p-adic L-Functions of Automorphic Forms

and Exceptional Zeros

Holger Deppe

Received: May 13, 2015

Communicated by Otmar Venjakob

Abstract. We construct p-adic L-functions for automorphic representations of GL_{2} of a number field F, and show that the corresponding p-adic L-function of a modular elliptic curve E over F has an extra zero at the central point for each prime above p at which E has split multiplicative reduction, a part of the exceptional zero conjecture.

2010 Mathematics Subject Classification: 11F41, 11F67, 11F70, 11G40 Keywords and Phrases: p-adic L-function, automorphic forms, exceptional zero conjecture, Mazur-Tate-Teitelbaum conjecture

Introduction

Let F be a number field (with adele ring 𝔸_{F}), and p a prime number. Let π = ⨂_{v} π_{v} be an automorphic representation of GL_{2}(𝔸_{F}). Attached to π is the complex L-function L(s, π), s ∈ ℂ, of Jacquet-Langlands [JL70]. Under certain conditions on π, we can also define a p-adic L-function L_{p}(s, π) of π, with s ∈ ℤ_{p}. It is related to L(s, π) by the interpolation property: For every character χ : G_{p} → ℂ^{∗} of finite order we have

L_{p}(0, π ⊗ χ) = τ(χ) ∏_{𝔭|p} e(π_{𝔭}, χ_{𝔭}) · L(\frac{1}{2}, π ⊗ χ),

where e(π_{𝔭}, χ_{𝔭}) is a certain Euler factor (see theorem 4.12 for its definition) and τ(χ) is the Gauss sum of χ. L_{p}(s, π) was defined by Haran [Har87] in the case where π has trivial central character and π_{𝔭} is an ordinary spherical principal series representation for all 𝔭|p. For a totally real field F, Spieß [Sp14] has given a new construction of L_{p}(s, π) that also allows for π_{𝔭} to be a special (Steinberg) representation for some 𝔭|p. In this article, we generalize Spieß’ construction of L_{p}(s, π) to automorphic representations π of GL_{2} over any number field, with arbitrary central character, and show that L_{p} has the conjectured number of exceptional zeros at the central point. We assume that π is ordinary at all primes 𝔭|p (cf. definition 2.3), that π_{v} is discrete of weight 2 at all real infinite places v, and is the principal series representation σ(| · |^{1/2}, | · |^{−1/2}) at the complex places. We define a p-adic measure μ_{π}, which heuristically is the image under the global reciprocity map of a product of certain local distributions μ_{π_{𝔭}} on F_{𝔭}^{∗} attached to π_{𝔭} for 𝔭|p and a Whittaker function times the Haar measure on the group of p-ideles 𝕀^{p} = ∏′_{v∤p} F_{v}^{∗}. Then we can define the p-adic L-function of π as an integral with respect to μ_{π} over the Galois group G_{p} of the maximal abelian extension that is unramified outside p and ∞; it is naturally a t-variable function, where t is the ℤ_{p}-rank of G_{p}:

L_{p}(\underline{s}, π) := L_{p}(s_{1}, …, s_{t}, π) := ∫_{G_{p}} ∏_{i=1}^{t} exp_{p}(s_{i}ℓ_{i}(γ))μ_{π}(dγ)

for s_{1}, …, s_{t} ∈ ℤ_{p}, where the ℓ_{i} are ℤ_{p}-valued homomorphisms corresponding to the t independent ℤ_{p}-extensions of F (cf. section 4.7 for their definition). For a modular elliptic curve E over F corresponding to π (i.e. the local Lfactors of the Hasse-Weil L-function L(E, s) and of the automorphic L-function L(s − \frac{1}{2}, π) coincide at all places v of F), our construction allows us to define the p-adic L-function of E as L_{p}(E, \underline{s}) := L_{p}(\underline{s}, π). The condition that π be ordinary at all 𝔭|p means that E must have good ordinary or multiplicative reduction at all places 𝔭|p of F. The exceptional zero conjecture (formulated by Mazur, Tate and Teitelbaum [MTT86] for F = ℚ, and by Hida [Hi09] for totally real F) states that

ord_{s=0} L_{p}(E, s) ≥ n,  (1)

where n is the number of 𝔭|p at which E has split multiplicative reduction, and gives an explicit formula for the value of the n-th derivative L_{p}^{(n)}(E, 0) as a multiple of certain L-invariants times L(E, 1). The conjecture was proved in the case F = ℚ by Greenberg and Stevens [GS93] and independently by Kato, Kurihara and Tsuji, and for totally real fields F by Spieß [Sp14]. In this article, we prove (1) for all number fields F.

p-adic L-Functions of Automorphic Forms 691

The structure of this article is as follows: In chapter 2, we describe the local distributions μ_{π_{𝔭}} on F_{𝔭}^{∗}; they are the image of a Whittaker functional under a map δ on the dual of π_{𝔭}. For constructing δ, we describe π_{𝔭} in terms of what we call the “Bruhat-Tits graph” of F_{𝔭}^{2}: the directed graph whose vertices (resp. edges) are the lattices of F_{𝔭}^{2} (resp. inclusions between lattices). Roughly speaking, it is a covering of the (directed) Bruhat-Tits tree of GL_{2}(F_{𝔭}) with fibres ≅ ℤ. When π_{𝔭} is the Steinberg representation, μ_{𝔭} can actually be extended to all of F_{𝔭}. In chapter 3, we attach a p-adic distribution μ_{ϕ} to any map ϕ(U, x^{p}) of an open compact subset U ⊆ F_{p}^{∗} := ∏_{𝔭|p} F_{𝔭}^{∗} and an idele x^{p} ∈ 𝕀^{p} (satisfying certain conditions). Integrating ϕ over all the infinite places, we get a cohomology class κ_{ϕ} ∈ H^{d}(F^{∗}′, D_{f}(ℂ)) (where d = r + s − 1 is the rank of the group of units of F, F^{∗}′ ≅ F^{∗}/μ_{F} is a maximal torsion-free subgroup of F^{∗}, and D_{f}(ℂ) is a space of distributions on the finite ideles of F). We show that μ_{ϕ} can be described solely in terms of κ_{ϕ}, and μ_{ϕ} is a (vector-valued) p-adic measure if κ_{ϕ} is “integral”, i.e. if it lies in the image of H^{d}(F^{∗}′, D_{f}(R)), for a Dedekind ring R consisting of “p-adic integers”. In chapter 4, we define a map ϕ_{π} by

ϕ_{π}(U, x^{p}) := ∑_{ζ∈F^{∗}} μ_{π_{p}}(ζU)W^{p} (\begin{matrix}ζx^{p} & 0 \\ 0 & 1 \end{matrix})

(U ⊆ F_{p}^{∗} compact open, x^{p} ∈ 𝕀^{p}). ϕ_{π} satisfies the conditions of chapter 3, and we show that κ_{π} := κ_{ϕ_{π}} is integral by “lifting” the map ϕ_{π} ↦ κ_{π} to a function mapping an automorphic form to a cohomology class in H^{d}(GL_{2}(F)^{+}, A_{f}), for a certain space of functions A_{f}. (Here GL_{2}(F)^{+} is the subgroup of M ∈ GL_{2}(F) with totally positive determinant.) For this, we associate to each automorphic form φ a harmonic form ω_{φ} on a generalized upper-half space H_{∞}, which we can integrate between any two cusps in ℙ^{1}(F). Then we can define the p-adic L-function L_{p}(\underline{s}, π) := L_{p}(\underline{s}, κ_{π}) as above, with μ_{π} := μ_{ϕ_{π}}. By a result of Harder [Ha87], H^{d}(GL_{2}(F)^{+}, A_{f})_{π} is onedimensional, which implies that L_{p}(\underline{s}, π) has values in a one-dimensional ℂ_{p}-vector space. Finally, we formulate an exceptional zero conjecture (conjecture 4.15) for all number fields F, and show that L_{p}(\underline{s}, π) satisfies (1).

Acknowledgements

This paper is based on my Ph.D. thesis “p-adic L-functions of automorphic forms” [De13], submitted at Bielefeld University in August 2013. I would like to thank Michael Spieß for suggesting and advising the thesis, and for many helpful discussions. I am also thankful to Werner Hoffmann for a useful discussion, to the referee for pointing out a few mistakes in an earlier version of this paper, and to the CRC 701, ‘Spectral Structures and Topological Methods in Mathematics’, for providing financial support during most of my studies.

Contents

Introduction  689
1 Preliminaries  692
1.1 p-adic measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
2 Local results  693
2.1 Gauss sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
2.2 Tamely ramified representations of GL_{2}(F) . . . . . . . . . . . 694
2.3 The Bruhat-Tits graph . . . . . . . . . . . . . . . . . . . . . . . 695
2.4 Distributions on the Bruhat-Tits graph . . . . . . . . . . . . . 701
2.5 Local distributions . . . . . . . . . . . . . . . . . . . . . . . . . 703
2.6 Semi-local theory . . . . . . . . . . . . . . . . . . . . . . . . . . 707
3 Cohomology classes and global measures  708
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
3.2 Global measures . . . . . . . . . . . . . . . . . . . . . . . . . . 710
3.3 Exceptional zeros . . . . . . . . . . . . . . . . . . . . . . . . . . 714
3.4 Integral cohomology classes . . . . . . . . . . . . . . . . . . . . 715
4 p-adic L-functions of automorphic forms  716
4.1 Upper half-space . . . . . . . . . . . . . . . . . . . . . . . . . . 717
4.2 Automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . 720
4.3 Cohomology of GL_{2}(F) . . . . . . . . . . . . . . . . . . . . . . 722
4.4 Eichler-Shimura map . . . . . . . . . . . . . . . . . . . . . . . . 725
4.5 Whittaker model . . . . . . . . . . . . . . . . . . . . . . . . . . 727
4.6 p-adic measures of automorphic forms . . . . . . . . . . . . . . 729
4.7 Vanishing order of the p-adic L-function . . . . . . . . . . . . . 732

1 Preliminaries

Let X be a totally disconnected locally compact topological space, R a topological Hausdorff ring. We denote by C(X , R) the ring of continuous maps X → R, and let C_{c}(X , R) ⊆ C(X , R) be the subring of compactly supported maps. When R has the discrete topology, we also write C^{0}(X , R) := C(X , R), C_{c}^{0}(X , R) := C_{c}(X , R). We denote by ℭ𝔬(X) the set of all compact open subsets of X, and for an Rmodule M we denote by Dist(X , M) the R-module of M-valued distributions on X, i.e. the set of maps μ : ℭ𝔬(X) → M such that μ(⋃_{i=1}^{n} U_{i}) = ∑_{i=1}^{n} μ(U_{i}) for any pairwise disjoint sets U_{i} ∈ ℭ𝔬(X). For an open set H ⊆ X, we let 1_{H} ∈ C(X , R) be the R-valued indicator function of H on X. Throughout this paper, we fix a prime p and embeddings ι_{∞} : \overline{ℚ} ↪ ℂ, ι_{p} : \overline{ℚ} ↪ ℂ_{p}. Let \overline{O} denote the valuation ring of \overline{ℚ} with respect to the p-adic valuation induced by ι_{p}.

We write G := GL_{2} throughout the article, and let B denote the Borel subgroup of upper triangular matrices, T the maximal torus (consisting of all diagonal matrices), and Z the center of G. For a number field F, we let G(F)^{+} ⊆ G(F) and B(F)^{+} ⊆ B(F) denote the corresponding subgroups of matrices with totally positive determinant, i.e. σ(det(g)) is positive for each real embedding σ : F ↪ ℝ. (If F is totally complex, this is an empty condition, so we have G(F)^{+} = G(F), B(F)^{+} = B(F) in this case.) Similarly, we define G(ℝ)^{+} and G(ℂ)^{+} = G(ℂ).

1.1 p-adic measures

Definition 1.1. Let X be a compact totally disconnected topological space. For a distribution μ : ℭ𝔬(X) → ℂ, consider the extension of μ to the ℂ_{p}-linear map C^{0}(X , ℂ_{p}) → ℂ_{p} ⊗\overline{ℚ} ℂ, f ↦ ∫ fdμ. If its image is a finitely-generated ℂ_{p}-vector space, μ is called a p-adic measure.

We denote the space of p-adic measures on X by Dist^{b}(X , ℂ) ⊆ Dist(X , ℂ). It is easily seen that μ is a p-adic measure if and only if the image of μ, considered as a map C^{0}(X , ℤ) → ℂ, is contained in a finitely generated \overline{O}-module. A p-adic measure can be integrated against any continuous function f ∈ C(X , ℂ_{p}).

2 Local results

For this chapter, let F be a finite extension of ℚ_{p}, O_{F} its ring of integers, ϖ its uniformizer and 𝔭 = (ϖ) the maximal ideal. Let q be the cardinality of O_{F} /𝔭, and set U := U^{(0)} := O_{F}^{×}, U^{(n)} := 1 + 𝔭^{n} ⊆ U for n ≥ 1. We fix an additive character ψ : F → \overline{ℚ}^{∗} with ker ψ ⊇ O_{F} and 𝔭^{−1} ⊈ ker ψ. ^{1} We let | · | be the absolute value on F^{∗} (normalized by |ϖ| = q^{−1}), ord = ord_{ϖ} the additive valuation, and dx the Haar measure on F normalized by ∫_{O_{F}} dx = 1. We define a (Haar) measure on F^{∗} by d^{×}x := \frac{q}{q−1} \frac{dx}{|x|} (so ∫_{O_{F}^{×}} d^{×}x = 1).

2.1 Gauss sums

Recall that the conductor of a character χ : F^{∗} → ℂ^{∗} is by definition the largest ideal 𝔭^{n}, n ≥ 0, such that ker χ ⊇ U^{(n)}, and that χ is unramified if its conductor is 𝔭^{0} = O_{F}.

Definition 2.1. Let χ : F^{∗} → ℂ^{∗} be a quasi-character with conductor 𝔭^{f}. The Gauss sum of χ (with respect to ψ) is defined by

τ(χ) := [U : U^{(f)}] ∫_{ϖ^{−f} U} ψ(x)χ(x)d^{×}x.

1Note that there is in general no ψ such that ker(ψ) = O_{F}, since 𝔭^{−1}/O_{F} has more than p points of order p if F |ℚ_{p} has inertia index > 1.

Documenta Mathematica 21 (2016) 689–734

For a locally constant function g : F^{∗} → ℂ, we define

∫_{F^{∗}} g(x)dx := lim_{n→∞} ∫_{x∈F^{∗},−n≤ord(x)≤n} g(x)dx,

whenever that limit exists.

Lemma 2.2. Let χ : F^{∗} → ℂ^{∗} be a quasi-character with conductor 𝔭^{f}. For f = 0, assume |χ(ϖ)| < q. Then we have

∫_{F^{∗}} χ(x)ψ(x)dx = \begin{cases}\frac{1−χ(ϖ)^{−1}}{1−χ(ϖ)q^{−1}} & if f = 0 \\ τ(χ) & if f > 0. \end{cases}

(Cf. [Sp14], lemma 3.4.)

2.2 Tamely ramified representations of GL_{2}(F)

For an ideal 𝔞 ⊂ O_{F}, let K_{0}(𝔞) ⊆ G(O_{F}) be the subgroup of matrices congruent to an upper triangular matrix modulo 𝔞. Let π : GL_{2}(F) → GL(V) be an irreducible admissible infinite-dimensional representation on a ℂ-vector space V, with central quasicharacter χ. It is wellknown (e.g [Ge75], Thm. 4.24) that there exists a maximal ideal 𝔠(π) = 𝔠 ⊂ O_{F}, the conductor of π, such that the space V^{K_{0}(𝔠),χ} = {v ∈ V |π(g)v = χ(a)v ∀g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ K_{0}(𝔠)\} is non-zero (and in fact one-dimensional). A representation π is called tamely ramified if its conductor divides 𝔭. If π is tamely ramified, then π is the spherical resp. special representation π(χ_{1}, χ_{2}) (in the notation of [Ge75] or [Sp14]): If the conductor is O_{F}, π is (by definition) spherical and thus a principal series representation π(χ_{1}, χ_{2}) for two unramified quasi-characters χ_{1} and χ_{2} with χ_{1}χ_{2}^{−1} ≠ | · |^{±1} ([Bu98], Thm. 4.6.4). If the conductor is 𝔭, then π = π(χ_{1}, χ_{2}) with χ_{1}χ_{2}^{−1} = | · |^{±1}. For α ∈ ℂ^{∗}, we define a character χ_{α} : F^{∗} → ℂ^{∗} by χ_{α}(x) := α^{ord(x)}. So let now π = π(χ_{1}, χ_{2}) be a tamely ramified irreducible admissible infinitedimensional representation of GL_{2}(F); in the special case, we assume χ_{1} and χ_{2} to be ordered such that χ_{1} = | · |χ_{2}. Set α_{i} := χ_{i}(ϖ)\sqrt{q} ∈ ℂ^{∗} for i = 1, 2. (We also write π = π_{α_{1},α_{2}} sometimes.) Set a := α_{1} + α_{2}, ν := α_{1}α_{2}/q. Define a distribution μ_{α_{1},ν} := μ_{α_{1}/ν} := ψ(x)χ_{α_{1}/ν}(x)dx on F^{∗}. For later use, we will need the following condition on the α_{i}:

Definition 2.3. Let π = π_{α_{1},α_{2}} be tamely ramified. π is called ordinary if a and ν both lie in \overline{O}^{∗} (i.e. they are p-adic units in \overline{ℚ}). Equivalently, this means that either α_{1} ∈ \overline{O}^{∗} and α_{2} ∈ q\overline{O}^{∗}, or vice versa.

Proposition 2.4. Let χ : F^{∗} → ℂ^{∗} be a quasi-character with conductor 𝔭^{f}; for f = 0, assume |χ(ϖ)| < |α_{2}|. Then the integral ∫_{F^{∗}} χ(x)μ_{α_{1}/ν}(dx) converges and we have

∫_{F^{∗}} χ(x)μ_{α_{1}/ν}(dx) = e(α_{1}, α_{2}, χ)τ(χ)L(\frac{1}{2}, π ⊗ χ),

where

e(α_{1}, α_{2}, χ) =\begin{cases}\frac{(1−α_{1}χ(ϖ)q^{−1})(1−α_{2}χ(ϖ)^{−1}q^{−1})(1−α_{2}χ(ϖ)q^{−1})}{(1−χ(ϖ)α_{2}^{−1})} , & f = 0 and π spherical, \\ \frac{(1−α_{1}χ(ϖ)q^{−1})(1−α_{2}χ(ϖ)^{−1}q^{−1})}{(1−χ(ϖ)α_{2}^{−1})} , & f = 0 and π special, \\ ( \frac{α_{1}}{ν} )^{−f} = ( \frac{α_{2}}{q} )^{f} , & f > 0, \end{cases}

and where we assume the right-hand side to be continuously extended to the potential removable singularities at χ(ϖ) = q/α_{1} or = q/α_{2}.

Proof. This follows immediately from lemma 2.2 and the definition of the (Jacquet-Langlands) L-function.

