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 underappropriate hypotheses. The problem of studying unitary completions has beenraised by Breuil in connection with his work on a possible “p-adic local Langlands
correspondence” for GL2 , and we relate our construction to certain conjectures of
Breuil for the group GL2 (Qp ). In particular, we show that the universal unitarycompletion of the locally analytic parabolic induction of a locally algebraic charactercoincides 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 GL2 (Qp ) obtained by p-adically completing the cohomology of classicalmodular curves. The mere existence of this representation implies that those locallyalgebraic parabolically induced representations of GL2 (Qp ) 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 inthis 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 givea simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture (in termsof 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 topologicallyirreducible 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 = GL2 (Qp ) 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 GL2 (Qp )” [4, 5].
In the second half of the paper we explain a representation theoretic point ofview on p-adic L-functions attached to modular forms. As an application, we provea 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] andOrton [18] in the weight two and higher weight cases respectively).
The two halves of the paper are more closely related than they might appear atfirst 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)
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MATTHEW EMERTON

argument of the second half. Conversely, the global results of the second half shedsome 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 reflexiveFréchet 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 completionof the contragredient G-representation on the strong dual Vb0 . (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 completionπ̂ 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 = GL2 (Qp ), and consider irreducible
admissible locally algebraic representations π whose smooth factor is principal seriesor special. In this case Breuil [4] has made a precise conjecture as to whether or notπ̂ 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 theJacquet module JB (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 andresults of [4]. The proof of Proposition 2.5 is given in Section 3.
Suppose now that the locally algebraic representation π of GL2 (Qp ) under consideration is of the form Wk−2 ⊗ πp , where Wk−2 is the contragredient to the (k − 2)nd
symmetric power of the standard representation of GL2 , and πp 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 aunitary Banach space representations H̃c1 (K p ) of GL2 (Qp ) (where K p is the tamelevel of the newform f giving rise to π), which has the property that π embeds intoH̃c1 (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 unitarycompletion of π, the existence of this embedding certainly implies that π̂ 6= 0. Furthermore, assuming that we can find a p-stabilized newform g attached to f (thenewform giving rise to π) that is of non-critical slope, Proposition 2.5 shows that theembeddings of π into H̃c1 (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).
In [5], Breuil has made a detailed investigation of the closure of the image of π inH̃c1 (K p ), in the case when πp 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

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

3

above). In Section 5 we give a proof of the Mazur-Tate-Teitelbaum exceptional zeroconjecture for such f using this description of the L-invariant, and the viewpointon 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 classicalautomorphic L-functions via automorphic distributions [19]. It is this lecture thatinspired 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 Qp -valued points of a connected, reductive, linear algebraic groupdefined over Qp . We also fix a p-adic local field K that will serve as our fieldof coefficients. All locally convex topological vector spaces will be assumed to be
defined over K (whether or not this is explicitly stated). We let OK denote thering of integers in K. We refer to [20] for an explanation of the terminology fromp-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 witha 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 aunitary 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 itexists, and (by abuse of language) sometimes refer to V̂ as the universal unitarycompletion of V . (There should be no confusion with the traditional notation forthe 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 propertyof 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 ,1 In the sense used in [20] – i.e. a lattice in a K-vector space is a spanning O -submodule.
K
Note that, with this usage, a lattice need not be separated.

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MATTHEW EMERTON

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).
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 topologyof V̂ ). Conversely, given such a lattice L, the completion of V with respect to thegauge of L is a universal unitary completion of V . 
It is quite possible for a G-representation on a non-zero convex space V toadmit 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 auseful necessary and sufficient condition for this to be the case.
Let C(G, K) denote the space of continuous K-valued functions on G, equippedwith its right regular G-action, and let C b (G, K) denote the G-invariant subspace ofC(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 endowedwith the topology induced by sup-norm).
Proof. Let u0 be a non-zero element of the dual space U 0 . By Frobenius reciprocity,u determines a non-zero continuous G-equivariant map U → C(G, K); explicitly, anelement u ∈ U maps to the function fu : g 7→ hu0 , gui 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 6= 0, then apply Lemma 1.4 to V̂ to obtain a non-zerocontinuous 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 seethat V admits a G-invariant norm. In particular, if V has a central character, thenwe 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 subgroupof G, with unipotent radical N and Levi quotient M , and let JP (V ) denote theJacquet 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 N0
be some compact open subgroup of N , let ZM denote the center of M , and write+
ZM:= {z ∈ ZM | zN0 z −1 ⊂ N0 }. Let δ denote the modulus character of P (which

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

5

is trivial on N , and so induces a character of M = P/N, which we also denote by
δ; concretely, δ(m) = [mN0 m−1 : N0 ]).
Lemma 1.6. Let χ be a locally analytic K-valued character of ZM . If, in the above
situation, the χ-eigenspace of JP (V ) is non-zero, and if V̂ 6= 0, then | χ(z)δ −1 (z) |≤+1 for all z ∈ ZM.
Proof. As already noted, the assumption that V̂ 6= 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 classesof G-invariant closed and bounded (equivalently compact, since V is reflexive [20,Prop. 15.3 (iii)]) OK -submodules of Vb0 contains a maximal element. (We equip Vb0
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 Vb0 is again continuous, and so wemay reverse the roles of V and Vb0 . If Vb0 admits a universal unitary completion,then Lemma 1.7 shows that V admits a compact G-invariant OK -submodule B
that is maximal (up to commensurability); equivalently, B absorbs any G-invariant
bounded OK -submodule of V .
Definition 1.8. In the preceding situation, define V b := K ⊗OK B, equipped withthe 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)

Vb0 → (Vb0 )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 Vb0 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)

Vb0 → (V b )0b .

The results of [22] show that (V b )0b is a Banach space, with unit ball the polar of B,
and that V b is naturally identified (via double duality) with ((V b )0b )0 . Thus (1.10)
and (1.12) are mutually dual. The preimage of the unit ball of (V b )0b under (1.12)
is the polar of B in Vb0 , which by construction of B (and the theory of bipolarity) coincides with the preimage of the unit ball of (Vb0 )ˆ under (1.9). Thus (1.12)
and (1.9) are naturally isomorphic. 

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MATTHEW EMERTON

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 oftranslates 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 equippedwith a continuous G-action. However, if G0 is a compact open subgroup of G, thenany continuous semi-norm on V is bounded above by a G0 -invariant continuoussemi-norm [10, Lem. 6.5.4]. Thus if V admits a generating semi-norm q, we may
assume that q is G0 -invariant, and so conclude (since G/G0 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 Vb0 admits a universal unitarycompletion (and so V b is defined).
Proof. Let q be a generating semi-norm, and set Bq = {v ∈ V | q(gv) ≤ 1 for all g ∈G}. By definition Bq is a bounded G-invariant OK -submodule of V . Since theG-action on V is continuous, Bq is evidently also closed. If B is any bounded
G-invariant OK -submodule of V , and we choose a ∈ K × so that | s(v) | ≤ | a |for all v ∈ B, then clearly a−1 B ⊂ Bq . Thus Bq absorbs all bounded G-invariant
OK -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 generatingsemi-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 thatdefines the topology of V restricts to a set of continuous semi-norms on W thatdefines 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 OK [G]-module.
Clearly V contains a finite type lattice if and only if it is finitely generated (as aK[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 latticeof 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 isadmissible 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 6= 0 is that this central character be unitary.

