Draft: January 2, 2006
ON THE INTERPOLATION OF SYSTEMS OF EIGENVALUESATTACHED TO AUTOMORPHIC HECKE EIGENFORMS

Matthew Emerton
Northwestern University
To Barry Mazur, for his 65th birthday

Contents
1. Continuous cohomology and Lie algebra cohomology
2. Cohomology of arithmetic quotients of symmetric spaces
3. p-adic automorphic representations
4. The case of GL2/Q
References

7
20
42
48
69

The goal of this paper is to illustrate how the techniques of locally analytic p-adic
representation theory (as developed in [28, 29, 30, 31] and [13, 14, 17]; see also [16]for a short summary of some of these results) may be applied to study arithmetic
properties of automorphic representations. More specifically, we consider the problem of p-adically interpolating the systems of eigenvalues attached to automorphic
Hecke eigenforms (as well as the corresponding Galois representations, in situations
where these appear in the étale cohomology of Shimura varieties). We can summarize our approach to the problem as follows: rather than attempting to directlyinterpolate the systems of eigenvalues attached to eigenforms, we instead attempt
to interpolate the automorphic representations that these eigenforms give rise to.
To be more precise, we fix a connected reductive linear algebraic group G defined over a number field F , and a finite prime p of F . We let Fp denote thecompletion of F at p, let E be a finite extension of Fp over which the group G
splits, let A denote the ring of adèles of F , and let Af denote the ring of finiteadèles of F . The representations that we construct are admissible locally analytic
representations of the group G(Af ) on certain locally convex topological E-vector
spaces. These representations are typically not irreducible; rather, they contain as
closed subrepresentations many locally algebraic representations of G(Af ) which
are closely related to automorphic representations of G(A) of cohomological type.
(It is for this reason that we regard the representations that we construct as forming
an “interpolation” of those automorphic representations.)
Once we have our locally analytic representations of G(Af ) in hand, we may
apply to them the Jacquet module functors of [14]. In this way we obtain p-adic
analytic families of systems of Hecke eigenvalues, which (under a suitable hypothesis, for which see the statement of Theorem 0.7 below) p-adically interpolate (in the
The author would like to acknowledge the support of the National Science Foundation (award
numbers DMS-0070711 and DMS-0401545)

1

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MATTHEW EMERTON

traditional sense!) the systems of Hecke eigenvalues attached to automorphic representations of cohomological type. More precisely, the resulting families interpolate
the systems of Hecke eigenvalues attached to those automorphic representations
which contribute to the cohomology of the arithmetic quotients associated to G,
and whose local factor at p embeds in a principal series representation. (This lastproperty is a representation theoretic interpretation of the finite slope condition
appearing in [11].)
Most of our efforts in this paper are devoted to constructing the representationsof G(Af ) alluded to above, and to establishing their basic properties; considerably
less space is devoted to the detailed analysis of the p-adic families that we obtainby applying the Jacquet module functors to these representations. However, we
do consider the case of GL2/Q in enough detail to explain how our results, in thatparticular case, lead to a new construction of the eigencurve of [11], valid for all
primes and arbitrary tame level. (In [11] the authors restrict themselves to odd
primes and trivial tame level.) We also indicate how the construction of this paper,
when combined with the results of the forthcoming paper [17], may be used toconstruct a two-variable p-adic L-function parameterized by the eigencurve.
The arrangement of the paper is as follows: In Section 1 we extend some of thedefinitions and results of [25, V.2.4] (which compares the group cohomology and Liealgebra cohomology of finite dimensional representations of locally p-adic analytic
groups) to the context of infinite dimensional representations. Section 2 presents
our main constructions and results. In Section 3 we describe how to produce locallyanalytic p-adic representations out of certain classical automorphic representations,thus providing a bridge between the classical theory of automorphic representations
and the locally analytic representation theory that underlies the methods of thispaper. Related to this, we interpret our results in the case of a group that iscompact at infinity in terms of the notion of “locally analytic p-adic automorphic
forms”. Section 4 is devoted to the example of GL2/Q . In the remainder of theintroduction, we briefly sketch the constructions and results of Section 2, and alsocomment on the relation between our work and other recent work related to thep-adic interpolation of automorphic forms.
Cohomology of arithmetic quotients. Let K∞ denote a fixed maximal compact◦
be its connected component ofsubgroup of the real group G(R ⊗Q F ), and let K∞◦
the identity. Let π0 denote the quotient K∞ /K∞ . For any compact open subgroup
Kf of G(Af ) we write◦
Y (Kf ) := G(F )\G(A)/K∞Kf .

If Kf is sufficiently small, and if Kf0 is a normal open subgroup of Kf , then thenatural map Y (Kf0 ) → Y (Kf ) is a Galois covering map, with group of deck transformations isomorphic to the quotient Kf /Kf0 .
Let Apf denote the prime-to-p part of the ring of finite adèles of F , so thatAf = Fp × Apf . We are particularly interested in Kf of the form Kf = Kp K p ,where Kp is a variable compact open subgroup in G(Fp ), and K p is a compact
open subgroup of G(Apf ), fixed for the moment. (We refer to such a subgroup K p
as a “tame level”.) If we allow Kp to shrink to the identity, while keeping the tamelevel fixed, the spaces Y (Kf ) form a projective system, equipped with an action of
π0 × G(Fp ). Passing to cohomology with coefficients in a ring A yields an inductive
system of A-modules, equipped with an A-linear action of π0 × G(Fp ).

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

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Taking A to be OE /ps for some natural number s > 0 (here OE denotes thering of integers of E) we define H n (K p , OE /ps ) := lim
H n (Y (Kp K p ), OE /ps ). We−→
Kpthen obtain an E-Banach space H̃ n (K p ), equipped with a continuous action of
π0 × G(Fp ), by passing to the projective limit in s, and tensoring with E:
H̃ n (K p ) := E ⊗OE lim
H n (K p , OE /ps ).
←−s

Let K p0 ⊂ K p be an inclusion of tame levels. We obtain a corresponding
π0 ×G(Fp )-equivariant closed embedding of E-Banach spaces H̃ n (K p ) → H̃ n (K p0 ).
Passing to the locally convex inductive limit over all tame levels yields a locally convex E-vector space H̃ n equipped in a natural way with an action of π0 × G(Af ).
In the following results, we will apply some terminology introduced in [13, §7] tothe group π0 × G(Af ). When applying this terminology, we will regard π0 × G(Af )as being factored as the product of the locally Fp -analytic group G(Fp ) and the
locally compact group π0 × G(Apf ).
The following result (which follows from Theorem 2.2.16 below) describes thebasic properties of the π0 × G(Af )-representation H̃ n .
Theorem 0.1. (i ) The representation H̃ n is an admissible continuous representation of π0 × G(Af ) (in the sense of [13, Def. 7.2.1]).
(ii ) For any tame level K p , the Banach space H̃ n (K p ) is recovered as the subspaceof K p -invariants in H̃ n .
Our next result describes the result of passing to the locally L-analytic vectorsof the representation H̃ n , for any local field L intermediate between Fp and Qp . It
is a consequence of Theorem 0.1 and the theory of [13, §7].
n
Theorem 0.2. (i ) The representation H̃L−lais an admissible locally L-analytic
representation of π0 × G(Af ) (in the sense of [13, Def. 7.2.7]).
pn)K from(ii ) For any tame level K p , the natural map H̃ n (K p )L−la → (H̃L−lathe space of locally L-analytic vectors of the G(Fp )-representation H̃ n (K p ) to thenspace of K p -invariants in H̃L−lais an isomorphism.

If W is a finite dimensional representation of G over E, then there is a local
system of E-vector spaces VW defined on each of the spaces Y (Kf ) (which is thetrivial local system when W is the trivial representation), and we may form theE-vector space H n (VW ) := lim
H n (Y (Kf ), VW ), which is equipped with a smooth−→Kfaction of π0 × G(Af ). It is in fact an admissible smooth representation of thisgroup (as is well known); indeed the space of Kf -invariants in H n (VW ) is precisely
H n (Y (Kf ), VW ), and so is finite dimensional.
Let W̌ denote the contragredient representation to W. We equip the tensor product H n (VW ) ⊗E W̌ with an action of π0 × G(Af ) induced by the tensor productaction of G(Fp ), and the action of π0 × G(Apf ) on the first factor. This tensor product is then an admissible locally algebraic representation of π0 × G(Af ) (by [13,Prop. 7.2.16]), and we show that there is a natural π0 × G(Af )-equivariant map(0.3)

H n (VW ) ⊗E W̌ → H̃ n .

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MATTHEW EMERTON

Since the π0 × G(Af )-action on H n (VW ) is smooth, giving such a map is equivalent
to giving a π0 × G(Af )-equivariant map(0.4)

n
H n (VW ) → Homg (W̌ , H̃L−la)(where g denotes the Lie algebra of G). When the map (0.4) is an isomorphism, ityields an intrinsic representation theoretic description of the smooth π0 × G(Af )nrepresentation H n (VW ), in terms of the locally analytic representation H̃L−laandthe given finite dimensional representation W . Furthermore, the map (0.3) thenidentifies H n (VW ) ⊗E W̌ with the subspace of locally W̌ -algebraic vectors in H̃ n .
We are able to give some examples for which (0.4) is an isomorphism (see thediscussion below); in general, we establish the following result.
Theorem 0.5. If W is a finite dimensional representation of G defined over E,with contragredient W̌ , then the map (0.4 ) (with L = Qp ) is the edge map of aπ0 × G(Af )-equivariant spectral sequenceE2i,j = Extig (W̌ , H̃Qj p −la ) =⇒ H i+j (VW ).
This result is a restatement of Corollary 2.2.18.
In this paper we consider two examples for which the higher Ext terms appearing in the spectral sequence of Theorem 0.5 can be shown to vanish, and thus forwhich (0.4) is an isomorphism (for any choice of L as above). One is the case when(the restriction of scalars to Q of) G satisfies the conditions of [19, Prop. 1.4] (thespaces Y (Kf ) are then the union of finitely many contractible connected components, and so we naturally take n = 0); the other is the case of GL2/Q (for which
we take n = 1). In such situations, we may regard the locally analytic representation H̃Fnp −la as providing a p-adic interpolation of the representations H n (VW )– or more precisely, of the locally algebraic representations H n (VW ) ⊗E W̌ , since
it is these representations that are then identified by (0.3) with locally algebraic
subrepresentations of H̃Fnp −la .
Let us remark that we also establish analogues of Theorems 0.1, 0.2 and 0.5 withcohomology replaced by compactly supported cohomology.
Eigenvarieties. Suppose now that G is quasi-split over Fp . We fix a Levi factor
T in a Borel subgroup B of G/Fp (so that T is a torus), and let T̂ denote the rigid
analytic space over E that parameterizes the locally Fp -analytic characters of theabelian locally Fp -analytic group T := T(Fp ). Fix a cohomological degree n, as
well as a tame level K p . Write H(K p ) := H(G(Apf )//K p ) to denote the prime-to-p
Hecke algebra of level K p (over E), and as in Subsection 2.3 below, factor H(K p )as H(K p ) := H(K p )ram ⊗E H(K p )sph . Note that H(K p )sph , the spherical Hecke
algebra of tame level K p , is a central subalgebra of H(K p ).
Fix an embedding of E into Qp . Suppose that πf is an irreducible π0 × G(Af )representation appearing as a subquotient of Qp ⊗E H n (VW ), for some irreducible
G-representation W (necessarily definable over E, since G is assumed to split overG(F )E), with the property that πp (the local factor at p of πf ) embeds into IndB(Fpp) θpfor some smooth Qp -valued character θ of T , and for which πfK 6= 0. Since πfpis irreducible, the spherical Hecke algebra H(K p )sph acts on πfK through a Qp valued character λ, which determines a Qp -valued point of Spec H(K p )sph . Let ψ

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

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denote the highest weight character of W̌ (with respect to the chosen Borel B),pregarded as a characterof T , and write χ := θψ. Let E(n, K )cl denote the set ofpoints (χ, λ) ∈ T̂ × Spec H(K p )sph (Qp ) constructed in this manner from such
irreducible subquotients πf .
Definition 0.6. We define the degree n cohomological eigenvariety of G of tamelevel K p to be the rigid analytic Zariski closure of E(n, K p )cl in T̂ ×Spec H(K p )sph ,and denote it by E(n, K p ).
The following theorem is a consequence of the results proved in Subsection 2.3
below.
Theorem 0.7. If the map (0.4 ) is an isomorphism for each irreducible representation W of G (and for our fixed choice of cohomological degree n), then the following
are true:(i ) Projection onto the first factor in the product T̂ ×Spec H(K p ) induces a finite
map E(n, K p ) → T̂ .
(ii ) The composite of the finite map E(n, K p ) → T̂ of (i ), and the map T̂ → ť
given by differentiating characters, has discrete fibres. (Here ť denotes the dual to
the Lie algebra t of T .) In particular, the dimension of E(n, K p ) is at most equal
to the dimension of T .
(iii ) If (χ, λ) is a point of E(n, K p ) in which the first factor χ is locally algebraic
and of non-critical slope (in the sense of [14, Def. 4.4.3]), then (χ, λ) is in fact a
point of E(n, K p )cl .
(iv ) There is a coherent sheaf M of right H(K p )ram [π0 ]-modules over E(n, K p ),whose fibre over any point (χ, λ) of E(n, K p )cl of non-critical slope is naturally isomorphic (as an H(K p )ram [π0 ]-module) to the E-linear dual of the (θ, λ)-eigenspaceof the Jacquet module (taken with respect to B(Fp )) of the smooth representation
H n (K p , VW̌ ). (Here we have factored χ = θψ as a product of a smooth and an
algebraic character, and written W to denote the irreducible representation of G
whose contragredient has highest weight ψ with respect to B. Also, we have written H(K p )ram [π0 ] to denote the group ring of π0 with coefficients in the algebraH(K p )ram .)Theorem 0.7 applies in particular to the two examples discussed above. In thecase when G satisfies the conditions of [19, Prop. 1.4], so that the only non-vanishingcohomology is in degree zero, we naturally take n = 0. This class of examples isalready extremely interesting; for example, taking G to be a totally definite quaternion algebra over a totally real field, one obtains a construction of eigenvarieties
parameterizing systems of Hecke eigenvalues attached to Hilbert modular forms
over F . In this case we are able to improve on Theorem 0.7 (ii), to show that infact E(n, K p ) is equidimensional of dimension equal to the dimension of T.
In the case of GL2/Q , the important cohomological degree is n = 1. We recover
the eigencurve of [11] (or more precisely, the surface obtained from it by allowing
twists by wild characters at p) as the eigenvariety of trivial tame level. Note that
our method imposes no constraints on the tame level, and so in this case we may
equally well construct an eigenvariety for arbitrary tame level. (We are also able toinclude the case p = 2, which is omitted in [11]. A construction of the eigencurve
for all primes and arbitrary tame level has been given independently by Buzzard[6], using a direct extension of the method of [11].) By working with compactly

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MATTHEW EMERTON

supported cohomology, and applying the results on Jacquet functors to appear in[17], we can use this point of view on the eigencurve to construct a two-variable padic L-function over the eigencurve. This L-function will be constructed as a certain
section of (a slight modification of) the coherent sheaf M of Theorem 0.7 (iv).
We close our summary with an observation that allows us in some instances tointerpolate Galois representations as well as systems of Hecke eigenvalues. Supposethat G is a group for which the arithmetic quotients Y (Kf ) (possibly modified in themanner discussed in Subsection 2.4 below) admit the structure of Shimura varieties,
with canonical models defined over some number field F 0 . Then throughout thepreceding discussion, we may take “cohomology” to mean “étale cohomology”, andthus endow the space H̃ n with a representation of Gal(Q/F 0 ), commuting with theG(Af )-action. In the cases when Theorem 0.7 applies, this Galois action will beinherited by the coherent sheaf M, and so M will be a coherent sheaf of Galois
representations over E(n, K p ). For example, in the case of GL2 over Q, for eachtame level we obtain a coherent sheaf of representations of Gal(Q/Q).
Other approaches. There are at least two other approaches to the study of eigenvarieties that appear in the literature. Coleman and Mazur’s original constructionof the eigencurve [11], via the geometric theory of modular forms, has been extended to the contexts of Shimura curves by Kassaei [22] and to the context ofHilbert modular varieties by Kisin and Lai [23].
There is also an approach, the basic idea of which is due to Stevens (generalizing
the approach of Hida [20] in the ordinary case), that has been developed by Ash
and Stevens (for the group GLn over Q [2]), Buzzard (for tori and for definite
quaternion algebras over Q [5], and for totally definite quaternion algebras overtotally real fields [6, §§8,9]) and Chenevier (for groups G over Q that are compact
over R and isomorphic to GLn over Qp [8]). Stevens’ method has in common withthe method of this paper that it uses cohomology of arithmetic quotients as ameans of achieving the desired p-adic interpolation. (In the case of [5], [6, §§8,9]
and [8], the arithmetic quotients that are considered are zero-dimensional. This is
the situation that we consider in Subsection 3.2 below.) Nevertheless, it is quite
different to the approach taken here.
Here is a sketch of Stevens’ method: Take an Iwahori subgroup I of G(Qp ), withimage Ω in the flag variety of G(Qp ), and consider the space of locally analytic
distributions on Ω. Out of this space construct a quasi-coherent sheaf D on “weightT
space” – that is, the space of locally analytic characters of the intersection I T .
Let Γ be an arithmetic subgroup of G(R ⊗Q F ) which embeds into I; then Γ acts on
D, and passing to Γ cohomology (in some chosen degree) yields a sheaf over weightspace on which analogues of the Up -operators act. Passing to the finite slope partof this sheaf yields the desired eigenvariety, and a coherent sheaf over it (similar tothe coherent sheaf M appearing in Theorem 0.7 (iv)). In practice, it can be difficult
to carry out this process sheaf theoretically over all of weight space, and one caninstead restrict to a sufficiently small affine open neighbourhood of a classical pointof weight space and work directly over the corresponding Tate algebra. (This isdone in [2], for example.)
The basic distinction between our approach and either of the approaches mentioned above is our emphasis on the point of view and methods of representation
theory. One advantage of this is that our approach is not specifically tailored tothe study of finite-slope automorphic forms. Spaces of overconvergent automorphic
forms (which form the basic objects in the geometric approach) and the arithmetic

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

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cohomology of the sheaf D (which forms the basic object in Stevens’ approach) bothseem to be hopelessly large as Hecke modules until one passes to the finite slope
parts. By contrast, the representations H̃ n that we construct, and their associatedspaces of analytic vectors, satisfy reasonable finiteness conditions – namely they
are admissible in the appropriate sense. (This is the pay-off for working with theentire group, and not just the Hecke algebra.) Thus it is reasonable to hope that
our approach will have applications to the construction of families of forms in theinfinite slope (i.e. non-principal series at p) case also.
The representations H̃ n that we construct are similar to certain representations
constructed by Shimura. In the paper [32] Shimura considers an inductive limitof cohomology of arithmetic quotients of Shimura curves attached to a quaternion
algebra over a totally real field F that is split at exactly one infinite place, andat all the primes lying over a fixed prime p (denoted ` in [32]), equipped with thenatural action of the group G(Qp ). (Here G denotes the restriction of scalars of
the quaternion algebra from F to Q.) Using a base-change argument, he is able toestablish a strong control theorem, which in the particular case when G = GL2 , isanalogous to the control theorem established by isomorphism (4.3.4) below. One
difference between the results of [32] and ours is that Shimura considers cohomology
with coefficients in Qp /Zp , whereas we consider cohomology with Qp -coefficients,
completed with respect to the norm induced by the cohomology with Zp -coefficients.
More importantly, in [32] there is no analogue of the passage to locally analytic
vectors that plays such an important role in this paper. (I would like to thankHaruzo Hida for drawing my attention to the details of [32].)
Let us close this discussion of alternative methods by pointing out that Stevens’
approach lends itself in a natural way to the construction of two-variable p-adic Lfunctions along the eigencurve (or at least, in the neighbourhood of classical points
on the eigencurve) [33]. As we explain in Section 4, our point of view also gives aconstruction of such two-variable p-adic L-functions. In that section, we will recall
Stevens’ result more precisely, and compare it with the construction yielded by ourrepresentation theoretic approach.
Notation and conventions. We follow closely the notational and terminologicalconventions introduced in [13] and [14]. In particular, we refer to those papers forthe meaning of any technical terms and pieces of notation that are used without
explanation in the present text.
Acknowledgments. During the course of preparing this paper I discussed my ideas
with many people, and I would like to thank Alexander Beilinson, David BenZvi, Christophe Breuil, Kevin Buzzard, Vladimir Drinfeld, Haruzo Hida, Robert
Kottwitz, Peter May, and Glenn Stevens for their helpful remarks arising fromthese discussions. I would also like to thank the referee, whose careful reading of
the paper and thoughtful suggestions helped to improve the clarity of the exposition.
1. Continuous cohomology and Lie algebra cohomology(1.1) In [13, §4.1] we studied some basic properties of the functor “pass tosmooth vectors”. In applications, it is necessary to consider not only this functor,
but also its derived functors. The object of this subsection is to introduce thesederived functors, and to establish those properties of them that will be required forthe applications that we have in mind.

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MATTHEW EMERTON

We begin by supposing that G is a compact locally Qp -analytic group. (Equivalently, a compact p-adic analytic group, in the terminology of [25].) We fix a finite
extension E of Qp , with ring of integers OE . Recall that the category of admissible
continuous representations of G over E (as defined and studied in [30] and [13,§6.2]; we employ the notation of the latter reference) is abelian.
Lemma 1.1.1. If G is a compact locally Qp -analytic group then the abelian category of admissible continuous representations of G over E has enough injectives.
Proof. Any admissible continuous representation V admits a closed G-equivariant
embedding into C(G, E)n for some n ≥ 0, and so it suffices to show that the admissible continuous representation C(G, E)n is injective. This easily follows from the factthat its dual space is equal to D(G, E)n , which is a free module over D(G, E). ∼

Since the inclusion E → C(G, E) induces an isomorphism E −→ C(G, E)G (where
the superscript G denotes “G-invariants”), we see that for any admissible continuous
G-representation V , the space V G is finite dimensional.
Definition 1.1.2. If G is a compact locally Qp -analytic group then we let H • (G, – )denote the derived functors of the functor “pass to G-invariants” from the categoryof admissible continuous representations of G over E to the category of finite dimensional E-vector spaces.
That these derived functors exist follows from Lemma 1.1.1. The following proposition guarantees that they agree with the usual continuous cohomology of G.
Proposition 1.1.3. The derived functors of Definition 1.1.2 agree with the usual
cohomology of G computed with continuous cochains.
Proof. We remark that cohomology with continuous cochains yields a δ-functor
on the category of admissible continuous representations of G. (To see that one
obtains the required long exact sequences, it is necessary to know that any surjection
between objects in our category can be split continuously as a map of topologicalspaces. However, such a surjection can even be split as a map of topological vector
spaces [27, Prop. 10.5].)
Since the derived functor of Definition 1.1.2 and continuous cohomology yield
the same H 0 term, to obtain the required isomorphism, it suffices to show that
continuous cohomology is effaceable. The argument of Lemma 1.1.1 then showsthat it suffices to prove that C(G, E) has trivial continuous cohomology. This is no
doubt standard; in any case, we recall the proof.
By definition, the continuous cohomology of C(G, E) is computed as the cohomology of G-invariants of the complex(1.1.4)0 → C(G, C(G, E)) → C(G2 , C(G, E)) → · · · ,with the usual boundary maps, on which the G-action is defined via the left regular action of G on the products Gn , together with the right-regular action of
G on C(G, E). The example of [27, pp. 111–112] yields natural isomorphisms∼∼
C(Gn , C(G, E)) −→ C(Gn × G, E) −→ C(G, C(Gn , E)), and [13, Lem. 3.2.11] yields
a G-equivariant isomorphism of C(G, C(Gn , E)) with itself, but regarded as a Grepresentation via the right regular action of G on itself, and the trivial G-action

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

9on C(Gn , E). Thus the space of G-invariants of the complex (1.1.4) is naturally
isomorphic to the complex
0 → C(G, E) → C(G2 , E) → · · ·with its usual boundary maps, which has no higher cohomology.



