π_{v}) the set of places where c(π_{v}) ≥ 2. Finally, let C_{v}(K, π, Ω) the local factor defined in [20, §4].

Theorem 1.2 (File-Martin-Pitale, Martin-Whitehouse). Assume Hypothesis 1.1, and that φ = ⊗φ_{v} ∈ π′ is test vector defined in [7, §7.1] (described below). Then,

\frac{|P_{Ω}^{B}(φ)|^{2}}{(φ, φ)} =\frac{1}{2} (\frac{Δ_{F}}{c(Ω)Δ_{K}} )^{\frac{1}{2}} L_{S(Ω)}(1, η)L_{S(π)∪S(Ω)}(1, 1_{F})L^{S(π)}(2, 1_{F})

× ∏_{v∈S(π)∩S(Ω)^{c}} e(K_{v}/F_{v}) × ∏_{v|∞} C_{v}(K, π, Ω) × \frac{L^{S_{2}(π)}(1/2, π × Ω)}{L^{S_{2}(π)}(1, π, ad)}.

Here, e(K_{v}/F_{v}) denotes the usual ramification index of the local extension K_{v}/F_{v}, and (·, ·) is the standard inner product on π′ with respect to the measure on B^{×}(A_{F}) given by the product of local Tamagawa measures.

Keeping with this k = 0 setup, let us now impose the following conditions at our fixed prime ideal 𝔭 ⊂ O_{F}. Note that the quaternion algebra B over F is split at 𝔭, whence we can and do fix an isomorphism B_{𝔭} := B ⊗_{F} F_{𝔭} ≈ M_{2}(F_{𝔭}). Note as well that there exists a nonzero decomposable vector φ ∈ π′ whose component φ_{𝔭} at 𝔭 is fixed by the action of the unit group R_{𝔭}^{×}, where R_{𝔭} is an Eichler order of level 𝔭^{δ} in B_{𝔭} for δ ∈ \{0, 1\}. Fix an embedding \overline{Q} → \overline{Q}_{p}.

Hypothesis 1.3. The local representation π′_{𝔭} is 𝔭-ordinary in the sense that the image under our fixed embedding \overline{Q} → \overline{Q}_{p} of its eigenvalue a_{𝔭} = a_{𝔭}(π) for the Hecke operator T_{𝔭} defined below is a 𝔭-adic unit.

The first aim of this paper is to give a general construction of p-adic interpolation series in this so-called ordinary setting for for the values appearing in Theorem 1.2,

L(1/2, π×Ω) := \frac{1}{2} (\frac{Δ_{F}}{c(Ω)Δ_{K}} )^{\frac{1}{2}} L_{S(Ω)}(1, η)L_{S(π)∪S(Ω)}(1, 1_{F})L^{S(π)}(2, 1_{F})

× ∏_{v∈S(π)∩S(Ω)^{c}} e(K_{v}/F_{v}) × ∏_{v|∞} C_{v}(K, π, Ω) × \frac{L^{S_{2}(π)}(1/2, π × Ω)}{L^{S_{2}(π)}(1, π, ad)}.

Let us remark that many of the conditions imposed in Hypothesis 1.1 can be lifted at the expense of clarity, and that Hypothesis 1.3 above is the main requisite condition to impose for this construction so long as the decomposable vector φ ∈ π′ can be chosen in such a way as to take values in a discrete valuation ring O. Assuming as we do that π is a holomorphic discrete series of parallel weight two, and hence that its Hecke eigenvalues define algebraic numbers by a theorem of Shimura [25], we let O denote the ring of integers of a fixed extension of Q_{p} containing the Hecke field Q(π). Consider the O-Iwasawa algebra of X,

O[[X]] = \underleftarrow{lim}_{n} O[X_{n}].

The elements of this group algebra O[[X]] can be identified with their corresponding O-valued measures on X (see e.g. [21, §7]). We shall make such an identification implicitly throughout the rest of this work.

Theorem 1.4 (Proposition 4.12, Corollary 4.14). Suppose that π satisfies Hypotheses 1.1 and 1.3, and that we are in the setting of k = 0 on the generic root number described in (5). There exists a nontrivial element 𝔏_{𝔭}(π′, K) ∈ O[[X]] whose specialization Ω(𝔏_{𝔭}(π′, K)) = ∫_{X_{n}} Ω(σ)d𝔏_{𝔭}(π′, K)(σ) to Ω any character of X of conductor 𝔭^{n} with exponent n ≥ 1 satisfies the following interpolation formula:

Ω(𝔏_{𝔭}(π′, K)) = α_{𝔭}^{2(δ−1−n)} · (\frac{h(O_{F})}{m(O_{𝔭^{n}})})^{2} · L(1/2, π × Ω) ∈ \overline{Q}_{p}.

Here, writing q_{𝔭} to denote the cardinality of the residue field of 𝔭, α_{𝔭} = α_{𝔭}(π) denotes the unit root of the Hecke polynomial t^{2}−a_{𝔭}t+q_{𝔭} if δ = 0 and otherwise the eigenvalue of the Hecke operators T_{𝔭}^{l} and T_{𝔭}^{u} defined below if δ = 1. As well, h(O_{F}) denotes the class number of F, and m(O_{𝔭^{n}}) the volume of Ô_{𝔭^{n}}^{×} in K^{×}\A_{K}^{×}/A_{F}^{×} with respect to our fixed choice of Haar measure.

This result extends those given by Bertolini-Darmon in [1] and [2] for the totally real field F = Q, as well as previous work of the author for the general totally real fields setting ([28], [29])^{2}. Our construction differs from these antecedent works though, and proceeds by choosing a sequence of local test vectors. This allows us to give a more precise interpolation formula. The novelty of this construction, apart from the scope in which it works (and the representation theoretic setup), is that it is completely local to 𝔭. This allows us for instance to use test vectors according to [20] and [7] to derive an even more precise interpolation formula than what was conjectured (in a special, classical case) by Bertolini-Darmon in [1]. The second aim of this work is to extend this construction of elements 𝔏_{𝔭}(π′, K) ∈ O[[X]] to the dichotomous setting of k = 1 on the generic root number, i.e. so that ϵ(1/2, π × Ω) = −1 for all but finitely many ring class characters Ω factoring through the profinite group X = \underleftarrow{lim}_{n} X_{n}. Since the central values L(1/2, π × Ω) in this setting vanish as by the symmetric functional equation (2), we propose to construct a p-adic interpolation series for the central first derivative values L′(1/2, π×Ω), where the issue of defining suitable periods is a subtler problem. Our construction here depends in a crucial way on the those given implicitly in the work of Yuan-Zhang-Zhang [35]. To state our main result for this second part, we shall first need to describe the central derivative formula of [35, Theorem 1.2] (Theorem 2.3 below). Let us for now give a preliminary sketch of their result, saving details for later. Let A be a principally polarized abelian variety of GL_{2}-type parametrized by a quaternionic Shimura curve M = {M_{H}}_{H} defined over F. Thus, A is defined over F, the endomorphism algebra End^{0}(A) = End(A) ⊗ Q defines a number field L of degree equal to the dimension of A, and there exists a nonconstant morphism M_{H} ⟶ A for H some compact open subgroup of the unit group defined by 𝔹̂^{×}. Here, 𝔹 is the ambient quaternion algebra defined over F associated to the Shimura curve M, which is ramified at each of the real places of F except for one fixed place τ, as well as each of the places dividing the inert level 𝔑^{−}. Hence, we can and do fix an embedding of K into 𝔹. Writing J_{H} to denote the Jacobian of the curve M_{H}, we consider the space defined by

2The existence of such an interpolation series is by no means new however, and has been given by completely different methods e.g. in Hida [13], [14] (cf. [22]) and Haran [12] (cf.

[19]).

π_{A} = Hom^{0}(J, A) = \underrightarrow{lim}_{H} Hom_{F} (J_{H}, A) ⊗_{Z} Q,

as constructed in [35, §3.2, Theorem 3.8] (cf. [35, §1.2.3]). To describe this briefly, π_{A} can be viewed as an automorphic representation of 𝔹̂^{×}, (see [35, §1.2.1]). This automorphic representation π_{A} admits a natural decomposition π_{A} = ⊗_{v}π_{A,v} into irreducible admissible representations π_{A,v} of the local unit groups 𝔹_{v}^{×} over F_{v}. Moreover (as explained in [35]), this representation can be viewed as a geometric realization of the nonarchimedean component of the Jacquet-Langands transfer of π. The conditions of Hypothesis 1.1 then imply that π_{A} is self-dual, with trivial central character. The main theorem of Yuan-Zhang-Zhang [35] relates the central derivative value L′(1/2, π_{A}×Ω) to a certain canonical generator α of the space P(π_{A}, Ω) ⊗ P(π_{A}, Ω^{−1}), where P(π_{A}, Ω) := Hom_{A_{K}^{×}} (π_{A} ⊗ Ω, L). We refer to the discussion in [35, §1] for more details. In brief, this generator α decomposes into a product ⊗_{v}α_{v} of local generators α_{v} in the analogously-defined local spaces P(π_{A,v}, Ω_{v}) ⊗ P(π_{A,v}, Ω_{v}^{−1}), and these local generators can be described more precisely as follows.^{3} Let us also for simplicity write π_{v} to denote the local representation π_{A,v} of 𝔹_{v}^{×}. Let us for each place v of F fix a Haar measure dt_{v} on K_{v}^{×}/F_{v}^{×} such that (i) the product measure defined over all places v of F is the Tamagawa measure on A_{K}^{×}/K^{×} and (ii) the maximal compact subgroup O_{K_{𝔭}}^{×}/O_{F_{𝔭}}^{×} is assigned as a volume some rational number. Then, for given vectors φ_{1,v} ∈ π_{v} and φ_{2,v} ∈ π_{v}^{∨}, the local component α_{v} is defined formally by the ratio of values

α(φ_{1,v}, φ_{2,v}) := \frac{L(1, η_{v})L(1, π_{v}, ad)}{ζ_{F_{v}}(2)L(1/2, π_{v} × Ω_{v})} ∫_{K_{v}^{×}/F_{v}^{×}}(π_{v}(t)φ_{1,v}, φ_{2,v})_{v}Ω_{v}(t)dt.

Fixing an embedding ι : L → C, this local integral can be seen to take values in L independently of the choice of embedding ι, as explained in [35, §1.3]. The local invariant pairings (·, ·)_{v} : π_{v} × π_{v}^{∨} ⟶ R also satisfy the compatibility relation (·, ·) = ⊗_{v}(·, ·)_{v}, where (·, ·) : π_{A} × π_{A}^{∨} ⟶ R denotes the perfect 𝔹^{×}(A_{F})-invariant pairing introduced below. As explained in [35] (and summarized below), there is for given a vector φ ∈ π_{A} an analogous notion of an associated automorphic period P_{Ω}^{𝔹}(φ) ∈ A(K^{ab}) ⊗_{R} L, where K^{ab} denotes the maximal abelian extension of K. The following formula, which is the main result of the [35], relates this period and the generator α = ⊗α_{v} to the central derivative values L′(1/2, π_{A} × Ω).

Theorem 1.5 (Yuan-Zhang-Zhang). Given decomposable vectors φ_{1} ∈ π_{A} and φ_{2} ∈ π_{A}^{∨} in the setup described above, we have the following identity in O_{L}⊗_{Q}C:

⟨P_{Ω}^{𝔹}(φ_{1}), P_{Ω^{−1}}^{𝔹}(φ_{2})⟩_{L} = \frac{ζ(2)L′(1/2, π_{A} × Ω)}{4L(1, η)^{2}L(1, π_{A}, ad)} · α(φ_{1}, φ_{2}).

Proof. The result is a consequence of [35, Theorem 1.2], see Theorem 2.3 below. □

Using this result, we construct the following measure. This construction requires the choosing a certain class x_{n} ∈ 𝔹_{𝔭}^{×}/R_{𝔭}^{×} for each integer n ≥ 1 to obtain suitable distribution relations. We commit a minor abuse of notation in also writing x_{n} to denote the associated adele class in 𝔹̂^{×}/H, for H = H^{𝔭}R_{𝔭}^{×} a suitably-defined compact open subgroup of 𝔹̂^{×}. Let us also now write O to denote the tensor product O = A(K^{ab}) ⊗_{Z} Z_{p}.

Theorem 1.6 (Proposition 4.17). Suppose that π satisfies Hypotheses 1.1 and 1.3, and that we are in the setting of k = 1 on the generic root number described in (5). Suppose as well that A/F is the abelian variety of GL_{2}-type for which the corresponding representation π_{A} is a Jacquet-Langlands transfer of the finite part π^{(∞)} = ⊗_{v∤}π_{v} of π to the quaternion algebra 𝔹, i.e. JL(π_{A}) = π^{(∞)}. There exists a nontrivial element D_{𝔭}^{(δ)}(φ, K) = (D_{𝔭}^{(δ)}(φ, K)(x_{n}))_{n≥1} in O[[X]], depending on the choice of sequence of local classes (x_{n})_{n≥1}, whose specialization after composition with the Néron-Tate height pairing ⟨ , ⟩_{L}, i.e.

