Draft: February 14, 2005

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS OF

REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS

Matthew Emerton

Northwestern University

Abstract. In the first half of the paper we introduce the notion of the universal unitary completion of a continuous representation of a p-adic reductive group on a locally convex p-adic vector space, and prove that such a completion exists under appropriate hypotheses. The problem of studying unitary completions has been raised by Breuil in connection with his work on a possible “p-adic local Langlands correspondence” for GL_{2}, and we relate our construction to certain conjectures of Breuil for the group GL_{2}(ℚ_{p}). In particular, we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction, provided that the character being induced satisfies a “non-critical slope” condition. (See Proposition 2.5 below.)

In the second half of the paper we consider a certain unitary Banach space representation of GL_{2}(ℚ_{p}) obtained by p-adically completing the cohomology of classical modular curves. The mere existence of this representation implies that those locally algebraic parabolically induced representations of GL_{2}(ℚ_{p}) that arise from classical finite slope newforms have a non-trivial universal unitary completion (verifying a conjecture of Breuil for these representations), while applying Proposition 2.5 in this context enables us to give a new construction of p-adic L-functions attached to p-stabilized newforms of non-critical slope. Combining our construction with a representation theoretic definition of L-invariants due to Breuil [5], we are able to give a simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture (in terms of Breuil’s definition of the L-invariant).

The object of this note is two-fold. In its first half we consider the following problem in the representation theory of p-adic groups: When does a topologically irreducible continuous representation of a p-adic reductive group G on a p-adic topological vector space (perhaps satisfying some additional hypotheses) admit an embedding into a unitary representation of G on a p-adic Banach space? If G = GL_{2}(ℚ_{p}) and the representation considered is admissible locally algebraic, this problem has been raised by Breuil in connection with his ideas on a “p-adic Local Langlands correspondence for GL_{2}(ℚ_{p})” [4, 5].

In the second half of the paper we explain a representation theoretic point of view on p-adic L-functions attached to modular forms. As an application, we prove a version of the Mazur-Tate-Teitelbaum exceptional zero conjecture [16, p. 46], using the representation theoretic definition of the L-invariant given by Breuil in [5] (which provides a reinterpretation of the definition given by Darmon [9] and Orton [18] in the weight two and higher weight cases respectively).

The two halves of the paper are more closely related than they might appear at first glance. Indeed, the local tools discussed in the first half play a vital role in the

The author would like to acknowledge the support of the National Science Foundation (award number DMS-0401545)

argument of the second half. Conversely, the global results of the second half shed some light on the local problems discussed in the first half.

Let us now describe the contents of the note in more detail. In Section 1 we present a general framework in which to consider the problem of unitary completions of G-representations. In particular, given a G-representation on a reflexive Fréchet space V that admits a generating semi-norm (in the sense of Definition 1.13) we construct an associated G-representation V^{b}, equipped with a continuous Gequivariant injection V^{b} → V, whose strong dual is the universal unitary completion of the contragredient G-representation on the strong dual V′_{b}. (See Lemma 1.11.) In the case when V is dual to a finitely generated admissible locally algebraic representation π of G, we observe (Proposition 1.17) that V^{b} is dual to the completion π̂ of π with respect to a finite type lattice. If furthermore π is irreducible and of the form “algebraic tensor absolutely cuspidal”, then we show (Proposition 1.18) that π̂ is non-zero if and only if the central character of π is unitary.

In Section 2 we restrict to the case G = GL_{2}(ℚ_{p}), and consider irreducible admissible locally algebraic representations π whose smooth factor is principal series or special. In this case Breuil [4] has made a precise conjecture as to whether or not π̂ is non-zero (see Conjecture 2.2). Briefly, this conjecture states the only possible obstruction to π̂ being non-zero is presented by the exponents appearing in the Jacquet module J_{B}(π) (where B is a Borel subgroup of G). (This conjecture is now known to be a theorem in many cases, thanks to recent work of Berger and Breuil [2] and Colmez [8].)

Associated to the admissible locally algebraic representation π one can construct, via locally analytic parabolic induction, an admissible locally analytic representation π^{la} that contains it. (There is one choice of π^{la} in the special case, while there are two choices in the principal series case.) Under a “non-critical slope” hypothesis, we show (Proposition 2.5) that the natural G-equivariant map π → π̂ extends uniquely to a G-equivariant map π^{la} → π̂. The remainder of Section 2 elaborates on the relation of Proposition 2.5 to Conjecture 2.2, and to certain other ideas and results of [4]. The proof of Proposition 2.5 is given in Section 3.

Suppose now that the locally algebraic representation π of GL_{2}(ℚ_{p}) under consideration is of the form W_{k−2} ⊗π_{p}, where W_{k−2} is the contragredient to the (k −2)nd symmetric power of the standard representation of GL_{2}, and π_{p} is the local factor at p of a newform f of weight k. In this case one can use global considerations to verify Conjecture 2.2 for π. Indeed, in Section 4 we recall the construction of a unitary Banach space representations H̃_{c}^{1}(K^{p}) of GL_{2}(ℚ_{p}) (where K^{p} is the tame level of the newform f giving rise to π), which has the property that π embeds into H̃_{c}^{1}(K^{p}). (In fact we get two embeddings, unique up to multiplication by a scalar, one for each choice of sign; see Proposition 4.3.) Since π̂ is the universal unitary completion of π, the existence of this embedding certainly implies that π̂ ≠ 0. Furthermore, assuming that we can find a p-stabilized newform g attached to f (the newform giving rise to π) that is of non-critical slope, Proposition 2.5 shows that the embeddings of π into H̃_{c}^{1}(K^{p}) extend to embeddings of π^{la} into this space (where the choice of π^{la} is adapted to the choice of g). Using these embeddings, we are able to give a new construction of the p-adic L-function attached to g (Proposition 4.9).

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS 3

In [5], Breuil has made a detailed investigation of the closure of the image of π in H̃_{c}^{1}(K^{p}), in the case when π_{p} is special. In particular, he has shown that this image is classified by a certain invariant L (the L-invariant of Darmon and Orton discussed above). In Section 5 we give a proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture for such f using this description of the L-invariant, and the viewpoint on p-adic L-functions developed in Section 4.

Acknowledgments. I would like to thank Christophe Breuil for helpful correspondence in which he explained the beautiful ideas underlying his preprints [4, 5], as well as a referee for suggestions that improved the discussion of Section 1. Finally, I would like to thank Wilfred Schmid for a wonderful lecture he gave at Northwestern in the Fall of 2002, in which he explained his approach to the study of classical automorphic L-functions via automorphic distributions [19]. It is this lecture that inspired the construction of p-adic L-functions provided by Proposition 4.9.

1. Unitary completions

In this section we generalize and axiomatize some constructions from [4, 5]. Throughout this section, we fix a p-adic reductive group G – more precisely, G

is the group of ℚ_{p}-valued points of a connected, reductive, linear algebraic group defined over ℚ_{p}. We also fix a p-adic local field K that will serve as our field of coefficients. All locally convex topological vector spaces will be assumed to be defined over K (whether or not this is explicitly stated). We let O_{K} denote the ring of integers in K. We refer to [20] for an explanation of the terminology from p-adic functional analysis that we use.

As in [4], a G-representation on a K-Banach space U will be called unitary if the topology of U may be defined by a G-invariant norm.

Definition 1.1. Let V be a locally convex topological K-vector space equipped with a continuous G-action, and let U be a K-Banach space equipped with a unitary G-action. We say that a given continuous K-linear G-equivariant continuous map V → U realizes U as a universal unitary completion of V if any continuous Klinear G-equivariant map V → W, where W is a K-Banach space equipped with a unitary G-action, factors uniquely through the given map V → U.

The usual arguments with universal properties shows that U (equipped with its map V → U) is unique up to unique isomorphism. We will denote U by V̂ if it exists, and (by abuse of language) sometimes refer to V̂ as the universal unitary completion of V. (There should be no confusion with the traditional notation for the completion of V, since all V that we have cause to consider will be complete.) We will refer to the universal continuous G-equivariant map

(1.2)  V → V̂

as the canonical map from V to V̂. If we take W in Definition 1.1 to be the closure of the image of V under the canonical map, and apply the universal mapping property of V̂, then we find that W = V̂, and thus that the canonical map (1.2) has dense image.

It is obvious that the formation of V̂ (when it exists) is functorial in V.

Lemma 1.3. The G-representation V admits a universal unitary completion if and only if the set of commensurability classes of G-invariant open lattices^{1} in V, ordered by inclusion, contains a minimal element (i.e. if V contains a G-invariant open lattice that is absorbed by any other G-invariant open lattice).

1In the sense used in [20] – i.e. a lattice in a K-vector space is a spanning O_{K}-submodule.

Note that, with this usage, a lattice need not be separated.

Proof. This is clear. Briefly, if V does admit a universal unitary completion V̂, then we take the required lattice to be the preimage under the canonical map of the unit ball of V̂ (with respect to some G-invariant norm defining the topology of V̂). Conversely, given such a lattice L, the completion of V with respect to the gauge of L is a universal unitary completion of V. □

It is quite possible for a G-representation on a non-zero convex space V to admit the zero space as its universal unitary completion. Indeed, it is a non-trivial question in general to determine if V̂ is non-zero. Proposition 1.5 below gives a useful necessary and sufficient condition for this to be the case.

Let C(G, K) denote the space of continuous K-valued functions on G, equipped with its right regular G-action, and let C^{b}(G, K) denote the G-invariant subspace of C(G, K) consisting of bounded functions. The sup norm makes C^{b}(G, K) a unitary Banach G-representation.

Lemma 1.4. If U is a non-zero unitary Banach G-representation, then there is a non-zero continuous G-equivariant map U → C^{b}(G, K) (where the target is endowed with the topology induced by sup-norm).

Proof. Let u′ be a non-zero element of the dual space U′. By Frobenius reciprocity, u determines a non-zero continuous G-equivariant map U → C(G, K); explicitly, an element u ∈ U maps to the function f_{u} : g ↦ ⟨u′, gu⟩ on G. If the G-action on U is unitary, one sees that this map factors through the inclusion C^{b}(G, K) ⊂ C(G, K), and that the resulting non-zero map U → C^{b}(G, K) is continuous. □

Proposition 1.5. If V is a convex K-vector space equipped with a continuous Gaction that admits a universal unitary completion, then V̂ is non-zero if and only if there is a non-zero continuous G-equivariant map V → C^{b}(G, K).

Proof. If such a map exist, then it must factor through (1.2), forcing V̂ to be non-zero. Conversely, if V̂ ≠ 0, then apply Lemma 1.4 to V̂ to obtain a non-zero continuous G-equivariant map V̂ → C^{b}(G, K). Composing this with (1.2) we obtain the required non-zero continuous G-equivariant map V → C^{b}(G, K). □

Suppose now that V is topologically irreducible as a G-representation. If V̂ exists and is non-zero, then the map (1.2) is non-zero (since it has dense image) and Gequivariant, and so must be injective (since its source is topologically irreducible and its target is Hausdorff). Pulling back a G-invariant norm from V̂ to V, we see that V admits a G-invariant norm. In particular, if V has a central character, then we see that this central character must be unitary.

The Jacquet module functors defined in [11] provide additional necessary conditions for V̂ (when it exists) to be non-zero. We suppose that V is a compact type locally analytic G-representation for which V̂ exists. Let P be a parabolic subgroup of G, with unipotent radical N and Levi quotient M, and let J_{P} (V) denote the Jacquet module (with respect to P) of V, as defined in [11, §3.4] – it is a compact type convex K-space equipped with a locally analytic action of M. Let N_{0} be some compact open subgroup of N, let Z_{M} denote the center of M, and write Z_{M}^{+} := {z ∈ Z_{M} | zN_{0}z^{−1} ⊂ N_{0}}. Let δ denote the modulus character of P (which is trivial on N, and so induces a character of M = P/N, which we also denote by δ; concretely, δ(m) = [mN_{0}m^{−1} : N_{0}]).

Lemma 1.6. Let χ be a locally analytic K-valued character of Z_{M}. If, in the above situation, the χ-eigenspace of J_{P} (V) is non-zero, and if V̂ ≠ 0, then | χ(z)δ^{−1}(z) |≤ 1 for all z ∈ Z_{M}^{+}.

Proof. As already noted, the assumption that V̂ ≠ 0 implies that V admits a G-invariant norm. The lemma thus follows from [11, Prop. 3.4.9, Lem. 4.4.2]. □

We now return to the general situation, focusing on the case when V is reflexive.

Lemma 1.7. If V is a reflexive convex K-space equipped with a continuous Grepresentation, then the set of commensurability classes of G-invariant open lattices in V contains a minimal element if and only if the set of commensurability classes of G-invariant closed and bounded (equivalently compact, since V is reflexive [20, Prop. 15.3 (iii)]) O_{K}-submodules of V′_{b} contains a maximal element. (We equip V′_{b} with the contragredient G-action.)

Proof. This follows immediately from the theory of bipolarity [20, §13]. □

In the context of Lemma 1.7, the G-action on V′_{b} is again continuous, and so we may reverse the roles of V and V′_{b}. If V′_{b} admits a universal unitary completion, then Lemma 1.7 shows that V admits a compact G-invariant O_{K}-submodule B that is maximal (up to commensurability); equivalently, B absorbs any G-invariant bounded O_{K}-submodule of V.

Definition 1.8. In the preceding situation, define V^{b} := K ⊗_{O_{K}} B, equipped with the finest convex topology for which the natural map B → V^{b} is continuous. (Since any two choices of B are commensurable, we see that V^{b} is well-defined, up to natural isomorphism, independent of the choice of B.)