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

7

Proposition 1.18. Let V be an irreducible admissible locally algebraic representa∼tion of G, and write V −→ U ⊗ W , where U is an admissible smooth representationof 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 6= 0 if and only if the central character of V is unitary. (Recall that U is said to be absolutely cuspidal if itcannot be embedded into the induction from P of a smooth M -representation, forany 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 andthat 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 fv : g 7→ hǔ ⊗ w̌, gvi. Note that (1.19) is
non-zero, since ǔ ⊗ w̌ 6= 0.
Under the hypothesis of the proposition, we claim that (1.19) factors through theinclusion of C b (G, K) into C(G, K), and thus induces a map V → C b (G, K) (which isnecessarily continuous, since V is equipped with its finest convex topology). Indeed,
it suffices to show that fv is bounded when v is of the form u ⊗ w for some u ∈ U
and w ∈ W . In this case we see that fv (g) = hǔ, guihw̌, gwi for all g ∈ G. Since U
is absolutely cuspidal, the matrix coefficient hǔ, gui is compactly supported modulo
the center of G [6, Thm. 5.3.1]. Thus the same is true of fv . Since V has a unitarycentral character, we conclude that fv 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 JP (V ) = 0 for all properparabolic subgroups P of G. The preceding proposition thus shows that in the extreme case when V has no non-vanishing Jacquet modules, the necessary conditionof 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 = GL2 (Qp ), will be considered in detail in thenext section.
Example B. In [4, 5], the spaces O(k) and O(k, L) are defined (for k ≥ 2 andL ∈ K). The first is the space of rigid analytic functions on the p-adic upper halfplane Ω := Cp \ Qp , equipped with a “weight k” action of GL2 (Qp ). The second
is the space of log-rigid analytic functions on Ω, equipped with a “weight 2 − k”
action of GL2 (Qp ); one takes the branch of the p-adic logarithm to be logL , wherelogL (p) = L. These are nuclear (and so reflexive) Fréchet spaces equipped with
continuous actions of GL2 (Qp ), and for fixed k and L, differentiating k − 1 times
induces a short exact sequence(1.20)

0 → Wk−2 ⊗K | det |−(k−2)/2 → O(k, L) → O(k) → 0;

2 As 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.

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MATTHEW EMERTON

here Wk−2 is the contragredient to the (k − 2)nd symmetric power of the standard
two dimensional representation of GL2 (Qp ), 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 Qp in Cp satisfying condition (20)of [4, p. 34], then the formula q(F ) := ||F|ΩU || (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)binduced by the surjection of (1.20) is a closed embedding;3 when k = 2 it is surjective∼(with kernel equal to W0 −→ 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
T in B, so that B = T N . Let B be the opposite Borel to B, chosen sothat B B = T . Let χ : T → K × be a locally analytic character of T , which we
also regard as a character of B via the natural projection B → T , and consider the
locally analytic parabolic inductionGχ := {f ∈ C la (G, K) | f (bg) = χ(b)f (g) for all b ∈ B, g ∈ G}.
IndB

The right regular G-action endows IndGχ with a continuous action of G.
B
Proposition 1.21. The G-representation IndG
χ is a strongly admissible locallyBanalytic representation of G (in the sense of [21]), whose strong dual (which isthus a nuclear, and hence reflexive, Fréchet space) admits a generating semi-norm.
Proof. If G0 is a hyperspecial maximal compact open subgroup of G, then G =BG0 , and so restriction of functions from G to G0 induces a G0 -equivariant closed
G
embedding IndB
χ → C la (G0 , K). Since C la (G0 , K) is a strongly admissible G0 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 IndGχ.
B
Since a locally analytic G-representation is strongly admissible if and only if itis strongly admissible as a G0 -representation (again by definition), we see thatIndG
χ is a strongly admissible locally analytic G-representation. In particular, it
B
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 anaffinoid rigid analytic space over Qp isomorphic to a closed polydisk, and φ is a∼locally analytic isomorphism φ : X −→ X(Qp ). 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 Ccla (N, K) (since C la (X, K) is a closed subspace of Ccla (N, K)). If t ∈ T , thenconjugation by t induces a locally analytic automorphism of N , which we denote
3 That one obtains an injection follows from the fact that W−(k−2)/2 admits nok−2 ⊗K | det |GL2 (Qp )-invariant norm, when k > 2. That this injection is in fact a closed embedding is provedin [5, Thm. 3.3.3].

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

9by σt . Thus (tXt−1 , X, φ ◦ σt−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-subspaceof Ccla (N, K) by C an (tXt−1 , K). We remark that the theory of roots shows thatthe set of neighbourhoods of the identity {tXt−1 }t∈T is cofinal in the set of allneighbourhoods of the identity.
The discussion of [13, §5] shows that Ccla (N, K) embeds as a closed subspace ofχ, and hence that C an (X, K) embeds as a BH-subspace of IndG
χ. One checks
IndG
B
B
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 theGχ.) The final sentence of the preceding paragraph,ambient G-representation IndB
together with (1.22) and the fact that G acts transitively on the space B\G, showsP
G
G
that IndB
χ = g∈G gC an (X, K). Thus the gauge of the polar in (IndBχ)0b of theGunit ball C an (X, OK ) of C an (X, K) is a generating semi-norm for (IndBχ)0b . 
2. Completing parabolically induced representations of GL2 (Qp ).
Take G = GL2 (Qp ), and K as in Section 1. Let B denote the standard upper
triangular Borel in GL2 (Qp ), let T denote the diagonal maximal torus in B, andT
let B denote the lower triangular Borel (so that B B = T ). Let χ be a locallyalgebraic K-valued character of T of the form χ = θψk−2 , for some k ≥ 2, where θ issmooth and ψk−2 is the highest weight of the irreducible representation Wk−2 of G(the contragredient to the (k−2)nd symmetric powerof the standard representationaof G) with respect to B. (Explicitly, ψk−2 :7→ d−(k−2) .) Let I(χ) denotedthe locally algebraic induction of χ from B to G; alternatively, I(χ) is the tensor
product over K of Wk−2 and thesmooth induction of θ from B to G.
a
Write θ in the form θ :7→ θ1 (a)θ2 (d), for two smooth characters θ1 andd×θ2 of Qp . Write α = θ1 (p) and β = pθ2 (p). The central character of I(χ) takesp
to αβp−(k−1) .
paLet θ0=| d/a | θ2 (a)θ1 (d), and define χ0 to be the locally algebraicdcharacter χ0 = θ0 ψk−2 . (Note that the α and β for χ0 are equal to the β and α
for χ.)
If θ1|Z×= θ2|Z×and α/β = p±1 , then we say that we are in the special case. Inppthis 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 Wk−2 , andthe quotient is isomorphic to a twist of Wk−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 Wk−2 , and the kernel of the
projection onto this quotient is isomorphic to π(χ0 ). 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 6= 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

10

MATTHEW EMERTON∼

intertwining operators shows that in this case there is an isomorphism π(χ) −→π(χ0 ).
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 |=| pk−1 | .
(ii) Each of α and β lie in OK .
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 isclear that (i) implies (ii). Thus we assume for the remainder of the proof that weare in the principal series case. We will compute JB (I(χ)), and apply Lemma 1.6.
∼θ)sm ⊗K Wk−2 (where the left-hand factor is the smooth
Since I(χ) −→ (IndG
B
parabolic induction of θ from B to G), it follows from [11, Props. 4.3.4 (ii), 4.3.6]∼θ)sm ) ⊗that there is an isomorphism of T -representations JB (I(χ)) −→ JB ((IndGBψk−2 . The theory of Jacquet modules of smooth representations [6] shows thatG
JB ((IndBθ)sm ) is two dimensional, and that it has a non-zero θδ-eigenspace, andalso a non-zero θ0 δ-eigenspace. (Below we will give a direct proof that these
eigenspaces are non-zero.) Thus JB (I(χ)) has non-zero χδ- and χ0 δ-eigenspaces.