3Note that we have already used the symbol α_{𝔭} to denote the unit root of the Hecke polynomial t^{2} − a_{𝔭}t + q_{𝔭} in the discussion above; we trust that the distinction between these notations should be clear.

⟨ , ⟩_{L} ∘ Ω(D_{𝔭}^{(δ)}(φ, K)) = ⟨ , ⟩_{L} ∘ ∫_{X} Ω(σ)dD_{𝔭}^{(δ)}(φ, K)(σ),

to any ring class character Ω of X of conductor 𝔭^{n} with n ≥ 1 satisfies the following interpolation formula:

⟨ , ⟩_{L} ∘ Ω(D_{𝔭}^{(δ)}(φ, K))

= α_{𝔭}(π_{A})^{−2(1−δ−n)} · (\frac{h(O_{F})}{m(O_{𝔭^{n}})})^{2} · \frac{ζ(2)L′(1/2, π_{A} × Ω)}{4L(1, η)^{2}L(1, π_{A}, ad)} · α(x_{n} · φ, x_{n} · φ).

Here, α_{𝔭}(π_{A}) denotes the unit root of the Hecke polynomial t^{2} − a_{𝔭}t + q_{𝔭}.

Outline of article. We first give some more details of the central value formulae described above in §2, and then the distribution relations of Cornut-Vatsal [5, §6] in §3, before giving the two main constructions of p-adic interpolation series in §4.

1.1. Notations. Given a prime v of O_{F}, we let F_{v} to denote the completion of F at v, and O_{F_{v}} the ring of integers of F_{v} if v is finite. If E is either a quadratic extension of F or a quaternion algebra over F, then we write E_{v} = E⊗_{F} F_{v}. If R is a module over the ring of integers O_{F} of F, then we write R_{v} = R ⊗_{O_{F}} O_{F_{v}}. We write A to denote the ring of adeles of Q, with A_{f} × R to denote the decomposition into the finite adeles A_{f}. We then write A_{F} = A ⊗_{Q} F and A_{K} = A ⊗_{Q} K. We also use the standard hat notation, e.g. F̂ = A_{f} ⊗_{Q} F. Thus, F̂ = Ô_{F} ⊗ Q, where Ô_{F} = O_{F} ⊗ Ẑ is the profinite completion of O_{F}. Given M a finitely generated Z-module, we also write M̂ = M ⊗ Ẑ to denote the profinite completion of M. Notations for L-values are adopted from the sources (namely [5], [20], [7], and [35]).

2. Central value formulae and algebraicity

We now include some details about the special value formulae mentioned above.

2.1. Central values for the case of k = 0. Suppose first that we are in the setting of k = 0 described above. Hence, ϵ(1/2, π × Ω) = +1 for all but finitely many ring class characters factoring through X, in which case the corresponding values L(1/2, π × Ω) are not forced to vanish by the functional equation. Recall that in this setting there exists a quaternion algebra B over F such that (i) K embeds into B, (ii) π = ⊗_{v}π_{v} transfers to an automorphic representation π′ = ⊗π′_{v} on B^{×}(A_{F}), and (iii) Hom_{A_{K}^{×}} (π′, Ω) is not identically zero. In this setting, we use the formulae of [20] and [7] described in Theorem 1.2 to relate the values L(1/2, π × Ω) to the automorphic periods P_{Ω}^{B}(φ). Let us thus define the test vectors and local factors appearing in the formula of Theorem 1.2 above to give a complete description, and then describe briefly the algebraicity properties satisfied by these values.

2.1.1. Choices of vectors. Given a prime v ∈ V_{F} which is inert in K, and for which c(π_{v}), c(Ω_{v}) > 0, let us assume that c(Ω_{v}) ≥ c(π_{v}). We shall later choose a sequence of local vectors φ_{𝔭,n} in π′_{𝔭} for our construction. Since we wish to use the result of Theorem 1.2 to derive a precise interpolation formula here, we shall have to ensure that our local vectors are test vectors in the sense of [7] (cf. [10]). According to [7, Theorem 1.7, §1.2.3], these test vectors can be characterized as follows.

Theorem 2.1 (File-Martin-Pitale). Given a prime ideal v ⊂ O_{F}, let 𝔓_{v} ⊂ O_{F_{v}} be the maximal ideal, and fix a uniformizer ϖ_{v}. Let π_{v} be an irreducible admissible representation of GL_{2}(F_{v}) of conductor 𝔓_{v}^{c(π_{v})} and trivial central character. Let Ω_{v} be a character of K_{v}^{×} for which the restriction Ω_{v}|_{F_{v}^{×}} to F_{v}^{×} is trivial. Assume that c(Ω_{v}) ≥ c(π_{v}) > 0. Viewing K_{v}^{×} as a torus of GL_{2}(F_{v}) via the embedding defined in [7, §2.3], we have that dim_{C}(Hom_{K_{v}^{×}}(π_{v}, Ω_{v})) = 1. Moreover, writing

K_{1}(𝔓_{v}^{n}) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(O_{F_{v}}) : c ∈ 𝔓_{v}^{n}, d ∈ 1 + 𝔓_{v}^{n}\} ,

and

h = ( \begin{matrix}ϖ_{v}^{c(Ω_{v})−c(π_{v})} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) = ( \begin{matrix}0 & ϖ_{v}^{c(Ω_{v})−c(π_{v})} \\ −1 & 0 \end{matrix} ) ,

the following is true: Given l_{v} ∈ Hom_{K_{v}^{×}}(π_{v}, Ω_{v}) a nonzero functional, the

subgroup hK_{1}(𝔓_{v}^{c(π_{v})})h^{−1} ⊂ GL_{2}(O_{F_{v}}) fixes a 1-dimensional subspace of π_{v} consisting of test vectors for l_{v}.

Taken with the construction of Gross-Prasad [10] (for c(Ω_{v}) ≥ c(π_{v}) = 0), this has the following interpretation (see [7, §1.2.3]). Let R_{v} be an order of M_{2}(F_{v}). Let d(R_{v}) be the exponent of its reduced discriminant. Let c(R_{v}) denote the exponent of its conductor, i.e. the smallest integer c ≥ 0 for which O_{F_{v}} + ϖ_{v}^{c}O_{K_{v}} ⊆ R_{v}. The local order R_{v} can fix a test vector only if c(R_{v}) ≥ c(Ω_{v}). Intuitively, it seems reasonable to expect that if such a local order R_{v} satisfies the conditions c(R_{v}) = c(Ω_{v}) and d(R_{v}) = c(π_{v}), then it might select a local test vector in π_{v}, i.e. so that the subgroup R_{v}^{×} of GL_{2}(F_{v}) fixes a 1-dimensional space of test vectors for any nonzero linear functional l_{v} ∈ Hom_{K_{v}^{×}}(π_{v}, Ω_{v}). If it is the case that either c(π_{v}) or c(Ω_{v}) equals 0, then it is shown in [10] that such an order R_{v} exists, and that is unique up to conjugation by K_{v}^{×}. Moreover, this order is maximal if c(π_{v}) = 0, and Eichler if c(π_{v}) > 0. If it is the case that c(Ω_{v}) ≥ c(π_{v}) > 0, then the invariants d(R_{v}) and c(R_{v}) no longer specify such an order R_{v} uniquely. However, the result of Theorem 2.1 can be used to give the following interpretation of the problem, as explained in File-Martin-Pitale [7, §1.2.3]. Namely, in the setting where c(Ω_{v}) ≥ c(π_{v}) > 0, there exists an Eichler order R_{v} of M_{2}(F_{v}) with c(R_{v}) = c(Ω_{v}) and d(R_{v}) = c(π_{v}) for which the subgroup R_{v}^{×} of GL_{2}(F_{v}) fixes a line or test vectors in π_{v}. Moreover, this order R_{v} can be expressed uniquely at the intersection or two maximal orders R_{1,v} and R_{2,v} for which c(R_{1,v}) = c(Ω_{v}) and c(R_{2,v}) = c(Ω_{v}) − c(π_{v}). We shall use these results of [10] and [7] below, specifically in the setting where v = 𝔭, n = c(Ω_{𝔭}) ≥ 0, and the exponent δ = c(π_{𝔭}) equals 0 or 1 respectively.

2.1.2. Algebraicity. We can deduce the following algebraicity result independently of the theorem Shimura [25]. Let π^{σ} denote the representation of GL_{2}(A_{F}) defined by the rule that sends the eigenvalues of π to their images under σ ∈ Aut(C). Let Ω^{σ} denote the character defined on nonzero ideals 𝔞 ⊂ O_{K} by the rule 𝔞 ↦ Ω(𝔞)^{σ}.

Corollary 2.2. Let L = Q(π, Ω) be the finite extension of Q obtained by adjoining the eigenvalues of π and the values of Ω. If B is totally definite, then the values

L(1/2, π × Ω) = \frac{1}{2} (\frac{Δ_{F}}{c(Ω)Δ_{K}} )^{\frac{1}{2}} L_{S(Ω)}(1, η)L_{S(π)∪S(Ω)}(1, 1_{F})L^{S(π)}(2, 1_{F})

× ∏_{v∈S(π)∩S(Ω)^{c}} e(K_{v}/F_{v}) × ∏_{v|∞} C_{v}(K, π, Ω) × \frac{L^{S_{2}(π)}(1/2, π × Ω)}{L^{S_{2}(π)}(1, π, ad)}

are algebraic, and moreover lie in the number field L ⊂ \overline{Q}. Moreover, there is a natural action of σ ∈ Aut(C) on these values given by the rule L(1/2, π×Ω)^{σ} = L(1/2, π^{γ} × Ω^{σ}), where γ denotes the restriction of σ to the Hecke field Q(π) of π.

Proof. By Theorem 1.2, we have that

(6)  L(1/2, π × Ω) = |P_{Ω}^{B}(φ)|^{2}/(φ, φ).

Now, we can view φ as an automorphic function on the set C(B; H) = B^{×}\B̂^{×}/H, for H some compact open subgroup of B̂^{×}. This function is determined uniquely up to multiplication by a nonzero complex scalar by the Jacquet-Langlands correspondence. Moreover, since B is totally definite, the space C(B; H) is finite. Using this fact, it is easy to see that we can fix a basis for the space of automorphic forms on C(B; H) taking values in Z(π). The result is then simple to deduce from (6), using the fact that P_{Ω}^{B}(φ) is a finite integral. □

2.2. Central derivative values for the case of k = 1. Let us now suppose that k = 1. Hence, ϵ(1/2, π × Ω) = −1 for all but finitely many ring class characters Ω factoring through X, and the corresponding central values L(1/2, π × Ω) are forced to vanish by the functional equation. In this setting, we have the following formula of Yuan-Zhang-Zhang [35] (generalizing Gross-Zagier [9]) for the derivative values L′(1/2, π × Ω). We now describe this in more detail, following [35, §1.2, 1.3, 3.2]. Fix a real place τ of F. Let 𝔹 denote the quaternion algebra defined over F which is split at τ, but ramified at each other real place, as well as the finite places dividing the inert level 𝔑^{−}. Again, we can and do fix an embedding of K into 𝔹.

2.2.1. Shimura curves. Let 𝔹̂^{×} denote the finite adelic points of 𝔹^{×}(A_{F}). Given H ⊂ 𝔹̂^{×} any compact open subgroup, let M_{H} denote the associated (compactified) Shimura curve over F, whose complex points determine a Riemann surface

M_{H,τ} (C) = 𝔹^{×}\ℌ^{±} × 𝔹̂^{×}/H × {cusps}.

Note that the set of cusps {cusps} is nonempty only if F = Q. Given compact open subgroups H_{1}, H_{2} ⊂ 𝔹̂^{×} for which the inclusion H_{1} ⊂ H_{2} holds, there is a natural and surjective morphism

π_{H_{1},H_{2}} : M_{H_{1}} ⟶ M_{H_{2}}.

We shall write M = {M_{H}}_{H} to denote the associated projective system. Note that each M_{H} can be identified with the quotient of M by the action of H.

2.2.2. Hodge classes. A Hodge class on a Shimura curve M_{H} is a line bundle L_{H} on Pic(M_{H})_{Q} whose global sections are holomorphic forms for parallel weight 2. We refer to [35, §1.2] for a more explicit description of these classes. Given a connected component β ∈ π_{0}(M_{H,\overline{F}}), we write L_{H,β} = L_{H}|_{M_{H,β}} to denote the restriction of L_{H} to the connected component M_{H,β} of M_{H} corresponding to β. We then view this L_{H,β} as a divisor on M_{H} via the pushforward under M_{H,β} ⟶ M_{H}. Definition Given a Hodge class L_{H} and a connected component β ∈ π_{0}(M_{H,β}), the normalized Hodge class on M_{H,β} is the weighted class ξ_{H,β} = L_{H,β}/ deg(L_{H,β}). The normalized Hodge class on M_{H} is the sum ξ_{H} = ∑_{β} ξ_{H,β}.