We now turn to defining the derived functors of the functor “pass to smooth
vectors”.
Definition 1.1.5. If G is a compact locally Qp -analytic group, then we denote by•
Hst(G, – ) the derived functors of the functor “pass to smooth vectors” from thecategory of admissible continuous representations of G over E to the category of
abstract E-vector spaces equipped with a G-action.
The subscript “st ” stands for stable, and is explained by the following proposition. (This choice of notation is taken from [25, V.2.4.10].)
Proposition 1.1.6. If G is a compact locally Qp -analytic group and V is an admissible continuous representation of G, then for any i ≥ 0 there is a natural∼i(G, V ), where the inductive
isomorphism of E-vector spaces lim
H i (H, V ) −→ Hst−→
Hlimit is taken over all the open subgroups H of G.
Proof. For any admissible continuous representation V of G, by definition there isa natural isomorphism lim
V H → Vsm . Passing to derived functors (and noting that−→H

any V that is injective as a G-representation is also injective as an H-representation,
since the forgetful functor from admissible continuous G-representations to admissible continuous H-representations is right adjoint to the exact functor U 7→
E[G] ⊗E[H] U ), the proposition follows. 
Corollary 1.1.7. If H is an open subgroup of the compact locally Qp -analytic
group G, and if V is an admissible continuous representation of G, then for eachii(H, V ) is an isomorphism.
(G, V ) → Hsti ≥ 0, the natural map Hst
Proof. This follows immediately from Proposition 1.1.6. Working directly with continuous cochains, we could use the inductive limit•(G, – ) on the category of
formula of Proposition 1.1.6 to define the functors Hstcontinuous G-representations for any topological group G. The reason that we
restrict our attention to admissible continuous representations of locally Qp -analytic•
groups, and define Hst(G, – ) as a derived functor rather than directly via continuouscochains, is because we then have the standard homological machinery available
with no expenditure of effort.
We now describe a construction that we will require in our subsequent applications, and which depends on this homological machinery. We suppose that G is acompact locally Qp -analytic group, and we let V • denote a bounded-below complexof admissible continuous G-representations, each of which is isomorphic to C(G, E)nfor some natural number n, and in which the boundary maps are continuous andG-equivariant. The cohomology spaces H • (V • ) are again admissible continuous
representations of G.
•
If we form the subcomplex Vsm
of V • then [13, Prop. 6.2.4] and [29, prop. 2.1]•
show that each term of Vsm
is a strongly admissible smooth representation of G.

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MATTHEW EMERTON•
If we regard Vsm
as a subcomplex of V • , and if we choose a norm that induces
the Banach space structure on each of the terms of V • , then these norms induce••
a norm on each term of Vsm, and the G-action on the terms of Vsm
is continuous•with respect to this norm. We thus regard Vsm as a complex of normed spaces eachequipped with a continuous strongly admissible smooth action of G, in which theboundary maps are continuous and G-equivariant.
•For each integer n the cohomology space H n (Vsm
) is thus naturally a seminormed space, equipped with a continuous strongly admissible smooth representa••tion of G. If Ĥ n (Vsm
) denotes the completion of H n (Vsm
) with respect to its semin•
norm then the continuous G-action on H (Vsm ) induces a continuous G-action on•••
Ĥ n (Vsm). There is a natural G-equivariant map H n (Vsm
) → Ĥ n (Vsm), which willn•
be injective precisely when the semi-norm on H (Vsm ) is in fact a norm.
•
The inclusion of Vsm
into V • induces for each integer n a G-equivariant map(1.1.8)•H n (Vsm) → H n (V • )sm .

Since H n (V • ) is complete with respect to its natural topology, the map (1.1.8)
extends to a continuous G-equivariant map(1.1.9)•Ĥ n (Vsm) → H n (V • ).

Proposition 1.1.10. (i ) For each integer n the natural map (1.1.9 ) is a closed
embedding.
(ii ) The map (1.1.8 ) is the edge map of a G-equivariant spectral sequencei•E2i,j = Hst(G, H j (V • )) =⇒ H i+j (Vsm).

Proof. Fix an integer n, let B n (V • ) denote the image of the boundary map from
V n−1 to V n , and let Z n (V • ) denote the kernel of the boundary map from V n toV n+1 . Both B n (V • ) and Z n (V • ) are closed subspaces of V n (the former by [13,Prop. 6.2.9], and the latter obviously, since the boundary maps in the complex•) denote the image ofV • are continuous by assumption). Similarly, let B n (Vsm
n−1
n
n
•
the boundary map from Vsm to Vsm , and let Z (Vsm ) denote the kernel of thenn+1
boundary map from Vsm
to Vsm.
n
The cohomology group H (V • ) is equal to the quotient of Z n (V • ) by B n (V • ),••) bywhile the cohomology group H n (Vsm
) is equal to the quotient of Z n (Vsm
n
•
n
•
n
•
n
n
•
B (Vsm ). If B̂ (Vsm ) denotes the closure of B (Vsm ) in V , and Ẑ (Vsm ) denotes••
the closure of Z n (Vsm
) in V n , then the completion Ĥ n (Vsm
) is isomorphic to then•n•quotient of Ẑ (Vsm ) by B̂ (Vsm ).
Since by assumption V n−1 is isomorphic to C(G, E)m for some integer m, we seen−1••that Vsm
is dense in V n−1 , and so B n (Vsm
) is dense in B n (V • ). Thus B̂ n (Vsm
)=
n
•
n
•
n
•
n
•
n
•
B (V ), Ẑ (Vsm ) contains B (V ), and Ĥ (V ) is equal to the quotient of Ẑ (Vsm
)
n
•
n
•
by B (V ). This certainly embeds as a closed subspace of H (V ), and so (i) is
proved.
•By assumption each term of V • is acyclic for Hst(G, – ), and so the existence ofthe spectral sequence of part (ii) is a consequence of standard homological algebra. ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

11•
We remark that in general Ẑ n (Vsm
) is a proper closed subspace of Z n (V • ), andso the map (1.1.9) is not a topological isomorphism. (We will see examples of thisin Subsection 4.2 below.)
We now extend Definition 1.1.5 to the case where G is a (not necessarily compact) locally Qp -analytic group. In this more general situation we cannot show theexistence of enough injectives, and so we resort to a more ad hoc definition.

Definition 1.1.11. If G is a locally Qp -analytic group, and if V is an E-Banach
space equipped with an admissible continuous representation of G, then for eachii ≥ 0 we define Hst(G, V ) = lim
H i (H, V ), where H runs over the directed set of−→H

all compact open subgroups of G.
Proposition 1.1.12. The construction of Definition 1.1.11 satisfies the following
properties:(i ) For any admissible continuous representation V of G, there is a natural∼
0isomorphism Hst(G, V ) −→ Vsm .
(ii ) For each i ≥ 0 and any admissible continuous representation V of G, theiE-vector space Hst(G, V ) is equipped with a natural admissible smooth G-action.
(iii ) If H is a compact open subgroup of G, and V an admissible continuous
representation of G, then H i (H, V ) maps isomorphically onto the subspace of Hi(G, V ).
invariants in Hst•(G, – ) forms a covariant δ-functor.
(iv ) The collection Hst(v ) Let V be an admissible continuous representation of G. If there exists a∼
compact open subgroup H of G and an H-equivariant isomorphism V −→ C(H, E)nifor some n ≥ 0, then Hst(G, V ) = 0 for all i > 0.
Proof. Part (i) is immediate.
If g is an element of G, and if H is an open subgroup of G, then conjugation∼
by g induces an isomorphism φg : H −→ gHg −1 . The action of g on V induces
an automorphism of V such that φg (h)gv = ghv, for h ∈ H and so we obtain an∼
induced isomorphism H i (H, V ) −→ H i (gHg −1 , V ). Passing to the inductive limiti(G, V ). It is
as H shrinks to the identity, we obtain an automorphism σg of Hstiimmediately verified that the association of σg to g defines a G-action on Hst(G, V ).
It is a standard fact that inner automorphisms act trivially on cohomology, implyingi(G, V ) is smooth. Tothat H i (H, V ) is fixed by H. Thus the G-action on Hst
complete the proof of (ii), we must also show that this action is admissible. This
follows from part (iii) (which we prove next).
If H is a compact open subgroup of G, then the set of normal open subgroupsof H is cofinal in the directed set of open subgroups of G. Thus there is a natural∼
isomorphism lim
H i (H1 , V ) −→ Hst (G, V ), where H1 runs over all normal open−→
H1

subgroups of H. To prove part (iii), it thus suffices to show that for each choice ofsuch a subgroup H1 , the natural map H i (H, V ) → H i (H1 , V ) identifies the sourcewith the H-invariants in the target.
The quotient H/H1 is finite, and so has trivial higher cohomology with coefficients in an E-vector space (E being of characteristic zero). The Hochschild-Serre
spectral sequence thus implies that restricting cohomology classes maps H i (H, V )isomorphically onto the H-invariants of H i (H1 , V ). This proves (iii).
Note also that since V is an admissible continuous G-representation (by assumption), it is also an admissible continuous H-representation (since H is compact open

12

MATTHEW EMERTON

in G). Thus H i (H, V ) is finite dimensional. This completes the proof of (ii).
•Part (iv) is an immediate consequence of the fact that the functors Hst(G, – )iare defined as the inductive limit of the δ-functors H (H, – ).
If H is a compact open subgroup of G then the proof of Lemma 1.1.1 showsthat C(H, E) is an injective object in the category of admissible continuous representations of G, and thus that H i (H, C(H, E)) vanishes for each i > 0. This
proves (v). 
We now observe that the functor V 7→ Vsm on the category of admissible
continuous representations of G admits an alternative description as the functor
V 7→ (Vla )g , where the subscript “la” denotes the space of locally analytic vectors in V (see [13, Def. 3.5.3]; the local field L is equal to Qp in the situation weare considering), and g denotes the Lie algebra of G. This suggests that there•
should be a relation between the functors Hst(G, V ) and the Lie algebra cohomol•
ogy H (g, Vla ). Indeed, there is such a relation, given by the following proposition.
The key ingredient is [31, Thm. 7.1], which shows that the functor V 7→ Vla is exact.
Theorem 1.1.13. If G is a locally Qp -analytic group, then there is a natural Gequivariant isomorphism∼•
Hst(G, – ) −→ H • (g, (– )la )of δ-functors on the category of admissible continuous G-representations.
Proof. As already noted, the functor V 7→ Vla is exact, by [31, Thm. 7.1]. As is
implicit in the proof of Lemma 1.1.1, any admissible continuous representation of
G has a resolution by a complex each of whose terms is a finite direct sum of copiesof the injective object C(G, E). Thus to prove the proposition, it suffices to showthat C(G, E)la has vanishing higher g-cohomology.
∼
By [13, 3.5.11], there is a natural isomorphism C la (G, E) −→ C(G, E)la . Thus welamust show that C (G, E) has trivial g-cohomology. We use the standard complex∧• g ⊗E C la (G, E) to compute this cohomology, and observe that this complex isnaturally isomorphic to the locally analytic de Rham complex on the locally Qp analytic manifold G. A simple locally analytic “Poincaré lemma” shows that thiscomplex has vanishing higher cohomology, and we are done. 
The preceding result is proved for finite dimensional G-representations in [25,V.2.4.10 (i)].
(1.2) Let G be a compact locally Qp -analytic group. Up to this point we haverestricted our attention to to E-Banach spaces with continuous G-action. However,
we will also require some more refined results for certain OE -modules with G-action.
If V is an E-Banach space, then any bounded open OE -lattice in V is a p-adicallyseparated and complete torsion free OE -module. Conversely, if M is a p-adicallyseparated and complete torsion free OE -module, then the tensor product E ⊗OE M
is naturally an E-Banach space. Indeed, the embedding M → E ⊗OE M identifies
M with an OE -lattice of E ⊗OE M, and the gauge of M is a complete norm on
E ⊗OE M . (More generally, if M is a p-adically complete and separated OE -module,
then the image of M in E ⊗OE M under the natural map becomes an OE -lattice,whose gauge is a complete norm on E ⊗OE M .)If M is any OE -submodule, we will let Mtor denote the torsion subgroup of M ,and let Mtf denote the quotient M/Mtor . Thus Mtor is the kernel of the natural
map M → E ⊗OE M, while Mtf is naturally isomorphic to the image of this map.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

13

We now introduce an analogue for OE [G]-modules of the notion of an admissible
continuous G-representation on an E-Banach space.
Definition 1.2.1. If M is an OE [G]-module, p-adically complete and separated
as an OE -module, then we say that M is admissible if it satisfies the following two
conditions:(i) The torsion subgroup Mtor of M has bounded exponent.
(ii) The induced G-action on E ⊗OE M makes this latter space an admissible
continuous representation of G (when equipped with its natural E-Banach spacestructure).
Our goal in this section is to establish some basic homological properties ofadmissible OE [G]-modules.
We begin with some lemmas. Before stating the first one, note that if M isan OE [G]-module, then Mtor is an OE [G]-submodule of M , and so Mtf is also
naturally an OE [G]-module.
Lemma 1.2.2. If M is an OE -module (respectively an OE [G]-module) such that
Mtor is of bounded exponent, then M is p-adically separated and complete (respectively admissible) if and only if Mtf is p-adically separated and complete (respectively admissible).
Proof. Both claims are immediate. (For the parenthetical claim, one should use
the fact that the natural map E ⊗OE M → E ⊗OE Mtf is an isomorphism.) 
Lemma 1.2.3. Let M be a p-adically separated and complete OE -module, andsuppose that Mtor is of bounded exponent. If N ⊂ M is an inclusion of OE modules, then the p-adic topology on M induces the p-adic topology on N if andonly if the p-adic topology on Mtf induces the p-adic topology on Ntf .
Proof. This is easily proved, and we leave the verification to the reader. 
Proposition 1.2.4. Let φ : M → N be an OE -linear and G-equivariant morphism
between admissible OE [G]-modules.
(i ) The OE [G]-module ker φ is admissible, and the natural map ker φ → M is a
closed embedding, when each of the source and target are endowed with their p-adic
topologies.
(ii ) The OE [G]-module im φ is admissible, and the natural map im φ → N is a
closed embedding, when each of the source and target are endowed with their p-adic
topologies.
(iii ) The OE [G]-module coker φ is again an admissible OE [G]-module.
Proof. Taking into account Lemmas 1.2.2 and 1.2.3, we see that we may replace M
and N by Mtf and Ntf in the proof of the lemma. Thus we assume that M and N
are both torsion free.
Tensoring with E, the map φ induces a continuous G-equivariant map id ⊗ φ :E ⊗OE M → E ⊗OE N of Banach spaces equipped with admissible continuous Grepresentations. The kernel of id ⊗ φ is a closed G-invariant Banach subspace ofE ⊗OE M , on which the resulting G-action is admissible continuous. Thus ker φ
is closed in M when equipped with its p-adic topology, and forms an admissible
OE [G]-module. This proves (i).
Any continuous G-equivariant map between admissible continuous representations of G is necessarily strict, with closed image [13, Prop. 6.2.9]. Thus im φ is

14

MATTHEW EMERTON

closed in N , when endowed with its p-adic topology, and is again an admissible
OE [G]-module. This proves (ii).
If M̃ denotes the saturation of im φ in N , then M̃ is a bounded subset of theimage of id ⊗ φ (being contained in N ), and so we see that pr M̃ ⊂ im φ for somesufficiently large integer r. This proves that the torsion subgroup of the cokernelof φ has bounded exponent. The cokernel of id ⊗ φ is again an E-Banach space,equipped with an admissible continuous representation of G [13, Cor. 6.2.16]. It
follows that the torsion free quotient of the cokernel of φ is an admissible OE [G]module. Thus coker φ is an extension of a torsion free admissible OE [G]-module
by a torsion subgroup of bounded exponent. By Lemma 1.2.2, the proof of (iii) is
complete. 
Lemma 1.2.5. Let 0 → M → N → P → 0 be a short exact sequence of OE modules. If M and P are p-adically separated and complete, and if Mtor and Ptor
are both of bounded exponent, then N is also p-adically separated and complete, andNtor is of bounded exponent. Furthermore, the p-adic topology on M coincides withthe topology obtained by regarding M as a subspace of N , equipped with its p-adic
topology.
Proof. The sequence 0 → Mtor → Ntor → Ptor is exact, and thus Ntor has bounded
exponent, since this is true of Mtor and Ptor , by assumption.
Consider now the exact sequence of projective systems{P [ps ]}s≥1 → {M/ps }s≥1 → {N/ps }s≥1 → {P/ps }s≥1 → 0.
Since Ptor is of bounded exponent, we see that the transition maps in the projective system {P [ps ]}s≥1 eventually vanish, and so lim
P [ps ] and R1 lim
P [ps ] are
←−
←−
s

s

both trivial. Since the transition maps in the projective systems {M/ps }s≥1 are
M/ps = 0. Passing to the projective limit, we
surjective, we also find that R1 lim←−sthus obtain a map of short exact sequences
s

0

M/p
/ lim
←−

0


/M

s

N/p
/ lim
←−
s

s

P/p
/ lim
←−

s

s


/N


/P

/0

/ 0.

The outer two vertical arrows are isomorphisms, by assumption, and thus so is the
middle vertical arrow. Thus N is p-adically separated and complete. The claim
about topologies also follows from the fact that the transition maps in the projective
system {P [ps ]}s≥1 are eventually trivial. 
Lemma 1.2.6. Any extension (in the category of OE [G]-modules) of admissible
OE [G]-modules is again an admissible OE [G]-modules.
Proof. Let 0 → M → N → P → 0 be an extension of OE [G]-modules, and supposethat M and P are admissible. Lemma 1.2.5 shows that N is p-adically separated
and complete, and that Ntor has bounded exponent. Tensoring this short exact
sequence with E over OE , we obtain a short exact sequence of Banach spaces
equipped with G-action
0 → E ⊗OE M → E ⊗OE N → E ⊗OE P → 0.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

15(Note that Lemma 1.2.5 shows that E ⊗OE M → E ⊗OE N is a closed embedding.) Since any extension of admissible continuous G-representations is again an
admissible continuous G-representation [13, Prop. 6.2.7], the lemma follows. 
Lemma 1.2.7. If M is a torsion free admissible OE [G]-module, then for somenatural number n we may find an embedding of OE [G]-modules M → C(G, OE )n .
Furthermore, if N denotes the cokernel of this embedding, then we may find an
OE -linear map φ : N → C(G, OE )n such that the composite of φ with the projection
C(G, OE )n → N is equal to multiplication by ps on N , for some natural number s.
Proof. Since M is torsion free, the natural map M → E ⊗OE M is an embedding.
Since by assumption E ⊗OE M is an admissible continuous G-representation, it
admits a closed embedding E ⊗OE M → C(G, E)n for some natural number n.
If we rescale sufficiently, then the restriction of this embedding yields a closed
embedding M → C(G, OE )n .
If N denotes the cokernel of this embedding, then we obtain a short exact sequence of E-Banach spaces
0 → E ⊗OE M → C(G, E)n → E ⊗OE N → 0.
By [27, Prop. 10.5] we may split this short exact sequence, say by a map σ :00
E ⊗OE N → C(G, E)n . If we take a suitably large power ps of p, then ps σ carries
Ntf (identified with the image of N in E ⊗OE N ) into C(G, OE )n . Since Ntor has
00
bounded exponent, multiplication by another suitably large power ps of p on N
factors through the natural map N → Ntf . Let s = s0 + s00 , and define φ to be thecomposite
ps σN −→ Ntf −→ C(G, OE )n .
By construction, the composite of φ with the projection C(G, OE ) → N induces
multiplication by ps on N . 
It will be useful to have an alternative description of the category of admissible
OE [G]-modules, in terms of projective systems. We begin by introducing somedefinitions and notation.
We let A0G denote the abelian category of OE [G]-modules, and let AG denotethe full subcategory of A0G consisting of admissible OE [G]-modules.
00
We let BG
denote the abelian category of projective systems {Ms }s≥1 of OE [G]modules, for which Ms is annihilated by ps for each s ≥ 1.
00
Definition 1.2.8. We say that a projective systems {Ms }s≥1 in BG
is essentially0null if for any s ≥ 1, the transition map Ms0 → Ms vanishes if s is sufficiently
large.

The full subcategory of essentially null projective systems is immediately checked
00
0
to be a Serre subcategory of BG. We let BG
denote the Serre quotient category.
00
There is a functor A0G → BG
, defined by sending an OE [G]-module M to the0
projective system {M/ps }s≥1 . We let S : A0G → BG
denote the composite of this000functor with the natural quotient functor BG→ BG. Passing to the projective
00
limit induces a functor BG→ A0G that factors through the natural quotient functor
000
0. We let T : BG
→ A0G denote the induced functor. The functor S is left
BG
→ BG
adjoint to the functor T . In particular, S is right exact and T is left exact.

16

MATTHEW EMERTON

0
Lemma 1.2.9. (i ) The functor S : A0G → BG
becomes exact when it is restricted
to AG .
(ii ) The natural transformation idAG → T ◦ S induced by the adjointness of S
and T becomes an isomorphism when restricted to AG .

Proof. As observed above, the functor S is right exact. Thus to prove part (i), wemust show that it is also left exact. For this, we must show that for any inclusion
M ⊂ N ofmap {M/ps }s≥1 →T admissible OE [G]-modules, the kernel of the natural
T{M/(M ps N )}s≥1 , that is, the projective system {(M ps N )/ps M }s≥1 , is essentially
In other words, we must show that for any s ≥ 0, there is an inclusion
T null.
s0
M p N → ps M for s0 sufficiently large. This follows from part (ii) of Proposition 1.2.4.
Part (ii) is just a rephrasing of the fact that objects of AG are p-adically separated
and complete. 0
Let BG denote the essential image of the functor AG → BG
obtained by restricting S. Thus the objects of BG are those projective systems {Ms }s≥1 of OE /ps [G]modules for which the projective limit M := lim
Ms is an admissible OE [G]-module,←−sand for which the natural map of projective systems {M/ps }s≥1 → {Ms }s≥1 hasessentially null kernel and cokernel.
Proposition 1.2.10. The functor S, when restricted to AG , yields an equivalenceof categories between AG and BG , with T providing a quasi-inverse.
Proof. This is general categorical nonsense, given the definition of BG as the essential image of S, the fact that T is right adjoint to S, and the conclusions ofLemma 1.2.9. 0is closed under passage to kernels
Proposition 1.2.11. The subcategory BG of BG
and cokernel of morphisms.

Proof. Since the restriction of S to AG is exact, and since AG is closed underpassing to kernels and cokernels in A0G (by Proposition 1.2.4), we see that BG is0. closed under passing to kernels and cokernels in BG
Proposition 1.2.12. If M • is a cochain complex in the category AG , then for each
integer n the natural map {H n (M • )/ps }s≥1 → {H n (M • /ps )}s≥1 is an isomorphism of objects in BG . Consequently, the natural map H n (M • ) → lim
H n (M • /ps )←−sis an isomorphism of objects in AG .
Proof. Propositions 1.2.4 and 1.2.11 show that AG and BG are each closed under0
passing to kernels and cokernels (when regarded as subcategories of A0G and BG
respectively), while Proposition 1.2.10 shows that the functors S and T induce
an equivalence of categories between AG and BG . Thus for each n the cohomology
module H n (M • ) lies in AG , the projective system H n (S(M • )) lies in BG , and there∼
is a natural isomorphism S(H n (M • )) −→ H n (S(M • )). This gives the required0
isomorphism in BG. Applying the quasi-inverse T we obtain the isomorphisms∼∼

H n (M • ) −→ T S(H n (M • )) −→ T (H n (S(M • )))in AG , completing the proof of the proposition. ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

17

In the remainder of this subsection we will analyze the continuous cohomology
of G with coefficients in an admissible OE [G]-module. The results that we obtainwill be applied only in the proof of Proposition 2.4.1, and the reader willing to takethat result on faith may safely pass on to Section 2.
We begin by remarking that the preceding definitions and results apply in particular when the group G is trivial. In this case we speak simply of admissible
OE -modules, and write A, A0 , B, and B 0 for the categories introduced above. Explicitly, the objects of A consist of those OE -modules M for which Mtor is ofbounded exponent, and Mtf is finitely generated. (Take into account Lemma 1.2.2
and the fact that any finitely generated OE -module is p-adically separated andcomplete.) In addition to our preceding results, in this case we have the following
lemmas.
Lemma 1.2.13. The category A of admissible OE -modules is a Serre subcategoryof the category A0 of all OE -modules.
Proof. Propositions 1.2.4 and 1.2.6 imply that A is closed under passing to extensions and cokernels in A0 . Thus it suffices to show that A is closed under passing
to subobjects. This follows directly from the fact that OE is Noetherian. Lemma 1.2.14. Suppose that we are given OE -modules M and N , and a pairof OE -linear morphisms M → N and N → M whose composition is equal tomultiplication by ps on M , for some natural number s. If N is an admissible
OE -module, then the same is true of M .
Proof. If X denotes the kernel of the map M → N, then X ⊂ M [ps ], and there isan exact sequence 0 → X → Mtor → Ntor . Thus if Ntor is of bounded exponent,
the same is true of Mtor . Also, the map M → N induces an embedding Mtf → Ntf ,and so if Ntf is of finite type over OE , the same is true of Mtf . 
If M is any topological OE -module equipped with a continuous G-action, then•we let Ccon
(G, M ) denote the complex of continuous cochains on G with values in••
M , and let Hcon
(G, M ) denote the cohomology of the complex Ccon
(G, M ) (thecontinuous cohomology of G with coefficients in M ). If M is an admissible OE [G]module, then when we speak of the continuous cochains or continuous cohomology
of G with coefficients in M , we will always regard M as being equipped with its
p-adic topology.
If 0 → M → N → P → 0 is a sequence of continuous OE -linear and Gequivariant maps of topological OE -modules equipped with continuous G-actions,that is short exact as a sequence of OE -modules, then we have an exact sequenceof complexes of OE -modules(1.2.15)•••
0 → Ccon(G, M ) → Ccon
(G, N ) → Ccon
(G, P ),

which need not be exact on the right in general. (This is one of the complications ofworking with continuous cohomology.) However, we do have the following lemma.
Lemma 1.2.16. If in the situation of the preceding paragraph, the topology on P
is the discrete topology, then (1.2.15 ) is exact on the right.
•
Proof. Since the topology on P is discrete, the maps in Ccon
(G, P ) are locally•constant, and so may be lifted to locally constant elements of Ccon
(G, N ) (since
N → P is surjective). 18

MATTHEW EMERTON

We begin by noting that the continuous cohomology of C(G, OE ) has the expected values.
Lemma 1.2.17. The natural embedding OE → C(G, OE ) induces an isomorphism∼
0i
OE −→ Hcon
(G, C(G, OE )). The continuous cohomology Hcon
(G, C(G, OE )) vanishes if i > 0.
0
Proof. The assertion regarding Hcon
(G, C(G, OE )) is evident. The assertion regarding the higher cohomology modules provides a refinement of the vanishing result
discussed in the proof of Proposition 1.1.3, and is proved in the same way. 