The G-action on B induces a continuous G-action on V^{b}. The formation of V^{b} (when it exists) is evidently functorial in V.

We have the canonical map

(1.9)  V′_{b} → (V′_{b})ˆ ,

while the inclusion of B in V induces a G-equivariant continuous K-linear map

(1.10)  V^{b} → V.

Lemma 1.11. Let V be a reflexive convex K-space equipped with a continuous G-action for which V′_{b} admits a universal unitary completion.

(i) The map (1.10) is continuous and injective.

(ii) The maps (1.9) and (1.10) are mutually dual.

Proof. Claim (i) is immediate. To prove (ii), consider the strong dual of (1.10):

(1.12)  V′_{b} → (V^{b})′_{b}.

The results of [22] show that (V^{b})′_{b} is a Banach space, with unit ball the polar of B, and that V^{b} is naturally identified (via double duality) with ((V^{b})′_{b})′. Thus (1.10) and (1.12) are mutually dual. The preimage of the unit ball of (V^{b})′_{b} under (1.12) is the polar of B in V′_{b}, which by construction of B (and the theory of bipolarity) coincides with the preimage of the unit ball of (V′_{b})ˆ under (1.9). Thus (1.12) and (1.9) are naturally isomorphic. □

Definition 1.13. If V is a K-Fréchet space equipped with a continuous G-action, then a continuous semi-norm q on V is called a generating semi-norm if the set of translates gq of q (as g ranges over all elements of G) serves to define the topology of V. (Here gq(v) := q(g^{−1}v) for all v ∈ V.)

This definition could be applied to arbitrary locally convex K-spaces equipped with a continuous G-action. However, if G_{0} is a compact open subgroup of G, then any continuous semi-norm on V is bounded above by a G_{0}-invariant continuous semi-norm [10, Lem. 6.5.4]. Thus if V admits a generating semi-norm q, we may assume that q is G_{0}-invariant, and so conclude (since G/G_{0} is countable) that the topology of V is definable by a countable set of semi-norms. Hence V is metrizable. If V is furthermore reflexive, then it is necessarily Fréchet [3, Cor., p. IV.23].

Proposition 1.14. If V is reflexive K-Fréchet space equipped with a continuous G-action that admits a generating semi-norm, then V′_{b} admits a universal unitary completion (and so V^{b} is defined).

Proof. Let q be a generating semi-norm, and set B_{q} = {v ∈ V | q(gv) ≤ 1 for all g ∈ G}. By definition B_{q} is a bounded G-invariant O_{K}-submodule of V. Since the G-action on V is continuous, B_{q} is evidently also closed. If B is any bounded G-invariant O_{K}-submodule of V, and we choose a ∈ K^{×} so that | s(v) | ≤ | a | for all v ∈ B, then clearly a^{−1}B ⊂ B_{q}. Thus B_{q} absorbs all bounded G-invariant O_{K}-submodules of V, and the proposition follows from Lemmas 1.3 and 1.7. □

Lemma 1.15. If V is a K-Fréchet space equipped with a continuous G-action and W is a closed G-invariant subspace of V, then the restriction of any generating semi-norm on V to W is a generating semi-norm on W.

Proof. This follows from the fact that any set of continuous semi-norms on V that defines the topology of V restricts to a set of continuous semi-norms on W that defines the topology of W (by definition of the subspace topology on W). □

Example A. We will show that universal unitary completions always exist for finitely generated G-representations equipped with their finest convex topology.

Definition 1.16. If V is a K-vector space equipped with a representation of G, then we say that a G-invariant lattice in V is of finite type if it is finitely generated as an O_{K}[G]-module.

Clearly V contains a finite type lattice if and only if it is finitely generated (as a K[G]-module). Furthermore, any two finite type lattices are then commensurable.

Proposition 1.17. If V is a finitely generated G-representation equipped with its finest convex topology, then the completion of V with respect to any finite type lattice of V is a universal unitary completion of V.

Proof. This follows from Lemma 1.3, the fact that any lattice in V is open (since V is equipped with its finest convex topology), and the fact that any G-invariant lattice in V contains a finite-type sublattice (since V is finitely generated). □

Recall that a locally algebraic representation V of G is called admissible if it is admissible as a locally analytic representation of G; see [10, Def. 6.3.9] for example. Such a representation V is necessarily equipped with its finest convex topology [10, Cor. 6.3.7]. If V is irreducible, then it admits a central character, and as remarked above, a necessary condition for V̂ ≠ 0 is that this central character be unitary.

Proposition 1.18. Let V be an irreducible admissible locally algebraic representation of G, and write V ⟶^{∼} U ⊗W, where U is an admissible smooth representation of G and W is a finite dimensional algebraic representation of G (as we may, by [10, Prop. 6.3.11]). If U is absolutely cuspidal, then V̂ ≠ 0 if and only if the central character of V is unitary. (Recall that U is said to be absolutely cuspidal if it cannot be embedded into the induction from P of a smooth M-representation, for any proper parabolic subgroup P of G with Levi factor M [6, §5].)

Proof. The “only if” direction has already been noted (and does not require U to be absolutely cuspidal). Suppose now that the central character of V is unitary and that U is absolutely cuspidal. If we fix non-zero elements ǔ and W̌ in the smooth dual to U and the dual to W respectively, then ǔ ⊗ w̌ is in the topological dual to V. By Frobenius reciprocity, the dual vector ǔ ⊗ w̌ induces a G-equivariant map

(1.19)  V → C(G, K);

explicitly, v ∈ V maps to the function f_{v} : g ↦ ⟨ǔ ⊗ w̌, gv⟩. Note that (1.19) is non-zero, since ǔ ⊗ w̌ ≠ 0.

Under the hypothesis of the proposition, we claim that (1.19) factors through the inclusion of C^{b}(G, K) into C(G, K), and thus induces a map V → C^{b}(G, K) (which is necessarily continuous, since V is equipped with its finest convex topology). Indeed, it suffices to show that f_{v} is bounded when v is of the form u ⊗ w for some u ∈ U and w ∈ W. In this case we see that f_{v}(g) = ⟨ǔ, gu⟩⟨w̌, gw⟩ for all g ∈ G. Since U is absolutely cuspidal, the matrix coefficient ⟨ǔ, gu⟩ is compactly supported modulo the center of G [6, Thm. 5.3.1]. Thus the same is true of f_{v}. Since V has a unitary central character, we conclude that f_{v} is bounded, as claimed. The proposition now follows from Proposition 1.5. □

It is immediate from the definition of absolutely cuspidal, together with [11, Prop. 4.3.4 (ii)], that the smooth factor of an irreducible admissible locally algebraic representation V is absolutely cuspidal if and only if J_{P} (V) = 0 for all proper parabolic subgroups P of G. The preceding proposition thus shows that in the extreme case when V has no non-vanishing Jacquet modules, the necessary condition of Lemma 1.6 is also sufficient.^{2}

If the smooth factor of V is not absolutely cuspidal, then the question of whether or not V̂ is non-zero is more subtle (as the necessary condition of Lemma 1.6 shows). This question, in the case when G = GL_{2}(ℚ_{p}), will be considered in detail in the next section.

Example B. In [4, 5], the spaces O(k) and O(k, L) are defined (for k ≥ 2 and L ∈ K). The first is the space of rigid analytic functions on the p-adic upper halfplane Ω := ℂ_{p} \ ℚ_{p}, equipped with a “weight k” action of GL_{2}(ℚ_{p}). The second is the space of log-rigid analytic functions on Ω, equipped with a “weight 2 − k” action of GL_{2}(ℚ_{p}); one takes the branch of the p-adic logarithm to be log_{L}, where log_{L}(p) = L. These are nuclear (and so reflexive) Fréchet spaces equipped with continuous actions of GL_{2}(ℚ_{p}), and for fixed k and L, differentiating k − 1 times induces a short exact sequence

(1.20)  0 → W_{k−2}⊗_{K} | det |^{−(k−2)/2}→ O(k, L) → O(k) → 0;

2As a referee has emphasized, in the context of the Proposition 1.18 it is easy to see that V̂ is not an admissible Banach space representation of G. In light of Breuil’s ideas regarding “p-adic Local Langlands”, one expects V̂ to admit a rich family of admissible quotients.

here W_{k−2} is the contragredient to the (k − 2)nd symmetric power of the standard two dimensional representation of GL_{2}(ℚ_{p}), embedded (with the indicated twist) into O(k, L) as the subspace of polynomial functions on Ω of degree at most k − 2.

One easily checks that if U is a covering of ℚ_{p} in ℂ_{p} satisfying condition (20) of [4, p. 34], then the formula q(F) := ||F_{|Ω_{U}} || (using the notation of [4]) defines a generating semi-norm (in fact norm) on each of O(k) and O(k, L), and thus that O(k)^{b} and O(k, L)^{b}, as defined above, coincide with the same spaces as defined in [4] and [5]. For k > 2 and any choice of L, the map

O(k, L)^{b} → O(k)^{b}

induced by the surjection of (1.20) is a closed embedding;^{3} when k = 2 it is surjective (with kernel equal to W_{0} ⟶^{∼} K).

Example C. Suppose that G is quasi-split, and let B be a Borel subgroup of G. Let N denote the unipotent radical of B, and fix a maximal torus T of G contained in B, so that B = T N. Let \overline{B} be the opposite Borel to B, chosen so that B ⋂ \overline{B} = T. Let χ : T → K^{×} be a locally analytic character of T, which we also regard as a character of \overline{B} via the natural projection \overline{B} → T, and consider the locally analytic parabolic induction

IndG ; \overline{B} χ := \{f ∈ C^{la}(G, K) | f(\overline{b}g) = χ(\overline{b})f(g) for all \overline{b} ∈ \overline{B}, g ∈ G\}.

The right regular G-action endows IndG ; \overline{B} χ with a continuous action of G.

Proposition 1.21. The G-representation IndG ; \overline{B} χ is a strongly admissible locally analytic representation of G (in the sense of [21]), whose strong dual (which is thus a nuclear, and hence reflexive, Fréchet space) admits a generating semi-norm.

Proof. If G_{0} is a hyperspecial maximal compact open subgroup of G, then G = \overline{B}G_{0}, and so restriction of functions from G to G_{0} induces a G_{0}-equivariant closed embedding IndG ; \overline{B} χ → C^{la}(G_{0}, K). Since C^{la}(G_{0}, K) is a strongly admissible G_{0}- representation (essentially by definition; see the discussion following [10, Def. 6.2.1], for example), we see that the same is true of its closed subrepresentation IndG ; \overline{B} χ. Since a locally analytic G-representation is strongly admissible if and only if it is strongly admissible as a G_{0}-representation (again by definition), we see that IndG ; \overline{B} χ is a strongly admissible locally analytic G-representation. In particular, it is of compact type, and so its strong dual is a nuclear Fréchet space.

Let (X, 𝕏, ϕ) be a chart centered at the identity of the p-adic analytic group N; that is, X is a compact open neighbourhood of the identity of N, 𝕏 is an affinoid rigid analytic space over ℚ_{p} isomorphic to a closed polydisk, and ϕ is a locally analytic isomorphism ϕ : X ⟶^{∼} 𝕏(ℚ_{p}). To save writing, we will also write simply 𝕏 to denote this chart. As in [10, §2.1], we let C^{an}(𝕏, K) denote the Banach space of rigid analytic K-valued functions on 𝕏; the isomorphism ϕ allows us to see C^{an}(𝕏, K) as a BH-subspace of C^{la}(X, K) (in the sense of [10, Def. 1.1.4]), and thus also of C_{c}^{la}(N, K) (since C^{la}(X, K) is a closed subspace of C_{c}^{la}(N, K)). If t ∈ T, then conjugation by t induces a locally analytic automorphism of N, which we denote by σ_{t}. Thus (tXt^{−1}, 𝕏, ϕ ∘ σ_{t^{−1}}) is also a chart of N. To save writing, we will denote this chart simply by t𝕏t^{−1}, and will denote the corresponding BH-subspace of C_{c}^{la}(N, K) by C^{an}(t𝕏t^{−1}, K). We remark that the theory of roots shows that the set of neighbourhoods of the identity {tXt^{−1}}_{t∈T} is cofinal in the set of all neighbourhoods of the identity.

3That one obtains an injection follows from the fact that W_{k−2}⊗_{K} | det |^{−(k−2)/2} admits no GL_{2}(ℚ_{p})-invariant norm, when k > 2. That this injection is in fact a closed embedding is proved in [5, Thm. 3.3.3].

The discussion of [13, §5] shows that C_{c}^{la}(N, K) embeds as a closed subspace of IndG ; \overline{B} χ, and hence that C^{an}(𝕏, K) embeds as a BH-subspace of IndG ; \overline{B} χ. One checks that if t ∈ T, then

(1.22)  tC^{an}(𝕏, K) = C^{an}(t𝕏t^{−1}, K).

(Here the action of t on the left-hand side is defined via the action of t on the ambient G-representation IndG ; \overline{B} χ.) The final sentence of the preceding paragraph, together with (1.22) and the fact that G acts transitively on the space \overline{B}\G, shows that IndG ; \overline{B} χ = ∑_{g∈G} gC^{an}(𝕏, K). Thus the gauge of the polar in (IndG ; \overline{B} χ)′_{b} of the unit ball C^{an}(𝕏, O_{K}) of C^{an}(𝕏, K) is a generating semi-norm for (IndG ; \overline{B} χ)′_{b}. □

2. Completing parabolically induced representations of GL_{2}(ℚ_{p}).

Take G = GL_{2}(ℚ_{p}), and K as in Section 1. Let B denote the standard upper triangular Borel in GL_{2}(ℚ_{p}), let T denote the diagonal maximal torus in B, and let \overline{B} denote the lower triangular Borel (so that \overline{B} ⋂ B = T). Let χ be a locally algebraic K-valued character of T of the form χ = θψ_{k−2}, for some k ≥ 2, where θ is smooth and ψ_{k−2} is the highest weight of the irreducible representation W_{k−2} of G (the contragredient to the (k−2)nd symmetric power of the standard representation

of G) with respect to B. (Explicitly, ψ_{k−2} : ( \begin{matrix}a \\ & d \end{matrix} ) ↦ d^{−(k−2)}.) Let I(χ) denote

the locally algebraic induction of χ from \overline{B} to G; alternatively, I(χ) is the tensor product over K of W_{k−2} and the smooth induction of θ from \overline{B} to G.