2.3 The Bruhat-Tits graph

Let Ṽ denote the set of lattices (i.e. submodules isomorphic to O_{F}^{2} ) in F^{2}, and let Ẽ be the set of all inclusion maps between two lattices; for such a map e : v_{1} ↪ v_{2} in Ẽ, we define o(e) := v_{1}, t(e) := v_{2}. Then the pair T̃ := (Ṽ, Ẽ) is naturally a directed graph, connected, with no directed cycles (specifically, Ẽ induces a partial ordering on Ṽ). For each v ∈ Ṽ, there are exactly q + 1 edges beginning (resp. ending) in v, each. Recall that the Bruhat-Tits tree T = (V, E⃗) of G(F) is the directed graph whose vertices are homothety classes of lattices of F^{2} (i.e. V = Ṽ/ ∼, where v ∼ ϖ^{i}v for all i ∈ ℤ), and the directed edges \overline{e} ∈ E⃗ are homothety classes of inclusions of lattices. We can define maps o, t : E⃗ → V analogously. For each edge \overline{e} ∈ E⃗, there is an opposite edge \overline{e}′ ∈ E⃗ with o(\overline{e}′) = t(\overline{e}), t(\overline{e}′) = o(\overline{e}); and the undirected graph underlying T is simply connected. We have a natural “projection map” π : T̃ → T , mapping each lattice and each homomorphism to its homothety class. Choosing a (set-theoretic) section s : V → Ṽ, we get a bijection V × ℤ ⟶^{≅} Ṽ via (v, i) ↦ ϖ^{i}s(v). The group G(F) operates on Ṽ via its standard action on F^{2}, i.e. gv = {gx|x ∈ v} for g ∈ G(F), and on Ẽ by mapping e : v_{1} → v_{2} to the inclusion map ge : gv_{1} → gv_{2}. The stabilizer of the standard vertex v_{0} := O_{F}^{2} is G(O_{F}). For a directed edge \overline{e} ∈ E⃗ of the Bruhat-Tits tree T , we define U(\overline{e}) to be the set of ends of \overline{e} (cf. [Se80]/[Sp14]); it is a compact open subset of ℙ^{1}(F), and we have gU(\overline{e}) = U(g\overline{e}) for all g ∈ G(F). For n ∈ ℤ, we set v_{n} := O_{F} ⊕ 𝔭^{n} ∈ Ṽ, and denote by e_{n} the edge from v_{n+1} to v_{n}; the “decreasing” sequence (π(e_{−n}))_{n∈ℤ} is the geodesic from ∞ to 0. (The geodesic from 0 to ∞ traverses the π(v_{n}) in the natural order of n ∈ ℤ.) We have U(π(e_{n})) = 𝔭^{−n} for each n. On T , we have the height function h : V → ℤ (cf. [BL95]) defined as follows: The geodesic ray from v ∈ V to ∞ must contain some π(v_{n}) (n ∈ ℤ), since it has non-empty intersection with A := {π(v_{n})|n ∈ ℤ}; we define h(v) := n − d(v, π(v_{n})) for any such v_{n}. This is easily seen to be well-defined, and satisfies h(π(v_{n})) = n for all n ∈ ℤ. We have the following lemma:

Lemma 2.5. (a) For all \overline{e} ∈ E, we have

h(t(\overline{e})) = \begin{cases}h(o(\overline{e})) + 1 & if ∞ ∈ U(\overline{e}), \\ h(o(\overline{e})) − 1 & otherwise. \end{cases}

(b) For a ∈ F^{∗}, b ∈ F, \overline{v} ∈ V we have

h ((\begin{matrix}a & b \\ 0 & 1 \end{matrix}) \overline{v}) = h(\overline{v}) − ord_{ϖ}(a).

(Cf. [Sp14], lemma 3.6) Let R be a ring, M an R-module. We let C(Ṽ, M) be the R-module of maps ϕ : Ṽ → M, and C(Ẽ, M) the R-module of maps Ẽ → M. Both are G(F)- modules via (gϕ)(v) := ϕ(g^{−1}v), (gc)(e) := c(g^{−1}e). We let C_{c}(Ṽ, M) ⊆ C(Ṽ, M) and C_{c}(Ẽ, M) ⊆ C(Ẽ, M) be the (G(F)-stable) submodules of maps with compact support, i.e. maps that are zero outside a finite set. We get pairings

⟨- , -⟩ : C_{c}(Ṽ, R) × C(Ṽ, M) → M, ⟨ϕ_{1}, ϕ_{2}⟩ := ∑_{v∈Ṽ} ϕ_{1}(v)ϕ_{2}(v)  (2)

and

⟨- , -⟩ : C_{c}(Ẽ, R) × C(Ẽ, M) → M, ⟨c_{1}, c_{2}⟩ := ∑_{e∈Ẽ} c_{1}(v)c_{2}(v).  (3)

We define Hecke operators T, N : C(Ṽ, M) → C(Ṽ, M) by

T ϕ(v) = ∑_{t(e)=v} ϕ(o(e)) and Nϕ := ϖϕ (i.e. Nϕ(v) = ϕ(ϖ^{−1}v))

for all v ∈ Ṽ. These restrict to operators on C_{c}(Ṽ, R), which we sometimes denote by T_{c} and N_{c} for emphasis. With respect to (2), T_{c} is adjoint to T N, and N_{c} is adjoint to its inverse operator N^{−1} : C_{c}(Ṽ, R) → C_{c}(Ṽ, R).

T and N obviously commute, and we have the following Hecke structure theorem on compactly supported functions on Ṽ (an analogue of [BL95], Thm. 10):

Theorem 2.6. C_{c}(Ṽ, R) is a free R[T, N^{±1}]-module (where R[T, N^{±1}] is the ring of Laurent polynomials in N over the polynomial ring R[T], with N and T commuting).

Proof. Fix a vertex v_{0} ∈ Ṽ. For each n ≥ 0, let C_{n} be the set of vertices v ∈ Ṽ such that there is a directed path of length n from v_{0} to v in Ṽ, and such that d(π(v_{0}), π(v)) = n in the Bruhat-Tits tree T . So C_{0} = {v_{0}}, and C_{n} is a lift of the ”circle of radius n around v_{0}” in T , in the parlance of [BL95]. One easily sees that ⋃_{n=0}^{∞} C_{n} is a complete set of representatives for the projection map π : Ṽ → V; specifically, for n > 1 and a given v ∈ C_{n−1}, C_{n} contains exactly q elements adjacent to v in Ṽ; and we can write Ṽ as a disjoint union ⋃_{j∈ℤ} ⋃_{n=0}^{∞} N^{j}(C_{n}). We further define V_{0} := {v_{0}} and choose subsets V_{n} ⊆ C_{n} as follows: We let V_{1} be any subset of cardinality q. For n > 1, we choose q − 1 out of the q elements of C_{n} adjacent to v′, for every v′ ∈ C_{n−1}, and let V_{n} be the union of these elements for all v′ ∈ C_{n−1}. Finally, we set

H_{n,j} := {ϕ ∈ C_{c}(Ṽ, R)| Supp(ϕ) ⊆ ⋃_{i=0}^{n} N^{j}(C_{i})} for each n ≥ 0, j ∈ ℤ,

H_{n} := ⋃_{j∈ℤ} H_{n,j}, and H_{−1} := H_{−1,j} := {0}. (For ease of notation, we identify v ∈ Ṽ with its indicator function 1_{{v}} ∈ C_{c}(Ṽ, R) in this proof.) Define T′ : C_{c}(Ṽ, R) → C_{c}(Ṽ, R) by

T′(ϕ)(v) := ∑_{{t(e)=(v), \atop o(e)∈N^{j} (Cn)}} ϕ(o(e)) for each v ∈ N^{j}(C_{n−1}), j ∈ ℤ;

T′ can be seen as the ”restriction to one layer” ⋃_{n=0}^{∞} N^{j}(C_{n}) of T. We have T′(v) ≡ T (v) mod H_{n−1} for each v ∈ H_{n}, since the ”missing summand” of T′ lies in H_{n−1}. We claim that for each n ≥ 0, the set X_{n,j} := ⋃_{i=0}^{n} N^{j}T^{n−i}(V_{i}) is an R-basis for H_{n,j}/H_{n−1,j}. By the above congruence, we can replace T by T′ in the definition of X_{n,j}. The claim is clear for n = 0. So let n ≥ 1, and assume the claim to be true for all n′ ≤ n. For each v ∈ C_{n−1}, the q points in C_{n} adjacent to v are generated by the q − 1 of these points lying in V_{n}, plus T′v (which just sums up these q points). By induction hypothesis, v is generated by X_{n−1,0}, and thus (taking the union over all v), C_{n} is generated by T′(X_{n−1,0}) ∪ V_{n} = X_{n,0}. Since the cardinality of X_{n,0} equals the R-rank of H_{n,0}/H_{n−1,0} (both are equal to (q + 1)q^{n−1}), X_{n,0} is in fact an R-basis. Analoguously, we see that H_{n,j}/H_{n−1,j} has N^{j}(X_{n,0}) = X_{n,j} as a basis, for each j ∈ ℤ. From the claim, it follows that ⋃_{j∈ℤ} X_{n,j} is an R-basis of H_{n}/H_{n−1} for each n, and that V := ⋃_{n=0}^{∞} V_{n} is an R[T, N^{±1}]-basis of C_{c}(Ṽ, R).

For a ∈ R and ν ∈ R^{∗} , we let B̃_{a,ν}(F, R) be the ”common cokernel” of T − a and N −ν in C_{c}(Ṽ, R), namely B̃_{a,ν}(F, R) := C_{c}(Ṽ, R)/(Im(T −a)+Im(N −ν)); dually, we define B̃^{a,ν}(F, M) := ker(T − a) ∩ ker(N − ν) ⊆ C(Ṽ, M).

For a lattice v ∈ Ṽ, we define a valuation ord_{v} on F as follows: For w ∈ F^{2}, the set {x ∈ F |xw ∈ v} is some fractional ideal ϖ^{m}O_{F} ⊆ F (m ∈ ℤ); we set ord_{v}(w) := m. This map can also be given explicitly as follows: Let λ_{1}, λ_{2} be a basis of v. We can write any w ∈ F^{2} as w = x_{1}λ_{1} + x_{2}λ_{2}; then we have ord_{v}(w) = min{ord_{ϖ}(x_{1}), ord_{ϖ}(x_{2})}. This gives a ”valuation” map on F^{2}, as one easily checks. We restrict it to F ≅ F × {0} ↪ F^{2} to get a valuation ord_{v} on F, and consider especially the value at e_{1} := (1, 0).

Lemma 2.7. Let α, ν ∈ R^{∗}, and put a := α + qν/α. Define a map ϱ = ϱ_{α,ν} : Ṽ → R by ϱ(v) := α^{h(π(v))}ν^{− ord_{v}(e_{1})}. Then ϱ ∈ B̃^{a,ν}(F, R).

Proof. One easily sees that (v ↦ ν^{− ord_{v}(e_{1})}) ∈ ker(N − ν). It remains to show that ϱ ∈ ker(T − a): We have the Iwasawa decomposition G(F) = B(F)G(O_{F}) = \{( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} )\}Z(F)G(O_{F}); thus every vertex in Ṽ can be written as ϖ^{i}v with

v = ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) v_{0}, with i ∈ ℤ, a ∈ F^{∗}, b ∈ F.

Now the lattice v = ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) v_{0} is generated by the vectors λ_{1} = ( \begin{matrix}a \\ 0 \end{matrix} ) and λ_{2} = ( \begin{matrix}b \\ 1 \end{matrix} ) ∈ O_{F}^{2} , so e_{1} = a^{−1}λ_{1} and thus ord_{v}(e_{1}) = ord_{ϖ}(a^{−1}) = − ord_{ϖ}(a). The q + 1 neighbouring vertices v′ for which there exists an e ∈ Ẽ with o(e) = v′, t(e) = v are given by N_{i}v, i ∈ \{∞\} ∪ O_{F} /𝔭, with N_{∞} := ( \begin{matrix}1 & 0 \\ 0 & ϖ \end{matrix}), and N_{i} := ( \begin{matrix}ϖ & i \\ 0 & 1 \end{matrix} ) where i ∈ O_{F} runs through a complete set of representatives mod ϖ. By lemma 2.5, h(π(N_{∞}v)) = h(π(v)) + 1 and h(π(N_{i}v)) = h(π(v)) − 1 for i ≠ ∞. By considering the basis {N_{i}λ_{1}, N_{i}λ_{2}} of N_{i}v for each N_{i}, we see that ord_{N_{∞}v}(e_{1}) = ord_{v}(e_{1}) and ord_{N_{i}v}(e_{1}) = ord_{v}(e_{1}) − 1 for i ≠ ∞. Thus we have

(T ϱ)(v) = ∑_{t(e)=v} α^{h(π(o(e)))}ν^{− ord_{o(e)}(e_{1})}
= α^{h(π(v))+1}ν^{− ord_{v} e_{1}} + q · α^{h(π(v))−1}ν^{1−ord_{v}(e_{1})}
= (α + qα^{−1}ν)α^{h(π(v))}ν^{− ord_{v} e_{1}} = aϱ(v),

and also (T ϱ)(ϖ^{i}v) = (T N^{−i}ϱ)(v) = N^{−i}(aϱ)(v) = aϱ(ϖ^{i}v) for a general ϖ^{i}v ∈ Ṽ, which shows that ϱ ∈ ker(T − a).

If a^{2} ≠ ν(q + 1)^{2} (the “spherical case”), we put B_{a,ν}(F, R) := B̃_{a,ν}(F, R) and B^{a,ν}(F, M) := B̃^{a,ν}(F, M).

In the “special case” a^{2} = ν(q + 1)^{2}, we need to assume that the polynomial X^{2} − aνX + qν^{−1} ∈ R[X] has a zero α′ ∈ R. Then the map ϱ := ϱ_{α′,ν} ∈ C(Ṽ, R) defined as above lies in B̃^{aν,ν^{−1}}(F, R) = ker(T N − a) ∩ ker(N^{−1} − ν) by Lemma 2.7, since aν = α′ + qν^{−1}/α′. In other words, the kernel of the map ⟨·, ϱ⟩ : C_{c}(Ṽ, R) → R contains Im(T − a) + Im(N − ν); and we define

B_{a,ν}(F, R) := ker (⟨·, ϱ⟩) / (Im(T − a) + Im(N − ν))

to be the quotient; evidently, it is an R-submodule of codimension 1 of B̃_{a,ν}(F, R). Dually, T −a and N −ν both map the submodule ϱM = {ϱ·m, m ∈ M} of C(Ṽ, M) to zero and thus induce endomorphisms on C(Ṽ, M)/ϱM; we define B^{a,ν}(F, M) to be the intersection of their kernels. In the special case, since ν = α^{2}, Lemma 2.7 states that ϱ(gv_{0}) =

χ_{α}(ad)ϱ(v_{0}) = χ_{α}(det g)ϱ(v_{0}) for all g = (\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ B(F), and thus for

all g ∈ G(F) by the Iwasawa decomposition, since G(O_{F}) fixes v_{0} and lies in the kernel of χ_{α} ∘ det. By the multiplicity of det, we have (g^{−1}ϱ)(v) = ϱ(gv) = χ_{α}(det g)ϱ(v) for all g ∈ G(F), v ∈ Ṽ. So ϕ ∈ ker⟨·, ϱ⟩ implies ⟨gϕ, ϱ⟩ = ⟨ϕ, g^{−1}ϱ⟩ = χ_{α}(det g)⟨ϕ, ϱ⟩ = 0, i.e. ker⟨·, ϱ⟩ and thus B_{a,ν}(F, R) are G(F)-modules. By the adjointness properties of the Hecke operators T and N, we have pairings coker(T_{c} − a) × ker(T N − a) → M and coker(N_{c} − ν) × ker(N^{−1} − ν) → M, which ”combine” to give a pairing

⟨- , -⟩ : B_{a,ν}(F, R) × B^{aν,ν^{−1}}(F, M) → M

(since ker(T N − a) ∩ ker(N^{−1} − ν) = ker(T − aν) ∩ ker(N − ν^{−1})), and a corresponding isomorphism B^{aν,ν^{−1}}(F, M) ⟶^{≅} Hom(B_{a,ν}(F, R), M).

Definition 2.8. Let G be a totally disconnected locally compact group, H ⊆ G an open subgroup. For a smooth R[H]-module M, we define the (compactly) induced G-representation of M, denoted Ind_{H}^{G} M, to be the space of maps f : G → M such that f(hg) = f(g) for all g ∈ G, h ∈ H, and such that f has compact support modulo H. We let G act on Ind_{H}^{G} M via g · f(x) := f(xg). (We can also write Ind_{H}^{G} M = R[G] ⊗_{R[H]} M, cf. [Br82], III.5.) We further define Coind_{H}^{G} M := Hom_{R[H]}(R[G], M). Finally, for an R[G]- module N, we write res_{H}^{G} N for its underlying R[H]-module (“restriction”).

By Theorem 2.6, T_{c} − a (as well as N_{c} − ν) is injective, and the induced map

N_{c} − ν : coker(T_{c} − a) = C_{c}(Ṽ, R)/ Im(T_{c} − a) → coker(T_{c} − a)

(of R[T, N^{±1}]/(T − a) = R[N^{±1}]-modules) is also injective. Now since G(F) acts transitively on Ṽ, with the stabilizer of v_{0} := O_{F}^{2} being K := G(O_{F}), we have an isomorphism C_{c}(Ṽ, R) ≅ Ind_{K}^{G(F)} R. Thus we have exact sequences

0 → Ind_{K}^{G(F)} R ⟶^{T −a} Ind_{K}^{G(F)} R ⟶ coker(T_{c} − a) ⟶ 0  (4)

and (for a, ν in the spherical case)

0 → coker(T_{c} − a) ⟶^{N−ν} coker(T_{c} − a) → B_{a,ν}(F, R) → 0,  (5)

with all entries being free R-modules. Applying Hom_{R}(·, M) to them, we get:

Lemma 2.9. We have exact sequences of R-modules

0 → ker(T N − a) → Coind_{K}^{G(F)} M ⟶^{T −a} Coind_{K}^{G(F)} M → 0

and, if B_{a,ν}(F, M) is spherical (i.e. a^{2} ≠ ν(q + 1)^{2}),

0 → B^{aν,ν^{−1}}(F, M) → ker(T N − a) ⟶^{N−ν} ker(T N − a) → 0.

For the special case, we have to work a bit more to get similar exact sequences: By [Sp14], eq. (22), for the representation St^{−}(F, R) := B_{−(q+1),1}(F, R) (i.e. ν = 1, α = −1) with trivial central character, we have an exact sequence of G-modules

0 → Ind_{KZ}^{G} R → Ind_{K′Z}^{G} R → St^{−}(F, R) → 0,  (6)

where K′ = ⟨W ⟩K_{0}(𝔭) is the subgroup of KZ generated by W := ( \begin{matrix}0 & 1 \\ ϖ & 0 \end{matrix} ) and the subgroup K_{0}(𝔭) ⊆ K of matrices that are upper-triangular modulo 𝔭. (Since W^{2} ∈ Z, K_{0}(𝔭)Z is a subgroup of K′ of order 2.) Now aany special representation (π, V) can be written as π = χ ⊗ St^{−} for some character χ = χ_{Z} (cf. the proof of lemma 2.12 below), and is obviously G-isomorphic to the representation π ⊗ (χ ∘ det) acting on the space V ⊗_{R} R(χ ∘ det), where R(χ ∘ det) is the ring R with G-module structure given via gr = χ(det(g))r for g ∈ G, r ∈ R. Tensoring (6) with R(χ ∘ det) over R gives an exact sequence of G-modules

0 → Ind_{KZ}^{G} χ → Ind_{K′Z}^{G} χ → V → 0.  (7)

It is easily seen that R(χ ∘ det) fits into another exact sequence of G-modules

0 → Ind_{H}^{G} R ⟶^{( \begin{matrix}ϖ & 0 \\ 0 & 1 \end{matrix} )−χ(ϖ) id} Ind_{H}^{G} R ⟶^{ψ} R(χ ∘ det) → 0,

where H := {g ∈ G| det g ∈ O_{F}^{×}} is a normal subgroup containing K,

( \begin{matrix}ϖ & 0 \\ 0 & 1 \end{matrix} ) (f)(g) := f(( \begin{matrix}ϖ & 0 \\ 0 & 1 \end{matrix} )^{−1} g) for f ∈ Ind_{H}^{G} R = \{f : G → R|f(Hg) = f(g)

for all g ∈ G}, g ∈ G, is the natural operation of G, and where ψ is the G-equivariant map defined by 1_{U} ↦ 1. Now since H ⊆ G is a normal subgroup, we have Ind_{H}^{G} R ≅ R[G/H] as Gmodules (in fact G/H ≅ ℤ as an abstract group). Let X ⊆ G be a subgroup such that the natural inclusion X/(X ∩H) ↪ G/H has finite cokernel; let g_{i}H, i = 1, … n be a set of representatives of that cokernel. Then we have a (noncanonical) X-isomorphism ⨁_{i=0}^{n} Ind_{X∩H}^{X} → Ind_{H}^{G} R defined via (1_{(X∩H)x})_{i} ↦ 1_{Hxg_{i}} for each i = 1, …, n (cf. [Br82], III (5.4)). Using this isomorphism and the “tensor identity” Ind_{H}^{G} M ⊗ N ≅ Ind_{H}^{G}(M ⊗ res_{H}^{G} N) for any groups H ⊆ G, H-module M and G-module N ([Br82] III.5, Ex. 2), we have

Ind_{KZ}^{G} R ⊗_{R} Ind_{H}^{G} R ≅ Ind_{KZ}^{G}(res_{KZ}^{G}(Ind_{H}^{G} R))
= Ind_{KZ}^{G}((Ind_{KZ∩H}^{KZ} R)^{2})
= (Ind_{KZ}^{G}(Ind_{K}^{KZ} R))^{2} = (Ind_{K}^{G} R)^{2}

(since KZ/KZ ∩ H ↪ G/H has index 2), and similarly

Ind_{K′Z}^{G} R ⊗_{R} Ind_{H}^{G} R ≅ (Ind_{K′}^{G} R)^{2}.