1 x
If we take the compact open subgroup N0 of Lemma 1.6 to be {
|x∈
0 1


p 0
Zp }, then we see that the matrix t =conjugates N0 into itself. Applying
0 1
Lemma 1.6 twice, with the character labelled χ in the statement of that result
taken to be χδ and χ0 δ in turn, we find that | α |=| χ(t) | ≤ 1 and also that| β |=| χ0 (t) | ≤ 1. This proves the lemma.
Gθ)sm ) is nonWe finish with a direct proof that the θδ-eigenspace of JB ((IndB
zero. 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 isG
a natural B-equivariant embedding Ccsm (N, K(θ)) → IndB
θ. (The target denotes
the locally analytic parabolic induction of θ from B to G.) Since θ is a smoothGcharacter of T , this embedding factors through the inclusion (IndBθ)sm → IndGθ
Bof the smooth parabolic induction into the locally analytic parabolic induction.
Applying the left-exact functor JB , and taking into account [11, Lemma 3.5.2], weG
see that the θδ-eigenspace of JB ((IndBθ)sm ) is non-zero. The theory of intertwining∼
operators provides an isomorphism (IndGθ)sm −→ (IndGθ0 )sm . Replacing θ by θ0BB0in the preceding argument shows that the θ δ-eigenspace of JB ((IndGθ)sm ) is alsoBnon-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.

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

11This conjecture has been proved in the special case by Teitelbaum [23] andGrosse-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 B to G is a unitary
Banach space representation of G containing I(χ). (Because χ is unitary, the
sup norm over B\G is a well-defined G-invariant norm on I cont (χ).) Similarly, if∼| α |=| pk−1 |, then β is a p-adic unit, and we may embed I(χ) −→ I(χ0 ) in thecont0unitary representation I(χ ).
Given χ as above, we let I la (χ) denote the locally analytic induction of χ from B
to G; by Proposition 1.21 this is a strongly admissible locally analytic representationof G. It contains I(χ) as a closed subrepresentation. We may identify I la (χ) withthe space of locally analytic functions f : Qp → K, satisfying an appropriate
regularity condition at infinity, with the G-action defined by

(2.3)

a b
c d


f (x) =

θ1 (a + cx)θ2 (ad − bc)
b + dx(a + cx)k−2f();θ2 (a + cx)(ad − bc)k−2 a + cx

I(χ) is then identified with the subspace of functions that are locally polynomialof 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 thatfurthermore | α |>| pk−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 inductionof χ – see p. 25, Def. 4.2.4, and Thm. 4.3.1 of [2]), using which they are able togive a direct proof of this result [2, p. 42]. Nevertheless, it seems worthwhile togive our original proof, since it is more representation theoretic in nature than thatof [2], and it exhibits an interesting relationship between the classical techniqueof Amice-Vélu and Vishik and the representation theory of parabolically induced
representations.

12

MATTHEW EMERTONIn 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 theother. 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 someunitary character ω of GL2 (Qp ). The results of Morita [17] then show that π la (χ) isdual to the representation O(k) ⊗K ω −1 , where O(k) is the GL2 (Qp )-representationconsidered in Section 1.
Dualizing the closed embedding (2.4) yields a surjection
O(k) ⊗K ω −1 → π(χ)0b ,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 GL2 (Qp ) considered in [23]. Proposition 2.5 implies that this surjection
induces an isomorphism∼

O(k)b ⊗K ω −1 −→ π(χ)0bb ,a result originally proved in [23] and [14]. As we noted above, in those papers it
is furthermore proved that O(k)b 6= 0, and thus that Conjecture 2.2 holds in thespecial 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 thesense of [22]). For each k > 2 and L ∈ K, dualizing the surjection of (1.20), andtwisting by ω, yields a closed embedding of strongly admissible locally analytic
GL2 (Qp )-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 GL2 (Qp )-representations∼π̂(χ) −→ (O(k)b ⊗K ω −1 )0 → (O(k, L)b ⊗K ω −1 )0 .
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 GL2 (Qp )-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 sourcewith the space of locally analytic vectors in the target.
(ii) The unitary Banach space GL2 (Qp )-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 topologicallyirreducible [4, Thm. 1.1.9]. In fact, Colmez [8] has recently proved that π̂(χ, L)

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

13

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 GQp with
Hodge-Tate weights 0, k − 1 that are not potentially crystalline. (Such representations are classified by their associated Weil group representation and Hodge-Tateweights, 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 toobtain for each L ∈ K a closed embedding of strongly admissible locally analytic
GL2 (Qp )-representations π la (χ) → π la (χ, L) and a map π̂(χ) → π̂(χ, L) of unitary
Banach space GL2 (Qp )-representations. In this case, though, this latter map is nota surjection, but rather is a closed embedding (since the map O(2, L)b → O(2)bis not a closed embedding, but is rather a surjection). When k = 2, the analogueof 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 aclosed 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 Qp ,having a pole of order at most k − 2 at infinity, and with the GL2 (Qp )-action being
given by the formula (2.3). We may identify π la (χ, L) with the space of locallyanalytic functions on Qp , whose singularity at infinity is at worst the sum of a poleof order k − 2 and a term of the form P (x) logL (x), where P (x) is a polynomialof degree at most k − 2 (with | x |>> 0). The GL2 (Qp )-action is given by amodification of the formula (2.3): if a function
 f (x) has a logarithmic singularitya bof the form P (x) logL x for | x |>> 0, thenf (x) is defined by adding thec dfollowing term to the expression in (2.3):
(2.7)

−

1 θ1 (a + cx)θ2 (ad − bc)
b + dxad − bc(a + cx)k−2
P() logL ().
2 θ2 (a + cx)(ad − bc)k−2 a + cx
(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(χ0 ) discussed above,shows that both π la (χ) and π la (χ0 ) map into π̂(χ). If π la (χ, χ0 ) denotes the sumof π la (χ) and π la (χ0 ) amalgamated along their common closed subspace π(χ), thenthese maps glue to give a map(2.8)

π la (χ, χ0 ) → 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.

14

MATTHEW EMERTON(ii) The unitary Banach space GL2 (Qp )-representation π̂(χ) is admissible.
Part (i) of this conjecture in particular implies that π̂(χ) 6= 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 theunitary 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 GQp for which the associatedrepresentation 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 unitarycompletion. 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 whichπ(χ) and π la (χ) are replaced by I(χ) and I la (χ). In the principal series case thisis simply a change of notation. In the special case, the representations π(χ) andπ 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 characterχ = θψk−2 of T as in Section 2,
I la (χ) := {f ∈ C la (G, K) | f (bg) = χ(b)f (g) for all b ∈ B, g ∈ G}.
(As always, we regard χ as a character of B via the natural projection B → T .)The support of any element of I la (χ) is clearly invariant under left-multiplication byelements of B, and so may be identified with a closed subset of the quotient 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 B\G.
If Ω is an open subset of B\G, then we let I la (χ)(Ω) denote the closed subspace ofI la (χ) consisting of functions whose support lies in Ω.
Let N denotethe unipotent radical of B; we identify N with Qp via the map
1 x
x 7→ nx :=. Write N0 := {nx | x ∈ Zp } ⊂ N . The embedding N → B\G
0 1
realizes the source as an open subset of the target, and thus realizes N0 as a compact
open subset of the target.
If for any integer r ≥ 1 we seta b
G0 (r) := {∈ GL2 (Zp ) | c ≡ 0 mod pr },c d
4 Actually we have omitted the crystalline representation with Hodge-Tate weights 0, k − 1 withk > 2 for which ap (the trace of Frobenius) is equal to p(k−2)/2 + pk/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].

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

15

then G0 (r) is a compact open subgroup of G that admits 
an Iwahori decomposition
1 0
with respect to B and B. Namely, if we write N (r) = {| x ∈ pr Zp } and
x
1


T
a 0
T0 := GL2 (Qp ) T = {| a, d ∈ Z×p }, then G0 (r) = N0 T0 N (r).