Write θ in the form θ : ( \begin{matrix}a \\ & d \end{matrix} ) ↦ θ_{1}(a)θ_{2}(d), for two smooth characters θ_{1} and

θ_{2} of ℚ_{p}^{×}. Write α = θ_{1}(p) and β = pθ_{2}(p). The central character of I(χ) takes

( \begin{matrix}p \\ & p \end{matrix} ) to αβp^{−(k−1)}.

Let θ′ ( \begin{matrix}a \\ & d \end{matrix} ) =| d/a | θ_{2}(a)θ_{1}(d), and define χ′ to be the locally algebraic

character χ′ = θ′ψ_{k−2}. (Note that the α and β for χ′ are equal to the β and α for χ.)

If θ_{1|ℤ_{p}^{×}} = θ_{2|ℤ_{p}^{×}} and α/β = p^{±1}, then we say that we are in the special case. In this case, the representation I(χ) is reducible. If α/β = p^{−1}, then I(χ) contains a unique finite dimensional subrepresentation, isomorphic to a twist of W_{k−2}, and the quotient is isomorphic to a twist of W_{k−2} tensored with St (the Steinberg representation), which we denote by π(χ). If α/β = p, then I(χ) has a unique finite dimensional quotient, isomorphic to a twist of W_{k−2}, and the kernel of the projection onto this quotient is isomorphic to π(χ′). From now on, when we are in the special case we always assume that χ is chosen so that α/β = p^{−1}.

If α/β ≠ p^{±1}, then we say that we are in the principal series case. In this case, the representation I(χ) is irreducible, and we write π(χ) = I(χ). The theory of intertwining operators shows that in this case there is an isomorphism π(χ) ⟶^{∼} π(χ′).

We let π̂(χ) denote the completion of π(χ) with respect to a finite type lattice. Proposition 1.17 shows that π̂(χ) is the universal unitary completion of π(χ).

Lemma 2.1. The following two conditions are necessary in order for π̂(χ) to be non-zero:

(i) The central character of π(χ) is unitary; that is, | αβ |=| p^{k−1} | .

(ii) Each of α and β lie in O_{K}.

Proof. It was observed in Section 1 that (i) is a necessary condition for π to admit a non-zero universal unitary completion. In the special case, since α/β = p^{−1}, it is clear that (i) implies (ii). Thus we assume for the remainder of the proof that we are in the principal series case. We will compute J_{B}(I(χ)), and apply Lemma 1.6.

Since I(χ) ⟶^{∼} (IndG ; \overline{B} θ)_{sm} ⊗_{K} W_{k−2} (where the left-hand factor is the smooth parabolic induction of θ from \overline{B} to G), it follows from [11, Props. 4.3.4 (ii), 4.3.6] that there is an isomorphism of T -representations J_{B}(I(χ)) ⟶^{∼} J_{B}((IndG ; \overline{B} θ)_{sm}) ⊗ ψ_{k−2}. The theory of Jacquet modules of smooth representations [6] shows that J_{B}((IndG ; \overline{B} θ)_{sm}) is two dimensional, and that it has a non-zero θδ-eigenspace, and also a non-zero θ′δ-eigenspace. (Below we will give a direct proof that these eigenspaces are non-zero.) Thus J_{B}(I(χ)) has non-zero χδ- and χ′δ-eigenspaces.

If we take the compact open subgroup N_{0} of Lemma 1.6 to be \{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) | x ∈

ℤ_{p}\}, then we see that the matrix t = ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) conjugates N_{0} into itself. Applying

Lemma 1.6 twice, with the character labelled χ in the statement of that result taken to be χδ and χ′δ in turn, we find that | α |=| χ(t) | ≤ 1 and also that | β |=| χ′(t) | ≤ 1. This proves the lemma.

We finish with a direct proof that the θδ-eigenspace of J_{B}((IndG ; \overline{B} θ)_{sm}) is nonzero. Let N denote the unipotent radical of B, and let K(θ) denote K regarded as a T -module via the character θ. The discussion of [13, §5] shows that there is a natural B-equivariant embedding C_{c}^{sm}(N, K(θ)) → IndG ; \overline{B} θ. (The target denotes the locally analytic parabolic induction of θ from \overline{B} to G.) Since θ is a smooth character of T, this embedding factors through the inclusion (IndG ; \overline{B} θ)_{sm} → IndG ; \overline{B} θ of the smooth parabolic induction into the locally analytic parabolic induction. Applying the left-exact functor J_{B}, and taking into account [11, Lemma 3.5.2], we see that the θδ-eigenspace of J_{B}((IndG ; \overline{B} θ)_{sm}) is non-zero. The theory of intertwining operators provides an isomorphism (IndG ; \overline{B} θ)_{sm} ⟶^{∼} (IndG ; \overline{B} θ′)_{sm}. Replacing θ by θ′ in the preceding argument shows that the θ′δ-eigenspace of J_{B}((IndG ; \overline{B} θ)_{sm}) is also non-zero. □

The following conjecture is due to Breuil [4].

Conjecture 2.2. The necessary conditions of Lemma 2.1 are also sufficient: if these conditions hold, then π̂(χ) is non-zero.

If we take into account the method of proof of Lemma 2.1, one can could rephrase the conjecture as saying that the necessary conditions of Lemma 1.6 are also sufficient; i.e. that the characters appearing in the Jacquet module of π(χ) are the only possible obstruction to its unitarizability.

This conjecture has been proved in the special case by Teitelbaum [23] and Grosse-Klönne [14], and recently in the unramified principal series case by Berger and Breuil [2]. We note that there is one subcase of the principal series case (what one might call the ordinary case) in which the conjecture is trivial to prove. Namely, if α is a p-adic unit, then the character χ is unitary, and so the continuous induction (as considered in [22], for example) I^{cont}(χ) of χ from \overline{B} to G is a unitary Banach space representation of G containing I(χ). (Because χ is unitary, the sup norm over \overline{B}\G is a well-defined G-invariant norm on I^{cont}(χ).) Similarly, if | α |=| p^{k−1} |, then β is a p-adic unit, and we may embed I(χ) ⟶^{∼} I(χ′) in the unitary representation I^{cont}(χ′).

Given χ as above, we let I^{la}(χ) denote the locally analytic induction of χ from \overline{B} to G; by Proposition 1.21 this is a strongly admissible locally analytic representation of G. It contains I(χ) as a closed subrepresentation. We may identify I^{la}(χ) with the space of locally analytic functions f : ℚ_{p} → K, satisfying an appropriate regularity condition at infinity, with the G-action defined by

(2.3)  ( \begin{matrix}a & b \\ c & d \end{matrix} ) f(x) = \frac{θ_{1}(a + cx)}{θ_{2}(a + cx)}(a + cx)^{k−2} \frac{θ_{2}(ad − bc)}{(ad − bc)^{k−2}} f(\frac{b + dx}{a + cx});

I(χ) is then identified with the subspace of functions that are locally polynomial of degree at most k − 2.

In the principal series case we write π^{la}(χ) = I^{la}(χ). In the special case (recall that in this case we assume that α/β = p^{−1}) we let π^{la}(χ) denote the quotient of I^{la}(χ) by the invariant subspace consisting of polynomials of degree at most k − 2. In both cases, the closed embedding I(χ) → I^{la}(χ) induces a closed embedding

(2.4)  π(χ) → π^{la}(χ).

The following result gives some hint as to the structure of π̂(χ), or more precisely, of its locally analytic vectors.

Proposition 2.5. Suppose that χ satisfies the conditions of Lemma 2.1, and that furthermore | α |>| p^{k−1} | . Then the following equivalent conditions hold:

(i) Any G-equivariant K-linear map π(χ) → U, where U is a unitary Banach space representation, extends uniquely to a continuous K-linear G-equivariant map π^{la}(χ) → U.

(ii) The natural map π(χ) → π̂(χ) extends uniquely to a continuous K-linear G-equivariant map π^{la}(χ) → π̂(χ).

(iii) The map (2.4) induces an isomorphism of universal unitary completions

π̂(χ) ⟶^{∼} π̂^{la}(χ).

We give our proof of this result in the following section. In [2] Berger and Breuil give an explicit description of π̂(χ) (as a quotient of a certain parabolic induction of χ – see p. 25, Def. 4.2.4, and Thm. 4.3.1 of [2]), using which they are able to give a direct proof of this result [2, p. 42]. Nevertheless, it seems worthwhile to give our original proof, since it is more representation theoretic in nature than that of [2], and it exhibits an interesting relationship between the classical technique of Amice-Vélu and Vishik and the representation theory of parabolically induced representations.

In the remainder of this section, we elaborate further on Conjecture 2.2, Proposition 2.5, and their relations to some of the ideas and results of [4, 5], in each of the two cases: special and principal series.

The special case. The special case has been discussed extensively by Breuil in his paper [4]. We recall the key points of that discussion.

In the special case, condition (i) of Lemma 2.1 implies condition (ii), and all the π(χ) satisfying (ii) (that are in the special case!) are unitary twists one of the other. The same is true of all the π^{la}(χ), and these representations are all unitary twists of the representation Σ(k) defined in [4]; write π^{la}(χ) = Σ(k) ⊗_{K} ω, for some unitary character ω of GL_{2}(ℚ_{p}). The results of Morita [17] then show that π^{la}(χ) is dual to the representation O(k)⊗_{K} ω^{−1}, where O(k) is the GL_{2}(ℚ_{p})-representation considered in Section 1.

Dualizing the closed embedding (2.4) yields a surjection

O(k) ⊗_{K} ω^{−1} → π(χ)′_{b},

which can be identified with (the twist by ω^{−1} of) the residue map from rigid analytic functions on Ω onto the space of harmonic cocycles on the Bruhat-Tits tree of GL_{2}(ℚ_{p}) considered in [23]. Proposition 2.5 implies that this surjection induces an isomorphism

O(k)^{b} ⊗_{K} ω^{−1} ⟶^{∼} π(χ)_{b}^{′b},

a result originally proved in [23] and [14]. As we noted above, in those papers it is furthermore proved that O(k)^{b} ≠ 0, and thus that Conjecture 2.2 holds in the special case. (Note that this conjecture is obviously invariant under unitary twists).

Breuil has noted [4, Prop. 4.6.5] that when k > 2, the unitary Banach space representation π̂(χ) is not an admissible Banach space G-representation (in the sense of [22]). For each k > 2 and L ∈ K, dualizing the surjection of (1.20), and twisting by ω, yields a closed embedding of strongly admissible locally analytic GL_{2}(ℚ_{p})-representations

π^{la}(χ) → π^{la}(χ, L)

(where the target is defined to be the strong dual of O(k, L) ⊗_{K} ω^{−1}). Dualizing the closed embedding O(k, L)^{b} ⊗_{K} ω → O(k)^{b} ⊗_{K} ω yields a surjection of unitary Banach space GL_{2}(ℚ_{p})-representations

π̂(χ) ⟶^{∼} (O(k)^{b} ⊗_{K} ω^{−1})′ → (O(k, L)^{b} ⊗_{K} ω^{−1})′.

Let π̂(χ, L) denote the target of this surjection. (This is equal to B(k, L) ⊗_{K} ω, where B(k, L) is the unitary Banach space GL_{2}(ℚ_{p})-representation defined in [4].)

Breuil makes the following conjecture [4, Conj. 1.1.8]:

Conjecture 2.6. (i) The natural map π^{la}(χ, L) → π̂(χ, L) identifies the source with the space of locally analytic vectors in the target.

(ii) The unitary Banach space GL_{2}(ℚ_{p})-representation π̂(χ, L) is admissible.

Note that part (i) of this conjecture in particular implies that π̂(χ, L) is nonzero. Together with part (ii), it furthermore implies that π̂(χ, L) is topologically irreducible [4, Thm. 1.1.9]. In fact, Colmez [8] has recently proved that π̂(χ, L) is non-zero, admissible, and topologically irreducible for all L (without, however, proving part (i) of the conjecture).

Thus for each π(χ) in the special case with k > 2, there exists a family of topologically irreducible quotients of π̂(χ) depending on the parameter L. According to Breuil’s p-adic local Langlands philosophy, these representations π̂(χ, L) correspond to those irreducible potentially semi-stable representations of G_{ℚ_{p}} with Hodge-Tate weights 0, k − 1 that are not potentially crystalline. (Such representations are classified by their associated Weil group representation and Hodge-Tate weights, encoded in the character χ, together with their L-invariant, which should be encoded in the parameter L on which π̂(χ, L) depends.)

When k = 2, one may make analogous constructions to those above so as to obtain for each L ∈ K a closed embedding of strongly admissible locally analytic GL_{2}(ℚ_{p})-representations π^{la}(χ) → π^{la}(χ, L) and a map π̂(χ) → π̂(χ, L) of unitary Banach space GL_{2}(ℚ_{p})-representations. In this case, though, this latter map is not a surjection, but rather is a closed embedding (since the map O(2, L)^{b} → O(2)^{b} is not a closed embedding, but is rather a surjection). When k = 2, the analogue of Conjecture 2.6 was proved by Breuil [4, §4.5]; note though that π̂(χ, L) is not topologically irreducible in this case (for example, because it contains π̂(χ) as a closed subrepresentation).