Proposition 1.2.18. If M is an admissible OE [G]-module, then for each naturali
number i the cohomology module Hcon
(G, M ) is an admissible OE -module.
Proof. Let ps be the exponent of Mtor . Then the endomorphism of M inducedby multiplication by ps factors as M → Mtf → M. Thus the endomorphism ofi
Hcon(G, M ) induced by multiplication by ps factors as
i
i
i
Hcon
(G, M ) → Hcon
(G, Mtf ) → Hcon
(G, M ).
i
(G, Mtf ) is an admissible OE -module,
By Lemma 1.2.14, it suffices to show that Hconand so we may restrict our attention to torsion free M .
0(G, C(G, OE )) = OE . If
We proceed by induction on i. Lemma 1.2.17 gives Hcon
M is torsion free, then by Lemma 1.2.7 we may find an embedding M → C(G, OE )n0n
for some natural number n, and hence an embedding Hcon
(G, M ) → OE
. Thisestablishes the case i = 0.
We now suppose the result is known for i − 1. Let M be torsion free, and again
use Lemma 1.2.7 to regard M as a submodule of C(G, OE )n , for some natural
number n. If N denotes the quotient C(G, OE )n /M, then by that same lemma we
may find a map φ : N → C(G, OE )n such that the compositeφN −→ C(G, OE )n −→ N(the second arrow being the natural projection) is equal to multiplication by ps , forsome natural number s.
We consider the diagram (1.2.15) attached to the short exact sequence 0 → M →•C(G, OE )n → N → 0. If C • denotes the cokernel of the natural map Ccon
(G, M ) →n•••
Ccon (G, C(G, OE ) ), then there is an injection C → Ccon (G, N ). Also, φ induces••
a map Ccon
(G, N ) → C • , and the composite C • → Ccon
(G, N ) → C • is equal tosmultiplication by p . Thus we obtain a sequence of mapsi−1
H i−1 (C • ) → Hcon
(G, N ) → H i−1 (C • )whose composite equals multiplication by ps . By induction, we may assume thati−1
Hcon
(G, N ) is an admissible OE -module, and Lemma 1.2.14 then implies that
H i−1 (C • ) is an admissible OE -module.
Consider now the long exact sequence of cohomology attached to the short exact
sequence of complexes••
0 → Ccon(G, M ) → Ccon
(G, C(G, OE )n ) → C • → 0.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

19

From this long exact sequence we may extract the exact sequenceii
H i−1 (C • ) → Hcon
(G, M ) → Hcon
(G, C(G, OE )n ).
i
Lemma 1.2.17 shows that Hcon
(G, C(G, OE )n ) is trivial for i > 0, and so we see thati
Hcon (G, M ) is a quotient of the admissible OE -module H i−1 (C • ). Lemma 1.2.13
i
implies that Hcon
(G, M ) is itself an admissible OE -module. This completes theproof of the proposition. 

Proposition 1.2.19. If M is an admissible OE [G]-module, then the natural map ofiiprojective systems {Hcon(G, M )/ps }s≥1 → {Hcon(G, M/ps )}s≥1 is an isomorphism
0in B . (That is, its kernel and cokernel are essentially null projective systems.)
Proof. If s0 is such that Mtor has exponent ps0 , then multiplication by ps0 factors
through the natural projection M → Mtf , say as
ps0

/ Mtf

M

φ

&
/ M,

for some map φ : Mtf → M. If s ≥ s0 , then write φs = ps−s0 φ. For each such valueof s, we have a short exact sequenceφs

0 −→ Mtf −→ M −→ M/ps −→ 0.
Passing to continuous cochains, we obtain a short exact sequence of complexes
φs0 −→ C • (G, Mtf ) −→ C • (G, M ) −→ C • (G, M/ps ) −→ 0.
(We have exactness on the right by Lemma 1.2.16.) Passing to cohomology, we
obtain for any natural number i a short exact sequence
i
i
i
i+1
0 → Hcon(G, M )/φs (Hcon(G, Mtf )) → Hcon
(G, M/ps ) → Hcon
(G, Mtf )[φs ] → 0.

(By abuse of notation, we have continued to denote by the same symbol the mapthat φs induces on cohomology.)
φs

The composite Mtf −→ M −→ Mtf is equal to multiplication by ps , and
i+1
i+1
i+1
thus Hcon(G, Mtf )[φs ] is contained in Hcon
(G, Mtf )[ps ]. Since Hcon
(G, Mtf ) is
an admissible OE -module, by Proposition 1.2.18, we see that the projective sysi+1(G, Mtf )[ps ]}s≥s0 (the transition maps being given by multiplication
tem {Hconby p) is essentially null, and hence the same is true of the projective system
i+1(G, Mtf )[φs ]}s≥s0 . Thus the natural injection of projective systems
{Hcon
i
i
i
{Hcon
(G, M )/φs (Hcon(G, Mtf ))}s≥s0 → {Hcon(G, M/ps )}s≥s0

has essentially null cokernel.
For any s ≥ s0 , we have inclusionsiiii
ps Hcon(G, M ) ⊂ φs (Hcon(G, Mtf )) = ps−s0 φ(Hcon(G, Mtf )) ⊂ ps−s0 Hcon
(G, M ).

Thus the natural surjection of projective systems
i
i
i
{Hcon
(G, M )/ps }s≥s0 → {Hcon(G, M )/φs (Hcon(G, Mtf ))}s≥s0

has essentially null kernel. Combining this conclusion with that of the preceding
paragraph, we find that the natural map of projective systems
i
i
{Hcon
(G, M )/ps }s≥1 → {Hcon(G, M/ps )}s≥1

has essentially null kernel and cokernel, as required.



20

MATTHEW EMERTON

Proposition 1.2.20. If M is an admissible OE [G]-module, then for any naturaliinumber i, the natural map E ⊗OE Hcon
(G, M ) → Hcon
(G, E ⊗OE M ) (where thecontinuous cohomology of E ⊗OE M is computed with respect to the Banach space
topology on this tensor product) is an isomorphism.
••
Proof. As G is compact, the natural map E ⊗OE Ccon
(G, M ) → Ccon
(G, E ⊗OE M )is an isomorphism. Passing to cohomology (and taking into account the fact that
E is flat over OE ) yields the proposition. 

2. Cohomology of arithmetic quotients of symmetric spaces(2.1) The goal of this subsection is to explain how one can obtain admissible
continuous representations of locally Qp -analytic groups by taking appropriate limits of cohomology in a tower of covering spaces. The main result is Theorem 2.1.5
below.
Throughout this subsection we assume that G is a compact locally Qp -analytic
group, and we fix a countable basis of open normal subgroups

G = G0 ⊃ G1 ⊃ . . . ⊃ Gr ⊃ . . . .

We suppose given a sequence of continuous maps of topological spaces· · · → Xr → · · · → X1 → X0 ,and assume that each of these spaces is equipped with a right action of G, suchthat(1) the maps in this sequence are G-equivariant;(2) the open subgroup Gr of G acts trivially on Xr ;(3) if 0 ≤ r0 ≤ r then the map Xr → Xr0 is a Galois covering map with decktransformations provided by the natural action of Gr0 /Gr on Xr .
(If X0 is connected with base point x0 , then it is equivalent to give a morphism
π1 (X0 , x0 ) → G. For any r ≥ 0, reduction modulo Gr yields a map π1 (X0 , x0 ) →G/Gr , and thus a (not necessarily connected) Galois covering space Xr → X0
whose group of deck transformations is isomorphic to G/Gr (acting on the right).
Precisely, Xr = X̃ ×π1 (X,x0 ) G/Gr , where X̃ is the universal cover of X, on which
π1 (X0 , x0 ) acts on the right as deck transformations (after furthermore fixing abase-point x̃0 of X̃ lying over x0 ).)We also suppose given a local system of free finite rank OE -modules V0 on X0(where OE is the ring of integers in a finite extension E of Qp ), and denote by Vr
the pullback of V0 to Xr for any r ≥ 0. The sheaf Vr is G/Gr -equivariant.
Let ∗ denote one of ∅ or c, so that H∗• denotes either cohomology or compactly

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

21

supported cohomology. DefineH∗n (V) := limH∗n (Xr , Vr ),−→r∼H∗n (V)E := E ⊗OE H∗n (V) −→ lim
H∗n (Xr , E ⊗OE Vr ),−→r

Ĥ∗n (V) := limH∗n (V)/ps ,←−s(2.1.1)Ĥ∗n (V)E := E ⊗OE Ĥ∗n (V),H̃∗n (V) := lim
lim H∗n (Xr , Vr /ps ),
←− −→
s

r

H̃∗n (V)E := E ⊗OE H̃∗n (V),Tp H∗n (V) := limH∗n (V)[ps ],←−s

Vp H∗n (V) := E ⊗OE Tp H∗n (V).
(Here r and s range over all non-negative integers.) Each of the OE -modules andE-vector spaces that we have defined is equipped with a natural G-action. (TheG-action on H∗n (V) and H∗n (V)E is smooth, but this will typically not be so for the
G-action on the other objects appearing in (2.1.1).)
If we fix for the moment non-negative integers r and s then the exact sequence ofps ·sheaves 0 −→ Vr −→ Vr −→ Vr /ps −→ 0 on Xr gives rise to a short exact sequence
0 → H∗n (Xr , Vr )/ps → H∗n (Xr , Vr /ps ) → H∗n+1 (Xr , Vr )[ps ] → 0.
Passing now to the inductive limit over r yields the short exact sequence
0 → H∗n (V)/ps → lim
H∗n (Xr , Vr /ps ) → H∗n+1 (V)[ps ] → 0.
−→r

Each OE /ps -module appearing in this exact sequence is equipped with a natural Gaction, and the exact sequence is obviously G-equivariant. Passing to the projective
limit over s, we obtain the short exact sequence of G-equivariant maps
(2.1.2)0 → Ĥ∗n (V) → H̃∗n (V) → Tp H∗n+1 (V) → 0.

(To see that this sequence is short exact, note that for each value of s the transition
map H∗n (V)/ps+1 → H∗n (V)/ps is surjective.) Tensoring with E over OE yields theG-equivariant short exact sequence(2.1.3)0 → Ĥ∗n (V)E → H̃∗n (V)E → Vp H∗n+1 → 0.

Lemma 2.1.4. The short exact sequence (2.1.2 ) is a short exact sequence of padically complete and separated OE -modules. Furthermore, the arrow Ĥ∗n (V) →H̃∗n (V) is a closed embedding, if we endow its source and target with their p-adic
topologies.
Proof. If M is any OE -module, then its p-adic completion lim
M/ps is a p-adically←−scomplete and separated OE -module. Its p-adic Tate module lim
M [ps ] is also p←−sadically complete and separated, and is furthermore torsion free. Reducing the

22

MATTHEW EMERTON

short exact sequence (2.1.2) modulo ps thus yields, for each natural number s, amorphism of short exact sequences0/ Ĥ n (V)∗/ H̃ n (V)∗/ Tp H n+1 (V)
∗

/0

0


/ Ĥ n (V)/ps/ H̃ n (V)/ps/ Tp H n+1 (V)/ps

/ 0.

∗

∗

∗

Passing to the projective limit in s, we obtain a morphism of short exact sequences0/ Ĥ n (V)∗/ H̃ n (V)∗/ Tp H n+1 (V)
∗

/0

0


n
lim
Ĥ
/ ←− ∗ (V)/ps


n
lim
H̃
/ ←− ∗ (V)/ps


n+1
lim
T
H
/ ←− p ∗ (V)/ps

/ 0.

s

s

s

The left-hand and right-hand vertical arrows of this diagram are isomorphisms, by
the remarks of the first paragraph. Thus the middle arrow is also an isomorphism.
We conclude that H̃∗n (V) is p-adically complete and separated, and also that themap Ĥ∗n (V) → H̃∗n (V) is a closed embedding. The preceding lemma implies that each of the tensor products Ĥ∗n (V)E , H̃∗n (V)E
and Vp H∗n (V) is naturally an E-Banach space on which G acts continuously, andthat (2.1.3) is a G-equivariant short exact sequence of E-Banach spaces.
Theorem 2.1.5. Take ∗ = ∅, so that we are discussing cohomology. If X0 is
homotopic to a finite simplicial complex then the following results hold:(i ) The p-adic locally analytic group G acts on each of the E-Banach spaces
appearing in the exact sequence (2.1.3 ) through an admissible continuous representation, and the exact sequence (2.1.3 ) is a short exact sequence in the abeliancategory of admissible continuous representations of G.
(ii ) The composite H n (V)E → Ĥ n (V)E → H̃ n (V)E (the first arrow being thenatural map and the second arrow being that which appears in the short exact sequence (2.1.3 )) has an alternative factorization as a composite
H n (V)E → (H̃ n (V)E )sm → H̃ n (V)E ,and the first of these maps is the edge map of a spectral sequenceiE2i,j = Hst(G, H̃ j (V)E ) =⇒ H i+j (V)E .

Proof. In order to prove the theorem, we introduce some notation. First note thatby replacing X0 by a finite simplicial complex homotopic to it, we may assume that
X0 is itself a finite simplicial complex. Let T• (X0 ) be a choice of finite triangulationof X0 (where Tn (X0 ) denotes the collection of n-dimensional simplices occuring in
the triangulation), and let T• (Xr ) denote the pullback of T• (X0 ) under the finite
covering map Xr → X0 . The set T• (Xr ) is equipped with a free action of G/Gr . If
∆ ∈ Tn (X0 ) for some n then we let Tn (Xr )/∆ denote the set of simplices in Tn (Xr )ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

23

lying over ∆; the set Tn (Xr )/∆ is then a principal homogeneous G/Gr -set. We
let T̂n/∆ denote the projective limit lim
Tn (Xr )/∆ ; this is a profinite set which is←−rprincipal homogeneous with respect to its natural G-action.
For any r ≥ 0, the cohomology of the sheaf Vr is computed by a complex S • (Vr )whose nth term isYYY∼
S n (Vr ) :=H 0 (∆0 , Vr ) −→H 0 (∆, V0 ).
∆0 ∈Tn (Xr )∆∈Tn (X0 ) ∆0 ∈Tn (Xr )/∆The complex S • (Vr ) is equipped with a natural action of G/Gr , and if 0 ≤ r0 ≤ rthen the natural (pullback) map S • (Vr0 ) → S • (Vr ) identifies S • (Vr0 ) with theGr0 /Gr -invariants of S • (Vr ). Setting S • (V) := lim
S • (Vr ), we see that S • (V) is−→requipped with an action of G. The nth term of S • (V) is given by
Y
Y
∼
lim
H 0 (∆, V0 )S n (V) −→−→∆∈Tn (X0 )r∆0 ∈Tn (Xr )/∆
∼

Y

−→

C sm (T̂n/∆ , OE ) ⊗OE H 0 (∆, V0 ).

∆∈Tn (X0 )We let S • (V)E denote the tensor product of the complex S • (V) with E over OE .
Since tensor products commute with inductive limits and finite products, the nthterm is given byY∼(2.1.6)
S n (V)E −→C sm (T̂n/∆ , E) ⊗E H 0 (∆, V0 ⊗OE E).
∆∈Tn (X0 )The nth cohomology module of the complex S • (V) is naturally isomorphic to
H n (V); the nth cohomology space of S • (V)E is naturally isomorphic to H n (V)E .
(We are using the fact that passage to the inductive limit is exact, and commuteswith tensor products.)
If s is a natural number then we can form similar constructions with V0 replacedS • (Vr /ps ), equipped with a Gby V0 /ps , and obtain a complex S • (V/ps ) = lim−→raction, whose nth term is∼

S n (V/ps ) −→Y

C sm (T̂n/∆ , OE /ps ) ⊗OE /ps H 0 (∆, V0 /ps ).

∆∈Tn (X0 )The nth cohomology module of S • (V/ps ) is naturally isomorphic to the inductive
limit limH n (Xr , Vr /ps ).
−→r

Taking the projective limit lim
S • (V/ps ) as s grows to ∞, we obtain a complex←−sthat we denote S̃ • (V), again equipped with a G-action, whose nth term is given byY∼(2.1.7) S̃ n (V) −→ lim
C(T̂n/∆ , OE /ps ) ⊗OE /ps H 0 (∆, V0 /ps )←−s∆∈Tn (X0 )∼−→Y∆∈Tn (X0 )C(T̂n/∆ , OE ) ⊗OE H 0 (∆, V0 ).

24

MATTHEW EMERTON

Tensoring with E over OE we obtain a complex S̃ • (V)E equipped with a G-action,
whose nth term is given byY∼(2.1.8)S̃ n (V)E −→C(T̂n/∆ , E) ⊗E H 0 (∆, V0 ⊗OE E).
∆∈Tn (X0 )Let ∆ be a simplex in Tn (X0 ). As already observed, the right action of G on T̂n/∆makes T̂n/∆ a principal homogeneous profinite G-set, and so C(T̂n/∆ , OE ) (respectively C(T̂n/∆ , E)) is isomorphic as a G-representation to C(G, OE ) (respectively
C(G, E)) equipped with its right regular G-action. In light of this, the descriptionof each of the terms of S̃ • (V) (respectively S̃ • (V)E ) afforded by (2.1.7) (respectively (2.1.8)) shows that S̃ • (V) (respectively S̃ • (V)E ) is a complex of admissible
OE [G]-modules, in the sense of Definition 1.2.1 (respectively admissible continuous
representations of G).
Passing to cohomology (and taking into account the natural isomorphism∼S̃ • (V)/ps −→ S • (V/ps )for each natural number s), Proposition 1.2.12 shows that for each value of n, there
is a natural isomorphism∼∼

H n (S̃ • (V)) −→ lim
H n (S • (V/ps )) −→ lim
lim H n (Xr , Vr /ps ) = H̃ n (V),
←−
←− −→
s

s

r

and hence, after tensoring with E, a natural isomorphism(2.1.9)∼H n (S̃ • (V)E ) −→ H̃ n (V)E .

•Since S̃E
is a complex of admissible continuous representations of G, the same istrue of its cohomology spaces, and thus we see that H̃ n (V)E , when endowed with
its natural Banach space structure, is an admissible continuous representation of G.
It was already observed that (2.1.3) is a short exact sequence of Banach spaces,
and we have seen that its middle term is equipped with an admissible continuous
representation of G. Any G-invariant closed subspace or Hausdorff quotient of an
E-Banach space equipped with an admissible continuous representation of G isagain an E-Banach space equipped with an admissible continuous representationof G, and so part (i) of the theorem follows.
The morphisms V0 → V0 /ps , as s ranges over all natural numbers, induce aG-equivariant morphism S • (V) → S • (V/ps ) for each s, and hence G-equivariant
morphisms S • (V) → S̃ • (V) and S • (V)E → S̃ • (V)E . A comparison of (2.1.6)
and (2.1.8) shows that this latter morphism identifies S • (V)E with the complexof G-smooth vectors in the complex S̃ • (V)E . Part (ii) now follows from Proposition 1.1.10, together with the isomorphism (2.1.9), upon taking the complex V • ofthat proposition to be the complex S̃ • (V)E . 

Using the results on continuous cohomology proved in Subsection 1.2 we can givean alternative construction of the spectral sequence of Theorem 2.1.5 (ii). Its only
application will be to the proof of Proposition 2.4.1.
For any fixed values of r and s, and any r0 ≥ r, we have the Hochschild-Serre
spectral sequenceE2i,j = H i (Gr /Gr0 , H j (Xr0 , V/ps )) =⇒ H i+j (Xr , V/ps ).
Passing to the inductive limit in r0 , we obtain a spectral sequence(2.1.10)i
E2i,j = Hcon
(Gr , limH j (Xr0 , V/ps )) =⇒ H i+j (Xr , V/ps ).
−→r0

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

25

Proposition 2.1.11. If we fix a value of r, then passing to the projective limit in sof the spectral sequences (2.1.10 ), followed by tensoring with E over OE , yields aspectral sequenceiE2i,j = Hcon
(Gr , H̃ j (V)E ) =⇒ H i+j (Xr , E ⊗OE V).

The inductive limit (with respect to r) of these spectral sequences yields a spectral
sequence that is naturally isomorphic to the spectral sequence of Theorem 2.1.5 (ii ).
Proof. The proof of Theorem 2.1.5 shows that H̃ j (V) is an admissible OE [G]module for each j ≥ 0, which can be computed as the degree j cohomology of
the complex of admissible OE [G]-modules S̃ • (V). Proposition 1.2.12 then implies
that for each value of j, the natural map of projective systems{H̃ j (V)/ps }s≥1 → {limH j (Xr0 , V/ps )}s≥1−→
r0

has essentially null kernel and cokernel. Thus the induced map of projective systemsii
H j (Xr0 , V/ps ))}s≥1{Hcon(Gr , H̃ j (V)/ps )}s≥1 → {Hcon(Gr , lim−→r0

also has essentially null kernel and cokernel. Taking into account Proposition 1.2.19,
we find that the map of projective systems
i
i
{Hcon
(Gr , H̃ j (V))/ps }s≥1 → {Hcon(Gr , limH j (Xr0 , V/ps ))}s≥1−→
r0

has essentially null kernel and cokernel. In the terminology of Subsection 1.2, thismap is an isomorphism of objects in the category B.
Since Xr is a homotopic to a finite simplicial complex, the natural map{H j (Xr , V)/ps }s≥1 → {H j (Xr , V/ps )}s≥1
is also an isomorphism in the category B. Thus we obtain a spectral sequenceiE2i,j = {Hcon(Gr , H̃ j (V))/ps }s≥1 =⇒ {H i+j (Xr , V)/ps }s≥1

in the category B. Since passing to the projective limit induces an equivalenceof categories between B and the category A of admissible OE -modules (Proposition 1.2.10), we obtain a spectral sequencei(Gr , H̃ j (V)) =⇒ H i+j (Xr , V).
E2i,j = Hcon∼ii
(Here we have used the isomorphism Hcon
(Gr , H̃ j (V)) −→ lim
Hcon(Gr , H̃ j (V))/ps ,←−swhich follows from the fact that H̃ j (V) is an admissible OE [G]-module, together∼
with Proposition 1.2.18, and the isomorphism H i+j (Xr , V) −→ lim
H i+j (Xr , V)/ps ,←−swhich is obvious, since H i+j (Xr , V) is finitely generated as an OE -module.) Tensoring this spectral sequence through by E over OE , and taking into account theisomorphism of Proposition 1.2.20, we obtain a spectral sequenceiE2i,j = Hcon
(Gr , H̃ j (V)E ) =⇒ H i+j (Xr , E ⊗OE V),

26

MATTHEW EMERTON

as in the statement of the proposition.
Passing to the inductive limit in r, and taking into account Propositions 1.1.3
and 1.1.6, yields a spectral sequenceiE2i,j = Hst(G, H̃ j (V)E ) =⇒ H i+j (V)E .