0 d
a
a 0
Write T + := {∈ T | ∈ Zp }. If t ∈ T + , then one sees that tN0 t−1 ⊂0 ddN0 , while t−1 N (r)t ⊂ N (r) for any r ≥ 1. Taking into account these formulas, theinclusion T0 ⊂ T + , and the fact that G0 (r) is a subgroup of G for each r ≥ 1, we
see that if for any r ≥ 1 weTwrite G+ (r) := N0 T + N (r), then G+ (r) is a submonoidof G. Write B + := G+ (r) B = N0 T + ; it is a submonoid of B, independent of the
choice of r, that generates B as a group.
laOne sees that BN0 G+ (1) ⊃ BN0 , and thus that IT(χ)(N0 ) is a G+ (1)-invariant
closed subspace of I la (χ). Write I(χ)(N0 ) := I(χ) I la (χ)(N0 ) (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 LG (I la (χ), U ) → LG+ (r) (I la (χ)(N0 ), U )and LG (I(χ), U ) → LG+ (r) (I(χ)(N0 ), 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 B\G, then the restriction F|Ω of F to Ω also lies in V . (The two
examples of Tsuch a V that we have in mind are I la (χ) itself and I(χ).) Then
V (N0 ) := V I la (χ)(N0 ) is G+ (1)-invariant, and we will prove that the restriction
map
(3.2)

LG (V, U ) → LG+ (r) (V (N0 ), U )is an isomorphism for any r ≥ 1.
Since the G-translates of N0 cover B\G, and since V is local, we see that V (N0 )generates V as a G-representation. The map (3.2) is thus certainly injective, andwe 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[N0 ]V (N0 ) → V (N ) is an isomorphism, and hence (taking into account the fact thatB + generates B as a group) restriction induces an isomorphism
(3.3)

∼

LB (V (N ), U ) −→ LB + (V (N0 ), U ).

Let LB (V (N ), U )◦ denote the subspace of the source of (3.3) consisting of maps
φ satisfying the following condition:(3.4) For each n ∈ N , there exists a neighbourhood Ω of the identity in N suchthat, for any F ∈ V whose support lies in Ω, the element nF lies in V (N ), andφ(nF ) = nφ(F ).
Note that LG+ (r) (V (N0 ), U ) is a subspace of the target of (3.3).
Claim 3.5. The image of LG+ (r) (V (N0 ), U ) under the inverse of the isomorphism (3.3) (which consists of elements of the source whose restrictions to V (N0 )are N (r)-equivariant) is contained in LB (V (N ), U )◦ .

16

MATTHEW EMERTON

Choose t ∈ T + such that n ∈ tN (r)t−1 , and take Ω = tN0 t−1 . Since tG0 (r)t−1 isa subgroup of G, we see that tN0 t−1 , regarded as a compact open subset of B\G, isinvariant under the action of tN (r)t−1 . Thus if F is any element of V with support
contained in Ω, the same is true of nF . In particular, we see that nF ∈ V (N ), as
required.
Now let φ be an element of LB (V (N ), U ) whose restriction to V (N0 ) is N (r)equivariant. If the support of F is contained in Ω, we compute thatφ(nF ) = tφ(t−1 ntt−1 F ) = t(t−1 nt)φ(t−1 F ) = nφ(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 N0 , since F was assumed to besupported on Ω, while t−1 nt lies in N (r), and φ is assumed to be N (r)-equivariant
on V (N0 ).) This proves the claim.
Claim 3.6. The image of the restriction map LG (V, U ) → LB (V (N ), U ) is equal
to LB (V (N ), U )◦ .
Since any element of LG (V, U ) is G+ (r)-equivariant when restricted to V (N0 ),it follows from Claim 3.5 that the restriction to V (N ) of any element of LG (V, U )lies in LB (V (N ), U )◦ . We must show that any element of LB (V (N ), U )◦ may be
extended to an element of LG (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 Weylgroup of G, then since N and N w together cover B\G, and since V is local, theclosed embeddings of V (N ) and wV (N ) into V induce a continuous surjection(3.7)

V (N )MwV (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 acontinuous K-linear surjectionM(3.8)
gV (N ) → V.
g∈G(More canonically, this is the G-equivariant map K[G] ⊗K V (N ) → V inducedby 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 ∈ LB (V (N ), U )◦ , and for each g ∈ G, define φg : gV (N ) → U via
φg (gF ) = gφ(F ). The maps φg induce a continuous K-linear map
(3.9)

M
g∈G

φg :

M

gV (N ) → U.

g∈G(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 thatPif (g1 F1 , . . . , gl Fl ) is
any finite sequence with gi ∈ G and Fi ∈ V (N ) such that i gi Fi = 0 (the sum

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

17

P?
taking place in V ), then i gi φ(Fi ) = 0. Since V is local, it suffices to show thateach point x of B\G has a compact open neighbourhood Ωx such that if Ω0x is aneighbourhood of x contained in Ωx thenX?
(3.10)gi φ(Fi|Ω0 g−1 ) = 0.
x iiIndeed, `we can then partition B\G intosay
Ps a disjoint union of such neighbourhoods,Ps
B\G = j=1 Ω0xj , and writing Fi = j=1 Fi|Ω0 g−1 , we conclude that i gi φ(Fi ) =
xj i
Ps P
P Ps
−1 ) =
g
φ(F
g
φ(F
0
0
i
i
i
j=1
j=1
i
i|Ω g
i|Ω g −1 ) = 0, as required.
xj

xj

i

i

If x = Bg for some g ∈ G, then replacing (g1 F1 , . . . , gl Fl ) by (gg1 F1 , . . . , ggl Fl ),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 → B\G. If e is
not in the domain of gi Fi , for some i, then taking the sought-after neighbourhoodΩe of e to be sufficiently small, we will have that (gi Fi )|Ωe = 0, or equivalently,
Fi|Ωe g−1 = 0, and hence the ith summand will contribute zero to the sum (3.10).
i
Thus we may furthermore assume that e is in the support of each gi Fi for each i.
Also, by choosing Ωe so as to be contained in N , we may assume without loss ofgenerality that gi Fi is supported on N .
Since the support of Fi lies in N by assumption, we see that gi ∈ BN = N B, andso we may write gi = ni bi for some ni ∈ N , bi ∈ B. Take Ωe to be the intersectionsof the neighbourhoods Ω corresponding to the elements n−1
in condition (3.4).
i
Then
−1
φ(Fi|Ωe g−1 ) = φ(gi−1 (gi Fi )|Ωe ) = φ(b−1i ni (gi Fi )|Ωe )
i

−1
−1
= b−1
i ni φ((gi Fi )|Ωe ) = gi φ((gi Fi )|Ωe ).

(Here the third equality follows from the fact that φ is B-equivariant, together withhypothesis (3.4).) ThusX
XX
gi φ(Fi |Ωe gi−1 ) =gi gi−1 φ((gi Fi )|Ωe ) = φ((gi Fi )|Ωe ) = 0,
i

i

i

P

since i gi Fi = 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, andfix some integer r ≥ 1. Restriction of continuous linear maps yields the following
commutative diagram:(3.11)LG (I la (χ), U )/ LG (I(χ), U )
LG+ (r) (I la (χ)(N0 ), U )/ LG+ (r) (I(χ)(N0 ), 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.