We conclude our discussion of the special case by noting that the space π^{la}(χ, L) has a concrete interpretation as a space of functions [4, Prop. 2.2.1]. As remarked above, we may identify π^{la}(χ) with the space of locally analytic functions on ℚ_{p}, having a pole of order at most k − 2 at infinity, and with the GL_{2}(ℚ_{p})-action being given by the formula (2.3). We may identify π^{la}(χ, L) with the space of locally analytic functions on ℚ_{p}, whose singularity at infinity is at worst the sum of a pole of order k − 2 and a term of the form P (x) log_{L}(x), where P (x) is a polynomial of degree at most k − 2 (with | x |>> 0). The GL_{2}(ℚ_{p})-action is given by a modification of the formula (2.3): if a function f(x) has a logarithmic singularity

of the form P (x) log_{L} x for | x |>> 0, then ( \begin{matrix}a & b \\ c & d \end{matrix} ) f(x) is defined by adding the

following term to the expression in (2.3):

(2.7)  −\frac{1}{2} \frac{θ_{1}(a + cx)}{θ_{2}(a + cx)}(a + cx)^{k−2} \frac{θ_{2}(ad − bc)}{(ad − bc)^{k−2}} P (\frac{b + dx}{a + cx}) log_{L}(\frac{ad − bc}{(a + cx)^{2}} ).

The principal series case. Suppose that π(χ) is in the principal series case, and that neither α nor β is a p-adic unit.

Proposition 2.5, together with the isomorphism π(χ) ⟶^{∼} π(χ′) discussed above, shows that both π^{la}(χ) and π^{la}(χ′) map into π̂(χ). If π^{la}(χ, χ′) denotes the sum of π^{la}(χ) and π^{la}(χ′) amalgamated along their common closed subspace π(χ), then these maps glue to give a map

(2.8)  π^{la}(χ, χ′) → π̂(χ).

The following conjecture is due to Breuil. It is extracted from the introduction to [4], and from a letter from Breuil to the author.

Conjecture 2.9. (i) The map (2.8) identifies the source with the locally analytic vectors in the target.

(ii) The unitary Banach space GL_{2}(ℚ_{p})-representation π̂(χ) is admissible.

Part (i) of this conjecture in particular implies that π̂(χ) ≠ 0, i.e. Conjecture 2.2 for π(χ). An argument analogous to that which proves [4, Thm. 1.1.9] shows that this conjecture also implies the topological irreducibility of π̂(χ). As was noted above, in the case when θ is unramified, Conjecture 2.2 has recently been proved [2]. In this reference Berger and Breuil furthermore prove that for such χ the unitary Banach space representation π̂(χ) is admissible and topologically irreducible (without, however, establishing part (i) of Conjecture 2.9).

According to Breuil’s local Langlands philosophy, the representations π̂(χ) (for χ in the principal series case with α and β both non-units) correspond to those irreducible potentially crystalline representations of G_{ℚ_{p}} for which the associated representation of the Weil group is abelian.^{4}

3. Proof of Proposition 2.5

We first note that Propositions 1.17, 1.21, and 1.14 (together with Lemma 1.15, in the special case) show that each of π(χ) and π^{la}(χ) admit a universal unitary completion. The equivalence of the conditions in the statement of Proposition 2.5 is then clear.

We turn to proving (i). In fact, we will prove the analogous statement in which π(χ) and π^{la}(χ) are replaced by I(χ) and I^{la}(χ). In the principal series case this is simply a change of notation. In the special case, the representations π(χ) and π^{la}(χ) are quotients of I(χ) and I^{la}(χ) respectively by the same finite dimensional subspace, and so the statement for I(χ) and I^{la}(χ) implies the corresponding statement for π(χ) and π^{la}(χ).

We begin by recalling (from Example C of Section 1) that for any character χ = θψ_{k−2} of T as in Section 2,

I^{la}(χ) := \{f ∈ C^{la}(G, K) | f(\overline{b}g) = χ(\overline{b})f(g) for all \overline{b} ∈ \overline{B}, g ∈ G\}.

(As always, we regard χ as a character of \overline{B} via the natural projection \overline{B} → T.) The support of any element of I^{la}(χ) is clearly invariant under left-multiplication by elements of \overline{B}, and so may be identified with a closed subset of the quotient \overline{B}\G. Since this quotient is compact, and since functions in I^{la}(χ) are locally analytic, we see that the support of any element of I^{la}(χ) is a compact open subset of \overline{B}\G. If Ω is an open subset of \overline{B}\G, then we let I^{la}(χ)(Ω) denote the closed subspace of I^{la}(χ) consisting of functions whose support lies in Ω.

Let N denote the unipotent radical of B; we identify N with ℚ_{p} via the map

x ↦ n_{x} := ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) . Write N_{0} := \{n_{x} | x ∈ ℤ_{p}\} ⊂ N. The embedding N → \overline{B}\G

realizes the source as an open subset of the target, and thus realizes N_{0} as a compact open subset of the target.

If for any integer r ≥ 1 we set

G_{0}(r) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℤ_{p}) | c ≡ 0 mod p^{r}\},

4Actually we have omitted the crystalline representation with Hodge-Tate weights 0, k −1 with k > 2 for which a_{p} (the trace of Frobenius) is equal to p^{(k−2)/2} + p^{k/2}, as well as all of its twists.

Breuil’s philosophy accounts for these representations as well, in a rather surprising way – see [4, Rem. 1.3.4].

then G_{0}(r) is a compact open subgroup of G that admits an Iwahori decomposition

with respect to B and \overline{B}. Namely, if we write \overline{N}(r) = \{( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) | x ∈ p^{r}ℤ_{p}\} and

T_{0} := GL_{2}(ℚ_{p}) ⋂ T = \{( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) | a, d ∈ ℤ_{p}^{×}\}, then G_{0}(r) = N_{0}T_{0}\overline{N}(r).

Write T^{+} := \{( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) ∈ T | \frac{a}{d} ∈ ℤ_{p}\}. If t ∈ T^{+}, then one sees that tN_{0}t^{−1} ⊂

N_{0}, while t^{−1}\overline{N}(r)t ⊂ \overline{N}(r) for any r ≥ 1. Taking into account these formulas, the inclusion T_{0} ⊂ T^{+}, and the fact that G_{0}(r) is a subgroup of G for each r ≥ 1, we see that if for any r ≥ 1 we write G^{+}(r) := N_{0}T^{+}\overline{N}(r), then G^{+}(r) is a submonoid of G. Write B^{+} := G^{+}(r) ⋂ B = N_{0}T^{+}; it is a submonoid of B, independent of the choice of r, that generates B as a group.

One sees that \overline{B}N_{0}G^{+}(1) ⊃ \overline{B}N_{0}, and thus that I^{la}(χ)(N_{0}) is a G^{+}(1)-invariant closed subspace of I^{la}(χ). Write I(χ)(N_{0}) := I(χ) ⋂ I^{la}(χ)(N_{0}) (the intersection taking place in the ambient space I^{la}(χ)).

Lemma 3.1. Let U be a K-Banach space equipped with a continuous G-action. For any integer r ≥ 1, the restriction maps L_{G}(I^{la}(χ), U) → L_{G^{+}(r)}(I^{la}(χ)(N_{0}), U) and L_{G}(I(χ), U) → L_{G^{+}(r)}(I(χ)(N_{0}), U) are both isomorphisms.

Proof. Let V be any closed subspace of I^{la}(χ) satisfying the following two conditions: (i) V is G-invariant; (ii) V is local: that is, if F ∈ V and if Ω is a compact open subset of \overline{B}\G, then the restriction F_{|Ω} of F to Ω also lies in V. (The two examples of such a V that we have in mind are I^{la}(χ) itself and I(χ).) Then V (N_{0}) := V ⋂ I^{la}(χ)(N_{0}) is G^{+}(1)-invariant, and we will prove that the restriction map

(3.2)  L_{G}(V, U) → L_{G^{+}(r)}(V (N_{0}), U)

is an isomorphism for any r ≥ 1.

Since the G-translates of N_{0} cover \overline{B}\G, and since V is local, we see that V (N_{0}) generates V as a G-representation. The map (3.2) is thus certainly injective, and we must show that it is also surjective.

Let V (N) denote the B-invariant closed subspace of V consisting of elements whose support is contained in N. Since V is local, the natural map K[N] ⊗_{K[N_{0}]} V (N_{0}) → V (N) is an isomorphism, and hence (taking into account the fact that B^{+} generates B as a group) restriction induces an isomorphism

(3.3)  L_{B}(V (N), U) ⟶^{∼} L_{B^{+}}(V (N_{0}), U).

Let L_{B}(V (N), U)^{∘} denote the subspace of the source of (3.3) consisting of maps ϕ satisfying the following condition:

(3.4) For each \overline{n} ∈ \overline{N}, there exists a neighbourhood Ω of the identity in N such that, for any F ∈ V whose support lies in Ω, the element \overline{n}F lies in V (N), and ϕ(\overline{n}F) = \overline{n}ϕ(F).

Note that L_{G^{+}(r)}(V (N_{0}), U) is a subspace of the target of (3.3).

Claim 3.5. The image of L_{G^{+}(r)}(V (N_{0}), U) under the inverse of the isomorphism (3.3) (which consists of elements of the source whose restrictions to V (N_{0}) are \overline{N}(r)-equivariant) is contained in L_{B}(V (N), U)^{∘}.

Choose t ∈ T^{+} such that \overline{n} ∈ t\overline{N}(r)t^{−1}, and take Ω = tN_{0}t^{−1}. Since tG_{0}(r)t^{−1} is a subgroup of G, we see that tN_{0}t^{−1}, regarded as a compact open subset of \overline{B}\G, is invariant under the action of t\overline{N}(r)t^{−1}. Thus if F is any element of V with support contained in Ω, the same is true of \overline{n}F. In particular, we see that \overline{n}F ∈ V (N), as required.

Now let ϕ be an element of L_{B}(V (N), U) whose restriction to V (N_{0}) is \overline{N}(r)- equivariant. If the support of F is contained in Ω, we compute that

ϕ(\overline{n}F) = tϕ(t^{−1}\overline{n}tt^{−1}F) = t(t^{−1}\overline{n}t)ϕ(t^{−1}F) = \overline{n}ϕ(F).

(The first and third equalities hold because ϕ is B-equivariant by assumption. The second equality holds because t^{−1}F is supported on N_{0}, since F was assumed to be supported on Ω, while t^{−1}\overline{n}t lies in \overline{N}(r), and ϕ is assumed to be \overline{N}(r)-equivariant on V (N_{0}).) This proves the claim.

Claim 3.6. The image of the restriction map L_{G}(V, U) → L_{B}(V (N), U) is equal to L_{B}(V (N), U)^{∘}.

Since any element of L_{G}(V, U) is G^{+}(r)-equivariant when restricted to V (N_{0}), it follows from Claim 3.5 that the restriction to V (N) of any element of L_{G}(V, U) lies in L_{B}(V (N), U)^{∘}. We must show that any element of L_{B}(V (N), U)^{∘} may be extended to an element of L_{G}(V, U).

For g ∈ G, let gV (N) denote the closed subspace of V obtained by translating V (N) by g. If w denotes a representative of the non-trivial element of the Weyl group of G, then since N and Nw together cover \overline{B}\G, and since V is local, the closed embeddings of V (N) and wV (N) into V induce a continuous surjection

(3.7)  V (N) ⨁ wV (N) → V,

which must be strict, since the source and target are of compact type. More generally, the closed embeddings of gV (N) into V for each g ∈ G together induce a continuous K-linear surjection

(3.8)  ⨁_{g∈G} gV (N) → V.

(More canonically, this is the G-equivariant map K[G] ⊗_{K} V (N) → V induced by the inclusion of V (N) into V and the G-action on V.) Since (3.7) is a strict surjection, the same is true of (3.8).

Now fix ϕ ∈ L_{B}(V (N), U)^{∘}, and for each g ∈ G, define ϕ_{g} : gV (N) → U via ϕ_{g}(gF) = gϕ(F). The maps ϕ_{g} induce a continuous K-linear map

(3.9)  ⨁_{g∈G} ϕ_{g} : ⨁_{g∈G} gV (N) → U.

(More canonically, this is the G-equivariant map K[G] ⊗_{K} V (N) → U induced by the map ϕ : V (N) → U and the G-action on U.) We must show that (3.9) factors through (3.8), to yield a map ϕ̃: V → U. The map ϕ̃ will then be a G-equivariant extension of ϕ by construction, and will also be continuous, since (3.8) is strict.

To prove the required factorization, we must show that if (g_{1}F_{1}, …, g_{l}F_{l}) is any finite sequence with g_{i} ∈ G and F_{i} ∈ V (N) such that ∑_{i} g_{i}F_{i} = 0 (the sum taking place in V), then ∑_{i} g_{i}ϕ(F_{i}) =^{?} 0. Since V is local, it suffices to show that each point x of \overline{B}\G has a compact open neighbourhood Ω_{x} such that if Ω′_{x} is a neighbourhood of x contained in Ω_{x} then

(3.10)  ∑_{i} g_{i}ϕ(F_{i|Ω′_{x}g_{i}^{−1}}) =^{?} 0.