We leave it to the reader to chase through the various isomorphisms used in theconstruction of this spectral sequence, and so verify that it coincides with thespectral sequence of Theorem 2.1.5 (ii). 
We now state an analogue of Theorem 2.1.5 for compactly supported cohomology.
For this, we assume that X0 is a topological manifold.
Theorem 2.1.12. Suppose that X0 is a topological manifold that is homotopic
to a finite simplicial complex. Then the analogue of Theorem 2.1.5 holds withcohomology replaced by compactly supported cohomology.
Proof. It suffices to prove the theorem with X0 replaced by a connected component; thus we may assume that X0 is equidimensional, say of dimension d. Poincaré
duality yields a canonical isomorphism between compactly supported cohomology
in degree n with coefficients in V and homology in degree d − n with coefficients
in V; pullback of compactly supported cohomology classes corresponds to pullbackof homology classes with respect to the finite covering maps Xr+1 → Xr . Thus wework from now on with homology rather than with compactly supported cohomology. As in the proof of Theorem 2.1.5 we replace X0 by a finite simplicial complex,
and we use the same notation T• (Xr ) as in that proof for the induced triangulation
on each Xr .
The homology of Xr with coefficients in Vr is computed by a complex S• (Vr )whose nth term isYYY∼
Sn (Vr ) :=H0 (∆0 , Vr ) −→H0 (∆, V0 ).
∆0 ∈Tn (Xr )∆∈Tn (X0 ) ∆0 ∈Tn (Xr )/∆The complex S• (Vr ) is equipped with a natural action of G/Gr , and if 0 ≤ r0 ≤r then the natural (pullback) map S• (Vr0 ) → S• (Vr ) identifies S• (Vr0 ) with theGr0 /Gr -invariants of S• (Vr ). Setting S• (V) := lim
S• (Vr ), we see that S• (V) is−→requipped with an action of G. The nth term of S• (V) is given by∼

Sn (V) −→Y∆∈Tn (X0 )

Y

lim
−→
r

H0 (∆, V0 )∆0 ∈Tn (Xr )/∆
∼

−→

Y

C sm (T̂n/∆ , OE ) ⊗OE H0 (∆, V0 ),∆∈Tn (X0 )and so is isomorphic as a topological G-module to S n (V) (in the notation of theproof of Theorem 2.1.5). Similarly, if we set S̃• (V) = lim
lim S̃• (Vr /ps ), then there
←− −→
∼

s

r

is an isomorphism S̃n (V) −→ S̃ n (V) for each n.
The chain complexes S• (V) and S̃• (V) are thus term-by-term isomorphic to thecochain complexes S • (V) and S̃ • (V); it is merely the boundary and coboundary

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

27

maps that differ. The proof of Theorem 2.1.5 is hence seen to carry over fromcohomology to homology, and so yields a proof of the present theorem. Let us note that in the context of the preceding theorem, the Hochschild-Serre
spectral sequence for compactly supported cohomology yields a spectral sequence
analogous to (2.1.10), with cohomology replaced by compactly supported cohomology. The proof of Proposition 2.1.11 then carries over in the compactly supported
case, and thus yields an analogue of that proposition for compactly supported cohomology.
(2.2) We return to the situation considered in the introduction. Thus we fix anumber field F ⊂ C, a reductive connected linear algebraic group G defined over∼F , an isomorphism ı : C −→ Qp , and hence a prime p of F lying over p. We let
Fp denote the completion Fp of F at p (equivalently, the closure of F in Qp ), and
let G denote the locally Fp -analytic group G(Fp ). We also let A denote the ring ofadèles of F , let Af denote the ring of finite adèles (which we may write as Fp × Apf ,where the second factor denotes the prime-to-p finite adèles), let F∞ denote R ⊗Q F
(so A = F∞ × Af ), and write G∞ := G(F∞ ). We fix once and for all a maximal
compact subgroup K∞ of G∞ .
We write H ◦ to denote the connected component of the identity of a real Liegroup H, and write π0 (H) to denote the group of connected components of H; thatis, the quotient H/H ◦ . The structure theory of real reductive groups (in particular,
the Iwasawa decomposition) shows that the natural map
π0 (K∞ ) → π0 (G∞ )is an isomorphism. We thus identity these two connected component groups, anddenote them simply by π0 .
If Kf is a compact open subgroup of G(Af ), then we write(2.2.1)◦Y (Kf ) := G(F )\G(A)/K∞Kf .

The quotient G∞ /K∞ is the symmetric space attached to the real Lie group G∞ ,◦and so the quotient G∞ /K∞
is a finite union of copies of this symmetric space,equipped with a natural action of π0 , which acts simply transitively on the set ofconnected components. Thus Y (Kf ) is a finite union of quotients of this symmetricspace by congruence subgroups. If Kf is sufficiently small, then G(F ) acts without◦
Kf , and hence Y (Kf ) is a smooth manifold.
fixed points on the quotient G(A)/K∞0
If Kf is a normal open subgroup of such a subgroup Kf , the natural map Y (Kf0 ) →Y (Kf ) realises Y (Kf0 ) as an unramified Galois cover of Y (Kf ), with Galois groupisomorphic to the quotient Kf /Kf0 (acting on the right). The natural continuous◦
action of π0 on the quotient G∞ /K∞
induces a natural continuous action of π0 oneach of the quotients Y (Kf ).
Let W be a finite dimensional complex vector space equipped with a representation of the algebraic group G/C . We define a π0 -equivariant local system of complex
vector spaces VW on the manifold Y (Kf ) (for Kf sufficiently small) as follows:(2.2.2)◦
VW := G(F )\(W × (G(A)/K∞Kf )).

(Here G(F ) acts on W through the given representation, and on the second factor
by left multiplication.) The local system VW on Y (Kf ) is locally isomorphic to theconstant sheaf defined by W .

28

MATTHEW EMERTONAs the compact open subgroup Kf shrinks down to the identity, the manifolds
Y (Kf ) form a projective system. The local systems VW are compatible with themaps in this projective system in an obvious sense: if Kf0 ⊂ Kf , then the pullbackof the local system VW on Y (Kf ) to Y (Kf0 ) is naturally isomorphic to the local
system VW on Y (Kf0 ). Since we intend to apply the results of Subsection 2.1 in thecontext of the arithmetic quotients Y (Kf ), let us recall that, although Y (Kf ) may
not be compact, and so may not be homeomorphic to a finite simplicial complex,
it does contain a deformation retract which is a finite simplicial complex.
Let us fix a degree n and a representation W , and consider the cohomology
spaces H∗n (Y (Kf ), VW ), as Kf varies over all compact open subgroups of G(Af ),and where as in the preceding subsection, ∗ denotes either ∅ or c (so that we areconsidering either cohomology or compactly supported cohomology). Passing to
the inductive limit as Kf shrinks down to the identity, we obtain a smooth representation H∗n (VW ) := limH∗n (Y (Kf ), VW ) of π0 ×G(Af ), which is well known to be−→Kfadmissible. In fact, passing to the Kf -fixed points exactly recovers the cohomologyH∗n (Y (Kf ), VW ).
Our intention is to “p-adically complete” the vector space H∗n (VW ) appropriately(using the isomorphism ı to regard W , and hence H∗n (VW ), as a Qp -vector space)so as to obtain an admissible continuous representation of G(Af ).
We first recall an alternative description of the local system VW (regarded as a
Qp -local system via ı).
Definition 2.2.3. Let Kf be a compact open subgroup of G(Af ) (chosen sufficiently small, so that Y (Kf ) is a manifold), and let G0 denote the projection of Kf
onto G. (Thus G0 is a compact open subgroup of G). If M is a G0 -module thenwe define◦
VM := (M × (G(F )\G(A)/K∞))/Kf ,
a π0 -equivariant local system over Y (Kf ). (The right action of Kf on the productis defined via m · k = kp−1 m on the first factor (where kp ∈ G0 is the pth componentof an element k ∈ Kf ) and via right multiplication on the second factor.)
If W is a complex representation of G, then via ı we may regard W as a representation of G defined over Qp , and so in particular as a representation of G, via
the inclusion G := G(Fp ) ⊂ G(Qp ). Let W 0 denote W regarded as a representationof G in this way.
Lemma 2.2.4. If Kf is a compact open subgroup of G(Af ) then there is a natural
isomorphism of π0 -equivariant local systems over Y (Kf ) between VW (as definedby (2.2.2 ), and then regarded as a Qp -local system via ı) and the local system VW 0
defined via Definition 2.2.3.
Proof. If g ∈ G(A) then let gp denote the pth component of g. The required
isomorphism is provided by the automorphism (w, g) 7→ (gp−1 w, g) of W ×G(A). 
The preceding lemma justifies the duplication of notation in (2.2.2) and Definition 2.2.3. For the duration of this subsection we will consider only local systems
arising from representations of compact open subgroups of G via Definition 2.2.3.
It is technically easier to work with vector spaces defined over a finite extensionof Qp , rather than over Qp . Thus we fix a finite extension E of Fp contained in
Qp , and from now on we let W denote a finite dimensional representation of G

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

29

defined over E (rather than over C or Qp ). If we choose E so that G is split overE, then any representation of G over Qp descends uniquely (up to an isomorphism)
to a representation of G defined over E, and so we do not lose any generality byconsidering only W that are defined over E.
In order to make our constructions as natural as possible, it helps to consider
certain directed sets. We let S denote the set of all compact open subgroups of G,
directed downward by inclusion. If W0 is a separated lattice in W , then we let SW0
denote the directed subset of S consisting of those compact open subgroups Kp of G
that leave W0 invariant. If G0 is the maximal subgroup of G leaving W0 invariant,then G0 is a compact open subgroup of G, and SW0 consists of all open subgroups
Kp of G0 . In particular, SW0 is cofinal
in S, and if W00 is a second separated latticeTin W , then the intersection SW0 SW00 is also cofinal in each of SW0 , SW00 , and S.
Let us temporarily fix a separated lattice W0 in W . If Kf is any open subgroup ofG(Af ) whose image under the projection onto G is contained in SW0 , then via Definition 2.2.3 we obtain on Y (Kf ) a π0 -equivariant local system of OE -submodules∼
VW0 of VW such that E ⊗OE VW0 −→ VW . If Kf0 ⊂ Kf , then the equivariant
0
local system so obtained on Y (Kf ) is naturally isomorphic to the pullback of thecorresponding local system on Y (Kf ).
We now fix a “tame level” K p (that is, a compact open subgroup K p of G(Apf )).
We form the projective system of arithmetic quotients {Y (Kp K p )}Kp ∈SW0 , eachmember of which is equipped with a local system VW0 . Let us fix for a moment a
sufficiently small element Kp of SW0 , chosen so that Y (Kp K p ) is a manifold. If Kp0
is a normal open subgroup of Kp , then the natural map Y (Kp0 K p ) → Y (Kp K p ) is
unramified and Galois, with Galois group Kp /Kp0 . Thus we are in the situation of
Subsection 2.1, if we take the group G of that subsection to be Kp , and we may
form the constructions introduced in that subsection. We writeH∗n (K p , VW0 ) :=lim−→
Kp ∈SW0H∗n (Y (Kp K p ), VW0 )
∼

−→

lim
−→
Kp ∈SW0lim
H∗n (Y (Kp K p ), VW0 /ps ),←−sH∗n (K p , VW0 )E := E ⊗OE H∗n (K p , VW0 ),H̃∗n (K p , VW0 ) := lim
←−
s

lim

−→
Kp ∈SW0H∗n (Y (Kp K p ), VW0 /ps ),H̃∗n (K p , VW0 )E := E ⊗OE H̃∗n (K p , VW0 ),Tp H∗n (K p , VW0 ) := limH∗n (K p , VW0 )[ps ],←−s

Vp H∗n (K p , VW0 ) := E ⊗OE Tp H∗n (K p , VW0 ).
(These correspond to the OE -modules and E-vector spaces that in Subsection 2.1
are denoted H∗n (V), H∗n (V)E , H̃∗n (V), H̃∗n (V)E , Tp H∗n (V), and Vp H∗n (V), respectively.)
The image of H∗n (K p , VW0 ) in H∗n (K p , VW0 )E is a lattice in this E-vector space,
and we let Ĥ∗n (K p , VW0 )E denote the E-Banach space obtained by completingH∗n (K p , VW0 )E with respect to the gauge of that lattice. Also, the image ofH̃∗n (K p , VW0 ) in H̃∗n (K p , VW0 )E is a lattice in this E-vector space, and we regard H̃∗n (K p , VW0 )E as a semi-normed space, the semi-norm being given by the

30

MATTHEW EMERTON

gauge of that lattice. Similarly, we use the gauge of Tp H∗n (K p , VW0 ) to regard
Vp H∗n (K p , VW0 ) as a semi-normed E-vector space. The results of Subsection 2.1
show that both these semi-normed spaces are in fact E-Banach spaces.
An inclusion W0 ⊂ W00 of separated lattices in W induces an injection of sheaves
VW0 → VW00 , and hence morphisms of OE -modules H∗n (K p , VW0 ) → H∗n (K p , VW00 ),H̃∗n (K p , VW0 ) → H̃∗n (K p , VW00 ), and Tp H∗n (K p , VW0 ) → Tp H∗n (K p , VW00 ). (Recall
that the intersection of the directed sets SW0 and SW00 is cofinal in each of them.)
Such an inclusion thus induces continuous maps of E-Banach spaces(2.2.5)Ĥ∗n (K p , VW0 )E → Ĥ∗n (K p , VW00 )E ,(2.2.6)H̃∗n (K p , VW0 )E → H̃∗n (K p , VW00 )E ,and(2.2.7)

Vp H∗n (K p , VW0 )E → Vp H∗n (K p , VW00 )E .

Lemma 2.2.8. If W0 ⊂ W00 is an inclusion of separated lattices in W then all
three morphisms (2.2.5 ), (2.2.6 ) and (2.2.7 ) are topological isomorphisms.
Proof. If the natural number r is chosen sufficiently large, then there is an inclusion
pr W00 ⊂ W0 . Thus we may embed (2.2.5) into the sequence of mapsĤ∗n (K p , pr VW00 )E −→ Ĥ∗n (K p , VW0 )E(2.2.5)−→ Ĥ∗n (K p , VW00 )E −→ Ĥ∗n (K p , p−r VW0 )E .

The composite of the first two arrows is obviously an isomorphism, as is the composite of the second two arrows. Thus the middle arrow is an isomorphism. A
similar argument applies to (2.2.6) and (2.2.7). 
Definition 2.2.9. We make the following definitions:Ĥ∗n (K p , VW0 )E .
(i) Ĥ∗n (K p , VW ) := lim−→
W0

H̃∗n (K p , VW0 )K .
(ii) H̃∗n (K p , VW ) := lim−→
W0(iii) Vp H∗n (K p , VW ) := lim
Vp H∗n (K p , VW0 ).
−→W0(In each case the locally convex inductive limit is taken over the directed set ofseparated lattices W0 in W , directed by inclusion).
Lemma 2.2.8 shows that the transition isomorphisms in each of the three inductive systems occuring in Definition 2.2.9 are topological isomorphisms. Thus
for any choice of W0 , each of the natural maps Ĥ∗n (K p , VW0 )E → Ĥ∗n (K p , VW ),H̃∗n (K p , VW0 )E → H̃∗n (K p , VW ) and Vp H∗n (K p , VW0 ) → Vp H∗n (K p , VW ) is a topological isomorphism. In particular, each of the topological E-vector spaces definedin Definition 2.2.9 is an E-Banach space.
If g ∈ G and W0 is a separated lattice in W then gW0 is again a separated latticein W . Thus we obtain an action of G on the directed set of all separated lattices
in W .

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

31

Lemma 2.2.10. The action of G on the directed set of separated lattices in W
lifts to a continuous actions of G on each of the inductive systems that occurs in
Definition 2.2.9. Thus each of the E-Banach spaces Ĥ∗n (K p , VW ), H̃∗n (K p , VW )and Vp H∗n (K p , VW ) is equipped with a continuous G-action.
If W is the trivial representation, then each of these E-Banach spaces admits anorm with respect to which the action of G is isometric.
Proof. If g ∈ G and W0 is a separated lattice in W then multiplication by g in∼∼duces isomorphisms Ĥ∗n (K p , VW0 )E −→ Ĥ∗n (K p , Vg−1 W0 )E , H̃∗n (K p , VW0 )E −→∼H̃∗n (K p , Vg−1 W0 )E , and Vp H∗n (K p , VW0 ) −→ Vp H∗n (K p , Vg−1 W0 ). Passing to the
inductive limit over all W0 yields the required action of G on each of Ĥ∗n (K p , VW ),H̃∗n (K p , VW ), and Vp H∗n (K p , VW ). There is a compact open subgroup G0 of G
that leaves any W0 -invariant, and this implies that the G-action on each of these
E-Banach spaces is continuous.
Note that when W is the trivial representation of G, any separated lattice W0
in W is invariant under all of G, and so in this case the G-action on Ĥ∗n (K p , VW )(respectively H̃∗n (K p , VW ), Vp H∗n (K p , VW )) is isometric with respect to the norm
defined as the gauge of the image of Ĥ∗n (K p , VW0 ) (respectively H̃∗n (K p , VW0 ),Vp H∗n (K p , VW0 )). 
Theorem 2.2.11. (i ) The group π0 × G acts on each of the E-Banach spacesĤ∗n (K p , VW ), H̃∗n (K p , VW ), and Vp H∗n (K p , VW ) via an admissible continuous representation.
(ii ) If W is the trivial local system, then each of these Banach spaces can be
topologized by a norm with respect to which π0 × G acts by isometries.
(iii ) If H∗n (K p , VW ) denotes the space of K p -invariants in the admissible smooth
π0 × G(Af )-representation H∗n (VW ), then there is a π0 × G-equivariant natural map
H∗n (K p , VW ) → Ĥ∗n (K p , VW ), whose image is dense.
(iv ) There is a short exact sequence in the category of admissible continuous
π0 × G-representations
0 → Ĥ∗n (K p , VW ) → H̃∗n (K p , VW ) → Vp H∗n+1 (K p , VW ) → 0.
(v ) The composite of the map of (iii ) with the closed embedding Ĥ∗n (K p , VW ) →(iv ) yields a map

H̃∗n (K p , VW ) ofH∗n (K p , VW ) → H̃∗n (K p , VW )sm ,which is the edge map of a π0 × G-equivariant spectral sequenceiE2i,j = Hst(G, H̃∗j (K p , VW )) =⇒ H∗i+j (K p , VW ).

Proof. Since all the local systems under consideration are π0 -equivariant, each of
the OK -modules and E-vector spaces that we have defined is equipped in a natural
way with an action of π0 , compatible with the various maps between them. The
theorem then follows by applying Theorems 2.1.5 and 2.1.12 to the sheaves VW0 foreach separated lattice W0 ⊂ W , passing to the limit with respect to the transition
isomorphisms (2.2.5), (2.2.6) and (2.2.7), and taking into account the naturality of
the G-action given by Lemma 2.2.10, and of the π0 -action. 32

MATTHEW EMERTON

Up to this point, we have kept fixed the tame level K p . We now consider what
happens when the tame level is allowed to change.
Suppose that K p0 ⊂ K p is an inclusion of compact open subgroups of G(Ap ),such that K p0 is normal in K p . If Kp is any compact open subgroup of G, then we
obtain a surjective map Y (Kp K p0 ) → Y (Kp K p ). Consequently, we obtain a mapof exact sequences in the category of admissible continuous π0 × G-representations(2.2.12)0/ Ĥ n (K p , VW )/ H̃ n (K p , VW )/ Vp H n+1 (K p , VW )

/0

0


/ Ĥ n (K p0 , VW )/ H̃ n (K p0 , VW )/ Vp H n+1 (K p0 , VW )

/ 0.

∗

∗

∗

∗

∗

∗

Since K p0 is normal in K p , each of the E-Banach spaces in the lower short exact
sequence is equipped with a natural continuous action of the quotient K p /K p0 , and(by virtue of the naturality of these actions) the arrows in this short exact sequence
are equivariant with respect to these actions.
Proposition 2.2.13. The vertical arrows of (2.2.12 ) are closed embeddings, andidentify the upper exact sequence with the K p /K p0 -invariants of the lower exact
sequence.
Proof. Let a denote the order of the quotient K p /K p0 . If U is any Hausdorff topological E-vector space equippedwith a continuous action of K p /K p0 , the averaging
P−1p
operator πK : u 7→ ak∈K p /K p0 ku induces a continuous projection of U onto
its closed subspace of K p -invariants.
Let W0 be a choice of separated lattice in W , and let W00 = a−1 W0 . Then
multiplication by a−1 is a well-defined map H∗n (K p0 , VW0 ) → H∗n (K p0 , VW00 ), while
there is a natural trace map tr : H∗n (K p0 , VW00 ) → H∗n (K p , VW00 ) (induced by thenatural maps Y (Kp K p0 ) → Y (Kp K p ) for each compact open subgroup Kp of G).
Also πK p projects H∗n (K p0 , VW ) onto H∗n (K p , VW ). These maps fit together into
the commutative diagram(2.2.14)H∗n (K p , VW0 )/ H∗n (K p , VW )H∗n (K p0 , VW0 )/ H∗n (K p0 , VW )



a−1 ·

H∗n (K p0 , VW00 )tr! H∗n (K p , VW00 )

id
πK p

 |
/ H∗n (K p , VW )(where the unlabelled arrows are the natural maps corresponding to the inclusions
W0 ⊂ W00 ⊂ W, and K p0 ⊂ K p ).
Lemma 2.2.8 shows that the E-Banach spaces Ĥ∗n (K p , VW ) and Ĥ∗n (K p0 , VW ))may be obtained as the completion of H∗n (K p , VW ) and H∗n (K p0 , VW ) respectively

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

33

with respect to the gauge of either the images of H∗n (K p , VW0 ) and of H∗n (K p0 , VW0 ),or the images of H∗n (K p , VW00 ) and of H∗n (K p0 , VW00 ). A consideration of (2.2.14)
thus shows that πKp extends to a continuous map Ĥ∗n (K p0 , VW ) → Ĥ∗n (K p , VW ),splitting the natural map Ĥ∗n (K p , VW ) → Ĥ∗n (K p0 , VW ) induced by the inclusion
K p0 ⊂ K p . Thus the natural map Ĥ∗n (K p , VW ) → Ĥ∗n (K p0 , VW ) is a closed embedding that identifies its source with the K p invariants of its target.
A similar argument, in which we pass to ps -torsion in the spaces on the left-handside of the diagram (2.2.14), pass to the limit in s, and then tensor with E, showsthat πK p induces a map Vp H∗n (K p0 , VW ) → Vp H∗n (K p , VW ), splitting the natural
map Vp H∗n (K p , VW ) → Vp H∗n (K p0 , VW ).
If instead of working with VW0 and VW00 , we work with the quotient sheaves
VW0 /ps and VW00 /ps , then after passing to the limit in s and tensoring with E, wesimilarly conclude that πK p induces a map H̃∗n (K p0 , VW ) → H̃∗n (K p , VW ), splitting
the natural map H̃∗n (K p , VW ) → H̃∗n (K p0 , VW ). This completes the proof of the
proposition. 
Definition 2.2.15. Write
Ĥ∗n (VW ) := lim
Ĥ∗n (K p , VW ),−→Kp

H̃∗n (K p , VW ),H̃∗n (VW ) := lim−→
Kp

Vp H∗n (VW ) := lim
Vp H∗n (K p , VW ).
−→Kp(Here, the locally convex inductive limits are taken with respect to the set of compact open subsets of G(Apf ), directed downward.)
We will apply the terminology of [13, §7] to the group π0 × G(Af ) = π0 × G ×
G(Apf ), taking the locally analytic group G of that reference to be the group G (inour current notation), and the auxiliary locally compact group Γ of that reference
to be the product π0 × G(Apf ).
Theorem 2.2.16. (i ) The group π0 × G(Af ) acts on each of the locally convex
topological E-vector spaces Ĥ∗n (VW ), H̃∗n (VW ), and Vp H∗n (VW ) via an admissible
continuous representation.
(ii ) For each compact open subgroup K p of G(Apf ) there are natural isomorphismsp∼of admissible continuous π0 × G-representations Ĥ∗n (K p , VW ) −→ Ĥ∗n (VW )K ,pp∼∼
H̃∗n (K p , VW ) −→ H̃∗n (VW )K , and Vp H∗n (K p , VW ) −→ Vp H∗n (VW )K .
(iii ) There is a natural π0 × G(Af )-equivariant map H∗n (VW ) → Ĥ∗n (VW ), whose
image is dense.
(iv ) There is a short exact sequence in the category of admissible continuous
π0 × G(Af )-representations
0 → Ĥ∗n (VW ) → H̃∗n (VW ) → Vp H∗n+1 (VW ) → 0.
Proof. If g ∈ G(Apf ), then right multiplication by g on the spaces Y (Kp K p ) (as Kp
ranges over the directed set of compact open subgroups of G) induces isomorphisms

34

MATTHEW EMERTON∼

of the directed systems of spaces {Y (Kp gK p g −1 )}Kp ∈S −→ {Y (Kp K p )}Kp ∈S , andhence topological isomorphisms∼Ĥ∗n (K p , VW ) −→ Ĥ∗n (gK p g −1 , VW ),∼H̃∗n (K p , VW ) −→ H̃∗n (gK p g −1 , VW ),and∼Vp H∗n (K p , VW ) −→ Vp H∗n (gK p g −1 , VW ),compatible with the π0 × G-action on these spaces. Thus we obtain a topological G(Apf )-action on each of the spaces Ĥ∗n (VW ), H̃∗n (VW ), and Vp H∗n (VW ) of
Definition 2.2.15, commuting with the π0 × G-action on these spaces. Propositions 2.2.13 shows that each of the locally convex inductive limits appearing in
Definition 2.2.15 is a strict inductive limit (in the sense of [4, p. II.33]), and thepsame proposition thus implies that the natural maps Hˆ∗n (K p , VW ) → Ĥ∗n (VW )K ,pp
H˜∗n (K p , VW ) → H̃∗n (VW )K , and Vp H∗n (K p , VW ) → Vp H∗n (VW )K are topologicalisomorphisms. In particular, the G(Apf )-action on each of the spaces of Definition 2.2.15 is strictly smooth, in the sense of [13, Def. 7.1.2]. Taking into account
Theorem 2.2.11 (i), we see that we have proved part (i) and (ii) of the current
theorem. Parts (iii) and (iv) are easy consequences of the corresponding parts ofTheorem 2.2.11, and so we are done. Note that by taking W to be the trivial representation (so that VW is the trivial
local system), this result yields Theorem 0.1 of the introduction.
To simplify notation, when W is the trivial representation of G on E we will
write simply H∗n , Ĥ∗n , H̃∗n , and Vp H∗n in place of H∗n (VW ), Ĥ∗n (VW ), H̃∗n (VW ), andVp H∗n (VW ), and will write H∗n (K p ), Ĥ∗n (K p ), H̃∗n (K p ), and Vp H∗n (K p ) in placeof H∗n (K p , VW ), Ĥ∗n (K p , VW ), H̃∗n (K p , VW ), and Vp H∗n (K p , VW ), for any choice oftame level K p .
Theorem 2.2.17. If W is any finite dimensional algebraic representation of G
defined over E, then there is a natural isomorphism of admissible continuous π0 ×∼G(Af )-representations H̃∗n (VW ) −→ H̃∗n ⊗E W . (The action of π0 × G(Af ) =p
π0 × G × G(Af ) on the target is via the diagonal action of G, and the action of
π0 × G(Apf ) on the first factor.)
Proof. Let us first fix a tame level K p . If W0 is a separated lattice in W and s isany natural number then there is a natural isomorphism∼H∗n (Y (Kp K p ), OE /ps ) ⊗OE /ps VW0 /ps .
limH∗n (Y (Kp K p ), VW0 /ps ) −→ lim
−→
−→
Kp

Kp

(If Kp is sufficiently small, it acts trivially on W0 /ps .) Thus we obtain a natural
isomorphism of inductive systems∼{H̃∗n (K p , VW0 )}W0 ⊂W −→ {H̃∗n (K p , OE ) ⊗OE W0 }W0 ⊂W(where these inductive systems are indexed by the collection of separated lattices
W0 in W ). The naturality of this isomorphism shows that it is π0 × G-equivariant.
Tensoring these OE -modules over OE with E, and then passing to the inductive∼
limit in W0 , we obtain a natural isomorphism H̃∗n (K p , VW ) −→ H̃∗n ⊗E W. Now
passing to inductive limit in K p yields the isomorphism of the theorem. (Theπ0 × G(Af )-equivariance follows from the naturality of the isomorphism.) In light of this result, we may rewrite the spectral sequence of Theorem 2.2.11 (v)in the following manner.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

35

Corollary 2.2.18. There is a π0 × G(Af )-equivariant mapnH∗n (VW ) → Homg (W̌ , H̃∗,Qp −la )which is the edge map of a π0 × G(Af )-equivariant spectral sequenceji+j
E2i,j = Extig (W̌ , H̃∗,Qp −la ) =⇒ H∗ (VW ).