Thus, we can resolve the first and second term of (7) into exact sequences

0 → (Ind_{K}^{G} R)^{2} → (Ind_{K}^{G} R)^{2} → Ind_{KZ}^{G} χ → 0,
0 → (Ind_{K′}^{G} R)^{2} → (Ind_{K′}^{G} R)^{2} → Ind_{⟨W ⟩K_{0}(𝔭)Z}^{G} χ → 0.

Dualizing (7) and these by taking Hom(·, M) for an R-module M, we get a “resolution” of B^{aν,ν^{−1}}(F, M) in terms of coinduced modules:

Lemma 2.10. We have exact sequences

0 →B^{aν,ν^{−1}}(F, M) → Coind_{K′Z}^{G} M(χ) → Coind_{KZ}^{G} M(χ) → 0,
0 → Coind_{KZ}^{G} M(χ) → (Coind_{K}^{G} R)^{2} → (Coind_{K}^{G} R)^{2} → 0,
0 → Coind_{K′Z}^{G} M(χ) → (Coind_{K′}^{G} R)^{2} → (Coind_{K′}^{G} R)^{2} → 0

for all special B_{a,ν}(F, R) (i.e. a^{2} = ν(q + 1)^{2}), where χ = χ_{Z} is the central character.

It is easily seen that the above arguments could be modified to get a similar set of exact sequences in the spherical case as well (replacing K′ by K everywhere), in addition to that given in lemma 2.9; but we will not need this.

2.4 Distributions on the Bruhat-Tits graph

For ϱ ∈ C(Ṽ, R) we define R-linear maps

δ̃_{ϱ} : C(Ẽ, M) → C(Ṽ, M), δ̃_{ϱ}(c)(v) := ∑_{v=t(e)} ϱ(o(e))c(e) − ∑_{v=o(e)} ϱ(t(e))c(e),

δ̃^{ϱ} : C(Ṽ, M) → C(Ẽ, M), δ̃^{ϱ}(ϕ)(e) := ϱ(o(e))ϕ(t(e)) − ϱ(t(e))ϕ(o(e)).

One easily checks that these are adjoint with respect to the pairings (2) and (3), i.e. we have ⟨δ̃_{ϱ}(c), ϕ⟩ = ⟨c, δ̃^{ϱ}(ϕ)⟩ for all c ∈ C_{c}(Ẽ, R), ϕ ∈ C(Ṽ, M). We denote the maps corresponding to ϱ ≡ 1 by δ := δ̃_{1}, δ^{∗} := δ̃^{1}. For each ϱ, the map δ̃_{ϱ} fits into an exact sequence

C_{c}(Ẽ, R) ⟶^{δ̃_{ϱ}} C_{c}(Ṽ, R) ⟶^{⟨·,ϱ⟩} R → 0

but it is not injective in general: e.g. for ϱ ≡ 1, the map Ẽ → R symbolized by

\xymatrix{· \ar[d]^{1} \ar[r]^{−1} & · \ar[d]^{−1} \\ · \ar[r]^{1} & ·}

(and zero outside the square) lies in ker δ. The restriction δ^{∗}|_{C_{c}(Ṽ,R)} to compactly supported maps is injective since T̃ has no directed circles, and we have a surjective map

coker (δ^{∗} : C_{c}(Ṽ, R) → C_{c}(Ẽ, R)) → C^{0}(ℙ^{1}(F), R)/R, c ↦ ∑_{e∈Ẽ} c(e)1_{U(π(e))}

(which corresponds to an isomorphism of the similar map on the Bruhat-Tits tree T ). Its kernel is generated by the functions 1_{{e}} − 1_{{e′}} for e, e′ ∈ Ẽ with π(e) = π(e′). For ϱ_{1}, ϱ_{2} ∈ C(Ṽ, R) and ϕ ∈ C(Ṽ, M) it is easily checked that

(δ̃_{ϱ_{1}} ∘ δ̃^{ϱ_{2}})(ϕ) = (T + T N)(ϱ_{1} · ϱ_{2}) · ϕ − ϱ_{2} · (T + T N)(ϱ_{1} · ϕ).

For a′ ∈ R and ϱ ∈ ker(T + T N − a′) , applying this equality for ϱ_{1} = ϱ and ϱ_{2} = 1 shows that δ̃_{ϱ} maps Im δ^{∗} into Im(T + T N − a′), so we get an R-linear map

δ̃_{ϱ} : coker (δ^{∗} : C_{c}(Ṽ, R) → C_{c}(Ẽ, R)) → coker(T_{c} + T_{c}N_{c} − a′).

Let now again α, ν ∈ R^{∗}, and a := α + qν/α. We let ϱ := ϱ_{α,ν} ∈ B̃^{a,ν}(F, R) as defined in lemma 2.7, and write δ̃_{α,ν} := δ̃_{ϱ}. Since δ̃_{α,ν} maps 1_{{e}} − 1_{{ϖe}} into Im(R − ν), it induces a map

δ̃_{α,ν} : C^{0}(ℙ^{1}(F), R)/R → B_{a,ν}(F, R)

(same name by abuse of notation) via the commutative diagram

\xymatrix{coker δ^{∗} \ar[r]^{δ̃_{α,ν}} \ar[d] & coker(T_{c} + T_{c}N_{c} − a′) \ar[d]^{mod (N−ν)} \\ C^{0}(ℙ^{1}(F), R)/R \ar[r]^{δ̃_{α,ν}} & B_{a,ν}(F, R)}

with a′ := a(1 + ν), since ϱ ∈ B^{a,ν}(F, R) ⊆ ker(T + T N − a′).

Lemma 2.11. We have ϱ (gv) = χ_{α}(d/a′)χ_{ν}(a′)ϱ(v), and thus

δ̃_{α,ν}(gf) = χ_{α}(d/a′)χ_{ν}(a′)gδ̃_{α,ν}(f),

for all v ∈ Ṽ, f ∈ C^{0}(ℙ^{1}(F), R)/R and g = (\begin{matrix}a′ & b \\ 0 & d \end{matrix}) ∈ B(F).

Proof. (a) Using lemma 2.5(b) and the fact that ord_{gv}(e_{1}) = − ord_{ϖ}(a′) + ord_{v}(e_{1}), we have

ϱ ((\begin{matrix}a′ & b \\ 0 & d \end{matrix}) v) = α^{h(v)−ord_{ϖ}(a′/d)}ν^{ord_{ϖ}(a′)−ord_{v}(e_{1})} = χ_{α}(d/a′)χ_{ν}(a′)ϱ(v)

for all v ∈ Ṽ. For f and g as in the assertion, we thus have

δ̃_{α,ν}(gf)(v) = ∑_{v=t(e)} ϱ(o(e))f(g^{−1}e) − ∑_{v=o(e)} ϱ(t(e))f(g^{−1}e)
= ∑_{g^{−1}v=t(e)} ϱ(o(ge))f(e) − ∑_{g^{−1}v=o(e)} ϱ(t(ge))f(e)

= χ_{α}(d/a′)χ_{ν}(a′)ϱ(v)( ∑_{g^{−1}v=t(e)} ϱ(o(e))f(e) − ∑_{g^{−1}v=o(e)} ϱ(t(e))f(e))
= χ_{α}(d/a′)χ_{ν}(a′)gδ̃_{α,ν}(f)(v).

We define a function δ_{α,ν} : C_{c}(F^{∗}, R) → B_{a,ν}(F, R) as follows: For f ∈ C_{c}(F^{∗}, R), we let ψ_{0}(f) ∈ C_{c}(ℙ^{1}(F), R) be the extension of x ↦ χ_{α}(x)χ_{ν}(x)^{−1}f(x) by zero to ℙ^{1}(F). We set δ_{α,ν} := δ̃_{α,ν} ∘ ψ_{0}. If α = ν, we can define δ_{α,ν} on all functions in C_{c}(F, R). We let F^{∗} operate on C_{c}(F, R) by (tf)(x) := f(t^{−1}x); this induces an action of the group T^{1}(F) := \{( \begin{matrix}t & 0 \\ 0 & 1 \end{matrix} ) |t ∈ F^{∗}\}, which we identify with F^{∗} in the obvious way. With respect to it, we have

ψ_{0}(tf)(x) = χ_{α}(t)χ_{ν}(t)^{−1}tψ_{0}(f)(x)

and

δ̃_{α,ν}(tf) = χ_{α}^{−1}(t)χ_{ν}(t)tδ̃_{α,ν}(f),

so δ_{α,ν} is T^{1}(F)-equivariant. For an R-module M, we define an F^{∗}-action on Dist(F^{∗}, M) by ∫ fd(tμ) := t ∫ (t^{−1}f)dμ. Let H ⊆ G(F) be a subgroup, and M an R[H]-module. We define an H-action on B^{aν,ν^{−1}}(F, M) by requiring ⟨ϕ, hλ⟩ = h · ⟨h^{−1}ϕ, λ⟩ for all ϕ ∈ B_{a,ν}(F, M), λ ∈ B^{aν,ν^{−1}}(F, M), h ∈ H. With respect to these two actions, we get a T^{1}(F) ∩ H-equivariant mapping

δ^{α,ν} : B^{aν,ν^{−1}}(F, M) → Dist(F^{∗}, M), δ^{α,ν}(λ) := ⟨δ_{α,ν}(·), λ⟩

dual to δ_{α,ν}.

2.5 Local distributions

Now consider the case R = ℂ. Let χ_{1}, χ_{2} : F^{∗} → ℂ^{∗} be two unramified characters. We consider (χ_{1}, χ_{2}) as a character on the torus T (F) of GL_{2}(F), which induces a character χ on B(F) by

χ (\begin{matrix}t_{1} & u \\ 0 & t_{2} \end{matrix}) := χ_{1}(t_{1})χ_{2}(t_{2}).

Put α_{i} := χ_{i}(ϖ)\sqrt{q} ∈ ℂ^{∗} for i = 1, 2. Set ν := χ_{1}(ϖ)χ_{2}(ϖ) = α_{1}α_{2}q^{−1} ∈ ℂ^{∗}, and a := α_{1} + α_{2} = α_{i} + qν/α_{i} for either i. When a and ν are given by the α_{i} like this, we will often write B_{α_{1},α_{2}}(F, R) := B_{a,ν}(F, R) and B^{α_{1},α_{2}}(F, M) := B^{aν,ν^{−1}}(F, M) (!) for its dual. In the special case a^{2} = ν(q + 1)^{2}, we assume the χ_{i} to be sorted such that χ_{1} = | · |χ_{2}. Let B(χ_{1}, χ_{2}) denote the space of continuous maps ϕ : G(F) → ℂ such that

ϕ ((\begin{matrix}t_{1} & u \\ 0 & t_{2} \end{matrix}) g) = χ_{α_{1}}(t_{1})χ_{α_{2}}(t_{2})|t_{1}|ϕ(g)  (8)

for all t_{1}, t_{2} ∈ F^{∗}, u ∈ F, g ∈ G(F). G(F) operates canonically on B(χ_{1}, χ_{2}) by right translation (cf. [Bu98], Ch. 4.5). If χ_{1}χ_{2}^{−1} ≠ | · |^{±1}, B(χ_{1}, χ_{2}) is a model of the spherical representation π(χ_{1}, χ_{2}); if χ_{1}χ_{2}^{−1} = | · |^{±1}, the special representation π(χ_{1}, χ_{2}) can be given as an irreducible subquotient of codimension 1 of B(χ_{1}, χ_{2}).^{2}

Lemma 2.12. We have a G-equivariant isomorphism B̃_{a,ν}(F, ℂ) ≅ B(χ_{1}, χ_{2}). It induces an isomorphism B_{a,ν}(F, ℂ) ≅ π(χ_{1}, χ_{2}) both for spherical and special representations.

Proof. We choose a “central” unramified character χ_{Z} : F^{∗} → ℂ satisfying χ_{Z}^{2}(ϖ) = ν; then we have χ_{1} = χ_{Z}χ_{0}^{−1}, χ_{2} = χ_{Z}χ_{0} for some unramified character χ_{0}. We set a′ := \sqrt{q} (χ_{0}(ϖ)^{−1} + χ_{0}(ϖ)), which satisfies a = χ_{Z}(ϖ)a′. For a representation (π, V) of G(F), by [Bu98], Ex. 4.5.9, we can define another representation χ_{Z} ⊗ π on V via

(g, v) ↦ χ_{Z}(det(g))π(g)v for all g ∈ G(F), v ∈ V,

and with this definition we have B(χ_{1}, χ_{2}) ≅ χ_{Z} ⊗B(χ_{0}^{−1}, χ_{0}). Since B(χ_{0}^{−1}, χ_{0}) has trivial central character, [BL95], Thm. 20 (as quoted in [Sp14]) states that B(χ_{0}^{−1}, χ_{0}) ≅ B_{a′,1}(F, ℂ) ≅ Ind_{KZ}^{G(F)} R/ Im(T − a′). Define a G-linear map ϕ : Ind_{K}^{G} R → χ_{Z} ⊗ Ind_{KZ}^{G} R by 1_{K} ↦ (χ_{Z} ∘ det) · 1_{KZ}. Since 1_{K} (resp. (χ_{Z} ∘ det) · 1_{KZ}) generates Ind_{K}^{G} R (resp. χ_{Z} ⊗ Ind_{KZ}^{G} R) as a ℂ[G]-module, ϕ is well-defined and surjective.

ϕ maps N1_{K} = ( \begin{matrix}ϖ & 0 \\ 0 & ϖ \end{matrix}) 1_{K} to

( \begin{matrix}ϖ & 0 \\ 0 & ϖ \end{matrix}) ((χ_{Z} ∘ det) · 1_{KZ}) = χ_{Z}(ϖ)^{2} · ((χ_{Z} ∘ det) · 1_{KZ}) = ν · ϕ(1_{K}).

Thus Im(N − ν) ⊆ ker ϕ, and in fact the two are equal, since the preimage of the space of functions of support in a coset KZg (g ∈ G(F)) under ϕ is exactly the space generated by the 1_{Kzg}, z ∈ Z(F) = Z(O_{F})\{( \begin{matrix}ϖ & 0 \\ 0 & ϖ \end{matrix})\}^{ℤ}. Furthermore, ϕ maps T 1_{K} = ∑_{i∈O_{F} /(ϖ)∪{∞}} N_{i}1_{K} (with the N_{i} as in Lemma 2.7) to

∑_{i} χ_{Z}(det(N_{i})) · ((χ_{Z} ∘ det) · N_{i}1_{KZ}) = χ_{Z}(ϖ) · (χ_{Z} ∘ det)T 1_{KZ}

(since det(N_{i}) = ϖ for all i),and thus Im(T − a) is mapped to Im (χ_{Z}(ϖ)T − a) = Im (χ_{Z}(ϖ)(T − a′)) = Im(T − a′).

2Note that [Bu98] denotes this special representation by σ(χ_{1}, χ_{2}), not by π(χ_{1}, χ_{2}).

Documenta Mathematica 21 (2016) 689–734

Putting everything together, we thus have G-isomorphisms

C_{c}(Ṽ, ℂ)/( Im(T − a) + Im(N − ν)) ≅ Ind_{K}^{G} R/( Im(T − a) + Im(N − ν))
≅ χ_{Z} ⊗ ( Ind_{KZ}^{G} R/ Im(T − a′))
≅ χ_{Z} ⊗ B(χ_{0}^{−1}, χ_{0}) ≅ B(χ_{1}, χ_{2}).

Thus, B_{a,ν}(F, ℂ) is isomorphic to the spherical principal series representation π(χ_{1}, χ_{2}) for a^{2} ≠ ν(q + 1)^{2}. In the special case, B_{a,ν}(F, ℂ) is a G-invariant subspace of B̃_{a,ν}(F, ℂ) of codimension 1, so it must be mapped under the isomorphism to the unique Ginvariant subspace of B(χ_{1}, χ_{2}) of codimension 1 (in fact, the unique infinitedimensional irreducible G-invariant subspace, by [Bu98], Thm. 4.5.1), which is the special representation π(χ_{1}, χ_{2}).

By [Bu98], section 4.4, there exists thus for all pairs a, ν a Whittaker functional λ on B_{a,ν}(F, ℂ), i.e. a nontrivial linear map λ : B_{a,ν}(F, ℂ) → ℂ such that λ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ϕ) = ψ(x)λ(ϕ). It is unique up to scalar multiples. From it, we furthermore get a Whittaker model W_{a,ν} of B_{a,ν}(F, ℂ):

W_{a,ν} := {W_{ξ} : GL_{2}(F) → ℂ | ξ ∈ B_{a,ν}(F, ℂ)},

where W_{ξ}(g) := λ(g · ξ) for all g ∈ GL_{2}(F). (see e.g. [Bu98], Ch. 3, eq. (5.6).) Now write α := α_{1} for short. Recall the distribution μ_{α,ν} = ψ(x)χ_{α/ν}(x)dx ∈ Dist(F^{∗}, ℂ). For α = ν, it extends to a distribution on F. We have the following generalization of [Sp14], Prop. 3.10:

Proposition 2.13. (a) There exists a unique Whittaker functional λ = λ_{a,ν} on B_{a,ν}(F, ℂ) such that δ^{α,ν}(λ) = μ_{α,ν}.

(b) For every f ∈ C_{c}(F^{∗}, ℂ), there exists W = W_{f} ∈ W_{a,ν} such that

∫_{F^{∗}}(af)(x)μ_{α,ν}(dx) = W_{f} (\begin{matrix}a & 0 \\ 0 & 1 \end{matrix}) .

If α = ν, then for every f ∈ C_{c}(F, ℂ), there exists W_{f} ∈ W_{a,ν} such that

∫_{F} (af)(x)μ_{α,ν}(dx) = W_{f} (\begin{matrix}a & 0 \\ 0 & 1 \end{matrix}) .

(c) Let H ⊆ U = O_{F}^{×} be an open subgroup, and write W_{H} := W_{1_{H}}. For every f ∈ C_{c}^{0}(F^{∗}, ℂ)^{H} we have

∫_{F^{∗}} f(x)μ_{α,ν}(dx) = [U : H] ∫_{F^{∗}} f(x)W_{H} (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) d^{×}x.

Proof. (a) By [Sp14], we have a Whittaker functional of the Steinberg representation given by the composite

St(F, ℂ) := C^{0}(ℙ^{1}(F), ℂ)/ℂ ⟶^{≅} C_{c}(F, ℂ) ⟶^{Λ} ℂ,  (9)

where the first map is the F -equivariant isomorphism

C^{0}(ℙ^{1}(F), ℂ)/ℂ → C_{c}(F, ℂ), ϕ ↦ f(x) := ϕ(x) − ϕ(∞),

(with F acting on C_{c}(F, ℂ) by (x · f)(y) := f(y − x), and on C^{0}(ℙ^{1}(F), ℂ)/ℂ by xϕ := ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ϕ), and the second is

Λ : C_{c}(F, ℂ) → ℂ, f ↦ ∫_{F} f(x)ψ(x)dx.