18

MATTHEW EMERTON

We first recall an important result due to Vishik and Amice-Vélu (Lemma 3.14
below).
Let A denote the Tate
P∞ algebra of rigid analytic functions on the closed unit disk
over K; thus A := { i=0 ai xiP| ai ∈ K, limi→∞ | ai |= 0}. Let ||–||A denote the∞usual Gauss norm on A; i.e. || i=0 ai xi ||A = sup{| ai |}i≥0 . With respect to thisnorm 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 (Zp , K) denote the closed subspace ofC la (Zp , K) consisting of functions that are locally polynomial of degree ≤ k − 2.
If F ∈ A, b ∈ Zp , and m ≥ 0, we write F|b+pm Zp to denote the element of
C la (Zp , K) defined by F on the compact open subset b + pm Zp of Zp , and by zeroat all other points of Zp . If F ∈ A≤k−2 then F|b+pm Zp lies in C lp≤k−2 (Zp , K).
Definition 3.12. If φ ∈ L(C la (Zp , K), U ) (respectively φ ∈ L(C lp≤k−2 (Zp , 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 ∈ Zp , and all m ≥ 0, we havex−b||φ F ( m ) ||U ≤ C|α|−m ||F (x)||Apfor some (equivalently any) choice of norm ||–||U defining the topology on U andsome positive constant C (both independent of the choice of F , b, or m). We let
L(C la (Zp , K), U )α−temp (respectively L(C lp≤k−2 (Zp , K), U )α−temp ) denote the subspace of L(C la (Zp , K), U ) (respectively of L(C lp≤k−2 (Zp , 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 (Zp , K) to C lp≤k−2 (Zp , K) induces a map(3.13)

laα−temp→ L(C lp≤k−2 (Zp , K), U )α−temp
L(C (Zp , K), U )Lemma 3.14. If α ∈ K × satisfies | α |>| pk−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], andalso 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 N0 induces isomorphisms(3.15)∼∼I la (χ)(N0 ) −→ C la (N0 , K) −→ C la (Zp , K)

(where the second isomorphism is induced via the identification of Zp and N0 given
by x 7→ nx ). Restricting (3.15) to I(χ)(N0 ) induces an isomorphism(3.16)∼

I(χ)(N0 ) −→ C lp≤k−2 (Zp , K)

The isomorphism (3.15) allows us to transport the G+ (1)-action on the source toaba G+ (1)-action on C la (Zp , K). Concretely, if F (x) ∈ C la (Zp , K) and g =∈c d
G+ (1), then one computes for any x ∈ Zp that

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

19(3.17)
b + dx
6∈ Zp
 0 ifa + cx(gF )(x) =b + dxb + dxθ1 (a + cx) θ2 (ad − bc) (a + cx)k−2
F() if∈ Zp .
θ2 (a + cx) (ad − bc)k−2 a + cx
a + cxSince I(χ) is a G-invariant closed subspace of I la (χ), the isomorphism (3.16) showsthat C lp≤k−2 (Zp , K) is a G+ (1)-invariant closed subspace of C la (Zp , 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 G0 (1) on A. Namely, if F (x) ∈ A anda b∈ G0 (1), then we defineg=c d(3.18)(gF )(x) = (a + cx)k−2

θ2 (ad − bc)
b + dxF()(ad − bc)k−2 a + cx

Note that the norm ||–||A is invariant under this G0 (1)-action, and that A≤k−2 isa G0 (1)-invariant
subspaceof A. Comparing formulas (3.17) and (3.18), we finda bthat if g =∈ G0 (1) and F (x) ∈ A, thenc d
(3.19)g(F (x)|Zp ) =θ1 (a + cx)(gF (x))|Zpθ2 (a + cx)

for any x ∈ Zp .
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 (Zp , K) or C lp≤k−2 (Zp , K); correspondingly, let AC 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|Zp ))||U ≤ C||F ||A for all
F ∈ AC and b ∈ B + . (Here ||–||U is some fixed choice of norm defining the topologyof U .)
 mp−b
Proof. Write B 0 := {| m ≥ 0, b ∈ Zp }; note that B 0 is a submonoid of
0
1
B + . If F ∈ AC , then we compute using (3.17) that
 m

x−b
p
−b
F (x)|Zp = αm F ( m )|b+pm Zp .
01p
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)
for all F ∈ A and b0 ∈ B 0 .

||φ(b0 (F|Zp ))||U ≤ C||F ||A20

MATTHEW EMERTON

Clearly (ii) implies condition (3.21). Conversely, suppose that (3.21) holds. Any0element of B + may be written in 
the form
 zb t0 , where z lies in the center of G,a 0b0 ∈ B 0 , and t0 ∈ T0 . Writing t0 =, and taking into account (3.21), (3.19),
0 d
the fact that T0 acts isometrically on A, and that χ|ZG is unitary by assumption,
we deduce that||φ(zb0 t0 (F|Zp ))||U = ||χ(z)θ1 (a)
φ(b0 (t0 F )|Zp )||U = ||φ(b0 (t0 F )|Zp )||U
θ2 (a)≤ C||t0 F ||A = ||F ||A

for any F ∈ AC . 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 aunitary B + -action. Then LB + (C, U ) ⊂ L(C, U )α−temp .
Proof. Fix a B + -invariant norm ||–||U defining the topology of U . If φ ∈ LB + (C, U )then the map AC → U defined by F 7→ φ(F|Zp ) is continuous, and so there is apositive constant C such that ||φ(F|Zp )||U ≤ C||F ||A for all F ∈ AC . For anyb ∈ B + , F ∈ AC , we compute that||φ(bF|Zp )||U = ||bφ(F|Zp )||U = ||φ(F|Zp )||U ≤ C||F |||Zp(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 witha unitary G-action, and that | α |<| pk−1 |. If r is chosen so that the conductor of each of θ1|Z×and θ2|Z×divides pr , then the map (3.13) (which is an isoppmorphism by Lemma 3.14) restricts to an isomorphism LG+ (r) (C la (Zp , K), U ) →LG+ (r) (C lp≤k−2 (Zp , K), U ).
Proof. Since B + ⊂ G+ (r), Lemma 3.22 gives the inclusions
lalaα−tempLG+ (r) (C (Zp , K), U ) ⊂ L(C (Zp , K), U )and
LG+ (r) (Clp≤k−2(Zp , K), U ) ⊂ L(C lp≤k−2 (Zp , K), U )α−temp .

The isomorphism of Lemma 3.14 thus restricts to an injection(3.24)

lalp≤k−2(Zp , K), U ),LG+ (r) (C (Zp , K), U ) → LG+ (r) (Cwhich 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 (Zp , 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 )p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

21

for all F ∈ C la (Zp , K). Since φ is G+ (r)-equivariant, both sides of (3.25) restrict toφ on C lp≤k−2 (Zp , K). Thus to show that they coincide, it suffices to show that
F 7→ g −1 φ̃(gF )(3.26)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||φ̃(bF|Zp )||U ≤ C||F ||A(3.27)

for all b ∈ B + , F ∈ A. If g ∈ G+ (r) and b in B + , then we may write gb = b0 n forsome b0 ∈ B + , n ∈ N (r), and so compute||g −1 φ̃(gbF|Zp )||U = ||φ̃(gbF|Zp )||U = ||φ̃(b0 nF|Zp )||U
= ||φ̃(b0 (nF )|Zp )||U ≤ C||nF ||A = C||F ||A .
(Here we have used the G-invariance of ||–||U , as well as (3.19) applied to n and F(taking into account our choice of r), the inequality (3.27) applied to nF , and thefact that N (r) acts isometrically on A.) Again appealing to Lemma 3.20, we seethat (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 arrowof (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 Af , Apf , have their usual meaning. Fix aprime-to-p natural number N , and let K p be the following compact open subgroupof GL2 (Apf ):

 Y
a b
K ={
∈
GL2 (Z` ) | c ≡ 0 mod N, d ≡ 1 mod N }.
c d
p

`6=p

Let us embed C× into GL2 (R) in the usual way, by choosing a real linear isomor∼
phism C −→ R2 , and regarding C× as the subgroup of the group of real linear
automorphisms of C consisting of complex linear automorphisms.
If Kp is a compact open subgroup of GL2 (Qp ) then the quotient
Y (K p Kp ) := GL2 (Q)\ GL2 (A)/C× K p Kp
is a classical modular curve. For example, if we take Kp to be the kernel of the
reduction map GL2 (Zp ) → GL2 (Z/pn ), then this quotient is isomorphic to themodular curve that classifies elliptic curves over C with a Γ1 (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 Kp to be a compact open subgroup of GL2 (Zp ),∼and use the isomorphism GL2 (R)/C× −→ C\R together with strong approximationfor SL2 , we obtain an isomorphism∼