Indeed, we can then partition \overline{B}\G into a disjoint union of such neighbourhoods, say \overline{B}\G = ∐_{j=1}^{s} Ω′_{x_{j}}, and writing F_{i} = ∑_{j=1}^{s} F_{i|Ω′_{x_{j}} g_{i}^{−1}}, we conclude that ∑_{i} g_{i}ϕ(F_{i}) =

∑_{i} ∑_{j=1}^{s} g_{i}ϕ(F_{i|Ω′_{x_{j}} g_{i}^{−1}}) = ∑_{j=1}^{s} ∑_{i} g_{i}ϕ(F_{i|Ω′_{x_{j}} g_{i}^{−1}}) = 0, as required.

If x = \overline{B}g for some g ∈ G, then replacing (g_{1}F_{1}, …, g_{l}F_{l}) by (gg_{1}F_{1}, …, gg_{l}F_{l}), we see that it suffices to treat the case when x equals the identity coset, which we identify with the identity e of N, under the open immersion N → \overline{B}\G. If e is not in the domain of g_{i}F_{i}, for some i, then taking the sought-after neighbourhood Ω_{e} of e to be sufficiently small, we will have that (g_{i}F_{i})_{|Ω_{e}} = 0, or equivalently, F_{i|Ω_{e}g_{i}^{−1}} = 0, and hence the ith summand will contribute zero to the sum (3.10). Thus we may furthermore assume that e is in the support of each g_{i}F_{i} for each i. Also, by choosing Ω_{e} so as to be contained in N, we may assume without loss of generality that g_{i}F_{i} is supported on N.

Since the support of F_{i} lies in N by assumption, we see that g_{i} ∈ \overline{B}N = \overline{N}B, and so we may write g_{i} = \overline{n}_{i}b_{i} for some \overline{n}_{i} ∈ \overline{N}, b_{i} ∈ B. Take Ω_{e} to be the intersections of the neighbourhoods Ω corresponding to the elements \overline{n}_{i}^{−1} in condition (3.4). Then

ϕ(F_{i|Ω_{e}g_{i}^{−1}}) = ϕ(g_{i}^{−1}(g_{i}F_{i})_{|Ω_{e}}) = ϕ(b_{i}^{−1}\overline{n}_{i}^{−1}(g_{i}F_{i})_{|Ω_{e}})

= b_{i}^{−1}\overline{n}_{i}^{−1}ϕ((g_{i}F_{i})_{|Ω_{e}}) = g_{i}^{−1}ϕ((g_{i}F_{i})_{|Ω_{e}}).

(Here the third equality follows from the fact that ϕ is B-equivariant, together with hypothesis (3.4).) Thus

∑_{i} g_{i}ϕ(F_{i}|Ω_{e}g_{i}^{−1}) = ∑_{i} g_{i}g_{i}^{−1}ϕ((g_{i}F_{i})_{|Ω_{e}}) = ϕ((∑_{i} g_{i}F_{i})_{|Ω_{e}}) = 0,

since ∑_{i} g_{i}F_{i} = 0 by assumption.

The surjectivity of (3.2) follows from Claims 3.5 and 3.6 taken together. □

Now let U be a K-Banach space equipped with a unitary G-representation, and fix some integer r ≥ 1. Restriction of continuous linear maps yields the following commutative diagram:

(3.11)  \xymatrix{L_{G}(I^{la}(χ), U) \ar[r] \ar[d] & L_{G}(I(χ), U) \ar[d] \\ L_{G^{+}(r)}(I^{la}(χ)(N_{0}), U) \ar[r] & L_{G^{+}(r)}(I(χ)(N_{0}), U).}

Our goal is to prove that top horizontal arrow is an isomorphism under the hypotheses of Proposition 2.5. Lemma 3.1 implies that both vertical arrows are isomorphisms, and so it suffices to prove that the bottom horizontal arrow is an isomorphism. We turn our efforts to this task.

We first recall an important result due to Vishik and Amice-Vélu (Lemma 3.14 below).

Let A denote the Tate algebra of rigid analytic functions on the closed unit disk over K; thus A := {∑_{i=0}^{∞} a_{i}x^{i} | a_{i} ∈ K, lim_{i→∞} | a_{i} |= 0}. Let ||–||_{A} denote the usual Gauss norm on A; i.e. || ∑_{i=0}^{∞} a_{i}x^{i}||_{A} = sup{| a_{i} |}_{i≥0}. With respect to this norm A becomes a Banach space over K.

We let A^{≤k−2} denote the finite dimensional subspace of A consisting of polynomials of degree ≤ k − 2, and let C^{lp≤k−2}(ℤ_{p}, K) denote the closed subspace of C^{la}(ℤ_{p}, K) consisting of functions that are locally polynomial of degree ≤ k − 2.

If F ∈ A, b ∈ ℤ_{p}, and m ≥ 0, we write F_{|b+p^{m}ℤ_{p}} to denote the element of C^{la}(ℤ_{p}, K) defined by F on the compact open subset b + p^{m}ℤ_{p} of ℤ_{p}, and by zero at all other points of ℤ_{p}. If F ∈ A^{≤k−2} then F_{|b+p^{m}ℤ_{p}} lies in C^{lp≤k−2}(ℤ_{p}, K).

Definition 3.12. If ϕ ∈ L(C^{la}(ℤ_{p}, K), U) (respectively ϕ ∈ L(C^{lp≤k−2}(ℤ_{p}, K), U)), then ϕ is called α-tempered (for some α ∈ K^{×}) if for all F (x) ∈ A (respectively F (x) ∈ A^{≤k−2}), all b ∈ ℤ_{p}, and all m ≥ 0, we have

||ϕ (F (\frac{x − b}{p^{m}})) ||_{U} ≤ C|α|^{−m}||F (x)||_{A}

for some (equivalently any) choice of norm ||–||_{U} defining the topology on U and some positive constant C (both independent of the choice of F, b, or m). We let L(C^{la}(ℤ_{p}, K), U)^{α−temp} (respectively L(C^{lp≤k−2}(ℤ_{p}, K), U)^{α−temp}) denote the subspace of L(C^{la}(ℤ_{p}, K), U) (respectively of L(C^{lp≤k−2}(ℤ_{p}, K), U)) consisting of αtempered maps.

We remark that α-tempered distributions are also said to be “tempered of order ≤ h”, if h is the p-adic valuation of α.

Restricting maps from C^{la}(ℤ_{p}, K) to C^{lp≤k−2}(ℤ_{p}, K) induces a map

(3.13)  L(C^{la}(ℤ_{p}, K), U)^{α−temp} → L(C^{lp≤k−2}(ℤ_{p}, K), U)^{α−temp}

Lemma 3.14. If α ∈ K^{×} satisfies | α |>| p^{k−1} |, then the restriction map (3.13) is an isomorphism.

Proof. This is the classical result of Vishik and Amice-Vélu. (See [1] and [24], and also the proof of [16, Thm., p. 13].) □

We return to the consideration of I^{la}(χ), for some character χ as in Section 2. Restricting functions in I^{la}(χ) to N_{0} induces isomorphisms

(3.15)  I^{la}(χ)(N_{0}) ⟶^{∼} C^{la}(N_{0}, K) ⟶^{∼} C^{la}(ℤ_{p}, K)

(where the second isomorphism is induced via the identification of ℤ_{p} and N_{0} given by x ↦ n_{x}). Restricting (3.15) to I(χ)(N_{0}) induces an isomorphism

(3.16)  I(χ)(N_{0}) ⟶^{∼} C^{lp≤k−2}(ℤ_{p}, K)

The isomorphism (3.15) allows us to transport the G^{+}(1)-action on the source to

a G^{+}(1)-action on C^{la}(ℤ_{p}, K). Concretely, if F (x) ∈ C^{la}(ℤ_{p}, K) and g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

G^{+}(1), then one computes for any x ∈ ℤ_{p} that (3.17)

(gF)(x) = \{ \begin{matrix}0 if \frac{b + dx}{a + cx} ∉ ℤ_{p} \\ (a + cx)^{k−2} \frac{θ_{1}(a + cx)}{θ_{2}(a + cx)} \frac{θ_{2}(ad − bc)}{(ad − bc)^{k−2}} F (\frac{b + dx}{a + cx}) if \frac{b + dx}{a + cx} ∈ ℤ_{p}. \end{matrix}

Since I(χ) is a G-invariant closed subspace of I^{la}(χ), the isomorphism (3.16) shows that C^{lp≤k−2}(ℤ_{p}, K) is a G^{+}(1)-invariant closed subspace of C^{la}(ℤ_{p}, K); this also may be checked directly from the formula (3.17) for the G^{+}(1)-action on this space.

It is useful to also define an action of G_{0}(1) on A. Namely, if F (x) ∈ A and

g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G_{0}(1), then we define

(3.18)  (gF)(x) = (a + cx)^{k−2} \frac{θ_{2}(ad − bc)}{(ad − bc)^{k−2}} F (\frac{b + dx}{a + cx})

Note that the norm ||–||_{A} is invariant under this G_{0}(1)-action, and that A^{≤k−2} is a G_{0}(1)-invariant subspace of A. Comparing formulas (3.17) and (3.18), we find

that if g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G_{0}(1) and F (x) ∈ A, then

(3.19)  g(F (x)_{|ℤ_{p}}) = \frac{θ_{1}(a + cx)}{θ_{2}(a + cx)}(gF (x))_{|ℤ_{p}}

for any x ∈ ℤ_{p}.

Lemma 3.20. Suppose that χ becomes unitary when restricted to the center of G (i.e. the subgroup of T consisting of diagonal matrices), or equivalently, that I^{la}(χ) has a unitary central character. As usual write α = χ(p), and let C denote either C^{la}(ℤ_{p}, K) or C^{lp≤k−2}(ℤ_{p}, K); correspondingly, let A_{C} denote either A or A^{≤k−2}, respectively.

If U is a K-Banach space and ϕ ∈ L(C, U) then the following are equivalent:

(i) ϕ is α-tempered.

(ii) There exists a positive constant C such that ||ϕ(b(F_{|ℤ_{p}}))||_{U} ≤ C||F ||_{A} for all F ∈ A_{C} and b ∈ B^{+}. (Here ||–||_{U} is some fixed choice of norm defining the topology of U.)

Proof. Write B′ := \{( \begin{matrix}p^{m} & −b \\ 0 & 1 \end{matrix} ) | m ≥ 0, b ∈ ℤ_{p}\}; note that B′ is a submonoid of

B^{+}. If F ∈ A_{C}, then we compute using (3.17) that

( \begin{matrix}p^{m} & −b \\ 0 & 1 \end{matrix} ) F (x)_{|ℤ_{p}} = α^{m}F (\frac{x − b}{p^{m}})_{|b+p^{m}ℤ_{p}}.

Thus (fixing a norm ||–||_{U} defining the topology of U) we find that ϕ is α-tempered if and only if there exists a positive constant C such that

(3.21)  ||ϕ(b′(F_{|ℤ_{p}}))||_{U} ≤ C||F ||_{A}

for all F ∈ A and b′ ∈ B′.

Clearly (ii) implies condition (3.21). Conversely, suppose that (3.21) holds. Any element of B^{+} may be written in the form zb′t_{0}, where z lies in the center of G,

b′ ∈ B′, and t_{0} ∈ T_{0}. Writing t_{0} = ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ), and taking into account (3.21), (3.19),

the fact that T_{0} acts isometrically on A, and that χ_{|Z_{G}} is unitary by assumption, we deduce that

||ϕ(zb′t_{0}(F_{|ℤ_{p}}))||_{U} = ||χ(z)\frac{θ_{1}(a)}{θ_{2}(a)}ϕ(b′(t_{0}F)_{|ℤ_{p}})||_{U} = ||ϕ(b′(t_{0}F)_{|ℤ_{p}})||_{U}

≤ C||t_{0}F ||_{A} = ||F ||_{A}

for any F ∈ A_{C}. Thus (3.21) is equivalent to condition (ii), and the lemma follows. □

Lemma 3.22. In the situation of Lemma 3.20, suppose that U is equipped with a unitary B^{+}-action. Then L_{B^{+}}(C, U) ⊂ L(C, U)^{α−temp}.

Proof. Fix a B^{+}-invariant norm ||–||_{U} defining the topology of U. If ϕ ∈ L_{B^{+}}(C, U) then the map A_{C} → U defined by F ↦ ϕ(F_{|ℤ_{p}}) is continuous, and so there is a positive constant C such that ||ϕ(F_{|ℤ_{p}})||_{U} ≤ C||F ||_{A} for all F ∈ A_{C}. For any b ∈ B^{+}, F ∈ A_{C}, we compute that

||ϕ(bF_{|ℤ_{p}})||_{U} = ||bϕ(F_{|ℤ_{p}})||_{U} = ||ϕ(F_{|ℤ_{p}})||_{U} ≤ C||F ||_{|ℤ_{p}}

(using the B^{+}-equivariance of ϕ and the B^{+}-invariance of ||–||_{U}.) Thus ϕ satisfies condition (ii) of Lemma 3.20, and so by that lemma is α-tempered. □

Lemma 3.23. In the situation of Lemma 3.20, suppose that U is equipped with a unitary G-action, and that | α |<| p^{k−1} |. If r is chosen so that the conductor of each of θ_{1|ℤ_{p}^{×}} and θ_{2|ℤ_{p}^{×}} divides p^{r}, then the map (3.13) (which is an isomorphism by Lemma 3.14) restricts to an isomorphism L_{G^{+}(r)}(C^{la}(ℤ_{p}, K), U) → L_{G^{+}(r)}(C^{lp≤k−2}(ℤ_{p}, K), U).

Proof. Since B^{+} ⊂ G^{+}(r), Lemma 3.22 gives the inclusions

L_{G^{+}(r)}(C^{la}(ℤ_{p}, K), U) ⊂ L(C^{la}(ℤ_{p}, K), U)^{α−temp}

and

L_{G^{+}(r)}(C^{lp≤k−2}(ℤ_{p}, K), U) ⊂ L(C^{lp≤k−2}(ℤ_{p}, K), U)^{α−temp}.