Let now λ : B_{a,ν}(F, ℂ) → ℂ be a Whittaker functional of B_{a,ν}(F, ℂ). By lemma

2.11, for u = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ∈ B(F),

(λ ∘ δ̃_{α,ν})(uϕ) = λ(uδ̃_{α,ν}(ϕ)) = ψ(x)λ(δ̃_{α,ν}(ϕ)),

so λ ∘ δ̃_{α,ν} is a Whittaker functional if it is not zero. To describe the image of δ̃_{α,ν}, consider the commutative diagram

\xymatrix{C_{c}(Ẽ, R) \ar[r]^{δ̃_{α,ν}} \ar[d]^{(10)} & C_{c}(Ṽ, R) \ar[d]^{ϕ↦ϕ·ϱ} \\ C_{c}(Ẽ, R) \ar[r]^{δ} & C_{c}(Ṽ, R) \ar[r]^{⟨·,1⟩} & R \ar[r] & 0}

where the vertical maps are defined by

C_{c}(Ẽ, R) → C_{c}(Ẽ, R), c ↦ (e ↦ c(e)ϱ(o(e))ϱ(t(e)))  (10)

resp. by mapping ϕ to v ↦ ϕ(v)ϱ(v); both are obviously isomorphisms. Since the lower row is exact, we have Im δ = ker⟨·, 1⟩ =: C_{c}^{0}(Ṽ, R) and thus Im δ̃_{α,ν} = ϱ^{−1} · C_{c}^{0}(Ṽ, R). Since λ ≠ 0 and B_{a,ν}(F, ℂ) is generated by (the equivalence classes of) the 1_{{v}}, v ∈ Ṽ, there exists a v ∈ Ṽ such that λ(1_{{v}}) ≠ 0. Let ϕ be this 1_{{v}}, and let u = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ∈ B(F) such that x ∉ ker ψ. Then

ϱ · (uϕ − ϕ) = ϱ · (1_{{u^{−1}v}} − 1_{{v}}) = ϱ(v)(1_{{u^{−1}v}} − 1_{{v}}) ∈ C_{c}^{0}(Ṽ, R)

by lemma 2.11, so 0 ≠ uϕ − ϕ ∈ Im δ̃_{α,ν}, but λ(uϕ − ϕ) = ψ(x)λ(ϕ) − λ(ϕ) ≠ 0. So λ ∘ δ̃_{α,ν} ≠ 0 is indeed a Whittaker functional. By replacing λ by a scalar multiple, we can assume λ ∘ δ̃_{α,ν} = (9). Considering λ as an element of B^{aν,ν^{−1}}(F, ℂ) ≅ Hom(B_{a,ν}(F, ℂ), ℂ), we have

δ^{α,ν}(λ)(f) = ⟨δ_{α,ν}(f), λ⟩
= Λ(χ_{α}χ_{ν}^{−1}f)
= ∫_{F^{∗}} χ_{α}(x)χ_{ν}^{−1}(x)f(x)ψ(x)dx
= μ_{α,ν}(f).

(b), (c) follow from (a) as in [Sp14].

2.6 Semi-local theory

We can generalize many of the previous constructions to the semi-local case, considering all primes 𝔭|p at once. So let F_{1}, …, F_{m} be finite extensions of ℚ_{p}, and for each i, let q_{i} be the number of elements of the residue field of F_{i}. We put \underline{F} := F_{1} × ⋯ × F_{m}. Let R again be a ring, and a_{i} ∈ R, ν_{i} ∈ R^{∗} for each i ∈ {1, …, m}. Put \underline{a} := (a_{1}, …, a_{m}), \underline{ν} := (ν_{1}, …, ν_{m}). We define B_{\underline{a},\underline{ν}}(\underline{F} , R) as the tensor product

B_{\underline{a},\underline{ν}}(\underline{F} , R) := ⨂_{i=1}^{m} B_{a_{i},ν_{i}}(F_{i}, R).

For an R-module M, we define B^{\underline{aν},\underline{ν}^{−1}}(\underline{F} , M) := Hom_{R}(B_{\underline{a},\underline{ν}}(\underline{F} , R), M); let

⟨·, ·⟩ : B_{\underline{a},\underline{ν}}(\underline{F} , R) × B^{\underline{aν},\underline{ν}^{−1}}(\underline{F} , M) → M  (11)

denote the evaluation pairing. We have an obvious isomorphism

⨂_{i=1}^{m} C_{c}^{0}(F_{i}^{∗}, R) → C_{c}^{0}(\underline{F}^{∗}, R), ⨂_{i} f_{i} ↦ ((x_{i})_{i=1,...,m} ↦ ∏_{i=1}^{m} f_{i}(x_{i})) . (12)

Now when we have α_{i,1}, α_{i,2} ∈ R^{∗} such that a_{i} = α_{i,1} + α_{i,2} and ν_{i} = α_{i,1}α_{i,2}q_{i}^{−1}, we can define the T^{1}(\underline{F})-equivariant map

δ_{\underline{α}_{1,2}} := δ_{\underline{α_{1}},\underline{ν}} : C_{c}^{0}(\underline{F} , R) → B_{\underline{a},\underline{ν}}(\underline{F} , R)

as the inverse of (12) composed with ⨂_{i=1}^{m} δ_{α_{i,1},ν_{i}}. Again, we will often write B_{\underline{α_{1}},\underline{α_{2}}}(F, R) := B_{\underline{aν},\underline{ν}^{−1}}(F, R) and B^{\underline{α_{1}},\underline{α_{2}}}(F, M) := B^{\underline{aν},\underline{ν}^{−1}}(F, M). If H ⊆ G(F) is a subgroup, and M an R[H]-module, we define an H-action on B^{\underline{aν},\underline{ν}^{−1}}(F, M) by requiring ⟨ϕ, hλ⟩ = h · ⟨h^{−1}ϕ, λ⟩ for all ϕ ∈ B_{\underline{a},\underline{ν}}(F, M), λ ∈ B^{\underline{aν},\underline{ν}^{−1}}(F, M), h ∈ H, and get a T^{1}(\underline{F}) ∩ H-equivariant mapping

δ^{\underline{α_{1}},\underline{α_{2}}} : B^{\underline{aν},\underline{ν}^{−1}}(F, M) → Dist(\underline{F}^{∗}, M), δ^{\underline{α_{1}},\underline{α_{2}}}(λ) := ⟨δ_{\underline{α_{1}},\underline{α_{2}}}(·), λ⟩.

Finally, we have a homomorphism

⨂_{i=1}^{m} B^{a_{i}ν_{i},ν_{i}^{−1}}(F_{i}, R) ⟶^{≅} ⨂_{i=1}^{m} Hom_{R}(B_{a_{i}ν_{i},ν_{i}^{−1}}(F_{i}, R), R)

→ Hom(B_{a_{1},ν_{1}}(F_{1}, R), Hom(B_{a_{2},ν_{2}}(F_{2}, R), Hom(…, R))...)

⟶^{≅} B^{\underline{aν},\underline{ν}^{−1}}(F, R).

(13)

where the second map is given by ⊗_{i}f_{i} ↦ (x_{1} ↦ (x_{2} ↦ (… ↦ ∏_{i} f_{i}(x_{i}))...), and the last map by iterating the adjunction formula of the tensor product.

3 Cohomology classes and global measures

3.1 Definitions

From now on, let F denote a number field, with ring of integers O_{F}. For each finite prime v, let U_{v} := O_{v}^{∗}. Let 𝔸 = 𝔸_{F} denote the ring of adeles of F, and 𝕀 = 𝕀_{F} the group of ideles of F. For a finite subset S of the set of places of F, we denote by 𝔸^{S} := {x ∈ 𝔸_{F} | x_{v} = 0 ∀v ∈ S} the S-adeles and by 𝕀^{S} the S-ideles, and put F_{S} := ∏_{v∈S} F_{v}, U_{S} := ∏_{v∈S} U_{v}, U^{S} := ∏_{v∉S} U_{v} (if S contains all infinite places of F), and similarly for other global groups. For ℓ a prime number or ∞, we write S_{ℓ} for the set of places of F above ℓ, and abbreviate the above notations to 𝔸^{ℓ} := 𝔸^{S_{ℓ}}, 𝔸^{p,∞} := 𝔸^{S_{p}∪S_{∞}}, and similarly write 𝕀^{p}, 𝕀^{∞}, F_{p}, F_{∞}, U^{∞}, U_{p}, U^{p,∞}, 𝕀_{∞} etc. Let F have r real embeddings and s pairs of complex embeddings. Set d := r + s − 1. Let {σ_{0}, …, σ_{r−1}, σ_{r}, …, σ_{d}} be a set of representatives of these embeddings (i.e. for i ≥ r, choose one from each pair of complex embeddings), and denote by ∞_{0}, …, ∞_{d} the corresponding archimedian primes of F. We let S_{∞}^{0} := {∞_{1}, …, ∞_{d}} ⊆ S_{∞}. For each place v, let dx_{v} denote the associated self-dual Haar measure on F_{v}, and dx := ∏_{v} dx_{v} the associated Haar measure on 𝔸_{F}. We define Haar measures d^{×}x_{v} on F_{v}^{∗} by d^{×}x_{v} := c_{v} \frac{dx_{v}}{|x_{v}|_{v}}, where c_{v} = (1 − \frac{1}{q_{v}})^{−1} for v finite, c_{v} = 1 for v|∞. For v|∞ complex, we use the decomposition ℂ^{∗} = ℝ_{+}^{∗} × S^{1} (with S^{1} = {x ∈ ℂ^{∗}; |x| = 1}) to write d^{×}x_{v} = d^{×}r_{v} dϑ_{v} for variables r_{v}, ϑ_{v} with r_{v} ∈ ℝ_{+}^{∗}, ϑ_{v} ∈ S^{1}. Let S_{1} ⊆ S_{p} be a set of primes of F lying above p, S_{2} := S_{p} − S_{1}. Let R be a topological Hausdorff ring.

Definition 3.1. We define the module of continuous functions

C(S_{1}, R) := C(F_{S_{1}} × F_{S_{2}}^{∗} × 𝕀^{p,∞}/U^{p,∞}, R);

and let C_{c}(S_{1}, R) be the submodule of all compactly supported f ∈ C(S_{1}, R). We write C^{0}(S_{1}, R), C_{c}^{0}(S_{1}, R) for the submodules of locally constant maps (or of continuous maps where R is assumed to have the discrete topology).We further define

C_{c}^{♭}(S_{1}, R) := C_{c}(∅, R) + C_{c}^{♭}(S_{1}, R) ⊆ C_{c}^{♭}(S_{1}, R)

to be the module of continuous compactly supported maps that are “constant near (0_{𝔭}, x^{𝔭})” for each 𝔭 ∈ S_{1}.

Definition 3.2. For an R-module M, let D_{f}(S_{1}, M) denote the R-module of maps

ϕ : ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}) × 𝕀_{F}^{p,∞} → M

that are U^{p,∞}-invariant and such that ϕ(·, x^{p,∞}) is a distribution for each x^{p,∞} ∈ 𝕀_{F}^{p,∞}.

Since 𝕀_{F}^{p,∞}/U^{p,∞} is a discrete topological group, D_{f}(S_{1}, M) naturally identifies with the space of M-valued distributions on F_{S_{1}} × F_{S_{2}}^{∗} × 𝕀_{F}^{p,∞}/U^{p,∞}. So there exists a canonical R-bilinear map

D_{f}(S_{1}, M) × C_{c}^{0}(S_{1}, R) → M, (ϕ, f) ↦ ∫ f dϕ,  (14)

which is easily seen to induce an isomorphism D_{f}(S_{1}, M) ≅ Hom_{R}(C_{c}^{0}(S_{1}, R), M). For a subgroup E ⊆ F^{∗} and an R[E]-module M, we let E operate on D_{f}(S_{1}, M) and C_{c}^{0}(S_{1}, R) by (aϕ)(U, x^{p,∞}) := aϕ(a^{−1}U, a^{−1}x^{p,∞}) and (af)(x^{∞}) := f(a^{−1}x^{∞}) for a ∈ E, U ∈ ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}), x^{·} ∈ 𝕀_{F}^{·}; thus we have ∫ (af) d(aϕ) = a ∫ f dϕ for all a, f, ϕ. When M = V is a finite-dimensional vector space over a p-adic field, we write D_{f}^{b}(S_{1}, V) for the subset of ϕ ∈ D_{f}(S_{1}, V) such that ϕ is even a measure on F_{S_{1}} × F_{S_{2}} × 𝕀_{F}^{p,∞}/U^{p,∞}.

Definition 3.3. For a ℂ-vector space V, define D(S_{1}, V) to be the set of all maps ϕ : ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}) × 𝕀^{p} → V such that:

(i) ϕ is invariant under F^{×} and U^{p,∞}.

(ii) For x^{p} ∈ 𝕀^{p}, ϕ(·, x^{p}) is a distribution of F_{S_{1}} × F_{S_{2}}.

(iii) For all U ∈ ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}), the map ϕ_{U} : 𝕀 = F_{p}^{×} × 𝕀^{p} → V, (x_{p}, x^{p}) ↦ ϕ(x_{p}U, x^{p}) is smooth, and rapidly decreasing as |x| → ∞ and |x| → 0.

We will need a variant of this last set: Let D′(S_{1}, V) be the set of all maps ϕ ∈ D(S_{1}, V) that are ”(S^{1})^{s}-invariant”, i.e. such that for all complex primes ∞_{j} of F and all ζ ∈ S^{1} = {x ∈ ℂ^{∗}; |x| = 1}, we have

ϕ(U, x^{p,∞_{j}}, ζx_{∞_{j}}) = ϕ(U, x^{p,∞_{j}}, x_{∞_{j}}) for all x^{p} = (x^{p,∞_{j}}, x_{∞_{j}}) ∈ 𝕀^{p}.

There is an obvious surjective map

D(S_{1}, V) → D′(S_{1}, V), ϕ ↦ ((U, x) ↦ ∫_{(S^{1})^{s}} ϕ(U, x)dϑ_{r} ⋯ dϑ_{r+s−1})

given by integrating over (S^{1})^{s} ⊆ (ℂ^{∗})^{s} ↪ 𝕀_{∞}.

Let F_{+}^{∗} denote the set of all x ∈ F ∗ that are totally positive, i.e. positive with respect to every real embedding of F. (For F totally imaginary, we have F^{∗} = F_{+}^{∗}.) Let F^{∗}′ ⊆ F_{+}^{∗} be a maximal torsion-free subgroup of F_{+}^{∗}. If F has at least one real embedding, we obviously have F^{∗}′ = F_{+}^{∗}; for totally imaginary F, F^{∗}′ is a subgroup of finite index of F^{∗} with F/F^{∗}′ ≅ μ_{F}, the roots of unity of F. We set

E′ := F^{∗}′ ∩ O_{F}^{×} ⊆ O_{F}^{×},

so E′ is a torsion-free ℤ-module of rank d. E′ operates freely and discretely on the space

ℝ_{0}^{d+1} := {(x_{0}, …, x_{d}) ∈ ℝ^{d+1}| ∑_{i=0}^{d} x_{i} = 0}

via the embedding

E′ ↪ ℝ_{0}^{d+1}
a ↦ (log |σ_{i}(a)|)_{i∈S_{∞}}

(cf. proof of Dirichlet’s unit theorem, e.g. in [Neu92], Ch. 1), and the quotient ℝ_{0}^{d+1}/E′ is compact. We choose the orientation on ℝ_{0}^{d+1} induced by the natural orientation on ℝ^{d} via the isomorphism ℝ^{d} ≅ ℝ_{0}^{d+1}, (x_{1}, …, x_{d}) ↦ (− ∑_{i=1}^{d} x_{i}, x_{1}, …, x_{d}). So ℝ_{0}^{d+1}/E′ becomes an oriented compact d-dimensional manifold. Let G_{p} be the Galois group of the maximal abelian extension of F which is unramified outside p and ∞; for a ℂ-vector space V, let Dist(G_{p}, V) be the set of V -valued distributions of G_{p}. Denote by ϱ : 𝕀_{F} /F^{∗} → G_{p} the projection given by global reciprocity.

3.2 Global measures

Now let V = ℂ, equipped with the trivial F^{∗}′-action. We want to construct a commutative diagram

\xymatrix{D(S_{1}, ℂ) \ar[dr]^{ϕ↦μ_{ϕ}} \ar[rr]^{ϕ↦κ_{ϕ}} &  & H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)) \ar[dl]^{κ↦μ_{κ}=κ∩∂(·)} \\  & Dist(G_{p}, ℂ)}  (15)

First, let R be any topological Hausdorff ring. Let \overline{E′} denote the closure of E′ in U_{p}. The projection map pr : 𝕀^{∞}/U^{p,∞} → 𝕀^{∞}/(\overline{E′} × U^{p,∞}) induces an isomorphism

pr^{∗} : C_{c}(𝕀^{∞}/(\overline{E′} × U^{p,∞}), R) → H^{0}(E′, C_{c}(𝕀^{∞}/U^{p,∞}, R)),

and the reciprocity map induces a surjective map \overline{ϱ} : 𝕀^{∞}/(\overline{E′} × U^{p,∞}) → G_{p}. Now we can define a map

ϱ^{♯} : H_{0}(F^{∗}′/E′, C_{c}(𝕀^{∞}/(\overline{E′} × U^{p,∞}), R)) → C(G_{p}, R),

[f] ↦ (\overline{ϱ}(x) ↦ ∑_{ζ∈F^{∗}′/E′} f(ζx) for x ∈ 𝕀^{∞}/(\overline{E′} × U^{p,∞})).

This is an isomorphism, with inverse map f ↦ [(f ∘ \overline{ϱ}) · 1_{F}], where 1_{F} is the characteristic function of a fundamental domain F of the action of F^{∗}′/E′ on 𝕀^{∞}/U^{∞}.

We get a composite map

\begin{aligned}C(G_{p}, R) & ⟶^{(ϱ^{♯})^{−1}} H_{0}(F^{∗}′/E′, C_{c}(𝕀^{∞}/(\overline{E′} × U^{p,∞}), R)) \\ & ⟶^{pr^{∗}} H_{0}(F^{∗}′/E′, H^{0}(E′, C_{c}(𝕀^{∞}/U^{p,∞}, R))) \\ & ⟶ H_{0}(F^{∗}′/E′, H^{0}(E′, C_{c}(S_{1}, R))), \end{aligned}  (16)

where the last arrow is induced by the “extension by zero” from C_{c}(𝕀^{∞}/U^{p,∞}, R) to C_{c}(S_{1}, R). Now let η ∈ H_{d}(E′, ℤ) ≅ ℤ be the generator that corresponds to the given orientation of ℝ_{0}^{d+1}. This gives us, for every R-module A, a homomorphism

\xymatrix{H_{0}(F^{∗}′/E′, H^{0}(E′, A)) \ar[r]^{∩η} & H_{0}(F^{∗}′/E′, H_{d}(E′, A))}

Composing this with the edge morphism

H_{0}(F^{∗}′/E′, H_{d}(E′, A)) → H_{d}(F^{∗}′, A)  (17)

(and setting A = C_{c}(S_{1}, R)) gives a map

H_{0}(F^{∗}′/E′, H^{0}(E′, C_{c}(S_{1}, R))) → H_{d}(F^{∗}′, C_{c}(S_{1}, R))  (18)

We define

∂ : C(G_{p}, R) → H_{d}(F^{∗}′, C_{c}(S_{1}, R))

as the composition of (16) with this map. Now, letting M be an R-module equipped with the trivial F^{∗}′-action, the bilinear form (14)

D_{f}(S_{1}, M) × C_{c}(S_{1}, R) → M
(ϕ, f) ↦ ∫ f dϕ

induces a cap product

∩ : H^{d}(F^{∗}′, D_{f}(S_{1}, M)) × H_{d}(F^{∗}′, C_{c}(S_{1}, R)) → H_{0}(F^{∗}′, M) = M. (19)

Thus for each κ ∈ H^{d}(F^{∗}′, D_{f}(S_{1}, M)), we get a distribution μ_{κ} on G_{p} by defining

∫_{G_{p}} f(γ) μ_{κ}(dγ) := κ ∩ ∂(f)  (20)

for all continuous maps f : G_{p} → R. Now let M = V be a finite-dimensional vector space over a p-adic field K, and let κ ∈ H^{d}(F^{∗}′, D_{f}^{b}(S_{1}, V)). We identify κ with its image in H^{d}(F^{∗}′, D_{f}(S_{1}, V)); then it is easily seen that μ_{κ} is also a measure, i.e. we have a map

H^{d}(F^{∗}′, D_{f}^{b}(S_{1}, V)) → Dist^{b}(G_{p}, V), κ ↦ μ_{κ}.  (21)

Let L|F be a ℤ_{p}-extension of F. Since it is unramified outside p, it gives rise to a continuous homomorphism G_{p} → Gal(L|F) via σ ↦ σ|_{L}. Fixing an isomorphism Gal(L|F) ≅ p^{ε_{p}}ℤ_{p} (where ε_{p} = 2 for p = 2, ε_{p} = 1 for p odd), we obtain a surjective homomorphism ℓ : G_{p} → p^{ε_{p}}ℤ_{p}. (Note that p^{ε_{p}}ℤ_{p} is the space of definition of the p-adic exponential function exp_{p}.)