Y (K p Kp ) = GL2 (Q)\ GL2 (A)/C× K p Kp −→ Γ1 (N )\(X × GL2 (Zp )/Kp ),22

MATTHEW EMERTON




a b
where X denotes the upper half-plane of C \ R, and as usual Γ1 (N ) = {∈c d
SL2 (Z) | a ≡ d ≡ 1 mod N, c ≡ 0 mod N }.
Define M (K p ) = lim
lim Hc1 (Y (K p Kp ), OK /ps ), where s and Kp range over the
←− −→
s

Kp

directed sets of non-negative integers and compact open subgroups of GL2 (Qp ) respectively, and define H̃c1 (K p ) = K ⊗OK M (K p ). By construction, M (K p ) is ap-adically complete and separated OK -module. Thus H̃c1 (K p ) is naturally a KBanach 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 GL2 (Qp ) acts on right on the
projective system of arithmetic quotients Y (K p Kp ), and hence on M (K p ). Thus
GL2 (Qp ) acts unitarily on the Banach space H̃c1 (K p ). This representation is further equipped with actions of the Hecke algebra Hp of K p in GL2 (Apf ) and of theconnected component group π0 = {±1} of GL2 (R), that commute with one anotherand with the GL2 (Qp )-action.
Proposition 4.1. The unitary Banach space representation H̃c1 (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 H̃c1 (K p )± denote the ±1-eigenspace for π0 on H̃c1 (K p ). Each of thesespaces is again an admissible unitary Banach space representation of GL2 (Qp ),equipped with a commuting Hp -action.
Let Γ denote the normalizer in GL2 (Q) of K p . (We regard GL2 (Q) as being
embedded diagonally in GL2 (Apf ).) There is a natural map Γ → π0 × (Hp )× ×
GL2 (Qp ): 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 GL2 (Apf )that normalizes K p to its double coset in Hp , and the projection onto the third
factor is given by the inclusion GL2 (Q) → GL2 (Qp ). In this way we may regardH̃c1 (K p ), and hence its dual H̃c1 (K p )0 , as a Γ-module.
The following proposition describes the theory of modular symbols, as it applies
to the completed cohomology space H̃c1 (K p ). Let D0 denote the OK -module ofdegree zero divisors on P1 (Q). The natural action of GL2 (Q) on D0 induces inparticular an action of Γ on D0 . Recall that P1 (Q) appears as the boundary of thestandard completion of the upper half-plane X. For any compact open subgroup
Kp of GL2 (Qp ), we let X(K pSKp ) denote the usual compactification of Y (K p Kp );thus X(K p Kp ) = Y (Kp K p ) C(Kp K p ), where C(Kp K p ) is the set of cusps. (IfKp ⊂ GL2 (Zp ), then C(Kp K p ) = Γ1 (N )\(P1 (Q) × GL2 (Zp )/Kp ).)Proposition 4.2. The standard pairings between one dimensional compactly supported cohomology classes on the open curves Y (K p Kp ), and one dimensional homology classes taken relative to the set of cusps C(K p Kp ) on the compactified curves
X(K p Kp ), induces a Γ-equivariant OK -linear map D0 → (H̃c1 (K p ))0 , whose image
lies in the closed unit ball of the target (taken with respect to norm that is dual to
the norm on H̃c1 (K p ) given by the gauge of M (K p )).
Proof. Given elements a, b ∈ P1 (Q), the geodesic in the upper half-plane that joinsa and b gives rise to an element of H1 (X(K p Kp ), C(K p Kp ); OK /ps ), for any choiceof Kp and s, and we obtain a map D0 → H1 (X(K p Kp ), C(K p Kp ); OK /ps ) bymapping a − b to this element. The pairing between this relative homology groupp-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

23

and the corresponding compactly supported cohomology group thus induces a OK bilinear pairing
D0 × Hc1 (GL2 (Q)\ GL2 (A)/C× K p Kp , OK /ps ) → OK /psfor each choice of K p and s. The functorial nature of this pairing shows that it isΓ-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
OK -bilinear pairing
D0 × M (K p ) → OK ,and consequently a Γ-equivariant OK -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 fgives rise to an automorphic representation π = π∞ ⊗K πp ⊗K π p of GL2 (A) =GL2 (R) × GL2 (Qp ) × GL2 (Apf ), where the infinite component π∞ is defined over C,
while the finite component πp ⊗ π p is defined over K. Since f is assumed to haveptame conductor N , the space (π p )K is thus a one dimensional Hp -representation,p
on which H acts through a character that we will denote by λ. If N denotes the
conductor-N
part of the nebentypus of f , then the central character of πp takesp
to N (p)pk−2 .
p
If V is any representation of GL2 (Qp ), then for each choice of sign ±1, we let
V ±,λ denote the GL2 (Qp )-representation V equipped with commuting actions of π0and Hp , defined as follows: the non-trivial element of π0 acts through the chosen
sign ±1, while Hp acts through the character λ.
Proposition 4.3. For each choice of sign, we obtain a π0 × Hp × GL2 (Qp )equivariant embedding (Wk−2 ⊗K πp )±,λ → H̃c1 (K p )± , unique up to scalar multiples.
Proof. For each choice of compact open subgroup Kp of GL2 (Qp ), write VW̌k−2 :=GL2 (Q)\(GL2 (A)/C× K p Kp × W̌k−2 ); this is a local system of K-vector spaceson the modular curve GL2 (Q)\ GL2 (A)/C× K p Kp . If we write Hc1 (K p , VW̌k−2 ) :=Hc1 (GL2 (Q)\ GL2 (A)/C× K p Kp , VW̌k−2 ), then [12, (4.3.7)] shows that there is
lim
−→
Kp

an isomorphism
(4.4)

∼

Wk−2 ⊗K Hc1 (K p , VW̌k−2 ) −→ H̃c1 (K p )Wk−2 −lalg ,while classical Eichler-Shimura theory shows that the spacep

HomGL2 (Qp ) (πp , Hc1 (K p , VW̌k−2 )H =λ )is two dimensional, and that under the action of π0 it decomposes as the direct sumof one dimensional ±-eigenspaces. The proposition follows. The homomorphism Γ → π0 × (Hp )× × GL2 (Qp ) described above allows us todefine a Γ-action on (Wk−2 ⊗K πp )±,λ , and hence a contragredient Γ-action on((Wk−2 ⊗K πp )±,λ )0b and on ((Wk−2 ⊗K πp )±,λ )0bb .
5 i.e. conductor equal to N times a power of p