2.2.3. Abelian varieties parametrized by Shimura curves. Let M = {M_{H}}_{H} be a Shimura curve defined over F, as above. A simple abelian variety A defined over F is said to be parametrized by M if for some compact open subgroup H ⊂ 𝔹̂^{×}, there exists a non constant morphism M_{H} ⟶ A defined over F. It is known by Eichler-Shimura theory that if such an abelian variety A parametrized by M is of strict GL_{2}-type in the sense that (i) R = End_{F} (A)⊗_{Z}Q is a field and (ii) Lie(A) is a free module of rank 1 over R ⊗_{Q} F by the induced action. Given such an abelian variety A, let us consider the space

π_{A} = Hom_{ξ}^{0}(M, A) = \underrightarrow{lim}_{H} Hom_{ξ_{U}}^{0} (M_{H}, A).

Here, each Hom_{ξ_{H}}^{0} (M_{H}, A) denotes the morphisms in the space

Hom^{0}(M_{H}, A) = Hom_{F} (M_{H}A) ⊗_{Z} Q

having basepoint equal to the Hodge class ξ_{H}. This π_{A} has the natural structure of a 𝔹̂^{×}-module. Moreover, as shown in [35, Theorem 3.8], this space π_{A} in fact determines an (A(\overline{F})_{Q} = A(\overline{F}) ⊗_{Z} Q-valued) automorphic representation of 𝔹̂^{×} over Q. The description of [35, §3.2.2] also shows that there is a natural identification R = End𝔹̂^{×}(π_{A}), as well as a decomposition π_{A} = ⊗_{v}π_{A,v} into absolutely irreducible representations π_{A,v} of 𝔹_{v}^{×} over R. Finally, since any morphism M_{H} ⟶ A factors through the Jacobian J_{H} of M_{H}, we can and do redefine this automorphic representation in the simpler form

π_{A} = Hom^{0}(J, A) = \underrightarrow{lim}_{H} Hom^{0}(J_{H}, A).

Here, in the same style as above, we have put Hom^{0}(J_{H}, A) = Hom_{F} (J_{H}, A)⊗_{Z} Q.

2.2.4. Dual abelian varieties. Let us write A^{∨} to denote the dual of an abelian variety A. If A is parametrized by a Shimura curve M = {M_{U}}_{U} in the sense defined above, then so too is A^{∨}. Moreover, writing R^{∨} = End_{F} (A^{∨}) ⊗_{Z} Q, there is a canonical isomorphism R ≈ R^{∨} obtained by sending a homomorphism r : A ⟶ A to its dual r^{∨} : A^{∨} ⟶ A^{∨}.

2.2.5. Pairings. Let us first consider the perfect, 𝔹^{×}(A_{F})-invariant pairing

(·, ·) : π_{A} × π_{A^{∨}} ⟶ R

defined on elements φ_{1,H} ∈ Hom(J_{H}, A) and φ_{2,H} ∈ Hom(J_{H}, A) by

(φ_{1}, φ_{2}) = (φ_{1,H} ∘ φ_{2,H}^{∨}) /vol(M_{H}).

Here, J_{H} denotes the Jacobian of the Shimura curve M_{H}, and φ_{2,H}^{∨} : A ⟶ J_{H} the dual of φ_{2,H} composed with the canonical isomorphism J_{H}^{∨} ≈ J_{H}. This description implies that π_{A^{∨}} is in fact the dual of π_{A} as a representation of 𝔹^{×}(A_{F}) over R. Let us also consider the Néron-Tate height pairing (as defined e.g. in [35, §7]), which recall is a Q-bilinear non-degenerate pairing

⟨·, ·⟩ : A(\overline{F})_{Q} × A^{∨}(\overline{F})_{Q} ⟶ R.

Here, we written A(\overline{F})_{Q} = A(\overline{F}) ⊗_{Z} Q and A^{∨}(\overline{F})_{Q} = A^{∨}(\overline{F}) ⊗_{Z} Q to denote the tensor products appearing throughout [35]. Of course, the Néron-Tate pairing is defined classically on the groups A(\overline{F}) and A^{∨}(\overline{F}), and then extended in a natural way to these tensor products. This extended Néron-Tate pairing in fact descends to a Q-linear map

⟨·, ·⟩ : A(\overline{F})_{Q} ⊗_{R} A^{∨}(\overline{F})_{Q} ⟶ R.

Some more explanation of this fact is given in [35, §1.2.3] (cf. [35, Proposition 7.3]). Now, given a ∈ R, x ∈ A(\overline{F})_{Q}, and y ∈ A^{∨}(\overline{F})_{Q}, the rule a ↦ ⟨ax, y⟩ defines an element of the space Hom(R, R). As explained in [35, §1.2.4], the trace map can be used to construct an isomorphism Hom(R, R) ≈ R ⊗_{Q} R, whence we write ⟨x, y⟩_{R} to denote the corresponding element of the space R ⊗_{Q} R. This gives us the construction of an R-bilinear pairing

⟨·, ·⟩_{R} : A(\overline{F})_{Q} ⊗_{R} A^{∨}(\overline{F})_{Q} ⟶ R ⊗_{Q} R

for which

⟨·, ·⟩ = Tr_{R⊗R/R} ⟨·, ·⟩_{R}.

We shall refer to this R-linear pairing as the R-linear Néron-Tate pairing.

2.2.6. CM points. Recall that we fix an embedding K → 𝔹. The adele group A_{K}^{×} acts on {M_{H}}_{H} by right multiplication via the embedding A_{K}^{×} → 𝔹^{×}(A_{F}). Let M^{K^{×}} denote the subscheme of M fixed by K^{×} under this action. By the theory of complex multiplication, each point of M^{K^{×}}(\overline{F}) is defined over the maximal abelian extension K^{ab} of K. Let us now fix a point Q ∈ M^{K^{×}}(K^{ab}), and hence a point Q_{H} ∈ M_{H}(K^{ab}) for each compact open subgroup H ⊂ 𝔹̂^{×}. We shall normalize

M_{H,τ} (C) = 𝔹^{×}\ℌ^{±} × 𝔹̂^{×}/H × {cusps}

in such a way that Q_{H} is represented by [z_{0}, 1], where z_{0} ∈ ℌ is the unique fixed point of K^{×} via the action induced by the embedding K → 𝔹. Note that for any vector φ ∈ π_{A}, we obtain a well-defined point φ(Q) ∈ A(K^{ab}).

2.2.7. Automorphic periods. Fix a ring class character Ω : A_{K}/K^{×} ⟶ C^{×} taking values in some subfield L of \overline{Q} containing the field R = End_{F} (A) ⊗_{Z} Q. Recall that given a simple abelian variety A/F parametrized by a Shimura curve M = {M_{H}}_{H} over F, we have some associated automorphic representation π_{A} = ⊗_{v}π_{A,v} of 𝔹̂^{×}. We shall assume (as throughout) that the central character of π_{A} is trivial. We then define for any choice of vector φ ∈ π_{A} the period integral

(7)  P_{Ω}^{𝔹}(φ) = ∫_{A_{K}^{×}/K^{×}} φ(Q^{t}) ⊗_{R} Ω(t)dt ∈ A(K^{ab}) ⊗_{R} L.

Here, dt is the Haar measure on A_{K}^{×}/K^{×} having total volume 1. An analogous definition is of course be made for the dual representation π_{A^{∨}}. It is also the case that we have the inclusion

P_{Ω}^{𝔹}(φ) ∈ A(Ω) := (A(K^{ab}) ⊗_{R} L_{Ω})^{G_{K}^{ab}} ,

where L_{Ω} denotes the R-vector space L with action of G_{K}^{ab} given by the character Ω, and that the correspondence φ ↦ P_{Ω}^{𝔹}(φ) defines an element of

Hom_{A_{K}^{×}} (π_{A} ⊗ Ω, L) ⊗_{L} A(Ω)

We refer again to the discussion in [35, §1.3] for more details.

2.2.8. Main formula. The theorem of Tunnel [26] and Saito [24] (cf. [35, Theorem 1.3]) implies that the space Hom_{A_{K}^{×}} (π_{A} ⊗ Ω, L) is at most 1-dimensional, with dimension 1 if and only if the set Ram(𝔹) of places of F where 𝔹 ramifies is given by

Ram(𝔹) = {v ∈ V_{F} : ϵ(1/2, π_{A,v} × Ω_{v}) ≠ Ω_{v}(−1)η_{v}(−1)}.

Recall that the global root number ϵ(1/2, π_{A} × Ω) in this setting must be equal to −1, in which case the corresponding central value L(1/2, π_{A}×Ω) must vanish. Let us now commit an abuse of notation in writing π = ⊗_{v}π_{v} to denote the automorphic representation π_{A} = Hom^{0}(J, A) of 𝔹̂^{×} introduced above. Let us also assume that the space Hom_{A_{K}^{×}} (π_{A} ⊗ Ω, L) has dimension 1. The main result of [35] relates the value L′(1/2, π ×Ω) to a certain generator of this space Hom_{A_{K}^{×}} (π ⊗ Ω, L). To be more precise, let

P(π, Ω) = Hom_{A_{K}^{×}} (π ⊗ Ω, L).

Let

P(π_{v}, Ω_{v}) = Hom_{K_{v}^{×}}(π_{v} ⊗ Ω_{v}, L)

for each place v of F, whence there is a decompositions P(π, Ω) = ⊗_{v}P(π_{v}, Ω_{v}), as well as an analogous decomposition P(π^{∨}, Ω^{−1}) = ⊗_{v}P(π_{v}^{∨}, Ω_{v}^{−1}) for the associated contragredient representations. The main idea of [35] is to find an explicit generator

α = ⊗_{v}α_{v} ∈ P(π, Ω) ⊗ P(π^{∨}, Ω^{−1}) = ⊗_{v}P(π_{v}, Ω_{v}) ⊗ P(π_{v}^{∨}, Ω_{v}^{−1}),

and then to use various calculations with the theta correspondence in the style of Waldspurger [33] (with their geometric description of the Jacquet-Langlands lift) to relate the this generator to the central derivative values L′(1/2, π × Ω). The generator α = ⊗α_{v} they obtain is defined formally as follows. For each place v of F, fix a Haar measure dt_{v} on K_{v}^{×}/F_{v}^{×} such that (i) the product measure over all places v gives the Tamagawa measure on A_{K}^{×}/K^{×} and (ii) the maximal compact subgroup O_{K_{v}}^{×}/O_{F_{v}}^{×} has a volume in Q for each nonarchimedean place v. Then, for vectors φ_{1,v} ∈ π_{v} and φ_{2,v} ∈ π_{v}^{∨}, the local component α_{v} is defined formally by

α(φ_{1,v}, φ_{2,v}) := \frac{L(1, η_{v})L(1, π_{v}, ad)}{ζ_{F_{v}}(2)L(1/2, π_{v} × Ω_{v})} ∫_{K_{v}^{×}/F_{v}^{×}}(π_{v}(t)φ_{1,v}, φ_{2,v})_{v}Ω_{v}(t)dt.

Fixing an embedding ι : L → C, this local integral can be seen to take values in L (independently of the choice of embedding ι), as explained in [35, §1.3]. The local invariant pairings (·, ·)_{v} : π_{v} × π_{v}^{∨} ⟶ R also satisfy the compatibility relation (·, ·) = ⊗_{v}(·, ·)_{v}, where (·, ·) : π_{A} × π_{A}^{∨} ⟶ R denotes the perfect 𝔹^{×}(A_{F})-invariant pairing introduced above. The following formula, which is the main result of the [35], relates this element α = ⊗α_{v} to the central derivative values L′(1/2, π_{A} × Ω).

Theorem 2.3. Given decomposable vectors φ_{1} ∈ π_{A} and φ_{2} ∈ π_{A}^{∨} in the setup above, we have the following identity in L ⊗_{Q} C :

(8)  ⟨P_{Ω}^{𝔹}(φ_{1}), P_{Ω^{−1}}^{𝔹}(φ_{2})⟩_{L} = \frac{ζ(2)L′(1/2, π_{A} × Ω)}{4L(1, η)^{2}L(1, π_{A}, ad)} · α(φ_{1}, φ_{2}).

Proof. See Yuan-Zhang-Zhang, [35, Theorem 1.2]. □

3. Distribution relations on quaternion algebras

We now recall some general distribution relations on split quaternion algebras, following the appendix of [5]. We shall use these results in our constructions below.

3.1. Local quadratic orders. Let K/F be any quadratic extension of number fields. Writing O_{F} and O_{K} to denote the rings of integers of F and K respectively, we consider the localizations O_{F_{𝔭}} and O_{K_{𝔭}} of these rings at a fixed prime 𝔭 of O_{F}.

Lemma 3.1. Let Z be any O_{F_{𝔭}}-order in K_{𝔭}. Then, there exists a unique integer n = l_{𝔭}(Z) ≥ 0 for which Z = O_{F_{𝔭}} + 𝔭^{n}O_{K_{𝔭}}, whence we write Z = Z_{n}.