Example 3.4. Let L be the cyclotomic ℤ_{p}-extension of F. Then we can take ℓ = log_{p} ∘N, where N : G_{p} → ℤ_{p}^{∗} is the p-adic cyclotomic character, defined by requiring γζ = ζ^{N (γ)} for all γ ∈ G_{p} and all p-power roots of unity ζ. It is well-known (cf. [Wa82], par. 5) that log_{p}(ℤ_{p}^{∗}) = p^{ε_{p}}ℤ_{p}.

It is well-known that F has t independent ℤ_{p}-extensions, where s + 1 ≤ t ≤ [F : ℚ]; the Leopoldt conjecture implies t = s+1. μ_{κ} defines a t-variable p-adic L-function as follows:

Definition 3.5. Let K be a p-adic field, V a finite-dimensional K-vector space, κ ∈ H^{d}(F^{∗}′, D_{f}^{b}(S_{1}, V)). Let ℓ_{1}, …, ℓ_{t} : G_{p} → p^{ε_{p}}ℤ_{p} be continuous homomorphisms.The p-adic L-function of κ is given by

L_{p}(\underline{s}, κ) := L_{p}(s_{1}, …, s_{t}, κ) := ∫_{G_{p}} (∏_{i=1}^{t} exp_{p}(s_{i}ℓ_{i}(γ))) μ_{κ}(dγ)

for all s_{1}, …, s_{t} ∈ ℤ_{p}.

Remark 3.6. Let Σ := {±1}^{r}, where r is the number of real embeddings of F. The group isomorphism ℤ/2ℤ ≅ {±1}, ε ↦ (−1)^{ε}, induces a pairing

⟨·, ·⟩ : Σ → {±1}, ⟨((−1)^{ε_{i}})_{i}, ((−1)^{ε′_{i}})_{i}⟩ := (−1)^{∑_{i} ε_{i}ε′_{i}}.

For a field k of characteristic zero, a k[Σ]-module V and \underline{μ} = (μ_{0}, …, μ_{r−1}) ∈ Σ, we put V\underline{μ} := \{v ∈ V | ⟨\underline{μ}, \underline{ν}⟩v = \underline{ν}v ∀\underline{ν} ∈ Σ\}, so that we have V = ⨁_{\underline{μ}∈Σ} V\underline{μ}. We write v\underline{μ} for the projection of v ∈ V to V\underline{μ}, and v_{+} := v_{(1,...,1)}. For r > 0, we identify Σ with F^{∗}/F^{∗}′ via the isomorphism Σ ≅ ∏_{i=0}^{r−1} ℝ^{∗}/ℝ_{+}^{∗} ≅ F^{∗}/F^{∗}′ = F^{∗}/F_{+}^{∗}. Then for each F^{∗}-module M, Σ acts on H^{d}(F^{∗}′, D_{f}(S_{1}, M)) and on H^{d}(F^{∗}′, D_{f}^{b}(S_{1}, M)). For r = 0, we let the trivial group Σ act on these groups as well for ease of notation. The exact sequence Σ ≅ ∏_{i=0}^{r−1} ℝ^{∗}/ℝ_{+}^{∗} = 𝕀_{∞}/𝕀_{∞}^{0} → G_{p} → G_{p}^{+} → 0 of class field theory (where 𝕀_{∞}^{0} is the maximal connected subgroup of 𝕀_{∞}) yields an action of Σ on G_{p}. We easily check that (21) is Σ-equivariant, and that the maps γ ↦ exp_{p}(sℓ_{i}(γ)) factor over G_{p} → G_{p}^{+} (since ℤ_{p}-extensions are unramified at ∞). Therefore we have L_{p}(\underline{s}, κ) = L_{p}(\underline{s}, κ_{+}).

For ϕ ∈ D(S_{1}, V) and f ∈ C^{0}(𝕀/F^{∗}, ℂ), let

∫_{𝕀/F^{∗}} f(x)ϕ(d^{×}x_{p}, x^{p}) d^{×}x^{p} := [U_{p} : U] ∫_{𝕀/F^{∗}} f(x)ϕ_{U}(x) d^{×}x,

where we choose an open set U ⊆ U_{p} such that f(x_{p}u, x^{p}) = f(x_{p}, x^{p}) for all (x_{p}, x^{p}) ∈ 𝕀 and u ∈ U; such a U exists by lemma 3.7 below. Since this integral is additive in f, there exists a unique V -valued distribution μ_{ϕ} on G_{p} such that

∫_{G_{p}} f dμ_{ϕ} = ∫_{𝕀/F^{∗}} f(ϱ(x))ϕ(d^{×}x_{p}, x^{p}) d^{×}x^{p}  (22)

for all functions f ∈ C^{0}(G_{p}, V).

Lemma 3.7. Let F : 𝕀/F^{∗} → X be a locally constant map to a set X. Then there exists an open subgroup U ⊆ 𝕀 such that f factors over 𝕀/F^{∗}U.

Proof. 𝕀_{∞} = ∏_{v|∞} F_{v} is connected, thus f factors over \overline{f} : 𝕀/F^{∗}𝕀_{∞} → X. Since 𝕀/F^{∗}𝕀_{∞} is profinite, \overline{f} further factors over a subgroup U′ ⊆ 𝕀^{∞} of finite index, which is open.

Let U_{∞}^{0} := ∏_{v∈S_{∞}^{0}} ℝ_{+}^{∗}; the isomorphisms U_{∞}^{0} ≅ ℝ^{d}, (r_{v})_{v} ↦ (log r_{v})_{v}, and ℝ^{d} ≅ ℝ_{0}^{d+1} give it the structure of a d-dimensional oriented manifold (with the natural orientation). It has the d-form d^{×}r_{1} · . . . · d^{×}r_{d}, where (by slight abuse of notation) we choose d^{×}r_{i} on F_{∞_{i}} corresponding to the Haar measure d^{×}x_{i} resp. d^{×}r_{i} on ℝ_{+}^{∗} ⊆ F_{∞_{i}}^{∗}. E′ operates on U_{∞}^{0} via a ↦ (|σ_{i}(a)|)_{i∈S_{∞}^{0}}, so the isomorphism U_{∞}^{0} ≅ ℝ_{0}^{d+1} is E′-equivariant. For ϕ ∈ D′(S_{1}, V), set

∫_{0}^{∞} ϕ d^{×}r_{0} : ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}) × 𝕀^{p,∞_{0}} → ℂ
(U, x^{p,∞_{0}}) ↦ ∫_{0}^{∞} ϕ(U, r_{0}, x^{p,∞_{0}}) d^{×}r_{0},

where we let r_{0} ∈ F_{∞_{0}} run through the positive real line ℝ_{+}^{∗} in F_{∞_{0}}. Composing this with the projection D(S_{1}, V) → D′(S_{1}, V) gives us a map

\begin{matrix}D(S_{1}, V) & → H^{0}(F^{∗}′, D_{f}(S_{1}, C^{∞}(U_{∞}^{0}, V))), \\ ϕ & ↦ ∫_{(S^{1})^{s}} (∫_{0}^{∞} ϕ d^{×}r_{0}) dϑ_{r} dϑ_{r+1} … dϑ_{r+s−1} \end{matrix}  (23)

(where C^{∞}(U_{∞}^{0}, V) denotes the space of smooth V -valued functions on U_{∞}^{0}), since one easily checks that ∫_{0}^{∞} ϕ d^{×}r_{0} is F^{∗}′-invariant. Define the complex C^{•} := D_{f}(S_{1}, Ω^{•}(U_{∞}^{0}, V)). By the Poincare lemma, this is a resolution of D_{f}(S_{1}, V). We now define the map ϕ ↦ κ_{ϕ} as the composition of (23) with the composition

H^{0}(F^{∗}′, D_{f}(S_{1}, C^{∞}(U_{∞}^{0}, V))) → H^{0}(F^{∗}′, C^{d}) → H^{d}(F^{∗}′, D_{f}(S_{1}, V)), (24)

where the first map is induced by

C^{∞}(U_{∞}^{0}, V) → Ω^{d}(U_{∞}^{0}, V), f ↦ f(r_{1}, …, r_{d})d^{×}r_{1} · . . . · d^{×}r_{d},  (25)

and the second is an edge morphism in the spectral sequence

H^{q}(F^{∗}′, C^{p}) ⇒ H^{p+q}(F^{∗}′, D_{f}(S_{1}, V)).  (26)

Specializing to V = ℂ, we now have:

Proposition 3.8. The diagram (15) commutes, i.e., for each ϕ ∈ D(S_{1}, ℂ), we have

μ_{ϕ} = μ_{κ_{ϕ}}.

Proof. Analoguously to [Sp14], proof of prop. 4.21, we define a pairing

⟨ , ⟩: D(S_{1}, ℂ) × C^{0}(G_{p}, ℂ) → ℂ

as the composite of (23) × (16) with

H^{0}(F^{∗}′, D_{f}(S_{1}, C^{∞}(U_{∞}^{0}, ℂ))) × H_{0}(F^{∗}′/E′, H^{0}(E′, C_{c}^{0}(S_{1}, ℂ)))

⟶^{∩} H_{0}(F^{∗}′/E′, H^{0}(E′, C^{∞}(U_{∞}^{0}, ℂ))) → H_{0}(F^{∗}′/E′, ℂ) ≅ ℂ, (27)

where ∩ is the cap product induced by (14), and the second map is induced by

H^{0}(E′, C^{∞}(U_{∞}^{0}, ℂ)) → ℂ, f ↦ ∫_{U_{∞}^{0}/E′} f(r_{1}, …, r_{d}) d^{×}r_{1} … d^{×}r_{d}. (28)

Then we can show that

κ_{ϕ} ∩ ∂(f) = ⟨ϕ, f⟩ = ∫_{G_{p}} f(γ) μ_{ϕ}(dγ) for all f ∈ C^{0}(G_{p}, ℂ),

by copying the proof for the totally real case (replacing F_{+}^{∗} by F^{∗}′, E_{+} by E′), using the fact that for a d-form on the d-dimensional oriented manifold M := ℝ_{0}^{d+1}/E′ ≅ U_{∞}^{0}/E′, integration over M corresponds to taking the cap product with the fundamental class η of M under the canonical isomorphism H_{dR}^{d}(M) ≅ H_{sing}^{d}(M) = H^{d}(E′, ℂ).

3.3 Exceptional zeros

Now let ℓ_{1}, …, ℓ_{t} : G_{p} → ℤ_{p} be continuous homomorphisms. Let again S_{1} = {𝔭_{1}, …, 𝔭_{n}} ⊆ S_{p} be a set of primes above p, of cardinality n := #S_{1}.

Proposition 3.9. For each \underline{x} = (x_{1}, …, x_{t}) ∈ ℕ_{0}^{t} set |\underline{x}| := ∑_{i=1}^{t} x_{i}. Then

∂(∏_{i=1}^{t} ℓ_{i}^{x_{i}}) = 0 for all \underline{x} with |\underline{x}| ≤ n − 1.

Proof. We can readily generalize the proof of Spieß’ result for the p-adic cyclotomic character (ℓ = log_{p} ∘N) in the totally real case ([Sp14], Prop. 4.6(a), Lemmas 4.1 and 4.7) to show that ∂(ℓ^{x}) = 0 for all 0 ≤ x ≤ n − 1, using the facts that we can write F^{∗}′ = E′ × T for some subgroup T ⊆ F^{∗}′ (since F^{∗}′/E′ = F^{∗}/O_{F}^{×} is a free ℤ-module), and that for each homomorphism ℓ : G_{p} → ℤ_{p}, the composition

ℓ̃: 𝕀^{∞} ⟶^{ϱ} G_{p} ⟶^{ℓ} ℤ_{p} ↪ ℚ_{p}.

is zero on 𝕀^{∞,p} (since the pro-q-part of G_{p} is finite for every prime q ≠ p and ℚ_{p} is torsion-free). Now for a ring R ⊇ ℚ, each monomial ∏_{i=1}^{t} X_{i}^{n_{i}} ∈ R[X_{1}, …, X_{t}] of degree n = ∑_{i} n_{i} can be written as a linear combination of n-th powers (X_{i}+r_{i,j}X_{j})^{n}. Therefore each product ∏_{i=1}^{t} ℓ_{i}^{x_{i}} of degree x = |\underline{x}| is a linear combination of x-th powers of the homomorphisms ℓ_{i,j} := ℓ_{i} + r_{i,j}ℓ_{j} : G_{p} → ℤ_{p}. This proves the proposition.

Definition 3.10. A t-variable p-adic analytic function f(\underline{s}) = f(s_{1}, …, s_{t}) (s_{i} ∈ ℤ_{p}) has vanishing order ≥ n at the point \underline{0} = (0, …, 0) if all its partial derivatives of total order ≤ n − 1 vanish, i.e. if

\frac{∂^{k}}{(∂\underline{s})\underline{k}} f(\underline{0}) := \frac{∂^{k}}{∂s_{1}^{k_{1}} ⋯ ∂s_{t}^{k_{t}}} f(\underline{0}) = 0

for all \underline{k} = (k_{1}, …, k_{t}) ∈ ℕ_{0}^{t} with k := |\underline{k}| ≤ n − 1. We write ord_{\underline{s}=\underline{0}} f(\underline{s}) ≥ n in this case.

Theorem 3.11. Let n := #(S_{1}), κ ∈ H^{d}(F^{∗}′, D_{f}^{b}(S_{1}, V)), V a finitedimensional vector space over a p-adic field. Then L_{p}(\underline{s}, κ) is a locally analytic function, and we have

ord_{\underline{s}=\underline{0}} L_{p}(\underline{s}, κ) ≥ n.

Proof. We have

\frac{∂^{k}}{(∂\underline{s})\underline{k}} L_{p}(\underline{0}, κ) = ∫_{G_{p}} (∏_{i=1}^{t} ℓ_{i}(γ)^{k_{i}}) μ_{κ}(dγ) = κ ∩ ∂ (∏_{i=1}^{t} ℓ_{i}(γ)^{k_{i}})

for all \underline{k} = (k_{1}, …, k_{t}) ∈ ℕ_{0}^{t}. Thus the theorem follows from proposition 3.9.

3.4 Integral cohomology classes

Definition 3.12. A nonzero cohomology class κ ∈ H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)) is called integral if κ lies in the image of

H^{d}(F^{∗}′, D_{f}(S_{1}, R)) ⊗_{R} ℂ → H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)) for some Dedekind ring R ⊆ \overline{O}. If, in addition, there exists a torsion-free Rsubmodule M ⊆ H^{d}(F^{∗}′, D_{f}(S_{1}, R)) of rank ≤ 1 (i.e. M can be embedded into R) such that κ lies in the image of M ⊗_{R} ℂ → H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)), then κ is integral of rank ≤ 1.

For κ as in def. 3.12 and R ⊆ ℂ, we let L_{κ,R} be the image of

H_{d}(F^{∗}′, C_{c}^{0}(S_{1}, R)) → H_{0}(F^{∗}′, ℂ) = ℂ, x ↦ κ ∩ x.

Proposition 3.13. Let κ ∈ H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)) be integral. Then

(a) μ_{κ} is a p-adic measure.

(b) There exists a Dedekind ring R ⊆ \overline{O} such that L_{κ,R} is a finitely generated R-module (resp. a torsion-free R-module of rank ≤ 1, if κ is integral of rank ≤ 1). For each such R, the map H^{d}(F^{∗}′, D_{f}(S_{1}, L_{κ,R})) ⊗ \overline{ℚ} → H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)) is injective and κ lies in its image.

Proof. The proofs of the corresponding results for totally real F ([Sp14], prop. 4.17 and cor. 4.18) also work in the general case.

Remark 3.14. Let κ be integral with Dedekind ring R as above. By (b) of the proposition, we can view κ as an element of H^{d}(F^{∗}′, D_{f}(S_{1}, L_{κ,R})) ⊗ \overline{ℚ}. Put V_{κ} := L_{κ,R} ⊗_{R} ℂ_{p}; let \overline{κ} be the image of κ under the composition

H^{d}(F^{∗}′, D_{f}(S_{1}, L_{κ,R})) ⊗_{R} \overline{ℚ} → H^{d}(F^{∗}′, D_{f}(S_{1}, L_{κ,R})) ⊗_{R} ℂ_{p}
→ H^{d}(F^{∗}′, D_{f}^{b}(S_{1}, V_{κ})),

where the second map is induced by D_{f}(S_{1}, L_{κ,R}) ⊗_{R} ℂ_{p} → D_{f}^{b}(S_{1}, V_{κ}). By [Sp14], lemma 4.15, \overline{κ} does not depend on the choice of R. Since μ_{κ} is a p-adic measure, μ\overline{κ} allows integration of all continuous functions f ∈ C(G_{p}, ℂ_{p}), and by abuse of notation, we write L_{p}(\underline{s}, κ) := L_{p}(\underline{s}, \overline{κ}) (cf. remark 3.6). So L_{p}(\underline{s}, κ) has values in the finite-dimensional ℂ_{p}-vector space V_{κ}.

4 p-adic L-functions of automorphic forms

We keep the notations from chapter 3; so F is again a number field with r real embeddings and s pairs of complex embeddings. For an ideal 0 ≠ 𝔪 ⊆ O_{F}, we let K_{0}(𝔪)_{v} ⊆ G(O_{F_{v}}) be the subgroup of matrices congruent to an upper triangular matrix modulo 𝔪, and we set K_{0}(𝔪) := ∏_{v∤∞} K_{0}(𝔪)_{v}, K_{0}(𝔪)^{S} := ∏_{v∤∞,v∉S} K_{0}(𝔪)_{v} for a finite set of primes S. For each 𝔭|p, let q_{𝔭} = N(𝔭) denote the number of elements of the residue class field of F_{𝔭}. We denote by |·|_{ℂ} the square of the usual absolute value on ℂ, i.e. |z|_{ℂ} = z\overline{z} for all z ∈ ℂ, and write |·|_{ℝ} for the usual absolute value on ℝ in context. We write |α| := |α|ℂ^{\frac{1}{2}} for the archimedian absolute value when α is given as a complex number in the context; whereas in the context of the p-adic characters, | · | denotes the p-adic absolute value, unless otherwise noted.

Definition 4.1. Let 𝔄_{0}(G, \underline{2}, χ_{Z}) denote the set of all cuspidal automorphic representations π = ⊗_{v}π_{v} of G(𝔸_{F}) with central character χ_{Z} such that π_{v} ≅ σ(| · |_{F_{v}}^{1/2}, | · |_{F_{v}}^{−1/2}) at all archimedian primes v. Here we follow the notation of [JL70]; so σ(| · |_{F_{v}}^{1/2}, | · |_{F_{v}}^{−1/2}) is the discrete series of weight 2, D(2), if v is real, and is isomorphic to the principal series representation π(μ_{1}, μ_{2}) with μ_{1}(z) = z^{1/2}\overline{z} ^{−1/2}, μ_{2}(z) = z^{−1/2}\overline{z}^{1/2} if v is complex (cf. section 4.5 below).