The isomorphism of Lemma 3.14 thus restricts to an injection

(3.24)  L_{G^{+}(r)}(C^{la}(ℤ_{p}, K), U) → L_{G^{+}(r)}(C^{lp≤k−2}(ℤ_{p}, K), U),

which we must show is again an isomorphism.

Given ϕ lying in the target of (3.24), let ϕ̃ denote its unique extension to an element of L(C^{la}(ℤ_{p}, K), U)^{α−temp}. We must show that ϕ̃ is again G^{+}(r)-equivariant. In other words, for any g ∈ G^{+}(r), we must show that

(3.25)  g^{−1}ϕ̃(gF) =^{?} ϕ̃(F)

for all F ∈ C^{la}(ℤ_{p}, K). Since ϕ is G^{+}(r)-equivariant, both sides of (3.25) restrict to ϕ on C^{lp≤k−2}(ℤ_{p}, K). Thus to show that they coincide, it suffices to show that

(3.26)  F ↦ g^{−1}ϕ̃(gF)

is also an α-tempered map.

Fix a G-invariant norm ||–||_{U} defining the topology of U. Since ϕ̃ is α-tempered, Lemma 3.20 (ii) gives a positive constant C such that

(3.27)  ||ϕ̃(bF_{|ℤ_{p}})||_{U} ≤ C||F ||_{A}

for all b ∈ B^{+}, F ∈ A. If g ∈ G^{+}(r) and b in B^{+}, then we may write gb = b′\overline{n} for some b′ ∈ B^{+}, \overline{n} ∈ \overline{N}(r), and so compute

||g^{−1}ϕ̃(gbF_{|ℤ_{p}})||_{U} = ||ϕ̃(gbF_{|ℤ_{p}})||_{U} = ||ϕ̃(b′\overline{n}F_{|ℤ_{p}})||_{U}

= ||ϕ̃(b′(\overline{n}F)_{|ℤ_{p}})||_{U} ≤ C||\overline{n}F ||_{A} = C||F ||_{A}.

(Here we have used the G-invariance of ||–||_{U}, as well as (3.19) applied to \overline{n} and F (taking into account our choice of r), the inequality (3.27) applied to \overline{n}F, and the fact that \overline{N}(r) acts isometrically on A.) Again appealing to Lemma 3.20, we see that (3.26) is α-tempered, as required. □

Taking into account the isomorphisms (3.15) and (3.16), we see that Lemma 3.23 implies that under the hypotheses of Proposition 2.5, the bottom horizontal arrow of (3.11) is an isomorphism, as required, and so Proposition 2.5 is proved.

4. p-adic L-functions

Let 𝔸 denote the adèles of ℚ, and let 𝔸_{f}, 𝔸_{f}^{p}, have their usual meaning. Fix a prime-to-p natural number N, and let K^{p} be the following compact open subgroup of GL_{2}(𝔸_{f}^{p}):

K^{p} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ ∏_{ℓ≠p} GL_{2}(ℤ_{ℓ}) | c ≡ 0 mod N, d ≡ 1 mod N\}.

Let us embed ℂ^{×} into GL_{2}(ℝ) in the usual way, by choosing a real linear isomorphism ℂ ⟶^{∼} ℝ^{2}, and regarding ℂ^{×} as the subgroup of the group of real linear automorphisms of ℂ consisting of complex linear automorphisms.

If K_{p} is a compact open subgroup of GL_{2}(ℚ_{p}) then the quotient

Y (K^{p}K_{p}) := GL_{2}(ℚ)\ GL_{2}(𝔸)/ℂ^{×}K^{p}K_{p}

is a classical modular curve. For example, if we take K_{p} to be the kernel of the reduction map GL_{2}(ℤ_{p}) → GL_{2}(ℤ/p^{n}), then this quotient is isomorphic to the modular curve that classifies elliptic curves over ℂ with a Γ_{1}(N)-level structure and a full level p structure (the Weil pairing of which is allowed to have arbitrary determinant). In general, if we take K_{p} to be a compact open subgroup of GL_{2}(ℤ_{p}), and use the isomorphism GL_{2}(ℝ)/ℂ^{×} ⟶^{∼} ℂ\ℝ together with strong approximation for SL_{2}, we obtain an isomorphism

Y (K^{p}K_{p}) = GL_{2}(ℚ)\ GL_{2}(𝔸)/ℂ^{×}K^{p}K_{p} ⟶^{∼} Γ_{1}(N)\(X × GL_{2}(ℤ_{p})/K_{p}),

where X denotes the upper half-plane of ℂ \ ℝ, and as usual Γ_{1}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

SL_{2}(ℤ) | a ≡ d ≡ 1 mod N, c ≡ 0 mod N}.

Define M(K^{p}) = lim_{\overleftarrow{s}} lim_{\overrightarrow{Kp}} H_{c}^{1}(Y (K^{p}K_{p}), O_{K}/p^{s}), where s and K_{p} range over the

directed sets of non-negative integers and compact open subgroups of GL_{2}(ℚ_{p}) respectively, and define H̃_{c}^{1}(K^{p}) = K ⊗_{O_{K}} M(K^{p}). By construction, M(K^{p}) is a p-adically complete and separated O_{K}-module. Thus H̃_{c}^{1}(K^{p}) is naturally a K-Banach space, if we equip it with the norm whose unit ball is equal to M(K^{p}) (in other words, the gauge of M(K^{p})). The group GL_{2}(ℚ_{p}) acts on right on the projective system of arithmetic quotients Y (K^{p}K_{p}), and hence on M(K^{p}). Thus GL_{2}(ℚ_{p}) acts unitarily on the Banach space H̃_{c}^{1}(K^{p}). This representation is further equipped with actions of the Hecke algebra H^{p} of K^{p} in GL_{2}(𝔸_{f}^{p}) and of the connected component group π_{0} = {±1} of GL_{2}(ℝ), that commute with one another and with the GL_{2}(ℚ_{p})-action.

Proposition 4.1. The unitary Banach space representation H̃_{c}^{1}(K^{p}) is admissible.

Proof. This is [5, Lem. 2.1.2]. It is also a special case of [12, Prop. 2.2.6]. □

We let H̃_{c}^{1}(K^{p})^{±} denote the ±1-eigenspace for π_{0} on H̃_{c}^{1}(K^{p}). Each of these spaces is again an admissible unitary Banach space representation of GL_{2}(ℚ_{p}), equipped with a commuting H^{p}-action.

Let Γ denote the normalizer in GL_{2}(ℚ) of K^{p}. (We regard GL_{2}(ℚ) as being embedded diagonally in GL_{2}(𝔸_{f}^{p}).) There is a natural map Γ → π_{0} × (H^{p})^{×} × GL_{2}(ℚ_{p}): the projection onto the first factor is given by the sign of the determinant, the projection onto the second factor is given by mapping an element in GL_{2}(𝔸_{f}^{p}) that normalizes K^{p} to its double coset in H^{p}, and the projection onto the third factor is given by the inclusion GL_{2}(ℚ) → GL_{2}(ℚ_{p}). In this way we may regard H̃_{c}^{1}(K^{p}), and hence its dual H̃_{c}^{1}(K^{p})′, as a Γ-module.

The following proposition describes the theory of modular symbols, as it applies to the completed cohomology space H̃_{c}^{1}(K^{p}). Let D^{0} denote the O_{K}-module of degree zero divisors on ℙ^{1}(ℚ). The natural action of GL_{2}(ℚ) on D^{0} induces in particular an action of Γ on D^{0}. Recall that ℙ^{1}(ℚ) appears as the boundary of the standard completion of the upper half-plane X. For any compact open subgroup K_{p} of GL_{2}(ℚ_{p}), we let X(K^{p}K_{p}) denote the usual compactification of Y (K^{p}K_{p}); thus X(K^{p}K_{p}) = Y (K_{p}K^{p}) ⋃ C(K_{p}K^{p}), where C(K_{p}K^{p}) is the set of cusps. (If K_{p} ⊂ GL_{2}(ℤ_{p}), then C(K_{p}K^{p}) = Γ_{1}(N)\(ℙ^{1}(ℚ) × GL_{2}(ℤ_{p})/K_{p}).)

Proposition 4.2. The standard pairings between one dimensional compactly supported cohomology classes on the open curves Y (K^{p}K_{p}), and one dimensional homology classes taken relative to the set of cusps C(K^{p}K_{p}) on the compactified curves X(K^{p}K_{p}), induces a Γ-equivariant O_{K}-linear map D^{0} → (H̃_{c}^{1}(K^{p}))′, whose image lies in the closed unit ball of the target (taken with respect to norm that is dual to the norm on H̃_{c}^{1}(K^{p}) given by the gauge of M(K^{p})).

Proof. Given elements a, b ∈ ℙ^{1}(ℚ), the geodesic in the upper half-plane that joins a and b gives rise to an element of H_{1}(X(K^{p}K_{p}), C(K^{p}K_{p}); O_{K}/p^{s}), for any choice of K_{p} and s, and we obtain a map D^{0} → H_{1}(X(K^{p}K_{p}), C(K^{p}K_{p}); O_{K}/p^{s}) by mapping a − b to this element. The pairing between this relative homology group and the corresponding compactly supported cohomology group thus induces a O_{K}- bilinear pairing

D^{0} × H_{c}^{1}(GL_{2}(ℚ)\ GL_{2}(𝔸)/ℂ^{×}K^{p}K_{p}, O_{K}/p^{s}) → O_{K}/p^{s}

for each choice of K^{p} and s. The functorial nature of this pairing shows that it is Γ-equivariant, and that it is compatible with change of K^{p} and s. Thus, passing to the inductive limit in K^{p} and the projective limit in s, we obtain a Γ-equivariant O_{K}-bilinear pairing

D^{0} × M(K^{p}) → O_{K},

and consequently a Γ-equivariant O_{K}-linear map as in the statement of the proposition. □

Let us fix for the duration of this section a newform f of weight k and tame conductor N^{5} that is defined over K. If we fix an embedding K → ℂ, then f gives rise to an automorphic representation π = π_{∞} ⊗_{K} π_{p} ⊗_{K} π^{p} of GL_{2}(𝔸) = GL_{2}(ℝ) × GL_{2}(ℚ_{p}) × GL_{2}(𝔸_{f}^{p}), where the infinite component π_{∞} is defined over ℂ, while the finite component π_{p} ⊗ π^{p} is defined over K. Since f is assumed to have tame conductor N, the space (π^{p})^{K^{p}} is thus a one dimensional H^{p}-representation, on which H^{p} acts through a character that we will denote by λ. If ϵ_{N} denotes the conductor-N part of the nebentypus of f, then the central character of π_{p} takes

( \begin{matrix}p \\ & p \end{matrix} ) to ϵ_{N}(p)p^{k−2}.

If V is any representation of GL_{2}(ℚ_{p}), then for each choice of sign ±1, we let V^{±,λ} denote the GL_{2}(ℚ_{p})-representation V equipped with commuting actions of π_{0} and H^{p}, defined as follows: the non-trivial element of π_{0} acts through the chosen sign ±1, while H^{p} acts through the character λ.

Proposition 4.3. For each choice of sign, we obtain a π_{0} × H^{p} × GL_{2}(ℚ_{p})- equivariant embedding (W_{k−2}⊗_{K} π_{p})^{±,λ} → H̃_{c}^{1}(K^{p})^{±}, unique up to scalar multiples.

Proof. For each choice of compact open subgroup K_{p} of GL_{2}(ℚ_{p}), write VW̌_{k−2} := GL_{2}(ℚ)\(GL_{2}(𝔸)/ℂ^{×}K^{p}K_{p} × W̌_{k−2}); this is a local system of K-vector spaces on the modular curve GL_{2}(ℚ)\ GL_{2}(𝔸)/ℂ^{×}K^{p}K_{p}. If we write H_{c}^{1}(K^{p}, VW̌_{k−2}) := lim_{\overrightarrow{Kp}} H_{c}^{1}(GL_{2}(ℚ)\ GL_{2}(𝔸)/ℂ^{×}K^{p}K_{p}, VW̌_{k−2}), then [12, (4.3.7)] shows that there is

an isomorphism

(4.4)  W_{k−2} ⊗_{K} H_{c}^{1}(K^{p}, VW̌_{k−2}) ⟶^{∼} H̃_{c}^{1}(K^{p})_{W_{k−2}−lalg},

while classical Eichler-Shimura theory shows that the space

Hom_{GL_{2}(ℚ_{p})}(π_{p}, H_{c}^{1}(K^{p}, VW̌_{k−2})^{H^{p}=λ})

is two dimensional, and that under the action of π_{0} it decomposes as the direct sum of one dimensional ±-eigenspaces. The proposition follows. □

The homomorphism Γ → π_{0} × (H^{p})^{×} × GL_{2}(ℚ_{p}) described above allows us to define a Γ-action on (W_{k−2} ⊗_{K} π_{p})^{±,λ}, and hence a contragredient Γ-action on ((W_{k−2} ⊗_{K} π_{p})^{±,λ})′_{b} and on ((W_{k−2} ⊗_{K} π_{p})^{±,λ})_{b}^{′b}.

5i.e. conductor equal to N times a power of p

Corollary 4.5. For each choice of sign we obtain a Γ-equivariant map D^{0} → ((W_{k−2} ⊗_{K} π_{p})^{±,λ})_{b}^{′b}.