Proof. The result is standard. See e.g. the proof given in [5, § 6.1]. □

3.2. Distribution relations on split quaternion algebras. Let us now consider any quaternion algebra B defined over F which (i) contains the quadratic extension K and (ii) splits at the fixed prime 𝔭. Let H be any compact open subgroup of B̂^{×} of the form H = H^{𝔭}R_{𝔭}^{×}, where R_{𝔭} is an Eichler order of the local quaternion algebra B_{𝔭} of level 𝔭^{δ} for some integer δ ≥ 0 . There is a left action of K_{𝔭}^{×} on the quotient B_{𝔭}^{×}/R_{𝔭}^{×} given by the rule

γ ⋆ x = [γb], γ ∈ K_{𝔭}^{×}, x = [b] ∈ B_{𝔭}^{×}/R_{𝔭}^{×}.

Given a class x = [b] in B_{𝔭}^{×}/R_{𝔭}^{×}, the stabilizer of this action is given by Z(x)^{×}, where Z(x) is the O_{F_{𝔭}}-order of K_{𝔭} determined by the intersection Z(x) = K_{𝔭}∩bR_{𝔭}b^{−1}. Using Lemma 3.1 above, we assign to this class x = [b] the uniquely determined integer l_{𝔭}(x) = l_{𝔭}(Z(x)) ≥ 0, i.e., so that Stab_{K_{𝔭}^{×}}(x) = Z_{l_{𝔭}(x)}^{×}. The main results of [5, Appendix § 6] use [5, Lemma 6.1] to relate the trace operator

Tr(x) = ∑_{γ∈Z_{l𝔭(x)−1}^{×}/Z_{l𝔭(x)}^{×}} γ ⋆ x = ∑_{γ∈Z_{l𝔭(x)−1}^{×}/Z_{l𝔭(x)}^{×}}[γb]

to the (local) Hecke operator

T_{𝔭}(x) = [R_{𝔭}^{×}ξ_{𝔭}R_{𝔭}^{×}](x).

To describe these relevant results in more detail, we shall divide into cases on the exponent δ ≥ 0 in the level of the local Eichler order R_{𝔭}.

3.2.1. The case of δ = 0. Suppose that R_{𝔭} has level 𝔭^{δ} = 1, whence R_{𝔭} is a maximal order in B_{𝔭}. Let V be a simple left B_{𝔭}-module for which V ≈ F_{𝔭}^{2} as an F_{𝔭}-vector space. The embedding K_{𝔭} ⟶ B_{𝔭} endows V with the structure of a free rank one (left) K_{𝔭}-module. Let L = L(V) denote the set of O_{F_{𝔭}}-lattices in V. Fix a base lattice L_{0} ∈ L such that {α ∈ B_{𝔭} : αL_{0} ⊂ L_{0}} = R_{𝔭}. Thus we may fix a bijection

(9)  B_{𝔭}^{×}/R_{𝔭}^{×} ⟶ L, b ⟼ bL_{0}.

The induced left action of K_{𝔭}^{×} on L is given by the rule

γ ⋆ L = γL, γ ∈ K_{𝔭}^{×}, L = bL_{0} ∈ L.

The induced function l_{𝔭} on L is given by the rule that sends a lattice L to the unique integer l_{𝔭}(L) ≥ 0 for which {γ ∈ K_{𝔭}^{×} : γL ⊂ L} equals Z_{l_{𝔭}(L)}.

Lemma 3.2. The induced function l_{𝔭} on L defines a bijection

K_{𝔭}^{×}\B_{𝔭}^{×}/R_{𝔭}^{×} ≈ K_{𝔭}^{×}\L ⟶ Z_{≥0}.

Proof. See [5, Lemma 6.2]. □

Let Z[L] denote the free abelian group generated by L. Definition Let L ∈ L be a lattice.

(i) The lower neighbours of L are the sublattices L′ ⊂ L for which we have L/L′ ≈ O_{F_{𝔭}}/𝔭O_{F_{𝔭}}. The upper neighbours of L are the superlattices L′ ⊃ L for which we have L′/L ≈ O_{F_{𝔭}}/𝔭O_{F_{𝔭}}.

(ii) The lower Hecke operator T_{𝔭}^{l} on Z[L] is the operator that sends a lattice L to the sum of its lower neighbours. The upper Hecke operator T_{𝔭}^{u} on Z[L] is the operator that sends a lattice L to the sum of its upper neighbours.

(iii) Given a lattice L ∈ L with l_{𝔭}(L) ≥ 1, the lower predecessor of L is defined by the lattice pr_{l}(L) = 𝔭Z_{l_{𝔭}(L)−1}L; the upper predecessor of L is defined by the lattice pr_{u}(L) = Z_{l_{𝔭}(L)−1}L.

Remark As explained in [5, Remark 6.3], the lower Hecke operator T_{𝔭}^{l} corresponds under the fixed bijection (9) to the double coset operator [R_{𝔭}^{×}α_{𝔭}R_{𝔭}^{×}], where we write α_{𝔭} ∈ R_{𝔭} ≈ M_{2}(O_{F_{𝔭}}) to denote some element of nrd(α_{𝔭}) = ϖ_{𝔭}. Similarly, the upper Hecke operator T_{𝔭}^{u} corresponds to the double coset operator [R_{𝔭}^{×}α_{𝔭}^{−1}R_{𝔭}^{×}]. Note also that [5, § 6] treats only the lower Hecke operators T_{𝔭}^{l} for simplicity, the analogous discussion for the upper Hecke operators T_{𝔭}^{u} being simple to deduce. Let us for simplicity of notation define

η_{𝔭} = \begin{cases}−1 & if 𝔭O_{K} = 𝔓 is inert \\ 0 & if 𝔭O_{K} = 𝔓^{2} is ramified \\ 1 & if 𝔭O_{K} = 𝔓𝔓^{∗} is split. \end{cases}

Lemma 3.3. Let L be a lattice in L.

(i) If l_{𝔭}(L) = 0, then there are precisely 1 + η_{𝔭} lower neighbours L′ ⊂ L for which l_{𝔭}(L′) = 0. Explicitly, these lower neighbours are given by

L′ = \begin{cases}∅ & if 𝔭O_{K} = 𝔓 is inert \\ 𝔓L & if 𝔭O_{K} = 𝔓^{2} is ramified \\ 𝔓L, 𝔓^{∗}L & if 𝔭O_{K} = 𝔓𝔓^{∗} is split. \end{cases}

(ii) If l_{𝔭}(L) > 0, then there exists a unique lower neighbour L′ ⊂ L for which l_{𝔭}(L′) ≤ l_{𝔭}(L). Explicitly, this lower neighbour L′ ⊂ L is given by the lower predecessor L′ = pr_{l}(L) = 𝔭Z_{l_{𝔭}(x)−1}L, which satisfies l_{𝔭}(L′) = l_{𝔭}(L) − 1.

(iii) In either case, the remaining lower neighbours L′ ⊂ L satisfy the property that l_{𝔭}(L′) = l_{𝔭}(L) + 1 . These remaining lower neighbours L′ are also permuted faithfully and transitively by the action of Z_{l_{𝔭}(L)}^{×}/Z_{l_{𝔭}(L)+1}^{×}, and moreover have L as their common upper predecessor.

Proof. See [5, Lemma 6.5], the result is deduced from [5, Lemma 6.1]. □

One can deduce from this the following result.

Corollary 3.4. Let x be a class in B_{𝔭}^{×}/R_{𝔭}^{×} with l_{𝔭}(x) ≥ 1. Then, we have

Tr(x) = T_{𝔭}^{l}(x′) − x′′,

where x′ = pr_{u}(x), and

x′′ = \begin{cases}0 & if l_{𝔭}(x) = 1 and 𝔭O_{K} = 𝔓 is inert \\ ϖ_{𝔓}x′ & if l_{𝔭}(x) = 1 and 𝔭O_{K} = 𝔓^{2} is ramified \\ (ϖ_{𝔓} + ϖ_{𝔓^{∗}})x′ & if l_{𝔭}(x) = 1 and 𝔭O_{K} = 𝔓𝔓^{∗} is split \\ pr_{l}(x′) & if l_{𝔭}(x) ≥ 2. \end{cases}

Here, for a prime 𝔓 of O_{K}, ϖ_{𝔓} denotes a uniformizer of 𝔓 (which corresponds under the reciprocity map rec_{K} to the geometric Frobenius at 𝔓).

Remark The discussion and results above do not depend upon the choice of base lattice L_{0} in L; see [5, § 6.2] for more explanation.

3.2.2. The case of δ = 1. Suppose now that R_{𝔭} is Eichler of level 𝔭^{δ} = 𝔭. Keep V as defined above. Let L_{1} = L_{1}(V) denote the set 1-lattices of L, i.e. the set of pairs of lattices L = (L(0), L(1)) with L(0), L(1) ∈ L such that L(1) ⊂ L(0) is a sublattice for which L(0)/L(1) ≈ O_{F_{𝔭}}/𝔭O_{F_{𝔭}}. The group B_{𝔭}^{×} ≈ GL(V) acts transitively on L_{1}, and so we can fix a 1-lattice L_{0} = (L_{0}(0), L_{0}(1)) whose stabilizer under this action is given by R_{𝔭}^{×}. Fixing such a 1-lattice, we may also fix a bijection

(10)  B_{𝔭}^{×}/R_{𝔭}^{×} ⟶ L_{1}, b ⟼ bL_{0}.

We can associate to any L = (L(0)), L(1)) ∈ L_{1} a pair of integers l_{𝔭,0} = l_{𝔭}(L(0)) and l_{𝔭,1} = l_{𝔭}(L(1)), whence we define l_{𝔭}(L) to be the maximum,

l_{𝔭}(L) = max(l_{𝔭,0}(L), l_{𝔭,1}(L)) = max(l_{𝔭}(L(0)), l_{𝔭}(L(1))).

This definition leads to the following possible orientations for a given 1-lattice L.

Definition Let L = (L(0), L(1)) be a 1-lattice in the set L_{1}.

(i) We say that L is of type I if l_{𝔭,0} < l_{𝔭,1} (whence l_{𝔭,1} = l_{𝔭,0} + 1), in which case the leading vertex of L is defined to be L(1). If l_{𝔭}(L) ≥ 2, then we also define the predecessor of L to be the 1-lattice in L_{1} defined by

pr(L) = (L(0), pr_{l}(L(0))) = (L(0), 𝔭Z_{l_{𝔭}(L)−2}L(0)).

(ii) We say that L is of type II if l_{𝔭,0} > l_{𝔭,1} (whence l_{𝔭,0} = l_{𝔭,1} +1), in which case the leading vertex of L is defined to be L(0). If l_{𝔭}(L) ≥ 2, then we also define the predecessor of L to be the 1-lattice in L_{1} defined by

pr(L) = (pr_{u}(L(1)), L(1)) = (Z_{l_{𝔭}(L)−2}L(1), L(1)).

(iii) We say that L is of type III if l_{𝔭,0} = l_{𝔭,1} (whence l_{𝔭}(L) = 0, η_{𝔭} ∈ {0, 1}, and L(1) = 𝔓L(0) or L(1) = 𝔓^{∗}L(0)). In this case, as a convention, we define the leading vertex of L to be L(0).

Remark The type of a 1-lattice L = (L(0), L(1)) in L_{1} is invariant under the action of γ ∈ K_{𝔭}^{×}. More precisely, L and γL = γ ⋆ L have the same type, and moreover pr(γL) = γpr(L) = γ ⋆ pr(L) if l_{𝔭}(L) ≥ 2. Let Z[L_{1}] denote the free abelian group generated by L_{1}. Definition (i) The lower Hecke operator T_{𝔭}^{l} on Z[L_{1}] is the rule that sends a 1-lattice L = (L(0), L(1)) to the sum of all 1-lattices L′ = (L′(0), L′(1)) for which L′(0) = L(0) but L′(1) ≠ L(1). (ii) The upper Hecke operator T_{𝔭}^{u} on Z[L_{1}] is the rule that sends a 1-lattice L = (L(0), L(1)) to the sum of all 1-lattices L′ = (L′(0), L′(1)) for which L′(1) = L(1) but L′(0) ≠ L(0). Remark These Hecke operators correspond to the following double coset operators, as explained in [5, Remark 6.10]. Suppose that we write the Eichler order R_{𝔭} as the intersection R_{𝔭} = R(0) ∩ R(1), where for i ∈ {0, 1}, we define

R(i) = {b ∈ B_{𝔭} : bL_{0}(i) ⊂ L_{0}(i)}.

Then, the lower Hecke operator T_{𝔭}^{l} corresponds to the double coset operator [R_{𝔭}^{×}αR_{𝔭}^{×}] for any element α ∈ R(0)^{×} − R(1)^{×}, and the upper Hecke operator T_{𝔭}^{u} corresponds to the double coset operator [R_{𝔭}^{×}βR_{𝔭}^{×}] for any element β ∈ R(1)^{×} − R(0)^{×}. Also, we have the decompositions R(0)^{×} = R_{𝔭}^{×} ∐ R_{𝔭}^{×}βR_{𝔭}^{×} and R(1)^{×} = R_{𝔭}^{×} ∐ R_{𝔭}^{×}αR_{𝔭}^{×} Definition The 𝔭-new quotient Z[L_{1}]^{𝔭−new} of Z[L_{1}] is the quotient of Z[L_{1}] by the Z-submodule spanned by elements of the form

∑_{{L′=(L′(0),L′(1)) \atop L′(0)=M}} L′, ∑_{{L′=(L′(0),L′(1)) \atop L′(1)=M}} L′,

where M is some lattice in L. Observe that by construction, we have the congruences

T_{𝔭}^{l} ≡ T_{𝔭}^{u} ≡ −1 on Z[L_{1}]^{𝔭−new}.