We will only consider automorphic representations that are p-ordinary , i.e π_{𝔭} is ordinary (in the sense of chapter 2) for every 𝔭|p. Therefore, for each 𝔭|p we fix two non-zero elements α_{𝔭,1}, α_{𝔭,2} ∈ \overline{O} ⊆ ℂ such that π_{α_{𝔭,1},α_{𝔭,2}} is an ordinary, unitary representation. By the classification of unitary representations (see e.g. [Ge75], Thm. 4.27), a spherical representation π_{α_{𝔭,1},α_{𝔭,2}} = π(χ_{1}, χ_{2}) is unitary if and only if either χ_{1}, χ_{2} are both unitary characters (i.e. |α_{𝔭,1}| = |α_{𝔭,2}| = \sqrt{q_{𝔭}}), or χ_{1,2} = χ_{0}| · |^{±s} with χ_{0} unitary and −\frac{1}{2} < s < \frac{1}{2}. A special representation π_{α_{𝔭,1},α_{𝔭,2}} = π(χ_{1}, χ_{2}) is unitary if and only if the central character χ_{1}χ_{2} is unitary. In all three cases, we have thus max\{|α_{𝔭,1}|, |α_{𝔭,2}|\} ≥ \sqrt{q_{𝔭}}. Without loss of generality, we will assume the α_{𝔭,i} to be ordered such that |α_{𝔭,1}| ≤ |α_{𝔭,2}| for all 𝔭|p. As in chapter 2, we define a_{𝔭} := α_{𝔭,1} + α_{𝔭,2}, ν_{𝔭} := α_{𝔭,1}α_{𝔭,2}/q_{𝔭}. Let \underline{α_{i}} := (α_{𝔭,i}, 𝔭|p), for i = 1, 2. We denote by 𝔄_{0}(G, \underline{2}, χ_{Z}, \underline{α_{1}}, \underline{α_{2}}) the subset of all π ∈ 𝔄_{0}(G, \underline{2}, χ_{Z}) such that π_{𝔭} = π_{α_{𝔭,1},α_{𝔭,2}} for all 𝔭|p. For later use we note that π^{∞} = ⊗_{v∤∞}π_{v} is known to be defined over a finite extension of ℚ, the smallest such field being the field of definition of π (cf. [Sp14]).

4.1 Upper half-space

For k ∈ {ℝ, ℂ}, let H_{m} := H_{k} := k × ℝ_{+}^{∗} be the upper half-space of dimension m := [k : ℝ] + 1. Each H_{m} is a differentiable manifold of dimension m. If we write x = (u, t) ∈ H_{m} with t ∈ ℝ_{+}^{∗}, u in ℝ or ℂ, respectively, it has a Riemannian metric ds^{2} = \frac{dt^{2}+du d\overline{u}}{t}, which induces a hyperbolic geometry on H_{m}, i.e. the geodesic lines on H_{m} are given by “vertical” lines {u} × ℝ_{+}^{∗} and half-circles with center in the line or plane t = 0. H_{ℝ} is naturally isomorphic to the complex upper half-plane {z ∈ ℂ| Im(z) > 0}. We have the decompositions GL_{2}(ℂ)^{+} = B′_{ℂ} · Z(ℂ) · K_{ℂ} and GL_{2}(ℝ)^{+} = B′_{ℝ} · Z(ℝ) · K_{ℝ}, where B′_{k} ⊆ GL_{2}(k) is the subgroup of matrices ( \begin{matrix}ℝ_{+}^{∗} & k \\ 0 & 1 \end{matrix} ) for k = ℝ, ℂ, Z is the center, and K_{ℝ} = SO(2), K_{ℂ} = SU(2) (cf. [By98], Cor. 43). Identifying B′_{k} with H_{k} via ( \begin{matrix}t & z \\ 0 & 1 \end{matrix} ) ↦ (z, t) gives natural projections

π_{ℝ} : GL_{2}(ℝ)^{+} ↠ GL_{2}(ℝ)^{+}/Z(ℝ) SO(2) ≅ H_{2},
π_{ℂ} : GL_{2}(ℂ) ↠ GL_{2}(ℂ)/Z(ℂ)K_{ℂ} ≅ H_{ℂ}

and corresponding left GL_{2}(k)-actions on cosets. A differential form ω on H_{m} is called left-invariant if it is invariant under the pullback L_{g}^{∗} of left multiplication L_{g} : x ↦ gx on H_{m}, for all g ∈ G.

Following [By98], eqs. (4.20), (4.24), we choose the following basis of leftinvariant differential 1-forms on H_{3}:

β_{0} := −\frac{dz}{t} , β_{1} := \frac{dt}{t}, β_{2} := \frac{d\overline{z}}{t} ,

and on H_{2} (writing z = x + iy ∈ H_{2} ⊆ ℂ):

β_{1} := \frac{dz}{y} , β_{2} := −\frac{d\overline{z}}{y} .

We note that a form f_{1}β_{1} + f_{2}β_{2} is harmonic on H_{2} if and only if f_{1}/y and f_{2}/y are holomorphic functions in z ([By98], lemma 60). The Jacobian J(g, (0, 1)) of left multiplication by g in (0, 1) ∈ H_{m} with respect to the basis (β_{i})_{i} gives rise to a representation

ϱ = ϱ_{k} : Z(k) · K_{k} → SL_{m}(ℂ)

with ϱ|_{Z(k)} trivial, which on K_{k} is explicitly given by

ϱ_{ℂ}(h) = (\begin{matrix}u^{2} & 2uv & v^{2} \\ −u\overline{v} & u\overline{u} − v\overline{v} & v\overline{u} \\ \overline{v}^{2} & −2\overline{uv} & \overline{u}^{2} \end{matrix}) for h = (\begin{matrix}u & v \\ −\overline{v} & \overline{u} \end{matrix}) ∈ SU(2),

resp.

ϱ_{ℝ} (\begin{matrix}cos(ϑ) & sin(ϑ) \\ − sin(ϑ) & cos(ϑ) \end{matrix}) = (\begin{matrix}e^{2iϑ} & 0 \\ 0 & e^{−2iϑ} \end{matrix})

([By98], (4.27), (4.21)). In the real case, we will only consider harmonic forms on H_{2} that are multiples of β_{1}, thus we sometimes identify ϱ_{ℝ} with its restriction ϱ_{ℝ}^{(1)} to the first basis vector β_{1},

ϱ_{ℝ}^{(1)} : SO(2) → S^{1} ⊆ ℂ^{∗}, κ_{ϑ} = (\begin{matrix}cos(ϑ) & sin(ϑ) \\ − sin(ϑ) & cos(ϑ) \end{matrix}) ↦ e^{2iϑ}.

For each i, let ω_{i} be the left-invariant differential 1-form on GL_{2}(k) which coincides with the pullback (π_{ℂ})^{∗}β_{i} at the identity. Write \underline{ω} (resp. \underline{β}) for the column vector of the ω_{i} (resp. β_{i}). Then we have the following lemma from [By98]:

Lemma 4.2. For each i, the differential ω_{i} on G induces β_{i} on H_{m}, by restriction to the subgroup B′_{k} ≅ H_{m}. For a function ϕ : G → ℂ^{m}, the form ϕ·\underline{ω} (with ℂ^{m} considered as a row vector, so · is the scalar product of vectors) induces f · \underline{β}, where f : H_{m} → ℂ^{m} is given by

f(z, t) := ϕ ((\begin{matrix}t & z \\ 0 & 1 \end{matrix})) .

(See [By98], Lemma 57.) To consider the infinite primes of F all at once, we define

H_{∞} := ∏_{i=0}^{d} H_{m_{i}} = ∏_{i=0}^{r−1} H_{2} × ∏_{i=r}^{d} H_{3}

(where m_{i} = 2 if σ_{i} is a real embedding, and m_{i} = 3 if σ_{i} is complex), and let H_{∞}^{0} := ∏_{i=1}^{d} H_{m_{i}} be the product with the zeroth factor removed. (The choice of the 0-th factor is for convenience; we could also choose any other infinite place, whether real or complex.)

For each embedding σ_{i}, the elements of ℙ^{1}(F) are cusps of H_{m_{i}}: for a given complex embedding F ↪ ℂ, we can identify F with F × {0} ↪ ℂ × ℝ_{≥0} and define the ”extended upper half-space“ as \overline{H_{3}} := H_{3}∪F ∪\{∞\} ⊆ ℂ×ℝ_{≥0}∪\{∞\}; similarly for a given real embedding F ↪ ℝ, we get the extended upper halfplane \overline{H_{2}} := H_{2} ∪ F ∪ \{∞\} . A basis of neighbourhoods of the cusp ∞ is given by the sets {(u, t) ∈ H_{m}|t > N}, N ≫ 0, and of x ∈ F by the open half-balls in H_{m} with center (x, 0). Let G(F)^{+} ⊆ G(F) denote the subgroup of matrices with totally positive determinant. It acts on H_{∞}^{0} by composing the embedding

G(F)^{+} ↪ ∏_{v|∞,v≠v_{0}} G(F_{v})^{+}, g ↦ (σ_{1}(g), …, σ_{d}(g)),

with the actions of G(ℂ)^{+} = G(ℂ) on H_{3} and G(ℝ)^{+} on H_{2} as defined above, and on Ω_{harm}^{d}(H_{∞}^{0}) by the inverse of the corresponding pullback, γ · \underline{ω} := (γ^{−1})^{∗}\underline{ω}. Both are left actions. For each complex v, we write the codomain of ϱ_{F_{v}} as

ϱ_{F_{v}} : Z(F_{v}) · K_{F_{v}} → SL_{3}(ℂ) =: SL(V_{v}),

for a three-dimensional ℂ-vector space V_{v}. We denote the harmonic forms on GL_{2}(F_{v}), H_{F_{v}} defined above by \underline{ω_{v}}, \underline{β_{v}} etc. Let V = ⨂_{v∈S_{ℂ}} V_{v} ≅ (ℂ^{3})^{⊗s}, Z_{∞} = ∏_{v|∞} Z(F_{v}), K_{∞} = ∏_{v|∞} K_{F_{v}}. Denoting by S_{ℂ} (resp. S_{ℝ}) the set of complex (resp. real) archimedian primes of F, we can merge the representations ϱ_{F_{v}} for each v|∞ into a representation

ϱ = ϱ_{∞} := ⨂_{v∈S_{ℂ}} ϱ_{ℂ} ⊗ ⨂_{v∈S_{ℝ}} ϱ_{ℝ}^{(1)} : Z_{∞} · K_{∞} → SL(V),

and define V -valued vectors of differential forms

\underline{ω} := ⨂_{v∈S_{ℂ}} \underline{ω_{v}} ⊗ ⨂_{v∈S_{ℝ}} ω_{v}^{1}, \underline{β} := ⨂_{v∈S_{ℂ}} \underline{β_{v}} ⊗ ⨂_{v∈S_{ℝ}}(β_{v})_{1}

on GL_{2}(F_{∞}) and H_{∞}, respectively.

4.2 Automorphic forms

Let χ_{Z} : 𝔸_{F}^{∗} /F^{∗} → ℂ^{∗} be a Hecke character that is trivial at the archimedian places. We also denote by χ_{Z} the corresponding character on Z(𝔸_{F}) under the isomorphism 𝔸_{F}^{∗} → Z(𝔸_{F}), a ↦ ( \begin{matrix}a & 0 \\ 0 & a \end{matrix}).

Definition 4.3. An automorphic cusp form of parallel weight \underline{2} with central character χ_{Z} is a map ϕ : G(𝔸_{F}) → V such that

(i) ϕ(zγg) = χ_{Z}(z)ϕ(g) for all g ∈ G(𝔸), z ∈ Z(𝔸), γ ∈ G(F).

(ii) ϕ(gk_{∞}) = ϕ(g)ϱ(k_{∞}) for all k_{∞} ∈ K_{∞}, g ∈ G(𝔸) (considering V as a row vector).

(iii) ϕ has “moderate growth“ on B′_{𝔸} := \{(\begin{matrix}y & x \\ 0 & 1 \end{matrix}) ∈ G(𝔸)\}, i.e. ∃C, λ ∀A ∈

B′_{𝔸} : ∥ϕ(A)∥ ≤ C · sup(|y|^{λ}, |y|^{−λ}) (for any fixed norm ∥·∥ on V);

and ϕ|_{G(𝔸_{∞})} · \underline{ω} is the pullback of a harmon ic form ω_{ϕ} = f_{ϕ} · \underline{β} on H_{∞}.

(iv) There exists a compact open subgroup K′ ⊆ G(𝔸^{∞}) such that ϕ(gk) = ϕ(g) for all g ∈ G(𝔸) and k ∈ K′.

(v) For all g ∈ G(𝔸_{F}),

∫_{𝔸_{F} /F} ϕ ((\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) g) dx = 0. (“Cuspidality”)

We denote by A_{0}(G, harm, \underline{2}, χ_{Z}) the space of all such maps ϕ.

For each g^{∞} ∈ 𝔸_{F}^{∞}, let ω_{ϕ}(g^{∞}) be the restriction of ϕ(g^{∞}, ·) · \underline{ω} from G(𝔸_{F}^{∞}) to H_{∞}; it is a (d + 1)-form on H_{∞}. We want to integrate ω_{ϕ}(g^{∞}) between two cusps of the space H_{m_{0}}. (We will identify each x ∈ ℙ^{1}(F) with its corresponding cusp in \overline{H_{m_{0}}} in the following.) The geodesic between the cusps x ∈ F and ∞ in \overline{H_{m_{0}}} is the line {x} × ℝ_{+}^{∗} ⊆ H_{m_{0}} and the integral of ω_{ϕ} along it is finite since ϕ is uniformly rapidly decreasing:

Theorem 4.4. (Gelfand, Piatetski-Shapiro) An automorphic cusp form ϕ is rapidly decreasing modulo the center on a fundamental domain F of GL_{2}(F)\ GL_{2}(𝔸_{F}); i.e. there exists an integer r such that for all N ∈ ℕ there exists a C > 0 such that

ϕ(zg) ≤ C|z|^{r}∥g∥^{−N}

for all z ∈ Z(𝔸_{F}), g ∈ F ∩SL_{2}(𝔸_{F}). Here ∥g∥ := max{|g_{i,j}|, |(g^{−1})_{i,j}|}_{i,j∈{1,2}}.

(See [CKM04], Thm. 2.2; or [Kur78], (6) for quadratic imaginary F.) In fact, the integral of ω_{ϕ}(g^{∞}) along {x} × ℝ_{+}^{∗} ⊆ H_{m_{0}} equals the integral of ϕ(g^{∞}, ·) · \underline{ω} along a path g_{t} ∈ GL_{2}(F_{∞_{0}}), t ∈ ℝ_{+}^{∗}, where we can choose

g_{t} = \frac{1}{\sqrt{t}} (\begin{matrix}t & x \\ 0 & 1 \end{matrix}) = ( 0 \sqrt{t}^{\frac{1}{\sqrt{t}} \frac{x}{\sqrt{t}}}) ,

and thus have ∥g_{t}∥ = \sqrt{t} for all t ≫ 0, ∥g_{t}∥ = C \frac{1}{\sqrt{t}} for t ≪ 1, so the integral ∫_{x}^{∞} ω_{ϕ}(g^{∞}) ∈ Ω_{harm}^{d}(H_{∞}^{0}) is well-defined by the theorem. For any two cusps a, b ∈ ℙ^{1}(F), we now define

∫_{a}^{b} ω_{ϕ}(g^{∞}) := ∫_{a}^{∞} ω_{ϕ}(g^{∞}) − ∫_{b}^{∞} ω_{ϕ}(g^{∞}) ∈ Ω_{harm}^{d}(H_{∞}^{0}).

Since ϕ is uniformly rapidly decreasing (∥g_{t}∥ does not depend on x, for t ≫ 0), this integral along the path (a, 0) → (a, ∞) = (b, ∞) → (b, 0) in \overline{H}_{m_{0}} is the same as the limit (for t → ∞) of the integral along (a, 0) → (a, t) → (b, t) → (b, 0); and since ω_{ϕ} is harmonic (and thus integration is path-independent within H_{m_{0}}) the latter is in fact independent of t, so equality holds for each t > 0, or along any path from (a, 0) to (b, 0) in H_{m_{0}}. Thus ∫_{a}^{b} ω_{ϕ}(g^{∞}) equals the integral of ω_{ϕ}(g^{∞}) along the geodesic from a to b, and we have

∫_{a}^{b} ω_{ϕ}(g^{∞}) + ∫_{b}^{c} ω_{ϕ}(g^{∞}) = ∫_{a}^{c} ω_{ϕ}(g^{∞})

for any three cusps a, b, c ∈ ℙ^{1}(F). Let Div(ℙ^{1}(F)) denote the free abelian group of divisors of ℙ^{1}(F), and let M := Div_{0}(ℙ^{1}(F)) be the subgroup of divisors of degree 0. We can extend the definition of the integral linearly to get a homomorphism

M → Ω_{harm}^{d}(H_{∞}^{0}), m ↦ ∫_{m} ω_{ϕ}(g^{∞}),

and easily check that

γ^{∗} (∫_{γm} ω_{ϕ}(γg)) = ∫_{m} ω_{ϕ}(g).  (29)

for all γ ∈ G(F)^{+}, g ∈ G(𝔸^{∞}), m ∈ M. Now let 𝔪 be an ideal of F prime to p, let χ_{Z} be a Hecke character of conductor dividing 𝔪, and \underline{α_{1}}, \underline{α_{2}} as above.

Definition 4.5. We define S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) to be the ℂ-vector space of all maps

Φ : G(𝔸^{p}) → B^{\underline{α_{1}},\underline{α_{2}}}(F_{p}, V) = Hom(B_{\underline{α_{1}},\underline{α_{2}}}(F_{p}, ℂ), V)

such that:

(a) ϕ is “almost” K_{0}(𝔪)-invariant (in the notation of [Ge75]), i.e. ϕ(gk) = ϕ(g) for all g ∈ G(𝔸^{p}) and k ∈ ∏_{v∤𝔪p} G(O_{v}), and ϕ(gk) = χ_{Z}(a)ϕ(g) for

all v|𝔪, k = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{0}(𝔪)_{v} and g ∈ G(𝔸^{p}).

(b) For each ψ ∈ B_{\underline{α_{1}},\underline{α_{2}}}(F_{p}, ℂ), the map

⟨Φ, ψ⟩ : G(𝔸) = G(F_{p}) × G(𝔸^{p}) → V, (g_{p}, g^{p}) ↦ Φ(g^{p})(g_{p}ψ)

lies in A_{0}(G, harm, \underline{2}, χ_{Z}).

Note that (a) implies that ϕ is K′-invariant for some open subgroup K′ ⊆ K_{0}(𝔪)^{p} of finite index ([By98]/[We71]).

4.3 Cohomology of GL_{2}(F)

Let M be a left G(F)-module and N an R[H]-module, for a ring R and a subgroup H ⊆ G(F). Let S ⊆ S_{p} be a set of primes of F dividing p; as above, let χ = χ_{Z} be a Hecke character of conductor 𝔪 prime to p.

Definition 4.6. For a compact open subgroup K ⊆ K_{0}(𝔪)^{S} ⊆ G(𝔸^{S,∞}), we denote by A_{f}(K, S, M; N) the R-module of all maps Φ : G(𝔸^{S,∞}) × M → N such that

1. Φ(gk, m) = Φ(g, m) for all g ∈ G(𝔸^{S,∞}), m ∈ M, k ∈ ∏_{v∤𝔪p} G(O_{v});

2. Φ(gk) = χ_{Z}(a)Φ(g) for all v|𝔪, k = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{0}(𝔪)_{v} and g ∈ G(𝔸^{S,∞}),

m ∈ M.

We denote by A_{f}(S, M; N) the union of the A_{f}(K, S, M; N) over all compact open subgroups K.