24

MATTHEW EMERTON

Corollary 4.5. For each choice of sign we obtain a Γ-equivariant map D0 →((Wk−2 ⊗K πp )±,λ )0bb .
Proof. Since the GL2 (Qp )-action on H̃c1 (K p )± is unitary, Lemma 1.11 shows thatthe dual of the map of Proposition 4.3 factors through the natural map±,λ 0((Wk−2 ⊗K πp )±,λ )0b)bb → ((Wk−2 ⊗K πp )of (1.10). Composing the resulting map (H̃c1 (K p )± )0 → ((Wk−2 ⊗K πp )±,λ )0b
b withthe map of Proposition 4.2 yields the corollary. In the case when πp is special and unramified, the maps of Corollary 4.5 wereconstructed by Breuil in [5, Lem. 4.2.2].
Suppose now that πp is principal series or special. Then we may choose χ asin Section 2 such that Wk−2 ⊗K πp = π(χ). We let α, β, θ1 , θ2 and χ0 have thesame meanings as in Section 2. A consideration of central characters shows thatαβ = N (p)pk−1 . Note that the necessary conditions of Lemma 2.1 are satisfied fora 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 H̃c1 (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 θ1 is unramified (by replacing χ by χ0 if necessary). If in fact
p divides the conductor of f and we are in the principal series case, then θ2 isramified, and so the condition that θ1 be unramified uniquely determines χ. If p
divides the conductor of f and we are in the special case, we again have a uniquechoice for χ, since in the special case we insist that α/β = p−1 . In either of thesecases, the Up -eigenvalue of f is equal to α. If the conductor of f is prime to p, thenboth θ1 and θ2 are unramified, and we have two choices for χ (namely a pair χ andχ0 , in the notation of Section 2). Fixing one of the two choices for χ corresponds
to choosing one of the two p-old Up -eigenforms attached to f ; namely, the one with
Up -eigenvalue equal to α.
We let g denote f in the case when p divides the conductor of f , and let g denotethe p-oldform attached to f with Up -eigenvalue α, in the case when the conductorof f is prime-to-p. In all cases g is a Up -eigenform with eigenvalue α.
Corollary 4.7. If | α |>| pk−1 |, then the embeddings
π(χ)±,λ = (Wk−2 ⊗K πp )±,λ → H̃ 1 (K p )±of Proposition 4.3 extend uniquely to (π0 , GL2 (Qp ), Hp )-equivariant maps
π la (χ)±,λ → H̃c1 (K p )± .
Proof. This follows from Proposition 2.5. p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

25

Dualizing the maps of Corollary 4.7, and taking into account Proposition 4.2, we
obtain Γ-equivariant maps D0 → (π la (χ)±,λ )0 (one for each choice of sign), lifting
the maps of Corollary 4.5. In particular, the element ∞ − 0 of D0 gives rise to apair of functionalslaµ±g : π (χ) → K.

(4.8)

If we identify I la (χ) as a space of locally analytic functions on Q×
p , as in Section 2,
lathen we obtain a closed embedding C la (Z×
,
K)
→
π
(χ),
by
identifying
elements ofp
the source as locally analytic functions on Qp that are supported on Z×.
p Restricting±la×
the functionals µg to C (Zp ) thus gives rise to a pair of distributions on Z×p.
±
Proposition 4.9. The distributions on Z×p obtained by restricting µg 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-functionof 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 evaluatedon 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 overm
is a positive power of p, andK of degree ≤ k − 2. Suppose that b ∈ Z×
p , that p≤k−2that F (x) is an element of A
. As in Section 3, we let F (x)|b+pm Zp denote theelement of C lp≤k−2 (Zp , K) defined by F (x) on b + pm Zp , and defined to be zerop0
1
elsewhere. Fix a choice of sign ±, let h– , –i±
1 : D × H̃c (K ) → K denote the±Γ-equivariant pairing provided by Proposition 4.2, and let φ1 : π(χ) → H̃c1 (K p )±denoteembeddingof Proposition 4.3. Using the formula (3.17) and the fact
 the

pm −b
that
, when thought of as an element of GL2 (Q), lies in Γ (in fact, even0
1
in K p ), we compute thatZ
(4.10)
b+pm Z×p±±
F (x)dµ±g = h∞ − 0, φ1 (F (x)|b+pm Zp )i1

 m

p
−b
±
m
= h∞ − 0, α−mφ±
1 (F (b + p x)|Zp )i1
0
1
 −m

p
bp−m
±
m
= α−m h(∞ − 0), φ±
1 (F (b + p x)|Zp )i1
0
1
b
±
m
= α−m h∞ − m , φ±
1 (F (b + p x)|Zp )i1 .
p
To analyze this expression, we must unwind the definition of the pairing h– , –i±1,which we do through a series of commutative diagrams.
6 We 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 × ZM associated to g may be constructed out
×of the Mazur-Tate-Teitelbaum distributions on Zp 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 anappropriate complex period Ω±
g .

26

MATTHEW EMERTON

We identify Wk−2 with a space of functions on G in the usual way: write w̌0 =∼(0, 1)k−2 ∈ Symk−2 K 2 −→ W̌k−2 , and define w(g) = hw̌0 , gwi for g ∈ G. If weGwrite I(θ) := (IndBθ)sm , then multiplication of functions induces an isomorphism(whose existence has been recalled several times already)(4.11)∼

Wk−2 ⊗K I(θ) −→ I(χ).

1 x
for x ∈
0 1
Zp , we also obtain an identification of Wk−2 with the space A≤k−2 . Multiplying
elements of Wk−2 , thought of as elements of A≤k−2 in this way, by locally constantfunctions on Zp induces an isomorphism
Restricting the functions w(g) to those g of the form nx :=(4.12)∼

Wk ⊗K C sm (Zp , K) −→ C lp≤k−2 (Zp , K).

(As in Section 3, we let C lp≤k−2 (Zp , K) denote the subspace of C la (Zp , K) consistingof functions that are locally polynomial of degree ≤ k − 2.)
We may apply the discussion of Section 3 to the representation I(θ) just as wellas to I(χ). The isomorphism (3.16) thus yields an isomorphism(4.13)∼

C sm (Zp , K) −→ I(θ)(N0 ),which fits in the commutative diagram of isomorphisms(4.14)

Wk−2 ⊗K C sm (Zp , K)∼
(4.12)/ C lp≤k−2 (Zp , K)

idWk−2 ⊗(4.13) ∼Wk−2 ⊗K I(θ)(N0 )(3.16) ∼∼(4.11)

/ I(χ)(N0 ).

We employ the notation introduced in the proof of Proposition 4.3. As recalledin that proof, for each choice of sign ±, classical Eichler-Shimura theory gives a±,λ(G(Qp ), Hp )-equivariant embedding φ±→ Hc1 (K p , VW̌k−2 )± , unique up
2 : (πp )1
pto multiplication by a scalar. (Here Hc (K , VW̌k−2 )± denotes the ±-eigenspace forthe action of π0 on Hc1 (K p , VW̌k−2 ).) By its construction, the embedding φ±
1 ofProposition 4.3 is obtained by composing the map (4.4) with idWk−2 ⊗ φ±.
Thus
2
we obtain a commutative diagram(4.15)

Wk−2 ⊗ I(θ)

Wk−2 ⊗ πp
idWk−2 ⊗φ±2Wk−2 ⊗ Hc1 (K p , VW )±∼(4.11)∼/ I(χ)
/ π(χ)φ±1
/ H̃ 1 (K p )± ,cin which the upper vertical arrows are the defining surjections (either isomorphisms
in the principal series case, or surjections with finite dimensional kernels in thespecial case), and the bottom horizontal arrow is induced by (4.4).

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

27

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 D0 ×
Hc1 (K p , VW̌k−2 ) → W̌k−2 , or equivalently, a pairing
h– , –i2 : D0 × Wk−2 ⊗K Hc1 (K p , VW̌k−2 ) → K.
Passing to ±-eigenspaces under the action of π0 , this yields pairings
01p±h– , –i±2 : D × Wk−2 ⊗K Hc (K , VW̌k−2 ) → K.