One can again use Lemma 3.3 to relate the trace operator

Tr(L) = ∑_{γ∈Z_{l𝔭(L)−1}^{×}/Z_{l𝔭(L)}^{×}} γ ⋆ L = ∑_{γ∈Z_{l𝔭(L)−1}^{×}/Z_{l𝔭(L)}^{×}}[γb]L_{0}

in Z[L_{1}] to the operators T_{𝔭}^{l}(L) and T_{𝔭}^{u}(L) to obtain the following result. Here, we extend the notion of types to classes of B_{𝔭}^{×}/R_{𝔭}^{×} in the natural way via (10).

Lemma 3.5. Let x be a class in B_{𝔭}^{×}/R_{𝔭}^{×} with l_{𝔭}(x) ≥ 2. Then, we have

Tr(x) = \begin{cases}T_{𝔭}^{l}(pr(x)) & if x is of type I \\ T_{𝔭}^{u}(pr(x)) & if x is of type II. \end{cases}

Moreover, in the 𝔭-new quotient of Z[B_{𝔭}^{×}/R_{𝔭}^{×}] corresponding to Z[L_{1}]^{𝔭−new},

Tr(x) = −pr(x).

Proof. See [5, Lemma 6.11]. □

Remark The constructions and results above do depend on the choice of base 1-lattice L_{0} = (L_{0}(0), L_{0}(1)). In particular, the definition of type depends upon the orientation of the underlying Eichler order R_{𝔭} = R(0) ∩ R(1).

4. Construction of p-adic interpolation series

We now construct p-adic interpolation series on the profinite group X for the values L^{(k)}(1/2, π × Ω), i.e. where Ω is a character factoring through X. We divide into cases k on the generic root number ϵ = ϵ(1/2, π × Ω) ∈ {±1}. Here, we shall also write ⋆ to denote the natural action of K̂^{×} by left multiplication on the set K^{×}\B̂^{×}/R̂^{×} induced by the choice of embedding K into B or 𝔹 (depending on whether k = 0 or k = 1 respectively). Remark Our constructions below works more generally whenever the vector in known φ ∈ π′ to take values in a discrete valuation ring and each prime dividing the level structure is known to split in K.

4.1. The case of k = 0. Let π = ⊗_{v}π_{v} be a cuspidal automorphic representation of GL_{2}(A_{F}) having trivial central character. Fix a prime 𝔭 of O_{F} with underlying rational prime p, and an embedding \overline{Q} ⟶ \overline{Q}_{p}. We assume Hypotheses 1.1 and 1.3.

Lemma 4.1. Let L be a finite extension of Q_{p} containing the eigenvalues of π. Let O = O_{L} denote its ring of integers. If the quaternion algebra B is totally definite, then we can and do choose φ ∈ π′ in such a way that φ takes values in the ring O.

Proof. By Hypothesis 1.1, the eigenvalues of π = JL(π′) of GL_{2}(A_{F}) are algebraic. Hence, Q(π) is a number field, and there exists a finite extension L of Q_{p} containing the eigenvalues of π. Any vector φ ∈ π′ has the same eigenvalues as π by Jacquet-Langlands, and is determined uniquely up to multiplication by nonzero complex scalar by this condition. Moreover, we can view any φ ∈ π′ as an automorphic function on the finite set C(B; H) = B^{×}\B̂^{×}/H, for H ⊂ B̂^{×} some compact open subgroup (with H_{𝔭} = R_{𝔭}^{×}). Fixing a basis for the space of functions C(B; H) ⟶ C, we then can choose φ ∈ π′ to take values in Q(π), or even Z(π). □

We now give the main construction of p-adic L-functions for the k = 0 setting.

4.1.1. The case of δ = 0. Let us first consider the case where R_{𝔭} of B_{𝔭} is Eichler of level 𝔭^{δ} = 1, i.e. that R_{𝔭} is maximal. Recall that we let V be a simple left B_{𝔭}-module for which V ≈ F_{𝔭}^{2} as an F_{𝔭}-vector space. Recall as well that we write L = L(V) to denote the set of O_{F_{𝔭}}-lattices in V, and that we fix a lattice L_{0} ∈ L in such a way that {α ∈ B_{𝔭} : αL_{0} ⊂ L_{0}} = R_{𝔭}. Thus we may fix a bijection

(11)  B_{𝔭}^{×}/R_{𝔭}^{×} ⟶ L, x ⟼ xL_{0}.

Using this bijection, we now fix the following sequence of local classes x_{n} ∈ B_{𝔭}^{×}/R_{𝔭}^{×}. Definition Fix a K_{𝔭}-basis e of V. For each integer n ≥ 0, let L_{n} = L_{n}(e) denote the lattice in L defined by L_{n} = Z_{n}e. We then define x_{n} = x_{n}(e) to be the class in B_{𝔭}^{×}/R_{𝔭}^{×} corresponding to L_{n} under the fixed bijection (11).

Lemma 4.2. The sequence x = x(e) of classes (x_{n})_{n≥0} = (x_{n}(e))_{n≥0} in B_{𝔭}^{×}/R_{𝔭}^{×} of Definition 4.1.1 satisfies the following properties for each integer n ≥ 0.

(i) l_{𝔭}(x_{n}) = n.

(ii) x_{n} = pr_{u}(x_{n+1}).

Proof. Property (i) is a direct consequence of the definition of l_{𝔭}(x_{n}) = l_{𝔭}(L_{n}). Property (ii) is then a direct consequence of the definition of upper predecessor pr_{u}(x_{n}), i.e. as pr_{u}(x_{n+1}) = pr_{u}(L_{n+1}) = Z_{n}L_{n+1} = Z_{n}Z_{n+1}e = Z_{n}e = L_{n}. □

Remark Given such a sequence of local classes (x_{n})_{n≥0} in B_{𝔭}^{×}/R_{𝔭}^{×}, we shall also write each x_{n} to denote its corresponding adele in B̂^{×}/R̂^{×}. Let us now fix a sequence of classes (x_{n})_{n≥0} as in Definition 4.1.1 above. Observe that Corollary 3.4 above implies that for each integer n ≥ 0, we have the relation

Tr(x_{n+1}) = T_{𝔭}^{l}(x_{n}) − x_{n}^{∗}

in Z[B_{𝔭}^{×}/R_{𝔭}^{×}], where for ϖ_{𝔓} a fixed uniformizer at 𝔓,

x_{n}^{∗} = \begin{cases}0 & if n = 0 and 𝔭O_{K} = 𝔓 is inert \\ ϖ_{𝔓}x_{n} & if n = 0 and 𝔭O_{K} = 𝔓^{2} is ramified \\ (ϖ_{𝔓} + ϖ_{𝔓^{∗}})x_{n} & if n = 0 and 𝔭O_{K} = 𝔓𝔓^{∗} is split \\ pr_{l}(x_{n}) & if n ≥ 1. \end{cases}

Corollary 4.3. Assume that π = JL(π′) has trivial central character, and let φ ∈ π′ be any decomposable vector whose component at 𝔭 is fixed by R_{𝔭}^{×}. Let (x_{n})_{n≥0} be the sequence of classes of Lemma 4.1.1. Then, for each integer n ≥ 1,

Tr (φ(x_{n+1})) := ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ(γ ⋆ x_{n+1}) = T_{𝔭}^{l}φ(x_{n}) − φ(x_{n−1}).

Proof. Extending by linearity, we obtain from Corollary 3.3 the relation

Tr (φ(x_{n+1})) := ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ(γ ⋆ x_{n+1}) = T_{𝔭}^{l}φ(x_{n}) − φ(pr_{l}(x_{n}))

for each integer n ≥ 1. Here, we have viewed each local class x_{n} ∈ B_{𝔭}^{×}/R_{𝔭}^{×} as its corresponding adele in B̂^{×}/R̂^{×}. We have also used the fact that the operators Tr and T_{𝔭}^{l} affect only the component at 𝔭. Now, recall that we write L_{n} = x_{n}L_{0} to denote the lattice in L corresponding to x_{n}, whence the lower predecessor pr_{l}(x_{n}) is by definition the class corresponding under (11) to the lattice pr_{l}(L_{n}) = 𝔭Z_{n−1}L_{n}. Observe that pr_{l}(L_{n}) = 𝔭pr_{u}(L_{n}), whence we argue that φ(pr_{l}(x_{n})) = φ(pr_{u}(x_{n})) as a consequence of the fact that φ has trivial central character. The desired relation thens follow from the property pr_{u}(x_{n}) = x_{n−1}. □

Definition Assume that π = JL(π′) is 𝔭-ordinary, with trivial central character. Let φ ∈ π′ be a decomposable vector whose local component at 𝔭 is fixed by R_{𝔭}^{×}, which we can and do normalize to take values in the ring O. Writing a_{𝔭} = a_{𝔭}(π) to denote the T_{𝔭}^{l}-eigenvalue of π, and q_{𝔭} the cardinality of the residue field at 𝔭, let α_{𝔭} = α_{𝔭}(π) denote the p-adic unit root of the Hecke polynomial t^{2} −a_{𝔭}t+q_{𝔭}. We then define the system {ϑ_{n}}_{n≥1} = {ϑ_{n}(x_{n}, φ)}_{n≥1} to be the sequence of mappings ϑ_{n} : X_{n} ⟶ O given by the rule

ϑ_{n} : X_{n} ⟶ O, A ⟼ α_{𝔭}^{1−n} · φ(A ⋆ x_{n}) − α_{𝔭}^{−n} · φ(A ⋆ x_{n−1}).

Before we continue, let us explain briefly that the points x_{n} here are welldefined. Recall that we fix a maximal order R ⊂ B and a function φ on the finite set C(R̂^{×}) = B^{×}\B̂^{×}/R̂^{×}. Recall as well that we fix an embedding K → B, a priori without any special conditions. Observe that this choice of embedding induces a natural map from the set Y (R̂^{×}) = K^{×}\B̂^{×}/R̂^{×} to C(R̂^{×}), and moreover that we may view φ as a function on Y (R̂^{×}) after composition with this natural map Y (R̂^{×}) → C(R̂^{×}). This setup gives rise to a natural definition of conductor of a class or “point” x = [b] in the set Y (R̂^{×}). It is easy to see that the points x_{n} = [b_{n}] of our chosen sequence in Definition 4.1.1 each have conductor 𝔠(x_{n}) equal to 𝔠𝔭^{n}, where 𝔠 is determined uniquely by the prime-to-𝔭-part of the conductor of the order R ∩ K of K. Hence, the values φ(A ⋆ x_{n}) and φ(A ⋆ x_{n−1}) are only well-defined if A ∈ Pic(O_{𝔠𝔭^{n}}). To proceed, we could either keep track of the 𝔠 explicitly throughout (at the expense of simplicity), or else we choose a specific embedding K → B to ensure that 𝔠 = 1. Let us for simplicity choose the latter option for the rest of this work, which we record as follows.

Corollary 4.4. Suppose (as we can) that we fix an embedding K → B for which the order K ∩ B has 𝔭-power conductor, in fact that K ∩ B = O_{K}. Then, each point x_{n} = [b_{n}] has conductor 𝔠(x_{n}) = 𝔭^{n}.

Proof. That such an embedding exists is known; the local criteria are described in [27, Ch. II, §3]. In brief, such an embedding exists in our setting if each prime dividing 𝔭^{δ}𝔑^{+} is split in K, which is the case by Hypothesis 1.1. □

Proposition 4.5. The sequence of mappings ϑ_{n} : X_{n} ⟶ O of Definition 4.1.1 above forms a distribution on the profinite group X = \underleftarrow{lim}_{n} X_{n}, and hence an O-valued measure on X.

Proof. It will suffice to show that for each sufficiently large integer n ≥ 1 and each class A ∈ X_{n}, the distribution relation

ϑ_{n}(A) = ∑_{{B∈X_{n+1} \atop π_{n+1,n}(B)=A}} ϑ_{n+1}(B)

holds, where π_{n+1,n} : X_{n+1} ⟶ X_{n} denotes the natural surjective homomorphism. Equivalently, it will suffice under the same conditions to show the relation

ϑ_{n}(A) = ∑_{C∈ker(π_{n+1,n})} ϑ_{n+1}(CA),

where A on the right hand side denotes any lift of A to X_{n+1}. Now, by definitions,

ker(π_{n+1,n}) = O_{𝔭^{n},𝔭}^{×}/O_{𝔭^{n}}^{×}O_{𝔭^{n+1},𝔭}^{×}.