Proof. Theorem 1.1.13 allows us to rewrite the spectral sequence given by part (v)of Theorem 2.2.11 in the form
E2i,j = H i (g, H̃∗j (K p , VW )Qp −la ) =⇒ H∗i+j (K p , VW ).
Passing to the inductive limit in K p and taking into account the fact that computing both locally analytic vectors and Lie algebra cohomology commutes with theinductive limit (the former by [13, Prop. 3.5.14]) yields a π0 × G(Af )-equivariant
spectral sequenceE2i,j = H i (g, H̃∗j (VW )Qp −la ) =⇒ H∗i+j (VW ).
Passing to locally Qp -analytic vectors in Theorem 2.2.17, and taking into account∼j[13, Prop. 3.6.15], yields an isomorphism H̃∗j (VW )Qp −la −→ H̃∗,Qp −la ⊗E W. Combining this with the preceding spectral sequence, we obtain the spectral sequenceof the corollary. This yields Theorem 0.5 of the introduction.
nRemark 2.2.19. The edge map H∗n (VW ) → Homg (W̌ , H̃∗,Qp −la ) of the preced-

ing corollary may be rewritten as a morphism H∗n (VW ) → (H̃∗n ⊗E W )sm , where
the subscript sm denotes the subspace of vectors on which G acts smoothly (asfollows from the degree 0 case of the isomorphism of Theorem 1.1.13). Giving thisn,morphism is in turn equivalent to giving a morphism H∗n (VW ) ⊗E W̌ → H̃∗,W̌ −lalgnin H̃∗n givesby [13, Prop. 4.2.4]. Composing this with the inclusion of H̃∗,W̌ −lalg
the map (0.3) of the introduction.
Since W̌ is an algebraic representation of G over Fp , the locally W̌ -algebraic
vectors of H̃∗n are not merely locally Qp -analytic, but are in fact locally Fp -analytic.
Thus the edge map of the preceding corollary factors asnnH∗n (VW ) → Homg (W̌ , H̃∗,Fp −la ) → Homg (W̌ , H̃∗, Qp −la ),where the second arrow is induced by the closed embedding of the locally Fp -analytic
vectors into the locally Qp -analytic vectors.
Remark 2.2.20. Theorem 2.2.17 shows that H̃∗n , equipped with the data of thespectral sequences of the preceding corollary, one for each choice of W , forms theprimitive object in our construction, from which all the other objects that we haveconsidered may be obtained. Indeed, we recover H̃∗n (VW ) as the tensor productH̃∗n ⊗E W . We may then recover Ĥ∗n (VW ) as the closure of the image of the map
H∗n (VW ) → (H̃∗n ⊗E W )sm → H̃∗n ⊗E W (where the first arrow is the edge map of
the spectral sequence of Corollary 2.2.18, reinterpreted as in Remark 2.2.19, andthe second arrow is the obvious inclusion), while Vp H∗n (VW ) arises as the cokernelof the embedding Ĥ∗n−1 (VW ) → H̃∗n−1 (VW ).
Our constructions are compatible with the cup-product on cohomology.

36

MATTHEW EMERTON

Proposition 2.2.21. Fix a pair of natural numbers m and n. Also fix ∗1 , ∗2 ∈{∅, c}, and take ∗ to be ∅ if ∗1 = ∗2 = ∅, and to be c otherwise.
(i ) The cup-product on cohomology induces continuous π0 × G(Af )-equivariant
ˆ E Ĥ∗n2 → Ĥ∗m+n , H̃∗m1 ⊗ˆ E H̃∗n2 → H̃∗m+n , and Vp H∗m1 ⊗ˆ E Vp H∗n2 →morphisms Ĥ∗m1 ⊗m+n
V p H∗, compatible with the exact sequences provided by Theorem 2.2.16 (iv ).
(ii ) If W1 and W2 are two finite dimensional representations of G defined overE, then the following diagram, in which the left-hand vertical arrow arises fromthe cup-product, the right-hand vertical arrow is that of part (i ) (restricted to theindicated spaces of locally algebraic vectors) and the horizontal arrows are givenby the edge-maps of the spectral sequences of Corollary 2.2.18, as reinterpreted in
Remark 2.2.19, is commutative:H∗m1 (VW1 ) ⊗ W̌1 ⊗ H∗n2 (VW2 ) ⊗ W̌2/ H̃ m∗1 , W̌1 −lalgH∗m+n (VW1 ⊗W2 ) ⊗ W̌1 ⊗ W̌2/ H̃ m+nˆ H̃ n⊗∗ , W̌ −lalg22∗, W̌1 ⊗W̌2 −lalg.

Proof. This is an easy consequence of the naturality of the cup-product, and of ourconstructions. In particular, taking ∗ = ∅ and m = n = 0, we find that H̃ 0 is naturally a
topological E-algebra, equipped with an action of π0 × G(Af ) as algebra automorphisms, and that for either choice of ∗ and each n ≥ 0, the space H̃∗n is naturally a
topological H̃ 0 -module, equipped with a compatible π0 × G(Af )-action.
We now pass from the admissible continuous G-representation H̃∗ to its associated space of locally analytic vectors. More precisely, if L is any local field intermediate between Fp and Qp , then we can consider the associated space of locallyL-analytic vectors.
nTheorem 2.2.22. (i ) The space H̃∗,
L−la is an admissible locally analytic representation of π0 × G(Af ) (in the sense of [13, Def. 7.2.7]).
(ii ) For any compact open subgroup K p of G(Apf ), the natural mapn
KH̃∗n (K p )L−la → (H̃∗,L−la )

p

is a π0 × G-equivariant isomorphism.
(iii ) If W is any finite dimensional representation of G over E, then there isna π0 × G(Af )-equivariant closed embedding of H̃∗,W −lalg , equipped with its finest
nconvex topology, into H̃∗, L−la .
Proof. Part (i) (respectively part (ii)) follows from the corresponding part of Theorem 2.2.16, together with Proposition 7.2.11 (respectively Proposition 7.2.5) of [13],while part (iii) follows from part (i) together with Propositions 7.2.13 and 7.2.14 of[13]. This yields Theorem 0.2 of the introduction.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

37

Proposition 2.2.23. If m and n are two natural numbers, and ∗1 , ∗2 , and ∗ are asin Proposition 2.2.21, then for any local field L intermediate between Fp and Qp ,the cup-product of cohomology classes induces a π0 × G(Af )-equivariant pairingm+n
ˆ E H̃∗n , L−la → H̃∗,H̃∗m1 , L−la ⊗L−la .
2
ˆ E H̃∗n → H̃∗m+n
Proof. This follows from [13, 3.5.15], applied to the pairing H̃∗m1 ⊗2
of Proposition 2.2.21. 
Since π0 is naturally identified with the group of connected components of G∞ ,we may regard π0 × G(Af ) as being a quotient of the group G(A) in a natural way,
and so regard the π0 × G(Af )-representations considered above as being G(A)representations. We have preferred to describe them explicitly as π0 × G(Af )representations, since this emphasizes that the real Lie group G∞ intervenes only
through the group of connected components π0 .
We close this subsection with the following simple propositions and their corollaries.
Proposition 2.2.24. If n = 0 or 1 then Vp H n vanishes.
Proof. It is a general fact that H 0 (–, OE ) and H 1 (–, OE ) are torsion free. Corollary 2.2.25. The embedding Ĥ 0 → H̃ 0 is an isomorphism. Furthermore,
for any representation W of G defined over E, the natural map H 0 (VW ) ⊗E W̌ →0
ĤW−lalg is an isomorphism.
Proof. The first claim of the corollary follows from the exact sequence of Theorem 2.2.16 (iv), and the case n = 1 of Proposition 2.2.24. The second claim follows
from an examination of the spectral sequence of Corollary 2.2.18 in the case wheni = j = 0. 
Proposition 2.2.26. (i ) If n = 0 or 1 then Vp Hcn vanishes.
(ii ) If d is the dimension of G∞ /K∞ , then Vp Hcd and Vp Hcd+1 both vanish.
Proof. Part (i) follows from the general fact that Hc0 (–, OE ) and Hc1 (–, OE ) are
torsion free. Since the dimension of each quotient Y (Kf ) is equal to d, we see that
Hcd (Y (Kf ), OE ) is the free OE -module spanned by the fundamental classes of theconnected components of Y (Kf ), while Hcd+1 (Y (Kf ), OE ) vanishes. Thus part (ii)holds. 
Corollary 2.2.27. The embeddings Ĥcd−1 → H̃cd−1 and Ĥcd → H̃cd are both isomorphisms.
Proof. The corollary follows from the exact sequence of Theorem 2.2.16 (iv), andProposition 2.2.26. (2.3) This subsection presents the proof of Theorem 0.7. We maintain thenotation of the preceding subsection, and suppose in addition that G is split overE (so that any irreducible representation of G over E is absolutely irreducible).
We also suppose that G is quasi-split over Fp , and choose a Borel subgroup B of G
defined over Fp , as well as a maximal torus T of B (which then splits over E, since
G is assumed to split over E). Write B := B(Fp ) and T := T(Fp ).
Let us fix a tame level K p .

38

MATTHEW EMERTON

Definition 2.3.1. We say that K p is unramified at a place v 6= p of F if:(i) G is unramified at v; that is, if G is quasi-split over Fv , and splits over anunramified extension of Fv ;T(ii) The compact open subgroup Kvp := K p G(Fv ) of G(Fv ) is a hyperspecial
maximal compact subgroup of G(Fv ).
Otherwise, we say that K p is ramified at v.
We let S denote the (finite) set of ramified primes of K p . We write Ap,S
tof
S
Q
denote the prime to S {p} finite adèles, and also write FS = v∈S Fv (so that
T
T
p
p
p,S
Apf = FS ×Ap,S:= K p G(Ap,Sf ). If we write KS := KQ G(FS ), and write K
f ),
p
p
p,S
p,S
p
S
then K = KS × K . Also, K= v6∈S {p} Kv .
If H is a compact open subgroup of a locally compact group G, then H(G//H)
will denote the Hecke algebra of H double cosets in G, with coefficients in E.
p,S
We will abbreviate H(G(Apf )//K p ), H(G(FS )//KSp ), and H(G(Ap,S) byf )//K
p
p ram
p sph
H(K ), H(K ) , and H(K )respectively. The product decomposition of K p
induces a corresponding tensor product decomposition of Hecke algebras:∼H(K p ) −→ H(K p )ram ⊗E H(K p )sph .
Similarly, the product decomposition of K p,S induces a decomposition of H(K p )sph
∼ N0
as a restricted tensor product: H(K p )sph −→H(G(Fv )//Kvp ). The Satake isomorphism shows that each of the algebras H(G(Fv )//Kvp ) is a commutative algebraof finite type over E. (See [19, §16] for a clear discussion of the Satake isomorphism.)Thus H(K p )sph is commutative, and forms a central subalgebra of H(K p ).
Since, by Theorem 2.2.22 (i), the space H̃ n (K p )Fp −la is naturally identified
with the space of K p -invariants in the admissible locally Fp -analytic π0 × G(Af )representation H̃Fnp −la , it is equipped with a locally Fp -analytic action of G, together
with commuting actions of π0 and H(K p ) by continuous operators. The Jacquet
module JB (H̃ n (K p )Fp −la ) (as defined in [14]) is then an essentially admissible locally Fp -analytic representation of the torus T , again equipped with commuting
actions of π0 and H(K p ) by continuous operators.
As in the introduction, let T̂ denote the rigid analytic variety over E that parameterizes the locally Fp -analytic characters of T . (We refer to [13, §6.4] for an
explanation of the construction of T̂ .) Let C an (T̂ , E) denote the E-Fréchet algebraof rigid analytic functions on T̂ . If F is a coherent rigid analytic sheaf on T̂ , thenthe space of global sections Γ(T̂ , F) is naturally a topological Fréchet module over
C an (T̂ , E), and the theory of quasi-Stein rigid analytic spaces shows that taking
global sections induces an equivalence of categories between the category of coherent rigid analytic sheaves on T̂ , and the category of coadmissible (in the sense of[31]) C an (T̂ , E)-modules. Evaluating characters at elements of T induces a continuous injection T → C an (T̂ , E)× . Thus if F is a coherent rigid analytic sheaf on T̂ ,then the space Γ(T̂ , F) is naturally equipped with a continuous T -representation,and hence so is its strong dual.
The following proposition then amounts to little more than the definition of
essentially admissible locally Fp -analytic T -representations. (See [13, §6.4].)
Proposition 2.3.2. There is an antiequivalence of categories between the categoryof coherent rigid analytic sheaves on T̂ , and the category of essentially admissible

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

39

locally Fp -analytic representations of T defined over E, given by associating to any
coherent sheaf F on T̂ the strong dual of its Fréchet space of global sections Γ(T̂ , F),
equipped with the T -action described above.
We let E denote the coherent rigid analytic sheaf on T̂ corresponding via theantiequivalence of categories of Proposition 2.3.2 to JB (H̃ n (K p )Fp −la ), and let
Exp(E) denote the support of E (the “set of exponents” appearing in E).
The action of the commutative E-algebra H(K p )sph on JB (H̃ n (K p )Fp −la ) induces an action of H(K p )sph on the coherent sheaf E, and thus gives rise to acoherent subsheaf of commutative rings A inside the endomorphism sheaf of E.
We form the relative Spec of A over T̂ ; by construction it admits a Zariski closed
embedding Spec A → T̂ × Spec H(K p )sph . Since A acts as endomorphisms of E, we
may localize E to a coherent sheaf M on Spec A.
The action of the group ring H(K p )ram [π0 ] on JB (H̃ n (K p )Fp −la ) commutes with
the H(K p )sph -action, and so the sheaf M is naturally a coherent sheaf of rightH(K p )ram [π0 ]-modules.
Proposition 2.3.3. (i ) The natural projection Spec A → T̂ is a finite morphism,
with set-theoretic image equal to Exp(E).
(ii ) The map Spec A → ť (where ť is the dual to the Lie algebra t of T ) obtainedby composing the projection of (i ) with differentiation of characters has discrete
fibres. In particular, the dimension of Spec A is at most equal to the dimension ofT.
(iii ) The fibre of M over a point (χ, λ) of T̂ × Spec H(K p )sph is dual to the(T = χ, H(K p )sph = λ)-eigenspace of JB (H̃ n (K p )Fp −la ). In particular, the point(χ, λ) lies in Spec A if and only if this eigenspace is non-zero.
Proof. Parts (i) and (iii) are immediate from the construction of A. Part (ii) follows
from [14, 0.11]. 
We let G̃, B̃, T̃ denote the restriction of scalars from Fp to Qp of G, B, and T
respectively. Since E contains Fp , there are natural projections(2.3.4)

G̃/E → G/E ,and(2.3.5)T̃/E → T/E .

Let W be an irreducible algebraic representation of G over E, whose highest weightwith respect to B is the character ψ of T. Then we may regard W as an irreducible
representation of G̃ via (2.3.4), whose highest weight with respect to B̃ is then ψ,
regarded as a character of T̃ via (2.3.5). Also, by definition T̃(Qp ) = T(Fp ) = T .
Thus if θ is a smooth character of T , the locally algebraic character θψ of T may
equally well be regarded as a locally algebraic character of T̃(Qp ), to which we may
apply Definition 4.4.3 of [14].
Proposition 2.3.6. If χ := θψ is of non-critical slope (in the sense of [14,Def. 4.4.3]), then the closed embedding JB (H̃ n (K p )W −lalg ) → JB (H̃ n (K p )Fp −la )induces an isomorphism on χ-eigenspaces.

40

MATTHEW EMERTON

Proof. This follows from [14, Thm. 4.4.5] and the fact that H̃ n (K p ) admits a G(Fp )invariant norm. 
Suppose now that the natural map(2.3.7)n
H n (VW̌ ) ⊗E W → H̃W−lalgprovided by Corollary 2.2.18 (see Remark 2.2.19) is an isomorphism for every choiceof irreducible representation W of G over E (and consequently for any such representation, since this map is compatible with the formation of direct sums ofrepresentations). Recall that Spec A and the eigenvariety E(n, K p ) (as defined in
Definition 0.6) are both Zariski closed subspaces of T̂ × Spec H(K p )sph .
Proposition 2.3.8. If (2.3.7 ) is an isomorphism for all W then Spec A contains
E(n, K p ) as a closed subspace.
Proof. Since E(n, K p ) is defined to be the Zariski closure in T̂ × Spec H(K p )sph ofE(n, K p )cl , it suffices to show that the latter set of points is contained in Spec A.
A typical point of E(n, K p ) is of the form (θψ, λ), where as above θ is a smoothcharacter of T and ψ is the highest weight of an irreducible G-representation W ,such that there exists an irreducible subquotient of H n (K p , VW̌ ) which may beG(F )embedded into IndB(Fpp) θ, and on which H(K p )sph acts via λ. (Here and in theensuing discussion, it is implicit that we base-change to a field of definition ofλ.) Since the Jacquet functor is exact when applied to smooth representations[7, Prop. 3.2.3], we see that θ appears in the λ-eigenspace of JB (H n (K p , VW̌ )),and so by [14, Prop. 4.3.6] the product θψ appears in JB (H n (K p , VW̌ ) ⊗E W ). By
assumption H n (K p , VW̌ )⊗E W maps isomorphically to the closed subrepresentation
n
n
n
H̃W
−lalg of H̃Fp −la , and so θψ also appears in the λ-eigenspace of JB (H̃Fp −la ). The
present proposition thus follows from Proposition 2.3.3 (iii). 
Theorem 0.7 follows from Propositions 2.3.3, 2.3.6, and 2.3.8 taken together.
In light of Proposition 2.3.8, it is natural to ask whether or not E(n, K p ) isequal to A, or equivalently, whether or not the points of E(n, K p )cl are Zariski
dense in A. We do not know the answer to this question in general (even under the
hypotheses of Theorem 0.7).
(2.4) The aim of this short subsection is to show that, when the arithmetic
quotients attached to G admit the structure of Shimura varieties, the constructionsof Subsection 3.3 are compatible with the comparison isomorphism between usualand étale cohomology, and hence that the various spaces considered are equippedwith Galois actions, and the various short exact and spectral sequences constructed
are Galois equivariant.
Let G0 denote the restriction of scalars of G from F to Q, let S denotes the
maximal split torus in the centre of G0 , and let S0 denote the maximal split torus
quotient of G0 . The natural map S → S0 is an isogeny, and so if S∞ := S(R) and
0
◦
0◦
S∞
:= S0 (R), then the induced map S∞→ S∞
is an isomorphism. Let Z∞ denote1
the centre of G∞ (so Z∞ contains S∞ , but is typically larger), and let Z∞
denote0
the kernel of the natural map Z∞ → S∞.
The spaces Y (Kf ) that we have defined are not the arithmetic quotients typicallyconsidered in the theory of automorphic forms; it is more usual to consider the quo◦tients Y ◦ (Kf ) := Y (Kf )/S∞
, which have the advantage of being of finite volume.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

41(These are the arithmetic quotients considered in [18], for example.) The projection◦
Y (Kf ) → Y ◦ (Kf ) makes the source an S∞-bundle over the target. Thus (since S∞
mis isomorphic to R for some m, and in particular is contractible) the local system
VW on Y (Kf ) attached to any algebraic G-representation W on Y (Kf ) may bedescended to Y ◦ (Kf ), and the cohomology and compactly supported cohomology
spaces of Y ◦ (Kf ) with coefficients in VW are isomorphic to the corresponding cohomology spaces of Y (Kf ) (up to a shift of degree, in the case of compactly supported
cohomology).
In the theory of Shimura varieties it is customary to consider a different quotient◦again, namely Ỹ (Kf ) := Y (Kf )/Z∞. The projection map Y ◦ (Kf ) → Ỹ (Kf ) makes
the source a d-dimensional torus bundle over the target (in the topological sense:
each fibre is a product of d circles), where d is the difference between the split
1
rank of Z∞ and the rank of S∞ (i.e. the split rank of Z∞). If W is an algebraic
representation of G, then the local system VW on Y (Kf ) descends to Ỹ (Kf ) if and1
only if Z∞
acts trivially on W . (Note that if this condition does not hold, then thelocal system VW has trivial cohomology on Y (Kf ); thus nothing is lost in any caseby restricting attention to those W that satisfy this condition.)
For the remainder of this section, we assume that S∞ is in fact the maximal split
torus in Z∞ , so that Y ◦ (Kf ) = Ỹ (Kf ). This allows us to replace the quotients
Y (Kf ) by the corresponding quotients Ỹ (Kf ) in the results of Subsection 2.2.
(Our assumption is a common one in the Shimura variety context, although itomits certain examples, such as Shimura curves and Hilbert modular varieties overtotally real fields of degree greater than one. We adopt it more for simplicity thanout of necessity, since we could get around it by working systematically with thequotients Ỹ (Kf ) in Subsection 2.2, provided that we restricted our attention to1
representations W on which Z∞
acts trivially.)
Suppose given a number field F 0 ⊂ C, a projective system of schemes Ỹ (Kf )/F 0
defined over F 0 indexed by the compact open subgroups Kf of G(Af ), and anaction of G(Af ) defined on this projective system, such that when we take theC-valued point of this projective system, the resulting projective system of spaces
Ỹ (Kf )/F 0 (C) with its G(Af )-action is isomorphic to the projective system of arithmetic quotients Ỹ (Kf ), with its G(Af )-action. (More precisely, we should allow
Ỹ (Kf )/F 0 to be a stack, in the case when Ỹ (Kf ) is an orbifold rather than a manifold. However, we will ignore this technicality, since we may restrict our attention
to those Kf that are sufficiently small, in which case Ỹ (Kf ) is a manifold.) If thegroup G0 admits a Shimura datum with reflex field equal to F 0 , then the theory of
canonical models of Shimura varieties shows that such schemes do exist.
Now suppose that M is a finite G0 -module, for some compact open subgroup
G0 of G, and assume that the G0 -action on M is continuous, when M is endowedwith its discrete topology. We may then follow the prescription of [24, §3] to define
a compatible system of étale sheaves VM,ét on the projective system of schemesỸ (Kf )/F 0 . More precisely, we can define the sheaf VM,ét on Ỹ (Kf )/F 0 provided
that the projection of Kf onto the pth factor is contained in G0 . (This will certainly
hold if Kf is small enough.) Assuming that Kf satisfies this hypothesis, we choose
a compact open normal subgroup Kf0 ⊂ Kf such that the projection of Kf0 onto
the pth factor is contained in the pointwise stabilizer of M, and define VM,ét to be
the étale sheaf represented by the étale cover M ×Kf Ỹ (Kf0 )/F 0 of Ỹ (Kf )/F 0 . The
resulting sheaf is, up to canonical isomorphism, independent of the choice of Kf0 .