A_{f}(S, M; N) is a left G(𝔸^{S,∞})-module via (γ · Φ)(g, m) := Φ(γ^{−1}g, m) and has a left H-operation given by (γ · Φ)(g, m) := γΦ(γ^{−1}g, γ^{−1}m), commuting with the G(𝔸^{S,∞})-operation. In contrast to our previous notation, we consider two subsets S_{1} ⊆ S_{2} ⊆ S_{p} in this section. We put (\underline{α_{1}}, \underline{α_{2}})_{S_{1}} := \{(α_{𝔭,1}, α_{𝔭,2})|𝔭 ∈ S_{1}\}, we set

A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S_{1}}, S_{2}, M; N) = A_{f}(S_{2}, M; B^{(\underline{α_{1}},\underline{α_{2}})_{S_{1}}} (F_{S_{1}}, N));

we write A_{f}(𝔪, (\underline{α_{1}}, \underline{α_{2}})_{S_{1}}, S_{2}, M; N) := A_{f}(K_{0}(𝔪), (\underline{α_{1}}, \underline{α_{2}})_{S_{1}}, S_{2}, M; N). If S_{1} = S_{2}, we will usually drop S_{2} from all these notations. We have a natural identification of A_{f}(𝔪, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; N) with the space of maps G(𝔸^{S,∞}) × M × B_{(\underline{α_{1}},\underline{α_{2}})_{S}}(F_{S}, R) → N that are “almost” K-invariant. Let S_{0} ⊆ S_{1} ⊆ S_{2} ⊆ S_{p} be subsets. The pairing (11) induces a pairing

A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S_{1}}, S_{2}, M; N) × B_{(\underline{α_{1}},\underline{α_{2}})_{S_{0}}} (F_{S_{0}}, R) → A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, S_{2}, M; N) which, when restricting to K-invariant elements, induces an isomorphism

A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S_{1}}, S_{2}, M; N) ≅ B^{(\underline{α_{1}},\underline{α_{2}})_{S_{1}−S_{0}}} (F_{S_{1}−S_{0}}, A_{f}(\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, S_{2}, M; N).

Putting S_{0} := S_{1} − {𝔭} for a prime 𝔭 ∈ S_{1}, we specifically get an isomorphism

A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S_{1}}, S_{2}, M; N) ≅ B^{α_{𝔭,1},α_{𝔭,2}}(F_{𝔭}, A_{f}(\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, S_{2}, M; N).

Lemmas 2.9 and 2.10 now immediately imply the following:

Lemma 4.7. Let S ⊆ S_{p}, 𝔭 ∈ S, S_{0} := S −{𝔭}. Let K ⊆ G(𝔸^{S,∞}) be a compact open subgroup.

(a) If π_{α_{𝔭,1},α_{𝔭,2}} is spherical, we have exact sequences

0 → A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; N) → Z ⟶^{N−ν_{𝔭}} Z → 0

and

0 → Z → A_{f}(K_{0}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; N) ⟶^{T −a_{𝔭}} A_{f}(K_{0}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; N) → 0

for a G(𝔸^{S_{0},∞})-module Z and a compact open subgroup K_{0} = K × K_{𝔭} of G(𝔸^{S_{0},∞}).

(b) If π_{α_{𝔭,1},α_{𝔭,2}} is special (with central character χ_{𝔭}), we have exact sequences

0 → A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; N) → Z′ → Z → 0

and

0 →Z → A_{f}(K_{0}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; N)^{2} → A_{f}(K_{0}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; N)^{2} → 0,
0 →Z′ → A_{f}(K′_{0}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; N)^{2} → A_{f}(K′_{0}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; N)^{2} → 0,

with Z^{(′)} := A_{f}(K_{0}^{(′)}, (\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, S, M; N(χ_{𝔭})), where K_{0}^{(′)} = K × K_{𝔭}^{(′)} are compact open subgroups of G(𝔸^{S_{0},∞}).

Proposition 4.8. Let S ⊆ S_{p} and let K be a compact open subgroup of G(𝔸^{S,∞}).

(a) For each flat R-module N (with trivial G(F)-action), the canonical map

H^{q}(G(F)^{+}, A_{f} (K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; R)) ⊗_{R} N → H^{q}(G(F)^{+}, A_{f} (K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; N))

is an isomorphism for each q ≥ 0.

(b) If R is finitely generated as a ℤ-module, H^{q}(G(F)^{+}, A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; R) is finitely generated over R.

Proof. We can copy the proof of [Sp14], Prop. 5.6, using lemma 4.7 instead of [Sp14], lemma 5.4 to reduce to the case S = ∅.

We define

H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; R)) := \underrightarrow{lim} H^{q}(G(F)^{+}, A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; R))

where the limit runs over all compact open subgroups K ⊆ G(𝔸^{S,∞}); and similarly define H_{∗}^{q}(B(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; R). The proposition immediately implies

Corollary 4.9. Let R → R′ be a flat ring homomorphism. Then the canonical map

H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; R)) ⊗_{R} R′ → H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; R′)

is an isomorphism, for all q ≥ 0.

If R = k is a field of characteristic zero, H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; R) is a smooth G(𝔸^{S,∞})-module, and we have

H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; k)^{K} = H^{q}(G(F)^{+}, A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; k).

We identify G(F)/G(F)^{+} with the group Σ = {±1}^{r} via the isomorphism

G(F)/G(F^{+}) ⟶^{det} F^{∗}/F_{+}^{∗} ≅ Σ

(with all groups being trivial for r = 0). Then Σ acts on H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; k) and H^{q}(G(F)^{+}, A_{f}(K, (\underline{α_{1}}, \underline{α_{2}})_{S}, M; k) by conjugation.For π ∈ 𝔄_{0}(G, \underline{2}) and \underline{μ} ∈ Σ, we write H_{∗}^{q}(G(F)^{+}, ·)_{π,\underline{μ}} := Hom_{G(𝔸^{S,∞})}(π^{S}, H_{∗}^{q}(G(F)^{+}, ·))\underline{μ}.

Proposition 4.10. Let π ∈ 𝔄_{0}(G, \underline{2}, χ_{Z}, \underline{α_{1}}, \underline{α_{2}}), S ⊆ S_{p}. Let k be a field which contains the field of definition of π. Then for every \underline{μ} ∈ Σ, we have

H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; k)_{π,\underline{μ}} = \begin{cases}k, if q = d; \\ 0, if q ∈ \{0, …, d − 1\} \end{cases}  (30)

Proof. The case S = ∅ is proved analogously to [Sp14], prop. 5.8, using the results of Harder [Ha87]. For S = S_{0} ∪ {𝔭} and π_{𝔭} spherical, lemma 4.7(a) and the statement for S_{0} give an isomorphism

H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S_{0}}, M; k))_{π,\underline{μ}} ≅ H_{∗}^{q}(G(F)^{+}, A_{f}((\underline{α_{1}}, \underline{α_{2}})_{S}, M; k))_{π,\underline{μ}}

since the Hecke operators T_{𝔭} and N_{𝔭} act on the left-hand side by multiplication with a_{𝔭} and ν_{𝔭}, respectively. If π_{𝔭} is special, we can similarly deduce the statement for S from that for S_{0}, using the first exact sequence of lemma 4.7(b), since the results of [Ha87] also hold when twisting k by a (central) character.

4.4 Eichler-Shimura map

From now on, let S_{1} ⊆ S_{p} be the set of places such that π_{𝔭} is the Steinberg representation (i.e. α_{𝔭,1} = ν_{𝔭} = 1, α_{𝔭,2} = q). Given a subgroup K_{0}(𝔪)^{p} ⊆ G(𝔸^{p,∞}) as above, there is a map

I_{0} : S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) → H^{0}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M; Ω_{harm}^{d}(H_{∞}^{0})))

given by

I_{0}(Φ) : (ψ, (g, m)) ↦ ∫_{m} ω_{⟨Φ,ψ⟩}(1_{p}, g),

for ψ ∈ B_{\underline{α_{1}},\underline{α_{2}}}(F_{p}, ℂ), g ∈ G(𝔸^{p,∞}), m ∈ M, where 1_{p} denotes the unity element in G(F_{p}). This is well-defined since both sides are “almost” K_{0}(𝔪)-invariant, and the G(F)^{+}-invariance of I_{0}(Φ) follows from a straightforward calculation, using (29). From the complex

A_{f}(m, \underline{α_{1}}, \underline{α_{2}}, M; ℂ) → C^{•} := A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M; Ω_{harm}^{•}(H_{∞}^{0}))

we get a map

S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) → H^{d}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M; ℂ))  (31)

by composing I_{0} with an edge morphism of the spectral sequence

H^{q}(G(F)^{+}, C^{p}) ⟹ H^{p+q}(G(F)^{+}, C^{•}).

Using the map δ^{\underline{α_{1}},\underline{α_{2}}} : B^{\underline{α_{1}},\underline{α_{2}}}(F, V) → Dist(F_{p}^{∗}, V) from section 2.6, we next define a map

Δ_{V}^{\underline{α_{1}},\underline{α_{2}}} : S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) → D(S_{1}, V)  (32)

by

Δ_{V}^{\underline{α_{1}},\underline{α_{2}}}(Φ)(U, x^{p}) = δ^{\underline{α_{1}},\underline{α_{2}}} (Φ (\begin{matrix}x^{p} & 0 \\ 0 & 1 \end{matrix})) (U)

for U ∈ ℭ𝔬(F_{S_{1}} × F_{S_{2}}), x^{p} ∈ 𝕀^{p}, and we denote by Δ^{\underline{α_{1}},\underline{α_{2}}} : S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) → D(S_{1}, ℂ) its (1,...,1)th coordinate function (i.e. corresponding to the harmonic forms ⨂_{v|∞}(ω_{v})_{1}, ⨂_{v|∞}(β_{v})_{1} in section 4.1):

Δ^{\underline{α_{1}},\underline{α_{2}}}(Φ)(U, x^{p}) = δ^{\underline{α_{1}},\underline{α_{2}}} (Φ (\begin{matrix}x^{p} & 0 \\ 0 & 1 \end{matrix}))_{(1,...,1)} (U).

Since for each complex prime v, S^{1} ≅ SU(2) ∩ T (ℂ) operates on Φ via ϱ_{v}, Δ^{\underline{α_{1}},\underline{α_{2}}} is easily seen to be S^{1}-invariant, i.e. it lies in D′(S_{1}, ℂ). We also have a natural (i.e. commuting with the complex maps of each C^{•}) family of maps

A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, Ω_{harm}^{i}(H_{∞}^{0})) → D_{f}(S_{1}, Ω^{i}(U_{∞}^{0}, ℂ))  (33)

for all i ≥ 0, and

A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, ℂ) → D_{f}(S_{1}, ℂ)  (34)

(the i = −1-th term in the complexes), by mapping Φ ∈ A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, ·) first to

(U, x^{p,∞}) ↦ Φ ((\begin{matrix}x^{p,∞} & 0 \\ 0 & 1 \end{matrix}) , ∞ − 0) (δ_{\underline{α_{1}},\underline{α_{2}}}(1_{U})) ∈ Ω_{harm}^{i}(H_{∞}^{0}) resp. ∈ ℂ,

and then for i ≥ 0 restricting the differential forms to Ω^{i}(U_{∞}^{0}) via

U_{∞}^{0} = ∏_{v∈S_{∞}^{0}} ℝ_{+}^{∗} ↪ ∏_{v∈S_{∞}^{0}} H_{v} = H_{∞}^{0}.

One easily checks that (33) and (34) are compatible with the homomorphism of “acting groups” F^{∗}′ ↪ G(F)^{+}, x ↦ ( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} ), so we get induced maps in cohomology

H^{0}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, Ω_{harm}^{d}(H_{∞}^{0}))) → H^{0}(F^{∗}′, D_{f}(S_{1}, Ω^{d}(U_{∞}^{0}, ℂ)))

(35)

and

H^{d}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, ℂ)) → H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)),  (36)

which are linked by edge morphisms of the respective spectral sequences to give a commutative diagram (given in the proof below).

Proposition 4.11. We have a commutative diagram:

\xymatrix{S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) \ar[r]^{(31)} \ar[d]^{Δ^{\underline{α_{1}},\underline{α_{2}}}} & H^{d}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, ℂ)) \ar[d]^{(36)} \\ D′(S_{1}, ℂ) \ar[r]^{ϕ↦κ_{ϕ}} & H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ))}

Proof. The given diagram factorizes as

\xymatrix{S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}) \ar[r]^{I_{0}} \ar[d]^{Δ^{\underline{α_{1}},\underline{α_{2}}}} & H^{0}(G(F)^{+}, A_{f} (Ω_{harm}^{d}(H_{∞}^{0}))) \ar[r] \ar[d]^{(35)} & H^{d}(G(F)^{+}, A_{f} (ℂ)) \ar[d]^{(36)} \\ D′(S_{1}, ℂ) \ar[r] & H^{0}(F^{∗}′, D_{f} (S_{1}, Ω^{d}(U_{∞}^{0}, ℂ))) \ar[r] & H^{d}(F^{∗}′, D_{f} (S_{1}, ℂ))}

(where we write A_{f}(·) instead of A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, ·) for brevity). The righthand square is the naturally commutative square mentioned above; the commutativity of the left-hand square can easily be checked by hand.

4.5 Whittaker model

We now consider an automorphic representation π = ⊗_{ν}π_{ν} ∈ 𝔄_{0}(G, \underline{2}, χ_{Z}, \underline{α_{1}}, \underline{α_{2}}). Denote by 𝔠(π) := ∏_{v finite} 𝔠(π_{v}) the conductor of π. Let χ : 𝕀^{∞} → ℂ^{∗} be a unitary character of the finite ideles; for each finite place v, set χ_{v} = χ|_{F_{v}^{∗}}. For each prime v of F, let W_{v} denote the Whittaker model of π_{v}. For each finite and each real prime, we choose W_{v} ∈ W_{v} such that the local L-factor equals the local zeta function at g = 1, i.e. such that

L(s, π_{v} ⊗ χ_{v}) = ∫_{F_{v}^{∗}} W_{v} (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) χ_{v}(x)|x|^{s− \frac{1}{2}} d^{×}x  (37)

for any unramified quasi-character χ_{v} : F_{v}^{∗} → ℂ^{∗} and Re(s) ≫ 0. This is possible by [Ge75], Thm. 6.12 (ii); and by loc.cit., Prop. 6.17, W_{v} can be chosen such that SO(2) operates on W_{v} via ϱ_{v} for real archimedian v, and is “almost” K_{0}(𝔠(π_{v}))-invariant for finite v. For complex primes v of F, we can also choose a W_{v} satisfying (37) and which behaves well with respect to the SU(2)-action ϱ_{v}, as follows: By [Kur77], there exists a function

\underline{W_{v}} = (W_{v}^{0}, W_{v}^{1}, W_{v}^{2}) : G(F_{v}) → ℂ^{3}

such that W_{v}^{i} ∈ W_{v} for all i, and such that SU(2) operates by the right via ϱ_{v} on \underline{W_{v}}; i.e. for all g ∈ G(F_{v}) and h ∈ SU(2), we have

\underline{W_{v}}(gh) = \underline{W_{v}}(g)ϱ_{ℂ}(h).

Note that W_{v}^{1} is thus invariant under right multiplication by a diagonal matrix

(\begin{matrix}u & 0 \\ 0 & \overline{u} \end{matrix}) with u ∈ S^{1} ⊆ ℂ. Since π_{v} has trivial central character for archi-

median v by our assumption, a function in W_{v} is also invariant under Z(F_{v}). Thus we have

W_{v}^{1} (g (\begin{matrix}u & 0 \\ 0 & 1 \end{matrix})) = W_{v}^{1}(g) for all g ∈ G(F_{v}), u ∈ S^{1}.

W_{v}^{1} can be described explicitly in terms of a certain Bessel function, as follows. The modified Bessel differential equation of order α ∈ ℂ is

x^{2} \frac{d^{2}y}{dx^{2}} + x\frac{dy}{dx} − (x^{2} + α^{2})y = 0.

Its solution space (on {Re z > 0}) is two-dimensional; we are only interested in the second standard solution K_{v}, which is characterised by the asymptotics

K_{v}(z) ∼ \sqrt{\frac{π}{2z}} e^{−z}

(cf. [We71]). By [Kur77],^{3} we have W_{v}^{1} (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) = \frac{2}{π} x^{2}K_{0}(4πx).

(W_{v}^{0} and W_{v}^{2} can also be described in terms of Bessel functions; they are linearly dependent and scalar multiples of x^{2}K_{1}(4πx).) By [JL70], Ch. 1, Thm. 6.2(vi), σ(| · |_{ℂ}^{1/2}, | · |_{ℂ}^{−1/2}) ≅ π(μ_{1}, μ_{2}) with

μ_{1}(z) = z^{1/2}\overline{z} ^{−1/2} = |z|_{ℂ}^{−1/2}z, μ_{2}(z) = z^{−1/2}\overline{z}^{1/2} = |z|_{ℂ}^{−1/2}\overline{z};

and the L-series of the representation is the product of the L-factors of these two characters:

L_{v}(s, π_{v}) = L(s, μ_{1})L(s, μ_{2}) = 2 (2π)^{−(s+ \frac{1}{2} )}Γ(s + \frac{1}{2}) · 2 (2π)^{−(s+ \frac{1}{2} )}Γ(s + \frac{1}{2})
= 4 (2π)^{−(2s+1)}Γ(s + \frac{1}{2})^{2}.

On the other hand, letting d^{×}x = \frac{dx}{|x|_{ℂ}} = \frac{dr}{r} dϑ (for x = re^{iϑ}), we have for Re(s) > −\frac{1}{2}:

∫_{ℂ^{∗}} W_{v}^{1} (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) |x|_{ℂ}^{s− \frac{1}{2}} d^{×}x = ∫_{S^{1}} ∫_{ℝ_{+}} W_{v}^{1} (\begin{matrix}re^{iϑ} & 0 \\ 0 & 1 \end{matrix}) |x|_{ℂ}^{s− \frac{1}{2}} \frac{dr}{r} dϑ
= 4 ∫_{0}^{∞} x^{2}K_{0}(4πx)x^{2s−1} \frac{dx}{x}

(invariance under SU(2) · Z(F_{v}) gives a constant integral w.r.t. ϑ)

= 4 (4π)^{−2s+1} ∫_{0}^{∞}K_{0}(x)x^{2s} dx
= 4 (4π)^{−2s+1} 2^{2s−1} Γ(s + \frac{1}{2})^{2}
= 4 (2π)^{−2s+1} Γ(s + \frac{1}{2})^{2}

by ([DLMF] 10.43.19). Thus we have

∫_{ℂ^{∗}} W_{v}^{1} (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) |x|_{ℂ}^{s− \frac{1}{2}} d^{×}x = (2π)^{2} L_{v}(s, π_{v})

for all Re(s) > −\frac{1}{2}.We set W_{v} := (2π)^{−2} W_{v}^{1}; thus (37) holds also for complex primes. Now that we have defined W_{v} for all primes v, we put W^{p}(g) := ∏_{v∤p} W_{v}(g_{v}) for all g = (g_{v})_{v} ∈ G(𝔸^{p}). We will also need the vector-valued function \underline{W}^{p} : G(𝔸_{F}) → V given by

\underline{W}^{p}(g) := ∏_{v∤p finite or v real} W_{v}(g_{v}) · ⨂_{v complex}(2π)^{−2}\underline{W_{v}}(g_{v}).

3Note that [Kur77] uses a slightly different definition of the K_{v}, which is \frac{2}{π} times our K_{v}.