If n is sufficiently large, then O_{𝔭^{n}}^{×} = O_{F}^{×} is contained in the local unit group O_{𝔭^{n+1},𝔭}^{×} (cf. the discussion in the proof of of [5, Lemma 2.8]). We thus obtain for n sufficiently large that ker(π_{n+1,n}) = O_{𝔭^{n},𝔭}^{×}/O_{𝔭^{n+1},𝔭}^{×} = Z_{n}^{×}/Z_{n+1}^{×}, whence it will suffice to show

(12)  ϑ_{n}(A) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} ϑ_{n+1}(γA).

To show this relation holds for n sufficiently large, we evaluate the right hand side of (12). More precisely, for n sufficiently large, we have by definition of ϑ_{n+1} that

∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} ϑ_{n+1}(γA) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} α_{𝔭}^{−n}φ(γA ⋆ x_{n+1}) − α_{𝔭}^{−(n+1)}φ(γA ⋆ x_{n})

= α_{𝔭}^{−n} ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ(γA ⋆ x_{n+1}) − q_{𝔭}α_{𝔭}^{−(n+1)}φ(A ⋆ x_{n}).

Extending by linearity, the result of Corollary 3.4 allows us to identify the right hand side of the last expression with the sum

α_{𝔭}^{−n} (T_{𝔭}^{l}φ(A ⋆ x_{n}) − φ(A ⋆ x_{n−1})) − q_{𝔭}α_{𝔭}^{−(n+1)}φ(A ⋆ x_{n}).

Here, we have used implicitly the easy-to-check fact that the map L ↦ pr_{u}(L) commutes with the action of K_{𝔭}^{×} (cf. [5, Lemma 6.5]). Thus rearranging terms, we have shown (for n ≥ 1) that

∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} ϑ_{n+1}(γA) = (α_{𝔭}^{−n}a_{𝔭} − q_{𝔭}α_{𝔭}^{−(n+1)})φ(A ⋆ x_{n}) − α_{𝔭}^{−n}φ(A ⋆ x_{n−1}).

Observe now that we have the elementary identity

(α_{𝔭}^{−n}a_{𝔭} − q_{𝔭}α_{𝔭}^{−(n+1)}) = α_{𝔭}^{1−n},

i.e. using the constraints imposed by the factorization

X^{2} − a_{𝔭}X + q_{𝔭} = (X − α_{𝔭})(X − β_{𝔭}).

Thus for n sufficiently large, we have shown the required relation

∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ_{n+1}(γA) = α_{𝔭}^{1−n}φ(A ⋆ x_{n}) − α^{−n}φ(A ⋆ x_{n−1}) = ϑ_{n}(A).

□

4.1.2. The case of δ = 1. Let us now suppose that the local order R_{𝔭} of B_{𝔭} is Eichler of level 𝔭^{δ} = 𝔭. Recall that we write L_{1} = L_{1}(V) to denote the set of 1lattices of L = L(V). Let us fix a 1-lattice L_{0} = (L_{0}(0), L_{0}(1)) whose stabilizer under the transitive action of B_{𝔭}^{×} is equal to R_{𝔭}^{×}. We then fix a bijection

(13)  B_{𝔭}^{×}/R_{𝔭}^{×} ⟶ L_{1}, x ⟼ xL_{0}.

Using this bijection, we now fix a sequence of classes x_{n} ∈ B_{𝔭}^{×}/R_{𝔭}^{×} as follows. Definition Fix a K_{𝔭}-basis e of V. For each integer n ≥ 0, let us write L_{n} = L_{n}(e) to denote the lattice in L = L(V) defined by L_{n} = 𝔭^{−⌊ \frac{n}{2} ⌋}Z_{n}e, where

⌊x⌋ = max{m ∈ Z : m ≤ x}

is the standard floor function. For each n ≥ 1, let us write M_{n} = M_{n}(e) to denote the 1-lattice in L_{1} = L_{1}(V) defined by

M_{n} = \begin{cases}(L_{n−1}, L_{n}) & if n ≡ 1 mod 2 \\ (L_{n}, L_{n−1}) & if n ≡ 0 mod 2 \end{cases}

Let x_{n} = x_{n}(e) be the class in B_{𝔭}^{×}/R_{𝔭}^{×} corresponding to M_{n} under (13).

Lemma 4.6. The sequence x = x(e) of classes (x_{n})_{n≥1} = (x_{n}(e))_{n≥1} in B_{𝔭}^{×}/R_{𝔭}^{×} of Definition 4.1.2 above satisfies the following properties for each integer n ≥ 1.

(i) l_{𝔭}(x_{n}) = n.

(ii) pr(x_{n+1}) = x_{n}.

Proof. Property (i) is a direct consequence of the definition of l_{𝔭}(x_{n}), i.e. as we have l_{𝔭}(x_{n}) = l_{𝔭}(M_{n}) = max(l_{𝔭,0}(M_{n}), l_{𝔭,1}(M_{n})) = n. Property (ii) is then seen by a direct calculation. To be more precise, since M_{n} is of type I if n is odd and of type II if n is even, we have by definition that

pr(x_{n+1}) = pr(M_{n+1}) = \begin{cases}(L_{n}, pr_{l}(L_{n})) = (L_{n}, 𝔭Z_{n−1}L_{n}) & if n ≡ 0 mod 2 \\ (pr_{u}(L_{n}), L_{n}) = (Z_{n−1}L_{n}, L_{n}) & if n ≡ 1 mod 2. \end{cases}

Suppose first that the index n is even. We then compute the lower predecessor pr_{l}(L_{n}) = 𝔭Z_{n−1}L_{n} = 𝔭^{1−⌊ \frac{n}{2} ⌋}Z_{n−1}e = 𝔭^{−⌊ \frac{n−1}{2} ⌋}Z_{n−1}e = L_{n−1} to verify the relation pr(M_{n+1}) = M_{n}, as required. Suppose now that the index n is odd. We then compute the lower predecessor pr_{u}(L_{n}) = Z_{n−1}L_{n} = 𝔭^{−⌊ \frac{n}{2} ⌋}Z_{n−1}e = 𝔭^{−⌊ \frac{n−1}{2} ⌋}Z_{n−1}e = L_{n−1} to verify the relation pr(M_{n+1}) = M_{n}, as required. □

Fix a sequence of classes (x_{n})_{n≥1} as in Definition 4.1.2. We obtain from Lemma 3.5 above the following result.

Corollary 4.7. Let φ ∈ π′ be a decomposable vector whose local component at 𝔭 is fixed by R_{𝔭}^{×}, which we can and do normalize to take values in O. Then, for each integer n ≥ 1,

Tr (φ(x_{n+1})) := ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ(γ ⋆ x_{n+1}) = \begin{cases}T_{𝔭}^{l}φ(x_{n}) & if n ≡ 0 mod 2 \\ T_{𝔭}^{u}φ(x_{n}) & if n ≡ 1 mod 2. \end{cases}

Proof. Given the sequence of classes (x_{n})_{n≥1} of Definition 4.1.2, along with the properties of Lemma 4.6, Lemma 3.5 implies that for each n ≥ 1, the relation

Tr(x_{n+1}) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} γ ⋆ x_{n+1} = \begin{cases}T_{𝔭}^{l}(x_{n}) & if n ≡ 0 mod 2 \\ T_{𝔭}^{u}(x_{n}) & if n ≡ 1 mod 2 \end{cases}

holds in Z[B_{𝔭}^{×}/R_{𝔭}^{×}]. Viewing each class x_{n} as its corresponding adele in B̂^{×}/R̂^{×}, and extending by linearity, we obtain for each n ≥ 1 the relation

Tr (φ(x_{n+1})) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ(γ ⋆ x_{n+1}) = \begin{cases}T_{𝔭}^{l}φ(x_{n}) & if n ≡ 0 mod 2 \\ T_{𝔭}^{u}φ(x_{n}) & if n ≡ 1 mod 2. \end{cases}

Here, we have used the fact that each of the operators Tr, T_{𝔭}^{l}, and T_{𝔭}^{u} affects only the components at 𝔭. The result follows. □

Definition Let φ ∈ π′ be a decomposable vector whose local component at 𝔭 is fixed by R_{𝔭}^{×} and which we normalize to take values in O. Assume that φ is a 𝔭-ordinary eigenform for both T_{𝔭}^{u} and T_{𝔭}^{l} with common (unit) eigenvalue α_{𝔭}. Fix a sequence of classes (x_{n})_{n≥1} as for Definition 4.1.1 above. Let {ϑ_{n}}_{n≥1} = {ϑ_{n}(Φ, x_{n})}_{n≥1} be the sequence of mappings ϑ_{n} : X_{n} ⟶ O defined by

ϑ_{n} : X_{n} ⟶ O, A ⟼ α_{𝔭}^{−n} · φ(A ⋆ x_{n}).

Proposition 4.8. The sequence of mappings ϑ_{n} : X_{n} ⟶ O of Definition 4.1.2 defines a distribution on the profinite group X = \underleftarrow{lim}_{n} X_{n}, and hence an O-valued measure on X.

Proof. As explained in the proof of Proposition 4.5 above, it will suffice to show for each sufficiently large integer n that

ϑ_{n}(A) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} ϑ_{n+1}(γA).

Thus, for the mappings {ϑ_{n}}_{n≥1} of Definition 4.1.2, it will suffice to show for each sufficiently large integer n that

α_{𝔭}^{−n} · φ(A ⋆ x_{n}) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} α_{𝔭}^{−(n+1)} · φ(γA ⋆ x_{n+1}),

which after multiplying out by α_{𝔭}^{n+1} is the same as

α_{𝔭} · φ(A ⋆ x_{n}) = ∑_{γ∈Z_{n}^{×}/Z_{n+1}^{×}} φ(γA ⋆ x_{n+1}).

Now, by our hypotheses on φ, we have that α_{𝔭} · φ(A ⋆ x_{n}) = T_{𝔭}φ(A ⋆ x_{n}) for T_{𝔭} denoting either operator T_{𝔭}^{l} or T_{𝔭}^{u}, and so the required relation is a direct consequence of Corollary 4.7 above. □

4.1.3. Definitions of p-adic L-functions. Recall that in either case on the exponent δ ∈ {0, 1} in the level 𝔭^{δ} of R_{𝔭} ⊂ B_{𝔭}, we construct a sequence x(e) = (x_{n}(e))_{n≥δ} of classes x_{n} ∈ B_{𝔭}^{×}/R_{𝔭}^{×} according to Definitions 4.1.1 and 4.1.2. These sequences depend on the choice of a fixed K_{𝔭}-basis e of the B_{𝔭}- module V in the following way.

Lemma 4.9. Suppose in either subcase on δ ∈ {0, 1} that we choose a different K_{𝔭}-basis e′ of V. Then, for some element σ = (σ_{n})_{n} of X = \underleftarrow{lim}_{n} X_{n}, we have

dϑ^{(δ)}(φ, x(e′)) = dϑ^{(δ)}(φ, σ ⋆ x(e)) = (ϑ_{n}^{(δ)}(φ, σ_{n} ⋆ x_{n}(e)))_{n≥1},

equivalently

θ_{φ}^{(δ)}(x(e′)) = σθ_{φ}^{(δ)}(x(e)) = (θ_{φ}^{(δ)}(σ_{n} ⋆ x_{n}(e))) in O[[X]].

Proof. Observe that K_{𝔭}^{×} acts simply transitively on the set of K_{𝔭}-bases of V. It follows that there exists a γ ∈ K_{𝔭}^{×} for which e′ = γe = γ ⋆e. We claim that this action commutes with each of the predecessor operations pr_{∗} defined above, i.e. that pr_{∗}(γ ⋆ x) = γ ⋆ pr_{∗}(x) for each γ ∈ K_{𝔭}^{×} and x ∈ B_{𝔭}^{×}/R_{𝔭}^{×}, as a direct consequence of the definitions. It is then easy to see that for each integer n ≥ 1, there exists an element σ_{n} ∈ X_{n} for which ϑ_{n}^{(δ)}(φ, x_{n}(e′)) = ϑ_{n}^{(δ)}(φ, σ_{n} ⋆ x_{n}(e)). To be more precise, we claim that this σ_{n} is determined by the image of γ in X_{n}. Since the σ_{n} come from one element γ ∈ K_{𝔭}^{×}, it is clear that (σ_{n})_{n} defines a compatible sequence, whence σ = (σ_{n})_{n} defines an element of the profinite limit X = \underleftarrow{lim}_{n} X_{n}. □

Thus, our elements θ_{Φ}^{(δ)}(e) are only well-defined up to multiplication by σ ∈ X. To correct this, we make the following modification. Let Λ = O[[X]]. Given λ ∈ Λ, let λ^{∗} denote the image of λ under the involution of Λ induced by the operation sending group elements σ ∈ X to their inverses σ^{−1} ∈ X. The elements θ_{φ}^{(δ)}(e)θ_{φ}^{(δ)}(e)^{∗} ∈ Λ are then well-defined. This leads us to make the following Definition Let φ ∈ π′ be a decomposable, O-valued, 𝔭-ordinary eigenvector as described above, i.e. so that the component of φ at 𝔭 is fixed by R_{𝔭}^{×}, where R_{𝔭} ⊂ B_{𝔭} is an Eichler order of level 𝔭^{δ} with δ ∈ {0, 1}. Let e be any K_{𝔭}-basis of V, and x(e) = (x_{n}(e))_{n≥δ} the sequence of classes described either by Definition 4.1.1 if δ = 0 or else by Definition 4.1.2 if δ = 1. We then define the imprimitive p-adic L-function associated to φ and the profinite group X to be the product

L_{𝔭}(φ, K) = L_{𝔭}^{(δ)}(φ, K) = θ_{φ}^{(δ)}(x(e))θ_{φ}^{(δ)}(x(e))^{∗} in Λ = O[[X]].