Proof. Since the GL_{2}(ℚ_{p})-action on H̃_{c}^{1}(K^{p})^{±} is unitary, Lemma 1.11 shows that the dual of the map of Proposition 4.3 factors through the natural map

((W_{k−2} ⊗_{K} π_{p})^{±,λ})_{b}^{′b} → ((W_{k−2} ⊗_{K} π_{p})^{±,λ})′_{b}

of (1.10). Composing the resulting map (H̃_{c}^{1}(K^{p})^{±})′ → ((W_{k−2} ⊗_{K} π_{p})^{±,λ})_{b}^{′b} with the map of Proposition 4.2 yields the corollary. □

In the case when π_{p} is special and unramified, the maps of Corollary 4.5 were constructed by Breuil in [5, Lem. 4.2.2].

Suppose now that π_{p} is principal series or special. Then we may choose χ as in Section 2 such that W_{k−2} ⊗_{K} π_{p} = π(χ). We let α, β, θ_{1}, θ_{2} and χ′ have the same meanings as in Section 2. A consideration of central characters shows that αβ = ϵ_{N}(p)p^{k−1}. Note that the necessary conditions of Lemma 2.1 are satisfied for a representation π(χ) arising in this way from a newform.

Proposition 4.6. Conjecture 2.2 is true for those representations π(χ) that arise from newforms that are principal series or special locally at p.

Proof. Proposition 4.3 shows that π(χ) embeds into the non-zero unitary representation H̃_{c}^{1}(K^{p})^{±} (for either choice of sign and an appropriate choice of K^{p}). □

Of course in the unramified principal series or special case, the results of [2], [14], and [23] establish Conjecture 2.2 without any global restriction on χ.

Now suppose furthermore that the newform f is of finite slope at p. Then we may assume that θ_{1} is unramified (by replacing χ by χ′ if necessary). If in fact p divides the conductor of f and we are in the principal series case, then θ_{2} is ramified, and so the condition that θ_{1} be unramified uniquely determines χ. If p divides the conductor of f and we are in the special case, we again have a unique choice for χ, since in the special case we insist that α/β = p^{−1}. In either of these cases, the U_{p}-eigenvalue of f is equal to α. If the conductor of f is prime to p, then both θ_{1} and θ_{2} are unramified, and we have two choices for χ (namely a pair χ and χ′, in the notation of Section 2). Fixing one of the two choices for χ corresponds to choosing one of the two p-old U_{p}-eigenforms attached to f; namely, the one with U_{p}-eigenvalue equal to α.

We let g denote f in the case when p divides the conductor of f, and let g denote the p-oldform attached to f with U_{p}-eigenvalue α, in the case when the conductor of f is prime-to-p. In all cases g is a U_{p}-eigenform with eigenvalue α.

Corollary 4.7. If | α |>| p^{k−1} |, then the embeddings

π(χ)^{±,λ} = (W_{k−2} ⊗_{K} π_{p})^{±,λ} → H̃^{1}(K^{p})^{±}

of Proposition 4.3 extend uniquely to (π_{0}, GL_{2}(ℚ_{p}), H^{p})-equivariant maps

π^{la}(χ)^{±,λ} → H̃_{c}^{1}(K^{p})^{±}.

Proof. This follows from Proposition 2.5. □

Dualizing the maps of Corollary 4.7, and taking into account Proposition 4.2, we obtain Γ-equivariant maps D^{0} → (π^{la}(χ)^{±,λ})′ (one for each choice of sign), lifting the maps of Corollary 4.5. In particular, the element ∞ − 0 of D^{0} gives rise to a pair of functionals

(4.8)  μ_{g}^{±} : π^{la}(χ) → K.

If we identify I^{la}(χ) as a space of locally analytic functions on ℚ_{p}^{×}, as in Section 2, then we obtain a closed embedding C^{la}(ℤ_{p}^{×}, K) → π^{la}(χ), by identifying elements of the source as locally analytic functions on ℚ_{p} that are supported on ℤ_{p}^{×}. Restricting the functionals μ_{g}^{±} to C^{la}(ℤ_{p}^{×}) thus gives rise to a pair of distributions on ℤ_{p}^{×}.

Proposition 4.9. The distributions on ℤ_{p}^{×} obtained by restricting μ_{g}^{±} coincide (up to a non-zero scalar multiple) with the Mazur-Tate-Teitelbaum distributions [16] that compute the p-adic L-function of g.^{6}

Proof. It follows from Lemma 3.22 that μ_{g}^{±} are α-tempered, while by their very construction, the Mazur-Tate-Teitelbaum distributions defining the p-adic L-function of g are α-tempered. (See [16, Thm., p. 13].) Thus it suffices to show that the distributions μ_{g}^{±} coincide with the Mazur-Tate-Teitelbaum distributions when evaluated on functions that are locally polynomial of degree ≤ k − 2.

As in Section 3, let A^{≤k−2} denote the space of polynomials in one variable over K of degree ≤ k − 2. Suppose that b ∈ ℤ_{p}^{×}, that p^{m} is a positive power of p, and that F (x) is an element of A^{≤k−2}. As in Section 3, we let F (x)_{|b+p^{m}ℤ_{p}} denote the element of C^{lp≤k−2}(ℤ_{p}, K) defined by F (x) on b + p^{m}ℤ_{p}, and defined to be zero elsewhere. Fix a choice of sign ±, let ⟨– , –⟩_{1}^{±} : D^{0} × H̃_{c}^{1}(K^{p}) → K denote the Γ-equivariant pairing provided by Proposition 4.2, and let ϕ_{1}^{±} : π(χ) → H̃_{c}^{1}(K^{p})^{±} denote the embedding of Proposition 4.3. Using the formula (3.17) and the fact

that ( \begin{matrix}p^{m} & −b \\ 0 & 1 \end{matrix} ), when thought of as an element of GL_{2}(ℚ), lies in Γ (in fact, even

in K^{p}), we compute that

(4.10) ∫_{b+p^{m}ℤ_{p}^{×}} F (x)dμ_{g}^{±} = ⟨∞ − 0, ϕ_{1}^{±}(F (x)_{|b+p^{m}ℤ_{p}})⟩_{1}^{±}

= ⟨∞ − 0, α^{−m} ( \begin{matrix}p^{m} & −b \\ 0 & 1 \end{matrix} ) ϕ_{1}^{±}(F (b + p^{m}x)_{|ℤ_{p}})⟩_{1}^{±}

= α^{−m}⟨( \begin{matrix}p^{−m} & bp^{−m} \\ 0 & 1 \end{matrix} ) (∞ − 0), ϕ_{1}^{±}(F (b + p^{m}x)_{|ℤ_{p}})⟩_{1}^{±}

= α^{−m}⟨∞ − \frac{b}{p^{m}} , ϕ_{1}^{±}(F (b + p^{m}x)_{|ℤ_{p}})⟩_{1}^{±}.

To analyze this expression, we must unwind the definition of the pairing ⟨– , –⟩_{1}^{±}, which we do through a series of commutative diagrams.

6We take the auxiliary integer “M” of that paper to be 1; this is no loss of generality, since the Mazur-Tate-Teitelbaum distributions on ℤ_{p}^{×} × ℤ_{M}^{×} associated to g may be constructed out of the Mazur-Tate-Teitelbaum distributions on ℤ_{p}^{×} attached to the collection of modular forms obtained by twisting g by all characters of ℤ_{M}^{×}. We have also scaled the distribution of [16] by an appropriate complex period Ω_{g}^{±}.

We identify W_{k−2} with a space of functions on G in the usual way: write w̌_{0} =

(0, 1)^{k−2} ∈ Sym^{k−2} K^{2} ⟶^{∼} W̌_{k−2}, and define w(g) = ⟨w̌_{0}, gw⟩ for g ∈ G. If we

write I(θ) := (IndG ; \overline{B} θ)_{sm}, then multiplication of functions induces an isomorphism (whose existence has been recalled several times already)

(4.11)  W_{k−2} ⊗_{K} I(θ) ⟶^{∼} I(χ).

Restricting the functions w(g) to those g of the form n_{x} := ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) for x ∈

ℤ_{p}, we also obtain an identification of W_{k−2} with the space A^{≤k−2}. Multiplying elements of W_{k−2}, thought of as elements of A^{≤k−2} in this way, by locally constant functions on ℤ_{p} induces an isomorphism

(4.12)  W_{k} ⊗_{K} C^{sm}(ℤ_{p}, K) ⟶^{∼} C^{lp≤k−2}(ℤ_{p}, K).

(As in Section 3, we let C^{lp≤k−2}(ℤ_{p}, K) denote the subspace of C^{la}(ℤ_{p}, K) consisting of functions that are locally polynomial of degree ≤ k − 2.)

We may apply the discussion of Section 3 to the representation I(θ) just as well as to I(χ). The isomorphism (3.16) thus yields an isomorphism

(4.13)  C^{sm}(ℤ_{p}, K) ⟶^{∼} I(θ)(N_{0}),

which fits in the commutative diagram of isomorphisms

(4.14)  \xymatrix{W_{k−2} ⊗_{K} C^{sm}(ℤ_{p}, K) \ar[r]^{∼}_{(4.12)} \ar[d]^{∼}_{id_{W_{k−2}} ⊗(4.13)} & C^{lp≤k−2}(ℤ_{p}, K) \ar[d]^{∼}_{(3.16)} \\ W_{k−2} ⊗_{K} I(θ)(N_{0}) \ar[r]^{∼}_{(4.11)} & I(χ)(N_{0}).}

We employ the notation introduced in the proof of Proposition 4.3. As recalled in that proof, for each choice of sign ±, classical Eichler-Shimura theory gives a (G(ℚ_{p}), H^{p})-equivariant embedding ϕ_{2}^{±} : (π_{p})^{±,λ} → H_{c}^{1}(K^{p}, VW̌_{k−2})^{±}, unique up to multiplication by a scalar. (Here H_{c}^{1}(K^{p}, VW̌_{k−2})^{±} denotes the ±-eigenspace for the action of π_{0} on H_{c}^{1}(K^{p}, VW̌_{k−2}).) By its construction, the embedding ϕ_{1}^{±} of Proposition 4.3 is obtained by composing the map (4.4) with id_{W_{k−2}} ⊗ ϕ_{2}^{±}. Thus we obtain a commutative diagram

(4.15)  \xymatrix{W_{k−2} ⊗ I(θ) \ar[d] \ar[r]^{∼}_{(4.11)} & I(χ) \ar[d] \\ W_{k−2} ⊗ π_{p} \ar[r]^{∼} \ar[d]_{id_{W_{k−2}} ⊗ϕ_{2}^{±}} & π(χ) \ar[d]_{ϕ_{1}^{±}} \\ W_{k−2} ⊗ H_{c}^{1}(K^{p}, V_{W})^{±} \ar[r] & H̃_{c}^{1}(K^{p})^{±},}

in which the upper vertical arrows are the defining surjections (either isomorphisms in the principal series case, or surjections with finite dimensional kernels in the special case), and the bottom horizontal arrow is induced by (4.4).

Pulling back a compactly compacted cohomology class to the upper half-plane, and then integrating over a geodesic joining two cusps, we obtain a pairing D^{0} × H_{c}^{1}(K^{p}, VW̌_{k−2}) → W̌_{k−2}, or equivalently, a pairing

⟨– , –⟩_{2} : D^{0} × W_{k−2} ⊗_{K} H_{c}^{1}(K^{p}, VW̌_{k−2}) → K.

Passing to ±-eigenspaces under the action of π_{0}, this yields pairings

⟨– , –⟩_{2}^{±} : D^{0} × W_{k−2} ⊗_{K} H_{c}^{1}(K^{p}, VW̌_{k−2})^{±} → K.

The diagram

(4.16)  \xymatrix{D^{0} × W_{k−2} ⊗_{K} H_{c}^{1}(K^{p}, VW̌_{k−2})^{±} \ar[dr]_{⟨– ,–⟩_{2}^{±}} \ar[r] & D^{0} × H̃_{c}^{1}(K^{p})^{±} \ar[d]^{⟨– ,–⟩_{1}^{±}} \\  & K}

in which the horizontal arrow is induced by (4.4), also commutes; indeed, the diagonal arrow is constructed by trivializing V_{W}, and then integrating over geodesics joining cusps, while an examination of the proof of [12, Thm. 2.2.19] shows that the horizontal arrow is similarly constructed by trivializing V_{W} (after choosing an integral model and reducing modulo powers of p), while the vertical arrow is again constructed by integrating over geodesics joining cusps.

Let 1_{|ℤ_{p}} denote the constant function 1 with domain ℤ_{p}. Using the commutative diagrams (4.14), (4.15), and (4.16), we compute that

α^{−m}⟨∞ − \frac{b}{p^{m}} , ϕ_{1}^{±}(F (b + p^{m}x)_{|ℤ_{p}})⟩_{1}^{±} = α^{−m}⟨∞ − \frac{b}{p^{m}} , F (b + p^{m}x) ⊗ ϕ_{2}^{±}(1_{|ℤ_{p}})⟩_{2}^{±}.

Combining this with (4.10), we see that

(4.17)  ∫_{b+p^{m}ℤ_{p}} F (x)dμ_{g}^{±} = α^{−m}⟨∞ − \frac{b}{p^{m}} , F (b + p^{m}x) ⊗ ϕ_{2}^{±}(1_{|ℤ_{p}})⟩_{2}^{±}

for any F (x) ∈ A^{≤k−2} ⟶^{∼} W_{k−2}.