42

MATTHEW EMERTON

After passing to C-valued points, the étale sheaf VM,ét corresponds to the sheaf VM
of Definition 2.2.3.
The comparison isomorphism between étale and classical cohomology yields a
canonical isomorphism
∼

•
Hét
(C ×F 0 Ỹ (Kf )/F 0 , VM,ét ) −→ H • (Ỹ (Kf ), VM ).

If Q denotes the algebraic closure of Q in C, then there is a natural action ofGal(Q/F 0 ) on the left-hand side of this isomorphism, which may then be transported to the right-hand side.
If W is a representation of G defined over E, then the spaces Ĥ • (VW ), H̃ • (VW ),and Vp H • (VW ) constructed in Subsection 2.2 may all be defined as limits withrespect to natural transition maps of various kinds of cohomology groups of theform H • (Ỹ (Kp K p ), VW0 /ps ), for some compact open subgroup Kp of G, some tamelevel K p , and some Kp -invariant sublattice W0 of W . Thus each of them is equipped(via the comparison isomorphism) with a continuous Gal(Q/F 0 )-action, commuting
with the G(Af )-action.
By virtue of the natural sheaf theoretic nature of its construction, the short exact sequence of Theorem 2.2.16 (iv) is compatible with respect to these Gal(Q/F 0 )actions. Also, by the very definition of the étale sheaf VW0 /ps , the isomorphismof Theorem 2.2.17 is compatible with respect to the comparison isomorphism,and hence with respect to the actions of Gal(Q/F 0 ) on its source and target (theGal(Q/F 0 )-action on H̃ • ⊗F 0 W being defined to be trivial on the second factor).
Since passing to locally analytic vector is functorial, the Galois action on H̃ • in•duces a continuous Gal(Q/F 0 )-action on H̃L−la, for each local field L intermediate
between Fp and Qp .
Proposition 2.4.1. In the situation under consideration, the spectral sequence ofCorollary 2.2.18 is Gal(Q/F 0 )-equivariant.
Proof. This spectral sequence is obtained by applying part (ii) of Theorem 2.1.5. Although the proof of that theorem uses simplicial methods, Proposition 2.1.11 gives
an alternative construction of the same spectral sequence using just the HochschildSerre spectral sequence for cohomology with coefficients in the torsion local systems
VW0 /ps . One has such a Hochschild-Serre spectral sequence also for étale cohomology, which is compatible with the comparison maps between singular and étale
cohomology with torsion coefficients. Thus the construction of Proposition 2.1.11
applies equally well in the étale cohomological context, and the present proposition
follows. Finally, suppose that G is quasi-split over Fp , so that we may form the eigenvariety E(n, K p ) for each cohomological degree n and each tame level K p . Suppose
furthermore that (2.3.6) is an isomorphism for some choice of n and K p , and for
all W . Theorem 0.7 yields a coherent sheaf M over E(n, K p ). Since this sheaf isconstructed by applying the functor JB to H̃ n (K p )Fp −la , we see that it inherits a
Galois action, and so forms a coherent sheaf of Gal(Q/F 0 )-modules on E(n, K p )(and this action is compatible with the right H(K p )ram -module structure on M).
3. p-adic automorphic representations(3.1) In this subsection we put ourselves in the situation of Subsection 2.2, andwe employ the notation introduced in that subsection. The given inclusion F ⊂ C

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

43

determines a map(3.1.1)

F∞ := R ⊗Q F → C,

and hence a map(3.1.2)

G∞ → G(C).

We let z(g∞ ) denote the centre of the universal enveloping algebra of the lie algebra
g∞ of G∞ .
Definition 3.1.3. (i) Let χ be a complex-valued character of z(g∞ ) . We say that
χ is allowable if there is an irreducible representation W of G over C such that the
representation of G∞ on W induced by the map (3.1.2) has infinitesimal character
equal to χ. (Note that W is unique up to isomorphism if it exists, since irreducible
representations of G/C are determined up to isomorphism by their infinitesimalcharacters.)(ii) Let π∞ be an irreducible admissible representation of the real reductive
group G∞ . We say that π∞ is allowable if the infinitesimal character of π∞ isallowable in the sense of (i). If we wish to be precise, then we will say that π∞
is W -allowable, where W is the representation of GC whose infinitesimal character(as a representation of G∞ ) coincides with that of π∞ .
(iii) Let π be an automorphic representation of G(A), and write π = π∞ ⊗ πf ,where π∞ is an admissible representation of G∞ , and πf is a smooth representationof G(Af ). We say that π is allowable if π∞ is allowable, and if πf can be definedover a finite extension of Q.
For example, if F = Q then the map (3.1.1) above is simply the inclusion R ⊂ C.
Thus in this case a character of z(g∞ ) is allowable if and only if it is the infinitesimalcharacter of an irreducible finite dimensional representation of G over C.
For general F , let G0 denote the restriction of scalars of G to Q, so that G∞ =0
G (R). A character of z(g∞ ) is allowable if it is the infinitesimal character of anirreducible algebraic representation of G0 of a certain kind, namely one pulled backfrom a finite dimensional representation of G over C by the surjection G0/C → G/C
induced by (3.1.1).
Note in particular that although we may regard an automorphic representation
π of G(AF ) equally well as an automorphic representation of G0 (AQ ) (where wehave written AF to denote the adèles of F and AQ to denote the adèles of Q), thecondition that π be allowable as a representation of G(AF ) is not equivalent to thecondition that it be allowable as a representation of G0 (AQ ). (The former implies
the latter, but not conversely, in general.)1
Since the infinitesimal character of any irreducible algebraic representation of G0
is regular, it follows that an allowable character of z(g∞ ) is necessarily regular. It
then follows from [10, Thm. 3.13] that for G = GLn , the representation π∞ being
allowable implies that πf is definable over a finite extension of Q.
We recall the following simple lemma.
1 For example, an automorphic representation of GL
1/F(that is, an idéle class character over×F ) is allowable if its restriction to the connected component of F∞(thought of as the component at
×× induced by (3.1.1) and an algebraic
infinity of A×
)
is
equal
to
the
composite
of
the
map
F
→
C
∞
F
character z → z n of C× . (Dirichlet’s unit theorem shows that there are no such characters with
n 6= 0 except in the case when F = Q or F is quadratic imaginary.) On the other hand, if G0
denotes the restriction of scalars of GL1/F from F to Q, then any algebraic Hecke character gives
an allowable automorphic representation of G0 (AQ ).

44

MATTHEW EMERTON

Lemma 3.1.4. Let V be an irreducible admissible smooth representation of a padic reductive group H defined over an algebraically closed field Ω. If V1 and V2
are two smooth representations of H defined over a subfield E of Ω, equipped with∼∼∼isomorphisms Ω ⊗E V1 −→ Ω ⊗E V2 −→ V, then there is an isomorphism of V1 −→V2 of H-representations over E, uniquely determined up to multiplication by a nonzero element of E.
Proof. Let U be a compact open subgroup of H such that V U 6= 0. Note that V1U∼∼
and V2U are then also non-zero; indeed V U −→ Ω ⊗E V1U −→ Ω ⊗E V2U . Each ofthese spaces of invariants is a representation of the Hecke algebra H(H//U ) definedover E of U double cosets in H. Furthermore, each of these spaces of invariants
is finite dimensional over its field of definition (either Ω or E), since V is assumed
admissible.
From [3, Cor. 3.9 (ii)] we see that we may (and do) choose U so that the functor
W 7→ W U induces an equivalence of categories between the category of smooth
H-representations W (defined over either Ω or E) that are generated over H by
W U , and the category of H(H//U )-modules (defined over either Ω or E). Our
assumption may then be rephrased as saying that we have an Ω-valued point of
the algebraic variety of H(H//U )-equivariant isomorphisms between V1U and V2U .
The Nullstellensatz assures us that we may then find a point of this variety definedover the algebraic closure of E, and thus we may assume that Ω is equal to thisalgebraic closure.
We now note that since V is irreducible, Schur’s Lemma implies that EndH (V )is isomorphic to Ω. The lemma thus follows from Hilbert’s Theorem 90. 
Let π be an automorphic representation of G(A) that is allowable in the sense of
Definition 3.1.3 (iii), and write πf = πp ⊗ π p , where πp is a smooth representationof G and π p is a smooth representation of G(Apf ). Let W be the finite dimensional
irreducible representation of G over C for which π∞ is W -allowable.
∼
Via the isomorphism ı : C −→ Qp , we may regard πf = πp ⊗ π p as an admissible
representation of G(Af ) on a Qp -vector space, and we may also regard W as a
Qp -vector space equipped with a finite dimensional representation of G/Qp , and so
in particular with a representation of G.
We may descend πf to a finite extension of Fp (since we assumed that it may
be descended even to a finite extension of Q). Let E be a finite extension of Fp towhich πf may be descended, and assume that E is chosen so large that G splits over
E. This implies that W may also be descended to a representation of G definedover E.
Let πf now denote a descent of πf to E (uniquely determined up to isomorphism
by Lemma 3.1.4) and W denote a descent of W to E (uniquely determined up toisomorphism by highest weight theory). Write π̃p := πp ⊗E W (equipped with the
diagonal action of G), and (for notational consistency) set π̃ p = πfp .
We set Γ = G(Apf ), so that G(Af ) = G × Γ is the product of a locally L-analytic
group and a locally compact group satisfying the hypothesis of [13, §7.2]. When
we apply the terminology and results of that reference to G(Af ) we always regard
it as being factored in this manner.
Definition 3.1.5. We write π̃ := π̃p ⊗E π̃ p = (πp ⊗E W ) ⊗E πfp , regarded as arepresentation of G(Af ) = G × Γ, and refer to π̃ as the classical p-adic automorphic

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

45

representation of G(Af ) over E attached to the allowable automorphic representation π.
If we equip π̃ with its finest convex topology then it becomes an admissible locallyalgebraic representation of the adèle group G(Af ), in the sense of [13, Def. 7.2.15].
Let us mention an initial motivation for the introduction of classical p-adic automorphic representations. Common conditions that arise in the p-adic theory of
automorphic representations, such as an automorphic representation π being ordinary or of non-critical slope, are not local at p; they depend on a comparison ofinvariants obtained from the local factors at the infinite places (typically, an infinity type) and from the local factor at p (such as a Satake parameter). However, if
we replace π by the associated classical p-adic automorphic representation π̃, thenthese properties often depend only on the local factor π̃p at p. (This is the casewith the two examples mentioned.) Thus they can be analyzed by local p-adic representation theoretic techniques. It is this phenomenon that allows us to apply the
local techniques of [13, 14] to the global problem of p-adic interpolation of systemsof Hecke eigenvalues.
The basic example of allowable automorphic representations that we have inmind are those automorphic representations that contribute to cohomology, as we
will now explain. As above, let G0 denote the restriction of scalars of G from F to Q.
As in Subsection 2.4, let S denote the maximal split subtorus in the centre of G0 , andS0 the the maximal split torus quotient of G0 ; the natural map S → S0 is an isogeny.
Thus if s denotes the Lie algebra of S, and g̃ denotes the Lie algebra ofL
the kernel∼of the the quotient map G0 → S0 , then there is a natural isomorphism g̃ s −→ g.
We write g̃∞ = R ⊗Q g̃. Also, as above, we fix a maximal compact subgroup K∞
of G∞ , and write k∞ to denote the Lie algebra of K∞ . We let A(G(F )\G(A))
denote the space of K∞ -finite automorphic forms on G(F )\G(A). Let W be a
finite dimensional representation of G (defined over C), let χ denote the character
through which z(g∞ ) acts on W̌ , let Iχ ⊂ z(g∞ ) denote the kernel of χ, and let◦(the group of connected componentsζ denote the character through which S∞
of S(R)) acts on W̌ (so ζ is obtained by exponentiating the restriction of χ to◦s ⊂ z(g∞ )). Let A(G(F )\G(A))[Iχ∞ ]S∞ =ζ denote the subspace of A(G(F )\G(A))◦
acts via ζ, and which are annihilated by someconsisting of vectors on which S∞
power of Iχ . The main result of [18] shows that there is a natural isomorphism(3.1.6)◦∼

H n (g̃∞ , k∞ ; W ⊗C A(G(F )\G(A))[Iχ∞ ]S∞ =ζ ) −→ H n (VW ).

Suppose that π is an irreducible automorphic representation of G(A), written◦
as a quotient π = U/V , where V ⊂ U ⊂ A(G(F )\G(A))[Iχ∞ ]S∞ =ζ are G(A)subrepresentations. (Note that since π is irreducible, z(g∞ ) must then act on π via
the character χ.) Suppose furthermore that the image of H n (g̃∞ , k∞ ; W ⊗C U ) in◦
H n (g̃∞ , k∞ ; W ⊗C A(G(F )\G(A))[Iχ∞ ]S∞ =ζ ) is a proper subspace of the image of◦H n (g̃∞ , k∞ ; W ⊗C V ) in H n (g̃∞ , k∞ ; W ⊗C A(G(F )\G(A))[Iχ∞ ]S∞ =ζ ). Then πf (thefinite component of π) appears as a subrepresentation of the quotient of the latter
image by the former, and so in particular as an irreducible subquotient of H n (VW ).
We say that π contributes to cohomology . Conversely, the isomorphism (3.1.6)
shows that any irreducible subquotient of H n (VW ) arises as the finite componentof such an automorphic representation π.

46

MATTHEW EMERTON

If an automorphic representation π contributes to the cohomology of VW , then
π∞ is W̌ -allowable. Since πf appears as a subquotient of H n (VW ) (for an appropriate choice of W ) it may be defined over a finite extension of Q, and thus we seethat π is in fact allowable. If we choose a finite extension E of Fp such that both W
and πf can be defined over E, then we find that the corresponding classical p-adic
automorphic representation π̃ := πf ⊗E W̌ appears as an irreducible subquotientof the admissible locally algebraic G(Af )-representation H n (VW ) ⊗E W̌ .
(3.2) As in Subsection 3.1, let G0 denote the restriction of scalars of G from F
to Q. In this subsection we suppose that G0 has the property that the maximal
split torus in the centre of G0 is a maximal split torus in G0 over R, so that we arein the situation of [19, Prop. 1.4]. We will construct a space of locally Fp -analytic
p-adic automorphic forms attached to G, in which the classical p-adic automorphic representations of G(Af ) constructed in the preceding subsection appear as
subrepresentations.
Our assumption implies that Y ◦ (Kf ) = Ỹ (Kf ) for any choice of level Kf . Also,
there is an equality K∞ Z∞ = G∞ (as in Subsection 2.4, we let Z∞ denote thecentre of G∞ ), so that (K∞ Z∞ )◦ = G◦∞ , and the quotient G∞ /(K∞ Z∞ )◦ is equal
to π0 , and hence finite. The quotients Ỹ (Kf ) are thus also finite, and their only
non-trivial cohomology is in degree zero. We will relate this cohomology to thenotion of algebraic automorphic form described in [19].
We choose E so that G splits over E, and make the following variation on thedefinition of [19, p. 68].
Definition 3.2.1. If W is a finite dimensional representation of G defined overE then an algebraic automorphic form on G(A) with coefficients in W is a locallyconstant function f : G(A) → W such that f (γg) = γf (g) for all γ ∈ G(F ) andg ∈ G(A). We denote this space by M(W ). It is endowed with a natural action ofG(A), induced by the action by right translation of G(A) on G(F )\G(A).
If K p is a tame level, then we let M(K p , W ) denote the K p -fixed functions inM(W ).
If W is the trivial representation of G on E then we omit reference to W , andwrite simply M and M(K p ) respectively.
∼

Proposition 3.2.2. There is a natural G(A)-equivariant isomorphism M(W ) −→H 0 (VW ). (Here we regard H 0 (VW ) as a G(A)-representation via the surjection
G(A) = G∞ × G(Af ) → π0 × G(Af ).)Proof. Any f ∈ M(W ) is constant on the right G0∞ -cosets in G(A). Fix f ∈ M(W ),and let Kf be a compact open subgroup of G(Af ) such that f is constant on
the right Kf -cosets in G(A). Then f gives rise to the section of G(A)/G0∞ Kf →W ×(G(A)/G0∞ Kf ) defined by g 7→ (f (g), g). The assumption that f is an algebraic
automorphic form implies that this section is G(F )-equivariant, and thus gives anelement of H 0 (Y (Kf ), VW ). Thus we obtain a G(A)-equivariant map M(W ) →H 0 (VW ), which is immediately checked to be an isomorphism. As a corollary we find that M(K p , W ) is isomorphic to H 0 (K p , VW ) for eachchoice of tame level K p . Taking W to be the trivial representation E of G, we seethat M(K p ) is isomorphic to H 0 (K p ), and thus that H̃ 0 (K p ) (being isomorphic
to Ĥ 0 (K p ) by Corollary 2.2.25) may be identified with the completion of the space

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

47

M(K p ) of algebraic automorphic forms on G(A) of tame level K p with trivial
coefficients, with respect to the sup norm.
Definition 3.2.3. For each choice of tame level K p , we refer to H̃ 0 (K p ) as thespace of p-adic automorphic forms on G(A) of tame level K p . It is an E-Banach
space, equipped with an admissible continuous representation of π0 × G by isometries. We refer to H̃ 0 as the space of p-adic automorphic forms on G(A). It isequipped with an admissible continuous action of π0 × G(Af ).
If L is a local field intermediate between Fp and Qp , then we refer to the space
H˜0 L−la as the space of locally L-analytic p-adic automorphic forms on G(Af ). It
is a locally convex p-adic vector space of compact type, equipped with a strongly
admissible locally L-analytic representation of π0 × G(Af ).
It seems worth remarking that H˜0 (respectively H˜0 L−la ) may be regarded as thespace of continuous E-valued functions on G(A) that are locally constant on G(Apf )cosets (respectively, that are locally analytic on G-cosets and locally constant on
G(Apf )-cosets) and invariant under left translation by elements of G(F ). This should
serve to justify our use of the adjective automorphic.
Corollary 2.2.25 and Proposition 3.2.2 together show that for any finite dimensional representation W of G defined over E, there is a natural isomorphism∼
0
M(W ) ⊗E W̌ −→ H̃W−lalg . Consequently, unless W is trivial, we do not find automorphic representations appearing as irreducible closed subrepresentations of H̃ 0 ;rather, as the following proposition shows, it is the classical p-adic automorphic
representations that so appear.
Proposition 3.2.4. The locally algebraic absolutely irreducible closed π0 × G(Af )subrepresentations of H̃ 0 are (if we forget the π0 -action) precisely the classical padic automorphic representations attached to allowable automorphic representationsof G(A) that are definable over E.
Proof. Let us begin by remarking that π0 is an abelian group [19, Prop. 2.4]. Thus
the tensor product of a π0 -representation and a G∞ -representation is naturally a
G∞ -representation, and also any absolutely irreducible π0 × G(Af )-representation
is absolutely irreducible as a G(Af )-representation. Thus we may ignore the π0 action, and we must show that the locally algebraic absolutely irreducible closed
G(Af )-subrepresentations of H̃ 0 coincide with the classical p-adic automorphic representations attached to allowable automorphic representations of G(A) that are
definable over E.
If W is an irreducible representation of G defined over E, then Corollary 2.2.25∼
0yields a natural isomorphism H 0 (VW ) ⊗E W̌ −→ H̃W̌
. Thus the locally alge−lalgbraic absolutely irreducible closed G(Af )-subrepresentations of H̃ 0 coincide withthe absolutely irreducible G(Af )-subrepresentations of H 0 (VW ) ⊗E W̌ , as W runs
over all irreducible representations of G defined over E.
∼
We use the isomorphism ı : C −→ Qp to regard C as an extension of E, and so
to form the base-changed representation W/C of G/C . Since G∞ is compact modulo
its centre, any automorphic representation π of G(A) whose infinity component π∞
is W̌/C -allowable contributes to H 0 (VW/C ); furthermore, this cohomology space issemi-simple as a G(Af )-representation. The discussion at the end of the preceding
subsection therefore shows that the absolutely irreducible G(Af )-subrepresentations

48

MATTHEW EMERTON

of H 0 (VW ) ⊗E W̌ are precisely the classical p-adic automorphic representations attached to the W̌/C -allowable automorphic representations of G(A) that are definable
over E. This proves the proposition. As we have already recalled, when n = 0, the map (2.3.7) is an isomorphism forany choice of W and any tame level K p . Thus, if G is quasi-split over Fp , then we
may form the eigenvariety E(0, K p ), for any choice of K p , and Theorem 0.7 applies.
Furthermore, it follows from [14, Thm. 4.2.36] that E(0, K p ) is equidimensional, ofdimension equal to the dimension of a maximal torus of G/Fp .
(3.3) So as to give a completely concrete example, in this subsection we consider
the case G = Gm over Q. In this context Fp = Qp is the only possible choice forthe local field L, and so we omit it from the notation.
∼
Let Ẑp denote the prime-to-p profinite completion of Z (so that Ẑ −→ Zp ×Ẑp ). Choose a natural number N prime to p, and denote by K(N ) the kernelof the natural map (Ẑp )× → (Z/N )× . Taking K p = K(N ) in the constructionsof Subsection 2.2, we find that H̃ 0 (K(N )) is an E-Banach algebra, canonically×
isomorphic to C(Z×p × (Z/N ) , E). Thus if we pass to the limit over all such K(N ),we obtain an isomorphism∼smH̃ 0 −→ C(Z×((Ẑp )× , E).
p , E) ⊗E C

The target is naturally isomorphic to the ring of E-valued functions on Ẑ that arecontinuous in the Zp -variable, and smooth in the Ẑp -variable. Passing to locallyanalytic vectors yields an isomorphism
∼

0
sm
H̃la
−→ C la (Z×((Ẑp )× , E).
p , E) ⊗E C

Since Gm is a torus, it is certainly quasi-split, and is equal to its own Borel subgroup, and its own maximal torus. Thus we may form the eigenvariety E(0, K(N ))for any choice of tame level N , and Theorem 0.7 applies (since n = 0). In fact,
we can described E(0, K(N )) explicitly: If we write W := (Z×p )ˆ (the space of lo×
cally analytic characters of Z×p ) and let ((Z/N ) )ˆdenote the space of characters of∼(Z/N )× , then E(0, K(N )) −→ W × ((Z/N )× )ˆ.
Z
×× ∼Since T = G = Gm , we have T = Q×
p , and the isomorphism Qp −→ Zp × p∼
induces an isomorphism T̂ −→ W × Gm . The finite map E(0, K(N )) → T̂ is the
product of the identity map on W and the map ((Z/N )× )ˆ→ Gm defined by taking
a character of (Z/N )× to its value on p−1 .
4. The case of GL2/Q(4.1) As the title of this section indicates, we now take F = Q and G = GL2 .
We fix a finite extension E of Qp as our coefficient field. This is a case in whichY ◦ (Kf ) = Ỹ (Kf ) for any choice of level Kf , and so we will work with the quotientsỸ (Kf ) throughout this section. As we will briefly recall, these are the familiar
modular curves.
∼
Indeed G(F∞ ) = GL2 (R) −→ AutR (C), the group of R-linear automorphismsof C, the subgroup Z∞ K∞ is equal to the group generated by C× (= AutC (C) ⊂AutR (C)) and complex conjugation (which we denote by c), and so (Z∞ K∞ )◦ isON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

49

equal to C× . The quotient GL2 (R)/(Z∞ K∞ )◦ is isomorphic to the space of orderedpairs of R-linearly independent complex numbers modulo scaling, which is in turn
isomorphic to C \ R. If X denotes the upper half-plane in C \ R, and if Kf is acompact open subgroup of GL2 (Ẑ), then strong approximation for SL2 , and thefact that Z is a PID, imply that GL2 (Af ) = GL2 (Q) GL2 (Ẑ), and hence that∼Ỹ (Kf ) −→ SL2 (Z)\(X × GL2 (Ẑ))/Kf(where of course SL2 (Z) acts on X by linear fractional transformations). (Compare
the discussion of [12, §1.2].) The group π0 has order two, and its non-trivial element
acts on Ỹ (Kf ) through the automorphism τ 7→ −τ c of X. As is well known, thearithmetic quotients Ỹ (Kf ) are affine algebraic curves over C, the so-called modular
curves.
Each finite dimensional representation W of GL2 over E defines a corresponding
family of local systems VW on the curves Ỹ (Kf ). The open Riemann surfacesỸ (Kf ) have non-trivial cohomology in degrees 0 and 1, and we will consider theconstructions of Subsection 2.2 in these cohomological degrees. Proposition 2.2.24,
and our remark concerning the cohomological dimension of Ỹ (Kf ), shows thatVp H n vanishes for all n; the short exact sequence of Theorem 2.2.16 (iv) then∼implies that Ĥ n −→ H̃ n for every n. Thus we have two spaces to consider: namely,Ĥ 0 and Ĥ 1 .
The compactly supported cohomology of Ỹ (Kf ) is supported in degrees one andtwo. Proposition 2.2.26 and Corollary 2.2.27 show that Vp Hcn vanishes, and thatthe map Ĥcn → H̃cn is an isomorphism, for n = 1, 2. Thus we have the spaces Ĥc1
and Ĥc2 to consider.
As well as their cohomology and compactly supported cohomology, it will also
be convenient to consider the parabolic cohomology of the modular curves.
Definition 4.1.1. For any finite dimensional representation W of GL2 over E,
1(VW ) denote the image of the natural π0 × G(Af )-equivariant map
we let Hpar1
Hc (VW ) → H 1 (VW ); it is a smooth π0 × G(Af )-representation. If K p is a choice
11(Kp , VW ) denote the K p -invariants of(Kp , VW ) and Hparof tame level, we let Hpar
1
H (VW ). As usual, if W is the trivial representation, we will omit “ VW ” from thenotation.
(4.2) If Kf is a compact open subgroup of GL2 (Af ), then det(Kf ) is a compact
open subgroup of Ẑ× , and it is well known (and follows directly from the adélic
description) that the set of connected components of the modular curve Ỹ (Kf ) isnaturally isomorphic to Ẑ× / det(Kf ). Thus we see that Ĥ 0 is isomorphic to thecorresponding space for the group Gm , considered in Subsection 3.3:∼sm
Ĥ 0 −→ C(Z×((Ẑp )× , E).
p , E) ⊗E C
det

The group π0 × GL2 (Af ) acts through the determinant map π0 × GL2 (Af ) −→{±1}×A×f . (As an aside, let us remark that if W is irreducible of dimension greaterthan one, then H 0 (VW ) = 0, and hence its completion Ĥ 0 (VW ) also vanishes. On
the other hand, H̃ 0 (VW ) = H̃ 0 ⊗E W 6= 0. This illustrates the possibility remarked
upon following the proof of Proposition 1.1.10.)