4.1.4. Interpolation. We now derive the interpolation properties of the imprimitive p-adic L-functions L_{𝔭}(φ, K) = L_{𝔭}^{(δ)}(φ, K) ∈ Λ = O[[X]]. This leads us to choose a certain normalization, i.e. to define primitive p-adic L-functions 𝔏_{𝔭}^{(δ)}(π′, K) which do not depend on our choice of vector φ ∈ π′. Let Ω be a character of X. We view such a character as a homomorphism Ω : X ⟶ \overline{Q}_{p} via our fixed embedding \overline{Q} → \overline{Q}_{p}, and moreover (enlarging O if necessary) as a homomorphism Ω : X ⟶ O. We shall use the fact that any character extends to an algebra homomorphism Ω : Λ ⟶ O via the specialization map λ ↦ ρ(λ) = ∫_{X} Ω(σ)dλ(σ).

Lemma 4.10. We have that Ω(λ^{∗}) = Ω^{−1}(λ) for any element λ ∈ X.

Proof. Since Ω : X ⟶ O is a group homomorphism, Ω(σ^{−1}) = Ω^{−1}(σ). Hence,

Ω(λ^{∗}) = ∫_{X} Ω(σ)dλ^{∗}(σ) = ∫_{X} Ω(σ^{−1})dλ(σ) = ∫_{X} Ω^{−1}(σ)dλ(σ) = Ω^{−1}(λ).

□

Lemma 4.11. In our constructions above of the measures dϑ_{φ}^{(δ)}(x) giving rise to the elements θ_{φ}^{(δ)}(x) of O[[X]], our choice of sequence (φ_{n})_{n≥1} = (x_{n} · φ)_{n≥1} in either case on δ defines a sequence of test vectors in the sense of [7, § 1.2.3].

Proof. Fix n ≥ 1. It suffices to check invariance properties at 𝔭 for each vector φ_{n} = x_{n} · φ ∈ π′ for each choice of δ ∈ {0, 1}. Let us first assume that δ = 0. Recall that we fix a class x_{n} ∈ B_{𝔭}^{×}/R_{𝔭}^{×} which is invariant by the action of Z_{n}^{×}, where Z_{n} = O_{F_{𝔭}} + ϖ_{𝔭}^{n}O_{K_{𝔭}}. We deduce that the local vector x_{n} · φ_{𝔭} ∈ π_{𝔭} is fixed by the action of R_{𝔭,n}^{×}, where R_{𝔭,n} ⊂ B_{𝔭} is the unique (up to K_{𝔭}^{×}-conjugacy) order for which c(R_{𝔭,n}) = c(Ω_{𝔭}) = n and d(R_{𝔭,n}) = c(π_{𝔭}) = δ = 0, whence the required local invariance follows from the work of Gross-Prasad [10], cf. [7, §1.2.3] or the summary given above. Let us assume now that δ = 1. Recall that we fix a class x_{n} ∈ B_{𝔭}^{×}/R_{𝔭}^{×} which corresponds under our fixed bijection (13) to a 1-lattice (L(0), L(1)) we have by Lemma 4.6 (i) that l_{𝔭}(x_{n}) = max(l_{𝔭}(L(0)), l_{𝔭}(L(1))) = n, equivalently that L(0) say is fixed by Z_{n−1}^{×} and L(1) by Z_{n}^{×} (as we can assume without loss of generality). We then deduce that the local vector x_{n} · φ_{𝔭} ∈ π_{𝔭} is fixed by the action of R_{𝔭,n}^{×}, where R_{𝔭,n} is the unique order of B_{𝔭} up to K_{𝔭}^{×}-conjugacy which can be expressed as the intersection of maximal order R_{𝔭,n} = R_{1,𝔭,n} ∩ R_{2,𝔭,n}, where c(R_{1,𝔭,n}) = c(Ω_{𝔭}) = n and c(R_{2,𝔭,n}) = c(Ω_{𝔭}) − δ = n − 1. The required local invariance is then deduced from [7], again cf. [7, §1.2.3] or the summary above. □

Equipped with this result, we now give the following main interpolation formula. Recall that for a given an integer n ≥ 1, we write m(O_{𝔭^{n}}) to denote the volume of Ô_{𝔭^{n}}^{×} in the space K^{×}\A_{K}^{×}/A_{F}^{×} with respect to our fixed choice of Haar measure (which assigns K^{×}\A_{K}^{×}/A_{F}^{×} volume one). Recall as well that we write the class number of F as h(O_{F}) = |Pic(O_{F})| = |F̂^{×}/F^{×}Ô_{F}^{×}|.

Proposition 4.12. Let Ω be a primitive character of X of conductor 𝔭^{n} with n ≥ 1. Then, for each choice δ ∈ {0, 1}, we have the following interpolation formula for the associated imprimitive p-adic L-function L_{𝔭}^{(δ)}(φ, K),

(14)  Ω(L_{𝔭}^{(δ)}(φ, K)) = α_{𝔭}^{2(δ−1−n)} · (\frac{h(O_{F})}{m(O_{𝔭^{n}})})^{2} · (φ, φ) · L(1/2, π × Ω).

Here, L(1/2, π ×Ω) denotes the algebraic L-value of Corollary 2.2 above, which we view as an element of \overline{Q}_{p} via our fixed embedding \overline{Q} ⟶ \overline{Q}_{p}.

Proof. Lemma 4.10 implies that for either choice of δ ∈ {0, 1}, we have

Ω(L_{𝔭}^{(δ)}(φ, K)) = Ω(θ_{φ}^{(δ)}(x))Ω(θ_{φ}^{(δ)}(x)^{∗}) = Ω(θ_{φ}^{(δ)}(x))Ω^{−1}(θ_{φ}^{(δ)}(x)).

Let us first assume that δ = 1, whence we are reduced to evaluating the sums

Ω(θ_{φ}^{(1)}(x)) = ∫_{X} Ω(σ)θ_{φ}^{(1)}(x)(σ) = α_{𝔭}^{−n} ∑_{A∈X_{n}} Ω(A)φ(A ⋆ x_{n}).

Here, the second equality follows by the definition of ϑ_{n} = ϑ_{n}(φ, x_{n}) of dϑ(φ, x). Since we know by Lemma 4.11 that the vectors defined by φ_{n} = x_{n} · φ are test vectors in the sense of [7, §7.1], we argue (cf. [32, (5)]) that we have the relation

Ω(θ_{φ}^{(1)}(x)) = \frac{h(O_{F})}{m(O_{𝔭^{n}})} · P_{Ω}^{B}(φ).

That is, we first argue that n that the natural map Pic(O_{F}) → Pic(O_{𝔭^{n}}) is injective. Since Ω factors through Pic(O_{𝔭^{n}})/Pic(O_{F}) ≈ K̂^{×}/F̂^{×}K^{×}Ô_{𝔭^{n}}^{×} by our assumptions, and since it is invariant on Ô_{𝔭^{n}}^{×}, we can express the period integral P_{Ω}^{B}(φ) as

P_{Ω}^{B}(φ) := ∫_{A_{K}^{×}/A_{F}^{×}K^{×}} φ(t)Ω(t)dt = m(O_{𝔭^{n}}) ∑_{t∈K̂^{×}/F̂^{×}Ô_{𝔭^{n}}^{×}} φ(t)Ω(t).

On the other hand, we can decompose our finite sum over X_{n} = Pic(O_{𝔭^{n}}) as

∑_{A∈X_{n}} Ω(A)φ(A) = ∑_{τ∈Pic(O_{F})} ∑_{t∈Pic(O_{𝔭^{n}})/Pic(O_{F})} Ω(t)φ(τt),

using that Ω is trivial on Pic(O_{F}). Now, observe that the inner sum here equals

∑_{τ^{−1}t∈Pic(O_{𝔭^{n}})/Pic(O_{F})} Ω(τ^{−1}t)φ(t) = ∑_{t∈Pic(O_{𝔭^{n}})/Pic(O_{F})} Ω(t)φ(t),

and hence that

∑_{A∈X_{n}} Ω(A)φ(A) = h(O_{F}) ∑_{t∈Pic(O_{𝔭^{n}})/Pic(O_{F})} Ω(t)φ(t) = \frac{h(O_{F})}{m(O_{𝔭^{n}})} · P_{Ω}^{B}(φ).

The stated interpolation formula is then easy to deduce from Theorem 1.2. Let us now assume that δ = 0, whence we are reduced to evaluating the sums

Ω(θ_{φ}^{(0)}(x)) = α_{𝔭}^{1−n} ∑_{A∈X_{n}} Ω(A)φ(A ⋆ x_{n}) − α_{𝔭}^{−n} ∑_{A∈X_{n}} Ω(A)φ(A ⋆ x_{n−1}).

Since the vectors φ_{n} = x_{n} · φ are again test vectors in the sense of [7, §7.1] by Lemma 4.11 above, we see that the second sum in this term must vanish, i.e. since the conductor of the test vector φ_{n−1} = x_{n−1} · φ appearing in this term does not have the same conductor as the character Ω. Indeed, observe that given any u ∈ A_{K}^{×}, which we embed into the finite set B^{×}\B̂^{×}/R̂^{×} via our choice of (optimal) embedding K → D, we must have that P_{Ω}(u · φ) = Ω^{−1}(u)P_{Ω}(φ), whence

Ω(θ_{φ}^{(0)}(x)) = α_{𝔭}^{1−n} ∑_{A∈X_{n}} Ω(A)φ(A ⋆ x_{n}),

which reduces us to the same style of proof as given for the δ = 1 case above. □

Although it is clear, let us state for the record the following direct consequence.

Corollary 4.13. The specialization value Ω(L_{𝔭}^{(δ)}(φ, K)) vanishes if and only if the complex central value L(1/2, π × Ω) vanishes when the conductor of Ω is nontrivial.

It is also clear that dividing out by (φ, φ) gives an interpolation formula which does not depend on the scaling factor of φ ∈ π′. We record this as follows.

Definition Let us in either case on δ ∈ {0, 1} define the primitive p-adic Lfunction 𝔏_{𝔭}(π′, K) = 𝔏_{𝔭}^{(δ)}𝔭(π′, K) associated to π′ and X to be the element of Λ = O[[X]] obtained by dividing out by the inner product (φ, φ), i.e.

(15)  𝔏_{𝔭}^{δ}(π′, K) = L_{𝔭}^{(δ)}(φ, K)/(φ, φ).

The following result is then easy to see from the discussion above.

Corollary 4.14. The element 𝔏_{𝔭}^{δ}(π′, K) ∈ Λ = O[[X]] does not depend on the choice of scaling factor of φ ∈ π′, and satisfies the following interpolation property: For each primitive character Ω of X of conductor 𝔭^{n} with n ≥ 1, we have that Ω(𝔏_{𝔭}^{δ}(π′, K)) = α_{𝔭}^{2(δ−1−n)} · (h(O_{F})/m(O_{𝔭^{n}}))^{2} · L(1/2, π × Ω) in \overline{Q}_{p}.

4.2. The case of k = 1. We now extend the construction given above to give a p-adic interpolation series for the setting of k = 1, i.e. where ϵ = ϵ(1/2, π ×Ω) is generically equal to −1. Keep all of the setup leading to Theorem 2.3 above, so that 𝔹 is the indefinite quaternion algebra which is ramified at all real places of F except for one fixed place τ, as well as the finite places corresponding to prime divisors of the inert level 𝔑^{−}. By Hypothesis 1.1, we know that 𝔹 is split at 𝔭. Hence, we can and do fix an isomorphism 𝔹_{𝔭}^{×} ≈ GL_{2}(F_{𝔭}). Let M = {M_{U}}_{U} be the Shimura curve over F associated 𝔹. Let A be an abelian variety defined over F which is parametrized by M. Recall that according to [35, §3.2.2], we have an an automorphic representation π_{A} = Hom^{0}(J, A) = \underrightarrow{lim}_{H} Hom^{0}(J_{H}, A) = \underrightarrow{lim}_{H} Hom_{F} (J_{H}, A) ⊗_{Z} Q of 𝔹^{×}(A_{F}) defined over Q, and moreover that this representation decomposes as a product π_{A} = ⊗_{v}π_{A,v} of absolutely irreducible representations π_{A,v} of 𝔹_{v}^{×}. Recall as well that for any choice of vector φ ∈ π_{A} and CM point Q ∈ M^{K^{×}}(K^{ab}), we have that φ(Q) is a well-defined element of the Mordell-Weil group A(K^{ab}), where K^{ab} denotes the maximal abelian extension of K. Let us define O in this setting to be the tensor product A(K^{ab})⊗_{Z}Z_{p}. We shall assume from now on that π_{A} has trivial central character, hence that π_{A} = π_{A}^{∨} is self-dual. Thus A is principally polarized, and we can fix an identification A ≈ A^{∨}. Granted Hypotheses 1.3 and 1.1 hold for π = π_{A}, the construction of the p-adic L-function L_{𝔭}(π, K) given above for k = 0 extends formally, as we now explain.