Documenta Mathematica 21 (2016) 689–734

4.6 p-adic measures of automorphic forms

Now return to our π ∈ 𝔄_{0}(G, \underline{2}, χ_{Z}, \underline{α_{1}}, \underline{α_{2}}). We fix an additive character ψ : 𝔸 → ℂ^{∗} which is trivial on F, and let ψ_{v} denote the restriction of ψ to F_{v} ↪ 𝔸, for all primes v. We further require that ker(ψ_{𝔭}) ⊇ O_{𝔭} and 𝔭^{−1} ⊈ ker ψ_{𝔭} for all 𝔭|p, so that we can apply the results of chapter 2. As in chapter 2, let μ_{π_{𝔭}} := μ_{α_{𝔭,1}/ν_{𝔭}} = μ_{q_{𝔭}/α_{𝔭,2}} denote the distribution χ_{q_{𝔭}/α_{𝔭,2}}(x)ψ_{𝔭}(x)dx on F_{𝔭}, and let μ_{π_{p}} := ∏_{𝔭|p} μ_{π_{𝔭}} be the product distribution on F_{p} := ∏_{𝔭|p} F_{𝔭}. Define ϕ = ϕ_{π} : ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}) × 𝕀^{p} → ℂ by

ϕ(U, x^{p}) := ∑_{ζ∈F^{∗}} μ_{π_{p}}(ζU)W^{p} (\begin{matrix}ζx^{p} & 0 \\ 0 & 1 \end{matrix}) .

By proposition 2.13(a), we have for each U ∈ ℭ𝔬(F_{S_{1}} × F_{S_{2}}^{∗}):

ϕ_{U}(x) := ϕ(x_{p}U, x^{p}) = ∑_{ζ∈F^{∗}} μ_{π_{p}}(ζx_{p}U)W^{p} (\begin{matrix}ζx^{p} & 0 \\ 0 & 1 \end{matrix})

= ∑_{ζ∈F^{∗}} W (\begin{matrix}ζx & 0 \\ 0 & 1 \end{matrix}) ,

where W (g) := W_{U}(g_{p})W^{p}(g^{p}) lies in the global Whittaker model W = W(π) for all g = (g_{p}, g^{p}) ∈ G(𝔸), putting W_{U} := W_{1_{U}}; so ϕ is well-defined and lies in D(S_{1}, ℂ) (since W is smooth and rapidly decreasing; distribution property, F^{∗}- and U^{p,∞}-invariance being clear by the definitions of ϕ and W^{p}). Let μ_{π} := μ_{ϕ_{π}} be the distribution on G_{p} corresponding to ϕ_{π}, as defined in (22), and let κ_{π} := κ_{ϕ_{π}} ∈ H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ)) be the cohomology class defined by (23) and (24).

Theorem 4.12. Let π ∈ 𝔄_{0}(G, \underline{2}, χ_{Z}, \underline{α_{1}}, \underline{α_{2}}); we assume the α_{𝔭,i} to be ordered such that |α_{𝔭,1}| ≤ |α_{𝔭,2}| for all 𝔭|p. (So χ_{𝔭,1} = | · |χ_{𝔭,2} for all special π_{𝔭}.)

(a) Let χ : G_{p} → ℂ^{∗} be a character of finite order with conductor 𝔣(χ). Then we have the interpolation property

∫_{G_{p}} χ(γ)μ_{π}(dγ) = τ(χ) ∏_{𝔭∈S_{p}} e(π_{𝔭}, χ_{𝔭}) · L(\frac{1}{2}, π ⊗ χ),

where

e(π_{𝔭}, χ_{𝔭}) = \begin{cases}\frac{(1 − α_{𝔭,1}x_{𝔭}q_{𝔭}^{−1})(1 − α_{𝔭,2}x_{𝔭}^{−1}q_{𝔭}^{−1})(1 − α_{𝔭,2}x_{𝔭}q_{𝔭}^{−1})}{(1 − x_{𝔭}α_{𝔭,2}^{−1})} , & ord_{𝔭}(𝔣(χ)) = 0 \\ & and π spherical, \\ \frac{(1 − α_{𝔭,1}x_{𝔭}q_{𝔭}^{−1})(1 − α_{𝔭,2}x_{𝔭}^{−1}q_{𝔭}^{−1})}{(1 − x_{𝔭}α_{𝔭,2}^{−1})} , & ord_{𝔭}(𝔣(χ)) = 0 \\ & and π special, \\ (α_{𝔭,2}/q_{𝔭})^{ord_{𝔭}(𝔣(χ))}, & ord_{𝔭}(𝔣(χ)) > 0 \end{cases}

Documenta Mathematica 21 (2016) 689–734

and x_{𝔭} := χ_{𝔭}(ϖ_{𝔭}).

(b) κ_{π} is integral (cf. definition 3.12). For \underline{μ} ∈ Σ, let κ_{π,\underline{μ}} be the projection of κ_{π} to H^{d}(F^{∗}′, D_{f}(S_{1}, ℂ))_{π,\underline{μ}}. Then κ_{π,\underline{μ}} is integral of rank ≤ 1.

Proof. (a) We consider χ as a character on 𝕀_{F} /F^{∗}, and choose a subgroup V = ∏_{𝔭|p} V_{𝔭} ⊆ U_{p} such that χ_{p}|_{V} = 1. Since π is unitary, we have |α_{𝔭,2}| ≥ \sqrt{q_{𝔭}} > 1 = |χ_{𝔭}(ϖ_{𝔭})| for all 𝔭, thus e(π_{𝔭}, χ_{𝔭}| · |_{𝔭}^{s}) is non-singular for all s ≥ 0, and we will be able to apply proposition 2.4 locally below. We have

∫_{G_{p}} χ(γ)μ_{π}(dγ) = [U_{p} : V] ∫_{𝕀_{F} /F^{∗}} χ(x)ϕ_{V} (x)d^{×}x,

and therefore we have to show that the equality

[U_{p} : V] ∫_{𝕀_{F} /F^{∗}}χ(x)|x|^{s}ϕ_{V} (x)d^{×}x = N(𝔣(χ))^{s}τ(χ) ∏_{𝔭|p} e(π_{𝔭}, χ_{𝔭}|·|_{𝔭}^{s})·L(s+ \frac{1}{2}, π⊗χ)

holds for s = 0. Since both the left-hand side and L(s+\frac{1}{2}, π⊗χ) are holomorphic in s (cf. [Ge75], Thm. 6.18), it suffices to show this for Re(s) ≫ 0. But for such s, we have

[U_{p} : V] ∫_{𝕀_{F} /F^{∗}} χ(x)|x|^{s}ϕ_{V} (x)d^{×}x = ∫_{𝕀_{F}} χ(x)|x|^{s}W (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) d^{×}x

= [U_{p} : V] ∫_{F_{p}^{∗}} χ_{p}(x)|x|^{s}W_{V} (\begin{matrix}x & 0 \\ 0 & 1 \end{matrix}) d^{×}x · ∫_{𝕀_{F}^{p}} χ^{p}(y)|y|^{s}W^{p} (\begin{matrix}y & 0 \\ 0 & 1 \end{matrix}) d^{×}y
= ∏_{𝔭|p} ∫_{F_{𝔭}^{∗}} χ_{𝔭}(x)|x|_{𝔭}^{s}μ_{π_{𝔭}}(dx) · L_{S_{p}}(s + \frac{1}{2}, π ⊗ χ)
= ∏_{𝔭|p} (e(π_{𝔭}, χ_{𝔭}| · |_{𝔭}^{s})τ(χ_{𝔭}| · |_{𝔭}^{s})) · L(s + \frac{1}{2}, π ⊗ χ)
= N(𝔣(χ))^{s}τ(χ) ∏_{𝔭|p} e(π_{𝔭}, χ_{𝔭}| · |_{𝔭}^{s}) · L(s + \frac{1}{2}, π ⊗ χ)

by propositions 2.13, 2.4 and equation (37).

(b) Let λ_{\underline{α_{1}},\underline{α_{2}}} ∈ B^{\underline{α_{1}},\underline{α_{2}}}(F_{p}, ℂ) be the image of ⊗_{v|p}λ_{a_{v},ν_{v}} under the map (13). For each ψ ∈ B_{\underline{α_{1}},\underline{α_{2}}}(F_{p}, ℂ), define

⟨Φ_{π}, ψ⟩(g^{p}, g_{p}) := ∑_{ζ∈F^{∗}} λ_{\underline{α_{1}},\underline{α_{2}}} ((\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix}) g_{p} · ψ) \underline{W}^{p} ((\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix}) g^{p})

=: ∑_{ζ∈F^{∗}} \underline{W_{ψ}} ((\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix}) g)

for a V -valued function \underline{W_{ψ}} whose every coordinate function is in W(π). This defines a map Φ_{π} : G(𝔸^{p}) → B^{\underline{α_{1}},\underline{α_{2}}}(F_{p}, V). In fact, Φ_{π} lies in S_{2}(G, 𝔪, \underline{α_{1}}, \underline{α_{2}}), where 𝔪 is the prime-to-p part of 𝔣(π): Condition (a) of definition 4.5 follows from the fact that the W_{v} are almost K_{0}(𝔠(π_{v}))-invariant, for v ∤ p, ∞. For condition (b), we check that ⟨Φ_{π}, ψ⟩ satisfies the conditions (i)-(v) in the definition of A_{0}(G, harm, \underline{2}, χ): Each coordinate function of ⟨Φ_{π}, ψ⟩ lies in (the underlying space of) π by [Bu98], Thm. 3.5.5, thus ⟨Φ, ψ⟩ fulfills (i) and (v), and has moderate growth. (ii) and (iv) follow from the choice of the W_{v} and \underline{W_{v}}. Now since π_{v} ≅

σ(| · |_{v}^{1/2}, | · |_{v}^{−1/2}) for v|∞, ⟨Φ, ψ⟩|_{B′_{Fv}} · \underline{β_{v}} = C ∑_{ζ∈F^{∗}} \underline{W_{v}} ( \begin{matrix}ζt & 0 \\ 0 & 1 \end{matrix} ) · \underline{β_{v}} is har-

monic for each archimedian place v of F: for real v, it is well-known that f(z)/y is holomorphic for f ∈ D(2), and thus f ·(β_{v})_{1} is harmonic; for complex v, harmonicity follows from the other conditions, see e.g. [Kur78], p. 546 or [We71]. An easy calculation shows that

λ_{\underline{α_{1}},\underline{α_{2}}} ((\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix}) δ_{\underline{α_{1}},\underline{α_{2}}}(1_{U})) = ∫_{ζU} ∏_{𝔭|p} χ_{α_{𝔭,2}}(−x)ψ_{𝔭}(−x)dx = μ_{π_{p}}(ζU)

for all ζ ∈ F^{∗}, and therefore we have

Δ^{\underline{α_{1}},\underline{α_{2}}}(Φ_{π})(U, x^{p}) = ∑_{ζ∈F^{∗}} λ_{\underline{α_{1}},\underline{α_{2}}} ((\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix}) δ_{\underline{α_{1}},\underline{α_{2}}}(1_{U})) W^{p} (\begin{matrix}ζx^{p} & 0 \\ 0 & 1 \end{matrix})

= ∑_{ζ∈F^{∗}} μ_{π_{p}}(ζU)W^{p} (\begin{matrix}ζx^{p} & 0 \\ 0 & 1 \end{matrix}) = ϕ_{π}(U, x^{p}).

Let R be the integral closure of ℤ[a_{𝔭}, ν_{𝔭}; 𝔭|p] in its field of fractions; thus R is a Dedekind ring ⊆ \overline{O} for which B_{\underline{α_{1}},\underline{α_{2}}}(F, R) is defined. Since ℂ is a flat R-module,

H^{d}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, R)) ⊗ ℂ → H^{d}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, ℂ))

is an isomorphism by proposition 4.8. The map (36) can be described as the ”R-valued” map

H^{d}(G(F)^{+}, A_{f}(𝔪, \underline{α_{1}}, \underline{α_{2}}, M, R)) → H^{d}(F^{∗}′, D_{f}(R))

tensored with ℂ. By proposition 4.11, κ_{π} lies in its image, and thus in H^{d}(F^{∗}′, D_{f}(R)) ⊗ ℂ; i.e. it is integral. Similarly, it follows from propositions 4.8 and 4.10 that κ_{π,\underline{μ}} is integral of rank ≤ 1.

Corollary 4.13. μ_{π} is a p-adic measure.

Proof. By proposition 3.8, μ_{π} = μ_{ϕ_{π}} = μ_{κ_{π}}. Since κ_{π} is integral, μ_{κ_{π}} is a p-adic measure by corollary 3.13.

4.7 Vanishing order of the p-adic L-function

Let L_{1}, …, L_{t} be independent ℤ_{p}-extensions of F, and let ℓ_{1}, …, ℓ_{t} : G_{p} → p^{ε_{p}}ℤ_{p} be the homomorphisms corresponding to them (as in section 3.2). Then we have the p-adic L-function

L_{p}(\underline{s}, π) := L_{p}(\underline{s}, κ_{π}) := L_{p}(s_{1}, …, s_{t}, κ_{π,+}) := ∫_{G_{p}} ∏_{i=1}^{t} exp_{p}(s_{i}ℓ_{i}(γ))μ_{π}(dγ)

of definition 3.5, with s_{1}, …, s_{t} ∈ ℤ_{p}. L_{p}(\underline{s}, π) is a locally analytic function with values in the one-dimensional ℂ_{p}-vector space V_{κ_{π,+}} = L_{κ,\overline{O},+} ⊗\overline{O} ℂ_{p}. By theorem 3.11, we have

Theorem 4.14. L_{p}(\underline{s}, π) is a locally analytic (t-variabled) function, and all partial derivatives of order ≤ n := #(S_{1}) vanish; i.e. we have

ord_{\underline{s}=\underline{0}} L_{p}(\underline{s}, π) ≥ n.

Now let E be a modular elliptic curve over F, corresponding to an automorphic representation π; by this we mean that the local L-factors of the Hasse-Weil L-function L(E, s) and of the automorphic L-function L(s − \frac{1}{2}, π) coincide at all places v of F. From the definition of the respective L-factors (cf. [Si86] for the Hasse-Weil L-function, [Ge75] for the automorphic L-function) we know that π has trivial central character. Moreover, for 𝔭|p, π_{𝔭} is a principal series representation iff E has good reduction at 𝔭, and in this case π_{𝔭} is ordinary iff E is ordinary (i.e. not supersingular) at 𝔭; π_{𝔭} is a special (resp. Steinberg) representation iff E has multiplicative (resp. split multiplicative) reduction at 𝔭. For v|∞, π_{v} is “of weight 2” as assumed before. We say that E is p-ordinary if it has good ordinary or multiplicative reduction at all places 𝔭|p of F. So E is p-ordinary iff π is ordinary at all 𝔭|p. In this case, we define the p-adic L-function of E by L_{p}(E, \underline{s}) := L_{p}(\underline{s}, π). For each i ∈ {1, …, t} and each prime 𝔭|p of F, we write ℓ_{𝔭,i} for the restriction of ℓ_{i} to F_{𝔭} ↪ 𝕀 ↠ G_{p}. Let q_{𝔭} be the Tate period of E|F_{𝔭} and ord_{𝔭} the normalized valuation on F_{𝔭}^{∗}. Defining the L-invariants of E|F_{𝔭} with respect to L_{i} by

L_{𝔭,i}(E) := ℓ_{𝔭,i}(q_{𝔭})/ ord_{𝔭}(q_{𝔭}),

we can generalize Hida’s exceptional zero conjecture to general number fields:

Conjecture 4.15. Let S_{1} be the set of 𝔭|p at which E has split multiplicative reduction, n := #S_{1}, S_{2} := S_{p} \ S_{1}. Then

ord_{\underline{s}=\underline{0}} L_{p}(E, \underline{s}) ≥ n,  (38)

and we have

\frac{∂^{n}}{∂s_{i}^{n}} L_{p}(E, \underline{s})|_{\underline{s}=\underline{0}} = n! ∏_{𝔭∈S_{1}} L_{𝔭,i}(E) ∏_{𝔭∈S_{2}} e(π_{𝔭}, 1) · L(E, 1),  (39)

for all i = 1, …, t, where e(π_{𝔭}, 1) = (1 − α_{𝔭,1}^{−1})^{2} if E has good ordinary reduction at 𝔭, and e(π_{𝔭}, 1) = 2 if E has non-split multiplicative reduction at 𝔭.

Note that the conjecture (when considered for all sets of independent ℤ_{p}- extensions of F) also determines the “mixed” partial derivatives \frac{∂^{k}}{∂\underline{n}\underline{s}}L_{p}(E, \underline{0}) of order n, since they can be written as ℚ-linear combinations of n-th “pure” partial derivatives \frac{∂^{n}}{∂s′_{i}^{n}} L_{p}(E, \underline{0}) with respect to other choices of independent ℤ_{p}-extensions of F (cf. the proof of proposition 3.9). Theorem 4.14 immediately implies the first part (38) of the conjecture:

Corollary 4.16. Let E be a p-ordinary modular elliptic curve over F. Let n be the number of places 𝔭|p at which E has split multiplicative reduction. Then we have

ord_{\underline{s}=\underline{0}} L_{p}(E, \underline{s}) ≥ n.

In future work, we hope to also establish formula (39) for a class of non-totally-real number fields.

References

[BL95] L. Barthel, R. Livne: Modular Representations of GL_{2} of a Local Field: The Ordinary, Unramified Case. Journal of Number Theory 55, 1-27 (1995).

[By98] J. Bygott: Modular Forms and Modular Sumbols over Imaginary Quadratic Fields. PhD thesis, 1998.

[Br82] K. Brown: Cohomology of Groups. Graduate Texts in Mathematics, Springer-Verlag, 1982.

[Bu98] D. Bump: Automorphic Forms and Representations. Cambridge University Press, 1998.

[CKM04] J. Cogdell, H. Kim, M. Murty: Lectures on automorphic L-functions. Fields Institute Monographs, 20. American Mathematical Society, Providence, RI, 2004.

[De13] H. Deppe: p-adic L-functions of automorphic forms, Dissertation (Ph.D. thesis) at Bielefeld University, 2013,

http://pub.uni-bielefeld.de/publication/2629389.

[DLMF] NIST Digital Library of Mathematical Functions,

http://dlmf.nist.gov/.

[Ge75] S. Gelbart: Automorphic forms on adele groups. Princeton University Press and University of Tokyo Press, 1975.

[GS93] R. Greenberg, G. Stevens: p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111 (1993), 407-447.

734 Holger Deppe

[Har87] Sh. Haran: p-adic L-functions for modular forms. Compositio Math. 62 (1987), no. 1, 3146.

[Ha87] G. Harder: Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89 (1987), 37-118.

[Hi09] H. Hida: L-invariants of Tate curves. Pure Appl. Math. Q. 5, 1343-1384 (2009)

[JL70] H. Jacquet; R. P. Langlands: Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970.

[Kud04] St. Kudla: From Modular Forms to Automorphic Representations. In: Bump, Cogdell et al.: An Introduction to the Langlands Program, Birkhäuser Boston, 2004.

[Kur77] P. F. Kurcanov: The zeta-function of elliptic curves over certain abelian extensions of imaginary quadratic fields. English translation in: Math. USSR-Sb. 31 (1977), no. 1, 49-62 (1978).

[Kur78] P. F. Kurcanov: Cohomology of discrete groups and Dirichlet series connected with Jacquet-Langlands cusp forms. English translation in: Math. USSR Izvestija, Vol 12 (1978), No. 3, 543-555.

[MTT86] B. Mazur, J. Tate and J. Teitelbaum: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84 (1986), 1-48.

[Neu92] J. Neukirch: Algebraische Zahlentheorie. Springer-Lehrbuch, 1992.

[Se71] J.-P. Serre: Cohomologie des groupes discrets. Ann. of Math. Studies 70 (1971), 77-169.

[Se80] J.-P. Serre: Trees. Springer Monographs in Mathematics, 1980.

[Si86] J. Silverman: Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Springer-Verlag, 1986.

[Sp14] M. Spieß: On special zeros of p-adic L-functions of Hilbert modular forms. Invent. Math. (2014), 196: 69-138.

[Wa82] L.C. Washington: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, Springer-Verlag, 1982.

[We71] A. Weil: Dirichlet Series and Automorphic Forms. Lecture Note in Mathematics, Springer-Verlag, 1971.

Holger Deppe

Mathematical Institute University of Cologne Weyertal 86-90 50931 Cologne Germany