50

MATTHEW EMERTON

The space Ĥ 1 is significantly more interesting then Ĥ 0 , since it contains information about p-power congruences of classical modular forms. Naturally, we cannot
give such an explicit description of it. The remainder of this section is devoted
to investigating some of the structure of this π0 × GL2 (Af )-representation (andthe various representations obtained from it by passing to locally analytic vectors,
locally algebraic vectors, and so on). In particular, we will study the associatedeigenvariety, and explain its relationship to the eigencurve of [11]. (See Proposition 4.4.15 below.)
We begin by considering the π0 × GL2 (Af )-equivariant map
ˆ E Ĥ 1 → Ĥ 1
Ĥ 0 ⊗(4.2.1)yielded by the cup-product and Proposition 2.2.21. This makes Ĥ 1 a topologicalmodule over the topological E-algebra Ĥ 0 .
Let T denote the maximal torus of GL2 consisting of diagonal matrices. As
usual, we write G = GL2 (Qp ) and T = T(Qp ). Before goingfurther in our analysis,u 0
it will help to introduce more notation. We write T0 = {| u ∈ Z×p } and
0
1






1 0
p 0
1 0
T1 = {| v ∈ Z×and ℘1 =. If h℘i ip }. We also write ℘0 =
0 v
0 1
0 p
denotes the infinite cyclic group generated by ℘i (i = 1, 2), then there is a natural
isomorphism∼

T0 × T1 × h℘0 i × h℘1 i −→ T.

(4.2.2)∼×
The determinant map det : G → Q×p induces an isomorphism det : T0 −→ Zp , andhence an embedding of rings(4.2.3)

∼

id⊗1

∼

×
sm
C(T0 , E) −→ C(Z×((Zp )× , E) −→ Ĥ 0 .
p , E) ,→ C(Zp , E) ⊗E C

Lemma 4.2.4. (i ) The image of the embedding (4.2.3 ) is a π0 × G(Af )-invariant
subalgebra of Ĥ 0 .
(ii ) The π0 × G(Af )-action on C(T0 , E) provided by (i ), when restricted to T0 ⊂π0 × G(Af ), agrees with the right regular action of T0 on C(T0 , E).
Proof. Both claims are straightforward. Thus the map (4.2.1) restricts to a π0 × GL2 (Af )-equivariant map(4.2.5)

ˆ E Ĥ 1 → Ĥ 1 .
C(T0 , E) ⊗We let (Ĥ 1 )T0 denote the subspace of Ĥ 1 consisting of T0 -fixed vectors. Clearly(Ĥ 1 )T0 is invariant under π0 × T × GL2 (Apf ).
Proposition 4.2.6. The map (4.2.5 ) restricts to a π0 × T × GL2 (Apf )-equivariant∼
ˆ E (Ĥ 1 )T0 −→topological isomorphism C(T0 , E) ⊗Ĥ 1 .
Proof. We have already observed that the map (4.2.5) has the stated equivarianceproperties, and thus so does its restriction(4.2.7)

ˆ E (Ĥ 1 )T0 → Ĥ 1 .
C(T0 , E) ⊗

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

51

It is a general fact that if G is a compact topological group, V is an E-Banach
space equipped with a continuous G-action, and V is furthermore equipped with aG-equivariant C(G, E)-Banach module structure (where G acts on C(G, E) throughˆ E V G → V induced by the topothe right regular action), then the map C(G, E) ⊗logical module structure on V is a topological isomorphism. Indeed, C(G, E) is naturally a Banach-Hopf-algebra, and our assumption on V implies that V is a BanachHopf-module over C(G, E). The main theorem on Hopf modules [34, Thm. 4.1.1,
p. 84], adapted to the Banach-Hopf setting, then yields the required isomorphism.2
p
Applying this with G = T0 and V = (Ĥ 1 )K for a compact open subgroup K p ofGL2 (Apf ) (taking into account Lemma 4.2.4 (ii)), and then passing to the inductive
limit over all such K p , we find that (4.2.7) is a topological isomorphism. The preceding proposition expresses the analogue, in our p-adically completed
situation, of the fact that cohomology classes in one of the spaces H 1 (VW ) may
always be twisted by elements of H 0 so as to be invariant under T0 . Together with
Proposition 4.2.8 below, it will be applied in Subsection 4.4 to obtain a corresponding decomposition of the eigenvariety for GL2 .
As observed in Proposition 2.2.23, the module structure (4.2.1) induces a module
structure
0 ˆ
1
1
Ĥla
⊗E Ĥla→ Ĥla.
(Since Qp is the only field available with respect to which we may consider locallyanalytic vectors, here and below we will abbreviate “Qp − la” by “la”.) We may repeat the above discussion in the locally analytic context, and so obtain the following
result.
Proposition 4.2.8. There is a natural π0 × T × GL2 (Apf )-equivariant topological1
1 T0 ∼
ˆ E (Ĥla.
isomorphism C la (T0 , E) ⊗) −→ Ĥla
Proof. This follows from the analogue for the compact type Hopf algebra C la (T0 , E)
1and its compact type topological Hopf module Ĥlaof the “main theorem on Hopf
modules” recalled in the proof of Proposition 4.2.6. 
Analogous results to Propositions 4.2.6 and 4.2.8 hold with Ĥ 1 replaced by Ĥc1 .
(4.3) We turn to analyzing the spectral sequences of Corollary 2.2.18. If we
fix a finite dimensional representation W of GL2 over E, then we obtain the π0 ×j
GL2 (Af )-equivariant spectral sequence E2i,j = Extigl2 (W̌ , Ĥla) =⇒ H i+j (VW ).
∼Let Z denote the centre of GL2 (so that Z −→ Gm ). Multiplication in G induces
a homomorphism Z(Qp ) × SL2 (Qp ) → G, which is locally an isomorphism (that is,
induces an isomorphism between sufficiently small neighbourhoods of the identity
in its source and target).
2 When applying the main theorem on Hopf modules, note that the Banach comodule structure∼ˆ V −→ C(G, V )cG : V → C(G, E) ⊗(the isomorphism following by the example of [27, pp. 111-112]) is defined by cG (v)(g) = gv forall g ∈ G and v ∈ V ; that is, cG (v) is the orbit map of v. Thusˆ v = the constant function v on G
cG (v) = 1 ⊗if and only v ∈ V G .

52

MATTHEW EMERTON

Proposition 4.3.1. If the centre Z of GL2 acts on W via a character χ, then for
any i ≥ 0, there is a natural isomorphism of π0 × Z(Qp ) × SL2 (Qp ) × GL2 (Apf )representations∼00
Homz (E(χ−1 ), Ĥla) ⊗E H i (sl2 , W ) −→ Extigl2 (W̌ , Ĥla).
0
(Here z denotes the Lie algebra of Z(Qp ) and Homz (E(χ−1 ), Ĥla
) denotes the space−1of z-equivariant maps from E(χ ) (the one-dimensional representation on which0Z(Qp ) acts by the character χ−1 ) to Ĥla
. It is a smooth π0 × Z(Qp ) × GL2 (Apf )representation, while H i (sl2 , W̌ ) is a smooth SL2 (Qp )-representation.)

Proof. The G-representation Ĥ 0 , when regarded as a representation of π0 ×Z(Qp )×
SL2 (Qp ) × GL2 (Apf ) via the natural map
π0 × Z(Qp ) × SL2 (Qp ) × GL2 (Apf ) → π0 × GL2 (Af ),may be regarded as the tensor product of itself, restricted to a representation of
π0 × Z(Qp ) × GL2 (Apf ), and of the trivial representation of SL2 (Qp ). Similarly, theGL2 (Qp )-representation W̌ , when regarded as a Z(Qp ) × SL2 (Qp )-representation,
may be regarded as the tensor product of χ−1 and of itself, restricted to a representation of SL2 (Qp ).
A version of the Künneth theorem (whose precise formulation and proof we leaveto the reader) shows that the cup-product induces an isomorphism
M

∼

0
0
),
) ⊗ Extbsl2 (W̌ , E) −→ Extigl2 (W̌ , Ĥla
Extaz (E(χ−1 ), Ĥlaa+b=iin which all the arrows are π0 × Z(Qp ) × SL2 (Qp ) × GL2 (Apf )-equivariant. (Weare using the fact that map Z(Qp ) × SL2 (Qp ) → G is a local isomorphism, and so
induces an isomorphism between the corresponding Lie algebras.)
Let H be a compact open subgroup of Z(Qp ), chosen so small that it maps
isomorphically onto its image in Z×
p under the determinant. For any tame levelK p , the representation Ĥ 0 (K p ) is isomorphic as an H-representation to a direct
sum of copies of C(Z×p , E). By Proposition 1.1.12 (v) and Theorem 1.1.13, thea−1spaces Extz (E(χ ), Ĥ 0 (K p )) vanish if a > 0. Taking into account the isomorphism∼
Extbsl2 (W̌ , E) −→ H b (sl2 , W ), the lemma follows. 
Corollary 4.3.2. If W is an irreducible finite dimensional representation W of0
GL2 , then the space Extigl2 (W̌ , Ĥla) vanishes except possibly when i = 0 and i = 3.
00Furthermore, Homgl2 (W̌ , Ĥla) and Ext3gl2 (W̌ , Ĥla
) are isomorphic as π0 × Z(Qp ) ×p
SL2 (Qp ) × GL2 (Af )-representations, and both vanish unless W is one-dimensional.
Proof. Since sl2 is a semi-simple Lie algebra, H 1 (sl2 , W ) vanishes for every finite dimensional representation W of SL2 over E. Since sl2 is three-dimensional, Poincaré
duality for Lie algebra cohomology implies that H 2 (sl2 , W ) also vanishes, that
H i (sl2 , W ) vanishes if i > 3, and that H 0 (sl2 , W ) and H 3 (sl2 , W̌ ) are dual to one
another.
If W is irreducible then H 0 (sl2 , W ) is non-zero if and only if W is the trivial representation of SL2 . The corollary is now seen to follow from Proposition 4.3.1. 

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

53

If we feed the information provided by Corollary 4.3.2 into the spectral sequenceof Corollary 2.2.18, and also take into account the fact that H n (VW ) vanishes forn > 1, we obtain the following natural isomorphisms:∼(4.3.3)0H 0 (VW ) −→ Homgl2 (W̌ , Ĥla),(4.3.4)1H 1 (VW ) −→ Homgl2 (W̌ , Ĥla),(4.3.5)1
0
Ext1gl2 (W̌ , Ĥla) −→ Ext3gl2 (W̌ , Ĥla
).

∼

∼

j
We also see that Extigl2 (W̌ , Ĥla) vanishes for all values of i and j not appearing in theabove list; that is, for (i, j) 6= (0, 0), (0, 1), (1, 1), (3, 0). Furthermore, Corollary 4.3.2
shows that when (i, j) equals one of (1, 1) or (3, 0), we get a non-vanishing expression
only when W is one-dimensional.
The isomorphism (4.3.4) shows that the hypotheses of Proposition 2.3.8 is satisfied for every representation W when n = 1. (The isomorphism (4.3.3) shows that
it is also satisfied for n = 0. Of course, this is a special case of Corollary 2.2.25.)
In [21, §1, ex. 3], Ihara considers the projective limit of the Zp -homology of
the connected components of the modular curves Ỹ (Kf ), as Kf ranges over asequence of groups of increasing p-power level and fixed prime-to-p level. This limit
is equipped with a natural action of SL2 (Zp ). This SL2 (Zp )-module is closely related
to the topological dual of the space Ĥ 1 , considered as a GL2 (Zp )-representation. It
may be that the methods of [21] could be of use in analyzing the structure of Ĥ 1
as a GL2 (Zp )-representation.
We now consider the case of compactly supported cohomology. The following
result shows that this case is particularly simple.

Proposition 4.3.6. The space Ĥc2 vanishes.
Proof. Fix a tame level K p . If Kp is a compact open subgroupof GL2 (QTp ), then Kp
Tcontains an open subgroup Kp0 such that the index [Kp SL2 (Qp ) : Kp0 SL2 (Qp )]is divisible by pr , for arbitrary r > 0. The map Ỹ (Kp0 K p ) → Ỹ (Kp K p ) thus
has degree divisible by pr on each connected component of the source, and so
the image of the pullback map Hc2 (Ỹ (Kp K p ), OE ) → Hc2 (Ỹ (Kp0 K p ), OE ) lies in
Hc2 (Ỹ (Kp K p ), OE )pr Hc2 (Ỹ (Kp0 K p ), OE ). Since r was arbitrary, it follows that lim−→
Kp

is p-divisible, and hence that its p-adic completion vanishes. This implies the proposition. 
From the spectral sequence of Corollary 2.2.18, we deduce the following isomorphisms:∼(4.3.7)1
Hc1 (VW ) −→ Homgl2 (W̌ , Ĥc,la),(4.3.8)1
Hc2 (VW ) −→ Ext1gl2 (W̌ , Ĥc,la),∼1
and also that Extigl2 (W̌ , Ĥc,la) vanishes for i > 1. Since Hc2 (VW ) vanishes for irreducible W , except in the case when W is one-dimensional, the isomorphism (4.3.8)
1
implies that when W is irreducible Ext1gl2 (W̌ , Ĥc,la) vanishes unless W is of dimension one.
We close this section with the following result, whose proof is similar to that of
Proposition 4.3.6.

54

MATTHEW EMERTON

Proposition 4.3.9. The natural map Ĥc1 → Ĥ 1 is surjective.
Proof. If Kf is a compact open subgroup of GL2 (Af ), let X(Kf ) denote the completion of the curve Ỹ (Kf ), and let C(Kf ) denote the complement of Ỹ (Kf ) in
X(Kf ) (the set of cusps). For any such Kf , the inclusion of Ỹ (Kf ) in X(Kf )∼1
induces a natural isomorphism H 1 (X(Kf ), E) −→ Hpar
(Kf ).
1
1
Fix a tame level K p , and denote by Ĥpar(K p ) the completion of Hpar
(Kp ) with1prespect to the gauge of its sublattice lim
Hpar (Ỹ (Kp K ), OE ). The proposition−→
Kp

1
will follow if we show that the map Ĥpar(K p ) → Ĥ 1 (K p ) is an isomorphism.

Aconsideration of the commutative diagram
0

0


1
p
lim
H
(X(K
p K ), OE )
−→


1
p
H
(X(K
p K ), E)
/ lim
−→

Kp

Kp


1
p
lim
H
(
Ỹ
(K
p K ), OE )
−→
Kp


1
p
H
(
Ỹ
(K
p K ), E)
/ lim
−→
Kp


lim
Div0 (C(Kp K p ), OE )−→∼1
/ Hpar(K p )


H 1 (K p )Div0 (C(Kp K p ), E)
/ lim
−→
Kp

Kp


0


0,

in which Div0 (C(Kp K p ), R) denotes the R-module of divisors of degree zero supported on the set of cusps C(Kf ) with coefficients in R, and whose first two columns
are exact, shows that it suffices to prove that the map lim
Div0 (C(Kf ), OE ) →
−→
Kp

0

lim
Div (C(Kf ), E) is an isomorphism. This in turn follows from the fact that for−→
Kp

any fixed compact open subgroup Kp of GL2 (Zp ), we may find a normal open subgroup Kp0 ⊂ Kp such that the map X(Kp0 K p ) → X(Kp K p ) is ramified over every
cusp, with ramification degree divisible by any given power of p. (4.4) We fix a choice of tame level K p of the form “Γ1 (M )”, for some natural
number M coprime to p ; more precisely, we assume that
a b
K p := {∈ GL2 (Ẑp ) | c ≡ 0 (mod M ), d ≡ 1 (mod M )}.
c d
We will write E(1, M ) rather than E(1, K p ) to denote the degree one cohomologicaleigenvariety of tame level K p , defined with respect to the Borel subgroup B of upper
triangular matrices in GL2 . Similarly we will write E(1, M )cl rather than E(1, K p )clto denote the set of classical points in E(1, M ).

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

55

We also define a variant of E(1, M )cl , to be denoted E(1, M )par,cl , by replac1
ing H 1 by Hparin the definition. We remark that E(1, M )par,cl is obviously asubset of E(1, M )cl . We further define E(1, M )cl,0 and E(1, M )par,cl,0 to be thesubsetsof E(1, M )cl and E(1, M )par,cl respectively consisting of pairs (χ, λ) ∈T̂ × Spec H(K p )sph (Qp ) such that χ is trivial when restricted to T0 . (Here andbelow we will use the notation introduced in Subsection 4.2.)
Let us give an explicit description of T̂ ×Spec H(K p )sph . The isomorphism (4.2.2)∼
yields an isomorphism T̂ −→ W0 ×W1 ×Gm,0 ×Gm,1 , where Gm,i denotes h℘i iˆ, and∼Wi denotes T̂i (i = 1, 2). The isomorphism det : Ti −→ Z×
p induces an isomorphism∼
Wi −→ W (where, as in Subsection 3.3, W denotes (Z×p )ˆ), while evaluation at∼℘i yields an isomorphism Gm,i −→ Gm . For each ` not dividing M p, the local
Hecke algebra H(GL2 (Q` )// GL2 (Z` )) is generated bythe Hecke
operators
T (`)

 ±1

`
0
`
0
±1
and T (`, `) , corresponding to the double cosets of
and
0 1
0 `±1
p sph
±1
respectively. Thus H(K )= E[{T (`), T (`, `) }(`,N p)=1 ].
We now describe the points of E(1, M )par,cl,0 in classical terms.
Definition 4.4.1. By a p-stabilized newform of weight k + 2 ≥ 2 we mean acuspform f of weight k + 2 satisfying one of the two following conditions:(i) The modular form f is a normalized newform on Γ1 (pr C) for some natural
numbers r > 0 and C prime to p.
(ii) There is a normalized newform g on Γ1 (C) for some natural number C
prime to p, such that f (τ ) = g(τ ) − αp g(pτ ) for one of the roots αp of the pth
Hecke polynomial of g.
We say that f has tame conductor C, and p-stabilized conductor Cpr (in case (i))
or Cp (in case (ii)). Note that in either case, if Cpr denotes the p-stabilized conductor of f , then f is a weight k + 2 modular form on Γ1 (Cpr ), and is an eigenform for
the Atkin-Lehner Up -operator. We say that f is of finite slope if its Up -eigenvalue
is non-zero.
For k ≥ 0, let Wk denote the dual to the kth symmetric power of the standard
two-dimensional representation of GL2 , and let ψk denote the highest weight of Wk
with respect to B. If (χ, λ) is a point of E(1, K)par,cl,0 , then we may factor χ as
χ = θψ, where θ is a smooth Qp -valued character of T , trivial on T0 , and ψ = ψkfor some k ≥ 0. Write α = pθ(℘0 ) and β = θ(℘1 ).
Proposition 4.4.2. The set E(1, M )par,cl,0 (Qp ) is in bijection with the set of pstabilized newforms of finite slope whose tame conductor divides M . Let (χ, λ) ∈E(1, M )par,cl,0 (Qp ), and let f be the corresponding p-stabilized newform. The asserted bijection is uniquely determined by the following properties:
Let θ, ψ, k, α, and β be attached to χ as in the preceding discussion. Let  denotethe nebentypus of f , and factor  = p M as the product of a character of p-power
conductor and a character of conductor dividing M .
(i ) The weight of f is equal to k + 2. 
1 0×(ii ) If we identify Zp and T1 via v 7→
, then θ|T1 and −1
p coincide.
0 v(iii ) If ` is a prime not dividing M p, then M (`) = λ(T (`, `))p (`)−1 `−k , whileM (p) = αβ/pk+1 .
(iv ) The Up -eigenvalue of f is equal to α.

56

MATTHEW EMERTON(v ) If ` is a prime not dividing M p, then the T` -eigenvalue of f is equal toλ(T (`)).

1 x
Proof. Write N0 := {| x ∈ Zp }, and write P0 := N0 T0 . A point (θψk , λ)0 1(for some k ≥ 0) appears in E(1, M )par,cl,0 if and only if the (T + = θ, H(K p )sph =1λ)-eigenspace of Hpar
(K p , VW̌k )P0 is non-zero. (Here T + := {t ∈ T | tN0 t−1 ⊂ N0 }acts via the Hecke operators πN0 ,t as defined in [14, Def. 3.4.2]; also we have implicitly extended scalars from E to a subfield of Qp over which θ and λ are defined.)
1
This follows from the relation between the Jacquet module JB (Hpar(K p , VW̌k )) andparabolic induction, given by [14, Prop. 4.3.4]; we are also taking into account thefact that θ and ψk are trivial on T0 .
∼1
1
There is a natural isomorphism Hpar
(K p , VW̌k )P0 −→ lim
Hpar(Y1 (pr M ), VW̌k ),−→rwhere as usual we have written Y1 (pr M ) to denote the modular curve corresponding
to the congruence subgroup Γ1 (pr M ). The proposition thus follows from the usual
relation between the cohomology of modular curves and modular forms (EichlerShimura theory), together with an interpretation of the Hecke operators πN0 ,t (fort = ℘0 , t = ℘0 ℘1 , or t ∈ T1 ), as well as T (`) and T (`, `) (for ` a prime not dividing
M p), in classical terms.
1(Y1 (pr M ), VW̌k ) as a direct sum of plus andMore precisely, we may write lim
Hpar
−→
r

minus eigenspaces under the action of π0 ; after extending scalars to C, the corresponding direct summands may be described by holomorphic and anti-holomorphic
modular forms respectively. We will briefly sketch the holomorphic case; the antiholomorphic case is treated in an identical fashion.
If Kf is a compact open subgroup of GL2 (Af ), with the property that the determinant mapping maps Kf onto Ẑ× , then the restriction of the projection∼

X × GL2 (Af ) → GL2 (Q)+ \(X × GL2 (Af ))/Kf −→ Y (Kf )to X × 1 (where 1 denotes the identity matrix in GL2 (Ẑ)) remains surjective. (HereGL2 (Q)+ denotes the subgroup of GL2 (Q) consisting of matrices with positive
determinant.) If τ is an element of X, then we may form the element (τ, 1) of
C2 , and hence the element (τ, 1)k of Symk (C2 ) (which equals W̌k , after extending
1(Y (Kf ), VW̌k ) is represented byscalars from E to C). A holomorphic class in Hpark2a holomorphic Sym (C )-valued one-form τ 7→ f (τ )dτ (τ, 1)k on X, where f (τ ) isa holomorphic function on X
vanishes at the cusps and satisfies the following
 that Ta binvariance property: if γ =∈ GL2 (Q)+ Kf , thenc df(aτ + b) = (cτ + d)k+2 f (τ );cτ + dTthat is, the function f (τ ) is a weight k +2 cuspform with respect to GL2 (Q)+ Kf .
1
Thus the holomorphic part of lim
Hpar(Y1 (pr M ), VW̌k ) may be identified with−→r

the inductive limit lim
Sk+2 (Γ1 (pr M )), where Sk+2 (Γ) denotes the C-vector space−→rof cuspforms of weight k + 2 on the congruence subgroup Γ. It remains to describe
the action of πN0 ,t , T (`), and T (`, `) in classical terms.