4.2.1. The case of δ = 0. Assume again that the local order R_{𝔭} ⊂ 𝔹_{𝔭} is Eichler of level 𝔭^{δ} = 1, i.e. that R_{𝔭} is maximal. Recall again that we fix V a simple left 𝔹_{𝔭}-module for which V ≈ F_{𝔭}^{2} as an F_{𝔭}-vector space, that we let L(V) denote the set of O_{F_{𝔭}}-lattices in V, and that we have the bijection (9). The arguments of Lemma 4.1.1, Corollary 4.3, and Proposition 4.5 above imply the following result.

Corollary 4.15. Let φ ∈ π_{A} be a nonzero 𝔭-ordinary decomposable vector whose component at 𝔭 is fixed by R_{𝔭}^{×}. Let (x_{n})_{n≥0} denote the sequence of classes in 𝔹_{𝔭}^{×}/R_{𝔭}^{×} as defined above. Writing T_{𝔭}^{l} to denote the lattice Hecke operator defined above, let a_{𝔭} = a_{𝔭}(π_{A}) denote the eigenvalue of T_{𝔭}^{l} acting on φ, with α_{𝔭} = α_{𝔭}(π_{A}) the unit root of the Hecke polynomial t^{2} − a_{𝔭}t + q_{𝔭}. Then, the system of mappings {ϕ_{n}}_{n≥1} = {ϕ_{n}(x_{n}, φ)}_{n≥1} defined by

(16)  ϕ_{n} : X_{n} ⟶ O, A ⟼ α_{𝔭}^{1−n} · φ(A ⋆ x_{n}) − α_{𝔭}^{−n} · φ(A ⋆ x_{n−1})

determines a distribution on the profinite group X = \underleftarrow{lim} X_{n}.

4.2.2. The case of δ = 1. Let us now assume that the order R_{𝔭} ⊂ 𝔹_{𝔭} is Eichler of level 𝔭^{δ} = 𝔭. Recall that in this case, we write L_{1} = L_{1}(V) to denote the set of 1-lattices of L = L(V). We then fix a 1-lattice L_{0} = (L_{0}(0), L_{0}(1)) whose stabilizer under the transitive action of 𝔹_{𝔭}^{×} is equal to R_{𝔭}^{×}, which allows us to define the bijection (10). Again, we fix a K_{𝔭}-basis e of V, and for each integer n ≥ 1 write L_{n} = L_{n}(e) to denote the lattice in L = L(V) defined by L_{n} = 𝔭^{−⌊ \frac{n}{2} ⌋/2}Z_{n}e. For each n ≥ 1, let J_{n} = J_{n}(e) to denote the 1-lattice in L_{1} = L_{1}(V) defined by

J_{n} = \begin{cases}(L_{n−1}, L_{n}) & if n ≡ 0 mod 2 \\ (L_{n}, L_{n−1}) & if n ≡ 1 mod 2 \end{cases}

Let x_{n} = x_{n}(e) ∈ 𝔹_{𝔭}^{×}/R_{𝔭}^{×} be the class corresponding to J_{n} under (10). The arguments of Lemma 4.6, Corollary 4.7, and Proposition 4.8 imply the following result.

Corollary 4.16. Let φ ∈ π_{A} be a nonzero decomposable vector whose component at 𝔭 is fixed by R_{𝔭}^{×}. Let (x_{n})_{n≥0} denote the sequence of classes in 𝔹_{𝔭}^{×}/R_{𝔭}^{×} defined by the 1-lattices J_{n} above. Suppose that φ is a 𝔭-ordinary eigenvector for both of the operators T_{𝔭}^{l} and T_{𝔭}^{u} with common (unit) eigenvalue α_{𝔭} = α_{𝔭}(π_{A}) Then, the system {ϕ_{n}}_{n≥1} = {ϕ_{n}(x_{n}, φ)}_{n≥1} of mappings

(17)  ϕ_{n} : X_{n} ⟶ O, A ⟼ α_{𝔭}^{−n} · φ(A ⋆ x_{n})_{n≥1}

defines a distribution on the profinite group X = \underleftarrow{lim} X_{n}.

4.2.3. Definition of the interpolation series. We have now defined in either case on the exponent δ ∈ {0, 1} an O-valued distribution

dϕ^{(δ)}(φ, x(e)) = {ϕ_{n}^{(δ)}(φ, x_{n}(e))}_{n≥1}

on X whose corresponding element in O[[X]] = \underleftarrow{lim}_{n} O[X_{n}] we denote by

Φ_{φ}^{(δ)} = Φ_{φ}^{(δ)}(x(e)) = (Φ_{φ}^{(δ)}(x_{n}(e))).

These elements depend on the choice of K_{𝔭}-basis e of V in the same way as described for Lemma 4.9. We again account for this ambiguity by considering the elements

Φ_{φ}^{(δ)}(x(e))Φ_{φ}^{(δ)}(x(e))^{∗} in O[[X]],

where Φ_{φ}^{δ}(x(e))^{∗} denotes the image of Φ_{φ}^{δ}(x(e)) under the involution of O[[X]] induced by sending elements σ ∈ X to their inverses σ^{−1} ∈ X. Definition Let φ ∈ π_{A} be a nonzero decomposable vector satisfying Hypotheses 1.1 and 1.3 above. Let e be any K_{𝔭}-basis of V. Let x(e) = (x_{n}(e))_{n≥δ} be either the sequence of classes defined above. Let us now define the well-defined elements

(18)  D_{𝔭}^{(δ)}(φ, K) = Φ_{φ}^{(δ)}(x(e))Φ_{φ}^{(δ)}(x(e))^{∗} ∈ O[[X]].

4.2.4. Interpolation. The elements defined in (18) satisfy the following interpolation property. Recall that we write ⟨·, ·⟩ to denote the L-linear Néron-Tate pairing

⟨·, ·⟩_{L} : A(\overline{F})_{Q} ⊗_{R} A(\overline{F})_{Q} ⟶ L ⊗_{Q} C

defined in [35, §1]. Note that this construction extends in a natural way to define an O_{L}-linear Néron-Tate height pairing

⟨·, ·⟩_{O_{L}} : A(\overline{F})_{Q} ⊗_{R} A(\overline{F})_{Q} ⟶ O_{L} ⊗_{Q} C.

We argue that these pairings also extend in a natural way by Z_{p}-linearity to define pairings on the elements in O we have considered above, and hence that we can derive the following straightforward consequence from the discussion above.

Proposition 4.17. Let Ω be a primitive ring class character of X of conductor 𝔭^{n} with n ≥ 1. Then, for each choice of exponent δ ∈ {0, 1}, we have the following interpolation formula for the elements D_{𝔭}^{(δ)}(φ, K) defined in (18) above composed with the O_{L}-linear Néron-Tate height pairing ⟨ , ⟩_{L},

⟨ , ⟩_{L} ∘ Ω(D_{𝔭}^{(δ)}(φ, K))

= α_{𝔭}(π_{A})^{−2(1−δ−n)} · (\frac{h(O_{F})}{m(O_{𝔭^{n}})})^{2} · \frac{ζ(2)L′(1/2, π_{A} × Ω)}{4L(1, η)^{2}L(1, π_{A}, ad)} · α(x_{n} · φ, x_{n} · φ).

Proof. The proof is formally identical to that of Proposition 4.12 above, using Theorem 2.3 in lieu of Theorem 1.2 for the interpolation values. □

Remark The recent work of Cai-Shu-Tian [3] should apply in this setting (as well as in the indefinite setting) to supply even more precise interpolation formulae for these constructions. Acknowledgements. I should like to thank Christophe Cornut first and foremost for extremely helpful comments about the main construction. I should also like to thank Masataka Chida, Henri Darmon, Daniel Disegni, Fred Diamond, Toby Gee, Min-Lun Hsieh, Kimball Martin, Udi de Shalit, Xinxi Yuan, Shouwu Zhang, Wei Zhang, and an anonymous referee for helpful comments and exchanges.

References

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

[2] M. Bertolini and H. Darmon, Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic Z_{p}-extensions, Ann. of Math. 162 (2005), 1 - 64.

[3] L. Cai, J. Shu, and Y. Tian, Explicit Gross-Zagier and Waldspurger Formulae, Algebra Number Theory 8 10 (2014), 2523 - 2572.

[4] M. Chida and M.-L. Hsieh, Special values of anticyclotomic L-functions for modular forms, preprint (2014).

[5] C. Cornut and V. Vatsal, Nontriviality of Rankin-Selberg L-functions and CM points, In “L-functions and Galois Representations”, Ed. Burns, Buzzard and Nekovář, Cambridge University Press (2007), 121 - 186

[6] C. Cornut and V. Vatsal, CM Points and Quaternion Algebras, Doc. Math. 10 (2005), 263-309.

[7] D. File, K. Martin, and A. Pitale, Test vectors and central L-values for GL(2), Algebra Number Theory (to appear)

[8] B.H. Gross, Heights and special values of L-series, in “Number Theory” (Montreal, Quebec, 1985), CMS Conference Proceedings 7, Providence RI, American Mathematical Society 1987 [9] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent.

math. 84 (1986), 225-320.

[10] B.H. Gross and D. Prasad, Test vectors for linear forms, Math. Ann. 291 (1991), 343-355

[11] A. Hamieh, Special values of anticyclotomic L-functions modulo λ, Manuscripta Math. 145 3 (2014) 449-472.

[12] S. Haran, p-adic L-functions for modular forms, Compos. Math. 62 no. 1 (1987), 31-46.

[13] H. Hida, A p-adic measure attached to the zeta functions associated to two elliptic modular forms I, Invent. math. 79, 159-195 (1985).

[14] H. Hida, On p-adic L-functions of GL(2) × GL(2) over totally real fields, Ann. de l’Institut Fourier, 41 no. 2 (1991), 311-391.

[15] P.-C. Hung, On the non-vanishing mod l of central values with anticyclotomic twists for Hilbert modular forms, preprint (2014).

[16] H. Jacquet, Automorphic forms on GL(2), Part II, Springer LNM, New York, 1972.

[17] H. Jacquet and N. Chen, Positivity of quadratic basechange L-functions, Bull. Soc. Math. France 129 (2001), no. 1, 33-40.

[18] H. Jacquet and R. Langlands, Automorphic forms on GL(2), Springer LNM 278, New York, 1970.

[19] D. Loeffler, P -adic integration on ray class groups and nonordinary p-adic L-functions, in “Iwasawa 2012”, available at

http://arxiv.org/abs/1304.4042.

[20] K. Martin and D. Whitehouse,Central L-values and Toric Periods for GL(2), IMRN 2009 no. 1 (2009), 141-191

[21] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. math. 25 (1974), 1-61.

[22] B. Perrin-Riou, Fonctions L p-adiques associées à une forme modulaire et à un corps quadratique imaginaire, J. of the London Math. Soc. (2) 38 (1988), 1 - 32.

[23] R. Pollack and T. Weston, On anticyclotomic μ-invariants of modular forms, Compos. Math. 147 (2011) no. 5, 1353-1381.

[24] H. Saito, On Tunnell’s formula for characters of GL(2), Compos. Math. 85 no. 1 (1993), 99-108.

[25] G. Shimura, The special values of zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), 637-679.

[26] J. Tunnell, Local ϵ-factors and characters of GL(2), Amer. J. Math. 105 no. 6 (1983), 1277-1307.

[27] M.-F. Vigneras, Arithmétique des Algèbres des Quaternions, Springer-Verlag Lecture Notes in Mathematics 800 (1980), New York.

[28] J. Van Order, On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms, Int. J. of Number Theory 8, No. 4 (2012), 1-35; Erratum, Int. J. Number Theory 12 No. 1 (2016), 1-7.

[29] J. Van Order, On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms, Canad. J. Math. (2013), 403-466.

[30] V. Vatsal, Uniform distribution of Heegner points, Invent. math. 148 1-46 (2002).

[31] V. Vatsal Special values of anticyclotomic p-adic L-functions, Duke Math. J. 116 (2) (2003), 219 - 261.

[32] V. Vatsal, Special value formulae for Rankin L-functions, in “Heegner points and Rankin L-series”, MRSI Publications 49 (2003), 165-190.

[33] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54 (1985), 173-242

[34] Y. Liu, S.-W. Zhang, and W. Zhang, On p-adic Waldspurger formula, preprint (2013).

[35] X. Yuan, S.-W. Zhang, and W. Zhang, The Gross-Zagier Formula for Shimura Curves, Ann. of Math. Studies 184, Princeton University Press 2013.

Jeanine Van Order Fakultät für Mathematik Universität Bielefeld Postfach 10031 Bielefeld 33501 Germany jvanorder@math.unibielefeld.de 1500

Documenta Mathematica 22 (2017)