If we fix an embedding K → ℂ, then the modular form g defines an element of c_{g} ∈ H_{c}^{1}(K^{p}, V_{W_{k−2}}) ⊗_{K} ℂ, defined to be the cohomology class of the holomorphic V_{W_{k−2}}-valued one-form g(τ)dτ(τ, 1)^{k−2}. (Here (τ, 1)^{k−2} is regarded as an element of Sym^{k−2} ℂ^{2} ⟶^{∼} W̌_{k−2} ⊗_{K} ℂ.) If we let c_{g}^{±} denote the ±-eigencomponent of c_{g}, then ϕ_{2}^{±}(1_{|ℤ_{p}}) = Ω^{±}c_{g}^{±} for some non-zero scalars Ω^{±}. (To see this, note that ϕ_{2}^{±}(1_{|ℤ_{p}}) is a U_{p}-eigenclass with eigenvalue α, and a H^{p}-eigenclass with character λ, while the same is true of c_{g}^{±}.) From (4.17) we compute

(4.18)  ∫_{b+p^{m}ℤ_{p}} F (x)dμ_{g}^{±} = Ω^{±}α^{−m}⟨∞ − \frac{b}{p^{m}} , F (b + p^{m}x) ⊗ c_{g}^{±}⟩_{2}^{±}.

If we don’t pass to ±-eigencomponents, then we compute that

(4.19) ⟨∞ − \frac{b}{p^{m}} , F (b + p^{m}x) ⊗ c_{g}⟩_{2} = ∫_{b/p^{m}}^{∞} g(τ)⟨(τ, 1)^{k−2}, F (b + p^{m}x)⟩dτ,

where the pairing ⟨(τ, 1)^{k−2}, F (b + p^{m}x)⟩ denotes the natural pairing W̌_{k−2} ⊗_{K} ℂ × W_{k−2} → ℂ. We furthermore compute that

⟨(τ, 1)^{k−2}, F (b + p^{m}x)⟩ = ⟨( \begin{matrix}1 & τ \\ 0 & 1 \end{matrix} ) w̌_{0}, F (b + p^{m}x)⟩

= ⟨w̌_{0}, ( \begin{matrix}1 & −τ \\ 0 & 1 \end{matrix} ) F (b + p^{m}x)⟩ = F (b − p^{m}τ)

(where the final equality follows from the way in which the isomorphism A^{≤k−2} ⟶^{∼} W_{k−2} was defined). Substituting this into (4.19), we find that

⟨∞ − \frac{b}{p^{m}} , F (b + p^{m}x) ⊗ c_{g}⟩_{2} = ∫_{b/p^{m}}^{∞} g(τ)F (b − p^{m}τ)dτ,

for any F ∈ A^{≤k−2}. Comparing this formula, along with (4.18), to the construction of [16], we find that μ_{g}^{±} does indeed coincide (up to multiplication by a non-zero scalar) with the corresponding distribution that computes the p-adic L-function of g. □

Let us remark that Proposition 4.9 and the maps (4.8) show that the distributions defining the p-adic L-function of g extend from distributions on ℤ_{p}^{×} to distributions on ℚ_{p} (or more precisely, to elements in the dual of I^{la}(χ)). In the case when π_{p} is special, this was noted in [5, §5.2].

5. L-invariants and the Mazur-Tate-Teitelbaum conjecture

In the discussion of the preceding section, suppose that the finite slope newform f is special at p. Corollary 4.7 yields a pair of Γ-equivariant morphisms

(5.1)  D^{0} → (π(χ)^{±,λ})_{b}^{′b} ⟶^{∼} (π^{la}(χ)^{±,λ})_{b}^{′b},

which up to twisting we may regard as a pair of Γ-equivariant morphisms D^{0} → O(k)^{b}. By applying the functor Hom_{Γ}(D^{0}, –) to the short exact sequence (1.20), and analyzing the result of applying the coboundary map

Hom_{Γ}(D^{0}, O(k)) → Ext_{Γ}^{1}(D^{0}, W_{k−2}⊗_{K} | det |^{−(k−2)/2})

to the morphisms (5.1), Breuil in [5] proves the following result:

Theorem 5.2. There is a unique L ∈ K so that each of the maps of (5.1) arises from a Γ-equivariant map D^{0} → (π^{la}(χ, L)^{λ,±})_{b}^{′b}.

Theorem 5.2 has a dual version, which states that there is a unique value of L for which one can find π_{0} × H^{p} × GL_{2}(ℚ)_{p}-equivariant maps

(5.3)  π^{la}(χ, L)^{±,λ} → H̃_{c}^{1}(K^{p})^{±}

extending the maps of Corollary 4.7. Furthermore, Breuil shows that these extensions (5.3) are unique.^{7}

7Breuil actually proves this statement with Γ replaced by the intersection GL_{2}(ℚ) ⋂ K^{p}. This weakens the equivariance claim, but strengthens the uniqueness claim.

Let L_{f} denote the unique value of L that satisfies the conclusion of Theorem 5.2. Theorem 5.2 then implies that the element ∞ − 0 ∈ D^{0} induces functionals

(5.4)  μ_{f}^{±} : π^{la}(χ, L_{f}) → K

extending the functionals (4.8). (We write the subscript as f rather than g, since in the special case f = g.)

Breuil observes in [5] that a result of Orton [18] proves that when k is even, f has trivial nebentypus, and the U_{p}-eigenvalue of f equals p^{(k−2)/2}, then the element −L_{f} satisfies the Mazur-Tate-Teitelbaum conjecture for f. We close this note by explaining how this follows formally from Theorem 5.2, even without the assumption of trivial nebentypus.

Recall that π^{la}(χ, L_{f}) can be identified with a certain space of locally analytic functions on ℚ_{p}^{×}; in particular, the function x^{i} log_{L_{f}} (x) supported on ℚ_{p} \ pℤ_{p} belongs to this space of functions, for 0 ≤ i ≤ k − 2.

Proposition 5.5. (i) Let χ be a locally algebraic character as in Section 2 with k even and for which π(χ) is in the special case, such that α = p^{(k−2)/2} (so that

β = p^{k/2}, and the central character of π(χ) is trivial on ( \begin{matrix}p \\ & p \end{matrix} )). If μ ∈ π^{la}(χ)′

is a continuous functional on π^{la}(χ) that is invariant under the action of ( \begin{matrix}p \\ & 1 \end{matrix} ) ,

then

∫_{ℤ_{p}^{×}} x^{(k−2)/2}dμ = 0.

(ii) If in addition to the hypotheses of (i) the functional μ extends in a ( \begin{matrix}p \\ & 1 \end{matrix} )-

invariant fashion to a continuous functional on π^{la}(χ, L) for some L ∈ K, then

∫_{ℤ_{p}^{×}} x^{(k−2)/2} log_{L} x dμ = −L ∫_{ℤ_{p}} x^{(k−2)/2} dμ.

Proof. If one computes ∫_{ℤ_{p}} x^{(k−2)/2}dμ by writing the domain of integration as the

union of ℤ_{p}^{×} and pℤ_{p}, one obtains the formula of (i). (Use the matrix ( \begin{matrix}p \\ & 1 \end{matrix} ) to

“scale” pℤ_{p} back to ℤ_{p}.)

Assume now that the hypothesis of (ii) holds. If one takes into account the formulas (2.3) and (2.7) giving the action of GL_{2}(ℚ_{p}) on elements of π^{la}(χ, L), as well as the fact that the polynomial x^{(k−2)/2} ∈ I(χ) has zero image in π(χ), and computes the value of

∫_{ℚ_{p}\pℤ_{p}} x^{(k−2)/2} log_{L} x dμ

by writing the domain of integration as the union of ℤ_{p}^{×} and ℚ_{p} \ ℤ_{p}, one finds that

(5.6)  ∫_{ℤ_{p}^{×}} x^{(k−2)/2} log_{L} x dμ = L ∫_{ℚ_{p}\pℤ_{p}} x^{(k−2)/2} dμ.

(Use the matrix ( \begin{matrix}1 \\ & p \end{matrix} ) to “scale” ℚ_{p} \ℤ_{p} back to ℚ_{p} \pℤ_{p}.) Finally, again taking

into account the fact that the element x^{(k−2)/2} ∈ I(χ) has zero image in π(χ), as well as the vanishing statement of (i), one finds that

∫_{ℚ_{p}\pℤ_{p}} x^{(k−2)/2} dμ = − ∫_{ℤ_{p}} x^{(k−2)/2} dμ.

Combined with (5.6), this gives the formula of (ii). □

Corollary 5.7. Suppose that f is a newform on Γ_{1}(N) ⋂ Γ_{0}(p) of even weight k, such that α = p^{(k−2)/2} (and hence such that ϵ_{N}(p) = 1). Then

(5.8)  ∫_{ℤ_{p}^{×}} x^{(k−2)/2}dμ_{f} = 0

and

(5.9)  ∫_{ℤ_{p}^{×}} x^{(k−2)/2} log_{L_{f}} x dμ_{f} = −L_{f} ∫_{ℤ_{p}} x^{(k−2)/2} dμ_{f}.

Proof. This follows immediately from Proposition 5.5 and the fact that ∞−0 ∈ D^{0}

is invariant under ( \begin{matrix}p \\ & 1 \end{matrix} ) (which lies in Γ). □

By proposition 4.9, the left-hand side of (5.8) is the value of the p-adic L-function of f at k/2. Thus (5.8) exhibits the “exceptional zero” of the p-adic L-function. Again by Proposition 4.9, the left-hand side of (5.9) gives the derivative of the p-adic L-function of f at k/2, while the proof of Proposition 4.9 shows that the integral on the right-hand side of (5.9) gives the (suitably normalized) special value of the classical L-function of f at k/2. Thus we have established the Mazur-Tate-Teitelbaum conjecture for f, if we define the L-invariant of f to be −L_{f}.

Note that this L-invariant is obviously “local” in the original sense of [16], in so far as it is invariant under twisting by characters of conductor prime to p that are trivial on p. In fact, it follows immediately from Breuil’s construction that L_{f} is invariant under arbitrary twisting of f by Dirichlet characters. If f is any newform of even weight k that is special at p, then the twist f_{ϵ} of f by a Dirichlet character ϵ will satisfy the hypothesis of Corollary 5.7 for some choice of ϵ. Assuming that we can choose ϵ such that furthermore L(f_{ϵ}, 1) ≠ 0, we see that the value of L_{f} can be “detected” via the Mazur-Tate-Teitelbaum formula (5.9) (applied to f_{ϵ}). This allows a comparison between the Breuil-Darmon-Orton definition of L_{f} and other definitions of L_{f} with respect to which the Mazur-Tate-Teitelbaum conjecture has been proved – namely, the “Coleman L-invariant”, in terms of which the Mazur-Tate-Teitelbaum conjecture has been proved by Stevens, and the “Fontaine L-invariant”, in terms of which the Mazur-Tate-Teitelbaum conjecture has been proved by Kato-Kurihara-Tsuji. (The introduction to [7] provides a useful survey of these definitions of the L-invariant, and the various means of comparing them.)^{8}

8In fact, it is an easy consequence of Colmez [8] and of Kato’s explicit reciprocity law for the p-adic L-function of f that L_{f}, as defined above, agrees with the Fontaine-Mazur definition of the L-invariant of f via the local-at-p Galois representation attached to f [15].

References

1. Y. Amice, J. Vélu, Distributions p-adiques associées aux séries de Hecke, Astérisque 24/25 (1975), 119–131.

2. L. Berger, C. Breuil, Représentations cristalline irréductible de GL_{2}(ℚ_{p}), preprint (2004).

3. N. Bourbaki, Elements of Mathematics. Topological Vector Spaces. Chapters 1-5, Springer-Verlag, 1987.

4. C. Breuil, Invariant L et série spéciale p-adique, preprint (2002).

5. C. Breuil, Série spéciale p-adique et cohomologie étale complétée, preprint (2003).

6. W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft May 7, 1993.

7. P. Colmez, Invariant L et dérivée de valeurs propres de Frobenius, preprint (2003).

8. P. Colmez, Une correspondance de Langlands locale p-adique pour les représentations semistable de dimension 2, preprint (2004).

9. H. Darmon, Integration on H_{p} × H and arithmetic applications, Ann. of Math. 154 (2001), 589–639.

10. M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, to appear in Memoirs of the AMS.

11. M. Emerton, Jacquet modules for locally analytic representations of reductive groups over p-adic local fields, preprint (2004).

12. M. Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, to appear in Invent. Math. (2004).

13. M. Emerton, Locally analytic representation theory of p-adic reductive groups: A summary of some recent developments, preprint (2004).

14. E. Grosse-Klönne, Integral structures in automorphic line bundles on the p-adic upper half plane, preprint (2003).

15. B. Mazur, On monodromy invariants occuring in global arithmetic, and Fontaine’s theory, p-adic monodromy and the Birch–Swinnerton-Dyer conjecture (Boston, MA, 1991), B. Mazur and G. Stevens eds., Contemp. Math., vol. 165, 1994, pp. 1–20.

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

17. Y. Morita, Analytic representations of SL_{2} over a 𝔭-adic number field II, Prog. Math., vol. 46, Birkhaüser, 1984, pp. 282–297.

18. L. Orton, An elementary proof of a weak exceptional zero conjecture, Canad. J. Math. 56 (2004), 373–405.

19. W. Schmid, Automorphic distributions for SL(2, ℝ), Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 345–387.

20. P. Schneider, Nonarchimedean functional analysis, Springer Monographs in Math., Springer-Verlag, 2002.

21. P. Schneider, J. Teitelbaum, Locally analytic distributions and p-adic representation theory, with applications to GL_{2}, J. Amer. Math. Soc. (2001).

22. P. Schneider, J. Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359–380.

23. J. Teitelbaum, Modular representations of PGL_{2} and automorphic forms for Shimura curves, Invent. Math. 113 (1993), 561–580.

24. M. Vishik, Nonarchimedean measures connected with Dirichlet series, Math. USSR Sb. (1976), 216–228.

Northwestern University Department of Mathematics 2033 Sheridan Rd. Evanston, IL 60208-2730, USA

E-mail address: emerton@math.northwestern.edu