The diagram(4.16)/ D0 × H̃ 1 (K p )±D0 × Wk−2 ⊗K Hc1 (K p , VW̌k−2 )±
c
VVVV
VVVV
VVVV
h– ,–i±1
VVVVVh– ,–i±
V

V
2
V*
K

in which the horizontal arrow is induced by (4.4), also commutes; indeed, the diagonal arrow is constructed by trivializing VW , and then integrating over geodesics
joining cusps, while an examination of the proof of [12, Thm. 2.2.19] shows thatthe horizontal arrow is similarly constructed by trivializing VW (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|Zp denote the constant function 1 with domain Zp . Using the commutative
diagrams (4.14), (4.15), and (4.16), we compute thatα−m h∞ −bb±−m, φ± (F (b + pm x)|Zp )i±h∞ − m , F (b + pm x) ⊗ φ±
1 =α
2 (1|Zp )i2 .
pm 1
pCombining this with (4.10), we see thatZb±−m
(4.17)
F (x)dµ±h∞ − m , F (b + pm x) ⊗ φ±g =α
2 (1|Zp )i2
pb+pm Zp∼

for any F (x) ∈ A≤k−2 −→ Wk−2 .
If we fix an embedding K → C, then the modular form g defines an element ofcg ∈ Hc1 (K p , VWk−2 ) ⊗K C, defined to be the cohomology class of the holomorphic
VWk−2 -valued one-form g(τ )dτ (τ, 1)k−2 . (Here (τ, 1)k−2 is regarded as an element of∼Symk−2 C2 −→ W̌k−2 ⊗K C.) If we let c±g denote the ±-eigencomponent of cg , then±± ±φ2 (1|Zp ) = Ω cg for some non-zero scalars Ω± . (To see this, note that φ±
2 (1|Zp ) isa Up -eigenclass with eigenvalue α, and a Hp -eigenclass with character λ, while thesame is true of c±g .) From (4.17) we compute
Z
b
±
± −m
(4.18)
F (x)dµ±h∞ − m , F (b + pm x) ⊗ c±g =Ω αg i2 .
pb+pm Zp
If we don’t pass to ±-eigencomponents, then we compute thatZ ∞bm
(4.19)
h∞ − m , F (b + p x) ⊗ cg i2 =g(τ )h(τ, 1)k−2 , F (b + pm x)idτ,
p
b/pm

28

MATTHEW EMERTON

where the pairing h(τ, 1)k−2 , F (b + pm x)i denotes the natural pairing W̌k−2 ⊗K C ×
Wk−2 → C. We furthermore compute that


1 τ
k−2
m
h(τ, 1), F (b + p x)i = hw̌0 , F (b + pm x)i
0 1


1 −τ
= hw̌0 ,F (b + pm x)i = F (b − pm τ )0 1∼(where the final equality follows from the way in which the isomorphism A≤k−2 −→Wk−2 was defined). Substituting this into (4.19), we find thatZ ∞bg(τ )F (b − pm τ )dτ,h∞ − m , F (b + pm x) ⊗ cg i2 =pmb/p
for any F ∈ A≤k−2 . Comparing this formula, along with (4.18), to the constructionof [16], we find that µ±g does indeed coincide (up to multiplication by a non-zeroscalar) with the corresponding distribution that computes the p-adic L-functionof 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
laon Qp (or more precisely, to elements in the dual of I (χ)). In the case when πp 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)

∼

la
±,λ 0bD0 → (π(χ)±,λ )0b)b ,b −→ (π (χ)

which up to twisting we may regard as a pair of Γ-equivariant morphisms D0 →O(k)b . By applying the functor HomΓ (D0 , –) to the short exact sequence (1.20),
and analyzing the result of applying the coboundary map
HomΓ (D0 , O(k)) → Ext1Γ (D0 , Wk−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 D0 → (π la (χ, L)λ,± )0bb .
Theorem 5.2 has a dual version, which states that there is a unique value of L
for which one can find π0 × Hp × GL2 (Q)p -equivariant maps
(5.3)

π la (χ, L)±,λ → H̃c1 (K p )±extending the maps of Corollary 4.7. Furthermore, Breuil shows that these extensions (5.3) are unique.7
7 Breuil actually proves this statement with Γ replaced by the intersection GL (Q)2weakens the equivariance claim, but strengthens the uniqueness claim.

TK p . Thisp-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

29

Let Lf denote the unique value of L that satisfies the conclusion of Theorem 5.2.
Theorem 5.2 then implies that the element ∞ − 0 ∈ D0 induces functionalslaµ±f : π (χ, Lf ) → K(5.4)

extending the functionals (4.8). (We write the subscript as f rather than g, sincein 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 Up -eigenvalue of f equals p(k−2)/2 , then theelement −Lf satisfies the Mazur-Tate-Teitelbaum conjecture for f . We close thisnote by explaining how this follows formally from Theorem 5.2, even without theassumption of trivial nebentypus.
Recall that π la (χ, Lf ) can be identified with a certain space of locally analyticifunctions on Q×
p ; in particular, the function x logLf (x) supported on Qp \ pZp
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−2)/2
k even and for which π(χ) is in the special case, such that(so that α =pp
β = pk/2 , and the central character of π(χ) is trivial on). If µ ∈ π la (χ)0pp
is a continuous functional on π la (χ) that is invariant under the action of
,
1
then
Z
x(k−2)/2 dµ = 0.
Z×p(ii) If in addition to the hypotheses of (i) the functional µ extends in ap1
invariant fashion to a continuous functional on π la (χ, L) for some L ∈ K, thenZZ×px(k−2)/2 logL x dµ = −LZ-x(k−2)/2 dµ.

Zpx(k−2)/2 dµ by writing the domain of integration as thep×union of Zp and pZp , one obtains the formula of (i). (Use the matrix
to1“scale” pZp back to Zp .)Assume now that the hypothesis of (ii) holds. If one takes into account theformulas (2.3) and (2.7) giving the action of GL2 (Qp ) 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), andcomputes the value ofZ
R

Proof. If one computes

Zpx(k−2)/2 logL x dµ

Qp \pZp

by writing the domain of integration as the union of Z×p and Qp \ Zp , one finds thatZ
(5.6)(k−2)/2xZ×p

ZlogL x dµ = L
Qp \pZpx(k−2)/2 dµ.

30

MATTHEW EMERTON



1



to “scale” Qp \Zp back to Qp \pZp .) Finally, again takingpinto 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 thatZZ(k−2)/2xdµ = −x(k−2)/2 dµ.
(Use the matrix

Qp \pZp

Zp

Combined with (5.6), this gives the formula of (ii). 
TCorollary 5.7. Suppose that f is a newform on Γ1 (N ) Γ0 (p) of even weight k,such that α = p(k−2)/2 (and hence such that N (p) = 1). ThenZ
(5.8)
x(k−2)/2 dµf = 0
Z×
p

and
Z
(5.9)
Z×
p

x(k−2)/2 logLf x dµf = −LfZx(k−2)/2 dµf .

Zp

Proof. This followsimmediatelyfrom Proposition 5.5 and the fact that ∞ − 0 ∈ D0p
is invariant under(which lies in Γ). 
1
By proposition 4.9, the left-hand side of (5.8) is the value of the p-adic L-functionof 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 thep-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 valueof the classical L-function of f at k/2. Thus we have established the Mazur-TateTeitelbaum conjecture for f , if we define the L-invariant of f to be −Lf .
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 Lf isinvariant under arbitrary twisting of f by Dirichlet characters. If f is any newformof even weight k that is special at p, then the twist f of f by a Dirichlet character will satisfy the hypothesis of Corollary 5.7 for some choice of . Assuming that
we can choose  such that furthermore L(f , 1) 6= 0, we see that the value of Lf
can be “detected” via the Mazur-Tate-Teitelbaum formula (5.9) (applied to f ).
This allows a comparison between the Breuil-Darmon-Orton definition of Lf andother definitions of Lf with respect to which the Mazur-Tate-Teitelbaum conjecture has been proved – namely, the “Coleman L-invariant”, in terms of which theMazur-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 surveyof these definitions of the L-invariant, and the various means of comparing them.)88 In fact, it is an easy consequence of Colmez [8] and of Kato’s explicit reciprocity law for thep-adic L-function of f that Lf , as defined above, agrees with the Fontaine-Mazur definition of theL-invariant of f via the local-at-p Galois representation attached to f [15].

p-ADIC L-FUNCTIONS AND UNITARY COMPLETIONS

31

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Northwestern University
Department of Mathematics
2033 Sheridan Rd.
Evanston, IL 60208-2730, USA
E-mail address: emerton@math.northwestern.edu