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

57

Let Kf be as above. If g ∈ GL2 (Af ), then multiplication on the right by g∼induces a homeomorphism Y (gKf g −1 ) −→ Y (Kf ), and hence an isomorphism
11−1
H (Y (Kf ), VW̌k ) → H (Y (gKf g ), VW̌k ), that we denote by g ∗ (pullback viag). (This is just the action of g on H 1 (VW̌k ), restricted to the subspace of Kf invariant vectors.) We will now describe explicitly the action induced by g ∗ on
the corresponding spaces of cuspforms. Although the necessary computations are
straightforward and well-known, at the suggestion of the referee we will give thedetails, since they are crucial for comparing our representation theoretic approachto p-adic interpolation with the more classical approaches in the literature.
Let cf be the cohomology class represented by a weight
 k + 2 cuspform f (τ ) onTa b
GL2 (Q)+ Kf . We wish to compute g ∗ cf . If γ =∈ GL2 (Q)+ is chosen soc dthat γg ∈ Kf , then γ −1 Kf γ = gKf g −1 , giving Y (γ −1 Kf γ) = Y (gKf g −1 ), and the∼homeomorphism Y (γ −1 Kf γ) = Y (gKf g −1 ) −→ Y (Kf ) induced by right multipli−1
cation by γ coincides with the homeomorphism induced by right multiplication
by g. Thus g ∗ cf = (γ −1 )∗ cf .
Let τ ∈ X. The image of the (GL2 (Q)+ , gKf g −1 )-double coset of (τ, 1) underright multiplication by γ −1 is equal to the (GL2 (Q)+ , Kf )-double coset of (τ, γ −1 ),which coincides with the (GL2 (Q)+ , Kf )-double coset of (γτ, 1). Similarly, theimage of the (GL2 (Q)+ , gKf g −1 )-double coset of (τ, γ, w̌) (where w̌ ∈ W̌k ; this isa typical element of VW̌k over Y (gKf g −1 )) is equal to the (GL2 (Q)+ , Kf )-double
coset of (γτ, 1, γ w̌) (which is an element of VW̌k over Y (Kf )). Thus, recalling that
cf is represented by the holomorphic one-form f (τ )dτ (τ, 1)k , and that γτ = (aτ +b)/(cτ + d), we compute that g ∗ cf = (γ −1 )∗ cf is represented by the holomorphicaτ + b)dτ (aτ + b, cτ + d)k , and hence (since
one-form (ad − bc)(cτ + d)−(k+2) f (cτ + daτ + bγ(τ, 1)k = (aτ + b, cτ + d)k ) that (ad − bc)(cτ + d)−(k+2) f () is the cuspformcτ + dT−1+
gKf g that corresponds to the cohomology class g ∗ cf .
on GL2 (Q)With this formula in hand we may now interpret our various representation
theoretically defined Hecke operators in classical terms. Naturally we take Kf torbe the open subgroup of GL 2 (Ẑ) corresponding to Γ1 (p M )-level structure for somea br ≥ 1; i.e. Kf := {∈ GL2 (Ẑ) | c ≡ 0 mod pr M, d ≡ 1 mod pr M }.
c d
We begin with the operatorπN0 ,℘0, which maps a cohomology class cf to the class
Pp−1
p i
p−1 i=0 gi∗ cf , where gi :=∈ GL2 (Qp ) ⊂ GL2 (Af ). (Here the inclusion
0 1
denotes
the embeddingof GL2 (Qp ) as the pth factor of GL2 (Af ).) If we set γi :=Pp−1 τ + i
1/p −i/p),∈ GL2 (Q)+ , then γi gi ∈ Kf . Since Up f := p−1 i=0 f (01p
the above formula allows us to compute that πN0 ,℘0 cf = p−1 cUp f , as claimed (thefactor p−1 arising from det γi = p−1 ).
p 0
The operator πN0 ,℘0 ℘1 acts via the element g :=∈ GL2 (Qp ). Choose
0 p



1 0
1 0
σ ∈ SL2 (Z) such that σ ≡(mod M ) and σ ≡(mod pr ). If
0
p
0
1
 −1

p
0
we set γ := σ∈ GL2 (Q)+ , then γg ∈ Kf , and computing with the
0
p−1

58

MATTHEW EMERTON

above formula shows that πN0 ,℘0 ℘1 cf = pk chpiM f (where hpiM denotes the diamond
operator corresponding to the residue classof p modulo
M ).

1 0
Similar computations show that if t =∈ T1 then πN0 ,t cf = chv−1 ip f
0 v
(where hv −1 ip denotes the mod p-power diamond operator corresponding to v −1 ∈k
Z×p ), that T (`)cf = cT` f , and that T (`, `)cf = ` ch`iM p f (where h`iM p denotes thediamond operator corresponding to ` modulo M times a power of p). The following corollary shows that our use of the term “critical slope” agrees
with the usual sense of this term, in the p-adic theory of modular forms.
Corollary 4.4.3. Let f be a p-stabilized newform of weight k+2 and Up -eigenvalue
α attached to a point (χ, λ) of E(1, M )par,cl,0 . The character χ ∈ T̂ (Qp ) is of noncritical slope (in the sense of [14, Def. 4.4.3]) if and only if ordp (α) < k + 1.
Proof. Write χ = θψk as above, and let α0 and α1 denote a basis for the characterlattice of T, chosen so that α0 denotes projection onto the upper left entry, and α1
projection onto the lower right entry, of a matrix in T. There is a unique positiveroot of T, namely α0 − α1 . The corresponding simple reflection s interchanges α0
and α1 . If h = ordp (α), then slope θ = (h − 1)α0 + (k − h + 1)α1 , while ψk = −kα1 .
(Here slope θ is defined as in [14, Def. 1.4.2].) Finally, ρ = (α0 − α1 )/2. Thus
slope θ + ρ + s(ψk + ρ)
1
1
1
1
= (h − )α0 + (k − h + )α1 + (−k − )α0 + α12222
= (h − k − 1)(α0 − α1 ),and so χ is of non-critical slope if and only if h < k + 1, as claimed. 
The points of E(1, M )cl,0 that don’t belong to E(1, M )par,0 correspond to various
classical Eisenstein series.
Recall from Definition 0.6 that E(1, M ) is defined to be the rigid analytic Zariski
closure of E(1, M )cl in T̂ × Spec H(K p )sph . Similarly, we define E(1, M )par to be
the rigid analytic Zariski closure of E(1, M )par,cl in T̂ × Spec H(K p )sph , and defineE(1, M )0 and E(1, M )par,0 to be the rigid analytic Zariski closures of E(1, M )cl,0and E(1, M )par,cl,0 respectively in T̂ × Spec H(K p )sph .
The isomorphism (4.3.4) shows that Proposition 2.3.8 applies to our current
situation. As in that subsection, let E denote the coherent sheaf on T̂ that corresponds to the dual of JB (H 1 (K p )la ), and let A denote the commutative subringof End(E) generated by H(K p )sph . Proposition 2.3.8 shows that Spec A (which
we regard as a closed subspace of T̂ × Spec H(K p )sph ) contains the eigenvarietyE(1, M ). Similarly, let Ec be the coherent sheaf on T̂ that corresponds to the dual of
JB (Hc1 (K p )la ), and let Ac denote the commutative subring of End(E) generated by
H(K p )sph . The isomorphism (4.3.7), and an obvious analogue of Proposition 2.3.8
in the compact supports case, shows that Spec Ac (regarded as a closed subspace ofT̂ × Spec H(K p )sph ) also contains the eigenvariety E(1, M ). (Here we are using thewell-known fact that the collection of irreducible GL2 (Af )-representations appearing in Hc1 (VW ) coincides with the collection of irreducible GL2 (Af )-representations
appearing in H 1 (VW ), for any algebraic representation W of GL2 . Thus if we replace cohomology by compactly supported cohomology in the definition of E(1, M ),we obtain exactly the same subset of T̂ × Spec H(K p )sph .)ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

59

In the remainder of this subsection we indicate the relation between our constructions and the construction of [11] (and its generalization in [6]).
The inclusion of the trivial character into W0 induces an embedding(4.4.4)∼W1 × Gm,0 × Gm,1 → W0 × W1 × Gm,0 × Gm,1 −→ T̂ .

We let E0 denote the pullback of E to W1 × Gm,0 × Gm,1 via this embedding. (Thus
E0 is a coherent sheaf on W1 × Gm,0 × Gm,1 .) We let A0 denote the image of A in
End(E0 ). Thus Spec A0 is equal to the pullback of Spec A along the embedding
W1 × Gm,0 × Gm,1 × Spec H(K p )sph → T̂ × Spec H(K p )sph
induced by (4.4.4).
We let W∆ denote the diagonal copy of W embedded in W0 × W1 . Since W isa rigid analytic group (the group structure given by multiplication of characters),
there is a natural isomorphism(4.4.5)∼W∆ × W1 −→ W0 × W1 ,given by (w, w) × w1 7→ (w, ww1 ).
Proposition 4.4.6. There are natural isomorphisms(4.4.7)∼E −→ OW∆  E0

and(4.4.8)∼A −→ OW∆  A0 .
∼(Here the exterior products OW∆  E0 and A −→ OW∆  A0 are regarded at firstas sheaves on W∆ × W1 × Gm,0 × Gm,1 , and are then transported to sheaves on
W0 × W1 × Gm,0 × Gm,1 via the isomorphism (4.4.5 ).)


1 x
0
Proof. Let N0 := {| x ∈ Zp }. Since N0 acts trivially on Ĥla
, we see that0 1
the isomorphism of Proposition 4.2.8 induces an isomorphism(4.4.9)∼

ˆ E (Ĥ 1 (K p )la )T0 N0 −→ (Ĥ 1 (K p )la )N0 .
C la (T0 , E) ⊗If we define T + as in the proof of Proposition 4.4.2, then the Hecke operators πN0 ,t(as defined in [14, Def. 3.4.2]) determine an action of T + on each of (Ĥ 1 (K p )la )T0 N0and (Ĥ 1 (K p )la )N0 . By Lemma 4.2.4 (i), the embedding (4.2.3) realises C la (T0 , E)
as a T -invariant subspace of Ĥ 0 (K p )la . The isomorphism (4.4.9) is T + -equivariant,
if we equip the source with the tensor product of the T + -action on the first factor(obtained by restricting the T -action) and the Hecke T + -action on the second factor,
and if we equip the target with the Hecke T + -action. By [14, Prop. 3.2.9] there is
an isomorphism of locally analytic T -representations(4.4.10)∼ˆ E ((Ĥ 1 (K p )la )T0 N0 )fs −→C la (T0 , E) ⊗((Ĥ 1 (K p )la )N0 )fs =: JB (Ĥ 1 (K p )la ).

60

MATTHEW EMERTON

It follows from [14, Prop. 3.2.11] that there is a natural isomorphism∼((Ĥ 1 (K p )la )T0 N0 )fs −→ JB (Ĥ 1 (K p )la )T0 ,and hence we may rewrite (4.4.10) as an isomorphism∼

ˆ E JB (Ĥ 1 (K p )la )T0 −→ JB (Ĥ 1 (K p )la ).
C la (T0 , E) ⊗Passing to duals, and reinterpreting this isomorphism sheaf theoretically, we obtain
the isomorphism (4.4.7). The isomorphism (4.4.8) follows directly. 
The isomorphism (4.4.8) induces an isomorphism of rigid analytic spaces(4.4.11)∼

Spec A −→ W × Spec A0 .

This isomorphism induces in an analogous fashion isomorphisms(4.4.12)∼E(1, M ) −→ W × E(1, M )0and
(4.4.13)∼E(1, M )par −→ W × E(1, M )par,0 .

Having eliminated one of the factors of W in the space T̂ from consideration,
let us explain how one can eliminate the corresponding factor of Gm . The group
××(Z/M
 −1 ) actson E in the following way: an element x ∈ (Z/M ) acts via the cosetx̃0
K p ∈ H(K p ), where x̃ is any lift of x to an element of Ẑp . (This0x̃−1
normalization of the action is chosen so that on classical points it corresponds to theaction of the diamond operator hxiM , via the correspondence of Proposition 4.4.2.)
×
Let us choose E so large that all thecharacters of (Z/M )× are defined
L Qp -valued
Mover E. Then we find that E =, where M runs over the collection of
M E
M
such
characters,
and(–)
denotestheM -eigenspace. Similarly, we have E0 =
L
M
E
.
M 0
Fix a character M of (Z/M )× , and let S denote the support of E0M , in the“scheme theoretic” sense, so that S is the closed rigid analytic subvariety of W1 ×
Gm,0 × Gm,1 , cut out by the annihilator ideal sheaf of E0M . Then the restriction of
E0M to S is a coherent rigid analytic sheaf whose pushforward to W1 × Gm,0 × Gm,1
is naturally isomorphic to E0M .
Proposition 4.4.14. (i ) The closed subvariety S of W1 × Gm,0 × Gm,1 maps
isomorphically onto its “scheme theoretic” image in W1 × Gm,1 .
(ii ) The projection map S → W1 has discrete fibres; in particular, S is at mostone-dimensional.
Proof. The global sections of E0M are equal to (JB (Ĥ 1 (K p )la )0 )T0 ,M . To provepart (i), it suffices to show that C an (W1 × Gm,0 × Gm,1 ) and C an (W1 × Gm,1 ) havethe same image in the ring of endomorphisms of this space.
Let us consider how the element ℘0 ∈ T acts on (JB (Ĥ 1 (K p )la )0 )T0 ,M . If wetake into account that Z(Q)>0 (by which we mean the group of scalar matrices withpositive rational number entries, regarded as a subgroup of Z(Af )) acts trivially on

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

61−1
this space, we find that ℘0 acts through the scalar M (p) times theaction of ℘
1 .
−1
x
0
(Here we write M (x) to denote M evaluated on the scalar matrix.)0
x−1
Thus we see that indeed C an (Gm,0 × Gm,1 × W0 ) and C an (Gm,0 × W0 ) have the samep
image in the ring of endomorphisms of (JB (Ĥ 1 (K p )la )0 )T0 K ,M , proving (i).
The claim of (ii) follows from Proposition 2.3.3 (ii), which shows that the supportof E maps onto W0 × W1 with discrete fibres. Together with Proposition 4.4.6, this
implies that the support of E0 , and so also the support of E0M , maps onto W1 withdiscrete fibres. The preceding proposition shows that E0M may be regarded as a coherent sheaf
on Gm,1 × W1 , whose support maps to W1 with discrete fibres.
Let A0M denote the image of A in End(E0M ). Then Spec A0M is a closed subspaceof Spec A0 , and the Proposition 4.4.14 allows us to regard Spec A0M as a closed
subspace of W1 × Gm,1 × Spec H(K p )sph . In fact if ` is a prime not dividing
M p, then T (`, `) acts on Spec E0M via M (`) times the action of the element `−1 ,regarded as a section of OW1 . Thus in fact Spec A0M embeds as a closed subspaceof the productY
W1 × Gm,1 ×Spec E[T (`)].
(`,M p)=1
M
If we let E(1,the the intersection of E(1, M )0 and Spec AM , then`M )0 denoteM
E(1, M )0 = M E(1, M )0 , and each E(1, M )M embeds as a closed subspace of

W1 × Gm,1 ×Y

Spec E[T (`)].

(`,M p)=1

In fact the unionof these embeddings is again an embedding of E(1, M )0 intoQ
W1 ×Gm,1 × (`,M p)=1 Spec E[T (`)]. This follows from the fact that the eigenvaluesof the Hecke operators T (`) parameterize the traces of Frobenius of a family of
Galois pseudorepresentations over E(1, M )0 , from whose determinant the tamenebentypus M may be recovered.
In short, a point of E(1, M )0 is determined by a “weight” (i.e. an element of
W1 ), a Up -eigenvalue (i.e. an element of Gm,1 – more precisely, if this element
is denoted a, then the corresponding Up -eigenvalue equals pM (p)/a, where Mdenotes the tame nebentypus, as one sees from the statement of Proposition 4.4.2),
and a collection of T` -eigenvalues, one for each prime ` not dividing M p.
It follows from its construction, and from Proposition 4.4.13 (ii), that E(1, M )0
is a reduced rigid analytic space whose connected components are either zerodimensional or equidimensional of dimension one. Since E(1, M )0 is equal to
the Zariski closure of E(1, M )cl,0 , we see that any zero-dimensional componentof E(1, M )0 must consist of classical points. Since “slope of the Up -eigenvalue” isa locally constant function on E(1, M )0 , we see that any point of E(1, M )cl that is
not isolated in E(1, M )0 can be written as the limit of a sequence of points of classical weight and of non-critical slope, which are hence themselves classical. Thus
we see that each one-dimensional irreducible component of E(1, M )0 contains a
Zariski dense set of classical points of non-critical slope.
As we will now explain, one can in fact show that E(1, M )0 is isomorphic to the
reduced eigencurve of [11] (or more precisely, to its tame level M generalization

62

MATTHEW EMERTON

considered in [6]). This will show in particular that every component of E(1, M )0
has dimension one (i.e. that E(1, M )0 contains no isolated points).
In Part II of [6] Buzzard applies his “eigenvariety machine” to spaces of overconvergent modular forms of tame level M (for any M ≥ 1 prime to p), equippedwith the action of the compact operator Up and the Hecke algebra T of prime-to-p
Hecke operators, to construct a rigid analytic curve that he calls “D”, which in thecase when M = 1 coincides with the reduced eigencurve constructed in [11].
In general, the curve that Buzzard constructs will not be reduced (because heworks with the full Hecke algebra T, and because of the presence of oldforms).
However, one may form the analogue of Buzzard’s construction after replacing T
by its subalgebra T0 of prime-to-N p Hecke operators, to obtain a curve that we will
denote by D(M ). Since we omit the T` for ` dividing M , the curve D(M ) will be
reduced. (Compare Proposition 3.9 and the proof of Proposition 4.8 of [9].)The curve D(M ) parameterizes prime-to-M systems of eigenvalues arising fromfinite slope overconvergent p-adic Hecke eigenforms of tame level M and arbitrary
weight. Thus a point of D(M ) is determined by giving a point of W (the weight),
a point of Gm (the non-zero Up -eigenvalue), and for each prime ` not dividing M p
a point of A1 (theQ `th Hecke eigenvalue). Thus we obtain an immersion of D(M )into W × Gm × (`,M p) A1 . We noted above that E(1, M )0 also embeds as a closed
subspace of this product.
Q
Proposition 4.4.15. The embedding of D(M ) in W×Gm × (`,M p)=1 A1 discussed
above induces an isomorphism of D(M ) with E(1, M )0 .
Proof. Proposition 4.4.2 (and its extension to the non-cuspidal case, the details of
which we leave to the reader) shows that D(M ) contains all the points of E(1, M )cl ,and these points are Zariski dense in D(M ) by an evident generalization of [11,Thm. F]. Thus the proposition will follows
if we show that D(M ) embeds as a
Q
Zariski closed subvariety of W ×Gm × (`,M p)=1 A1 . This follows from the fact that(by its very construction) the projection D(M ) → W ×Gm is a finite morphism. Corollary 4.4.16. The rigid analytic space E(1, M )0 is equidimensional of dimension one.
Proof. This follows from the corresponding statement for D(M ) (which is a particular case of [6, Lem. 5.8]). 
Since the isomorphisms (4.4.12) and (4.4.13) show that the varieties E(1, M ) andE(1, M )par are obtained from E(1, M )0 and E(1, M )par,0 by allowing all wild twists,
we deduce from this corollary that E(1, M ) and E(1, M )par are equidimensional ofdimension two.
We let M denote the localization of E over Spec A. It is a coherent sheaf on
E. It will be useful to describe explicitly its fibre over a point of E(1, M )par,cl,0 of
non-critical slope and tame conductor M .
Definition 4.4.17. If f is a p-stabilized newform of weight k + 2 and p-stabilized
conductor M pr , defined over a finite extension E 0 of E, then let Hf denote thesubspace of E 0 ⊗E H 1 (Y1 (M pr ), VW̌k ) on which the classical Hecke and diamond
operators of level M pr act through the corresponding eigenvalues of f .
Let us remark that if in the preceding definition we were to replace H 1 by either
1
Hc1 or Hparthen we could analogously define spaces Hc,f and Hpar,f . However,ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

63

since f is a cuspform, the natural maps
Hc,f → Hpar,f → Hf
are both isomorphisms. Thus we write Hf to denote any of these canonicallyisomorphic spaces.
Proposition 4.4.18. Suppose that x = (χ, λ) ∈ E(1, M )par,cl,0 (Qp ) corresponds(via Proposition 4.4.2 ) to a p-stabilized newform f of weight k + 2, of tame conductor M , and of Up -eigenvalue α. If χ is of non-critical slope (equivalently, by
Corollary 4.4.3, if ordp (α) < k + 1), then the fibre Mx of M over x is canonicallydual to the space Hf ⊗E (Wk )N . (Here N denotes the unipotent radical of B, sothat (Wk )N is the (one-dimensional ) highest weight space of Wk with respect toN .) It is two-dimensional, and the ±-eigenspaces under the action of π0 are each
one-dimensional.
Proof. Extending scalars from E to a finite extension over which x is defined, we
may assume that x is defined over E. As in the statement and proof of Proposition 4.4.2, write χ = θψk with θ smooth. The fibre Mc,x is by Proposition 2.3.3 (iii)
dual to the (T = χ, H(K p ) = λ)-eigenspace of JB (Ĥ 1 (K p )la ). This eigenspace is
identified via the canonical lifting of [14, (3.4.8)] with the subspace of Ĥ 1 (K p )P0
on which H(K p )sph acts through λ, and the Hecke operators πN0 ,t act throughχ(t), for t ∈ T + . (Here we are using the notation introduced in the proof of
Proposition 4.4.2.) Since χ is of non-critical slope, this eigenspace consists of locally Wk -algebraic vectors [14, Thm. 4.4.5], and so (by the isomorphism (4.3.4))+p
is isomorphic to H 1 (K p )P0 ,T =θ,H(K )=λ ⊗E (Wk )N . By assumption (χ, λ) lies in+p
1(K p )P0 ,T =θ,H(K )=λ ⊗E (Wk )N .
E(1, M )par,cl,0 , and so this space is equal to Hpar
If α = pθ(℘0 ), then α is equal to the Up -eigenvalue of f , and the discussion in
the proof of Proposition 4.4.2 (together with strong multiplicity one for newforms)+p
1(K p )P0 ,T =θ,H(K )=λ with the space Hf . Eichler-Shimura theory
identifies Hparshows that this space is two-dimensional, and that it splits as the direct sum of±-eigenspaces under the action of π0 , each one-dimensional. 
If we take into account the fact that the modular curves Ỹ (Kf ) admit canonicalmodels defined over Q, then the discussion of Subsection 2.4 shows that that M
is equipped with a natural Gal(Q/Q)-action, commuting with its Hfs (K p )-module
structure.
Proposition 4.4.19. If x is a point of E(1, M )par,cl,0 of non-critical slope, corresponding (via Proposition 4.4.2 ) to a p-stabilized newform f of weight k + 2 andtame conductor M , then the fibre Mx is “the” two-dimensional Galois representation attached to f . More precisely, this action is unramified at primes not dividing
M p, and if ` is such a prime, then the characteristic polynomial of Frob` (thearithmetic Frobenius at `) is equal to X 2 − a` X + (`)`k+1 , where a` denotes the
eigenvalue of T` on f , and  denotes the nebentypus character of f .
Proof. This follows from Proposition 4.4.18 and the known properties of the Ga1
lois representations on the spaces Hpar
(K p , VW̌k ), together with the fact that theisomorphism of Proposition 4.4.18 is Galois equivariant (as follows from Proposition 2.4.1). 64

MATTHEW EMERTON

Proposition 4.4.20. The coherent sheaf M is locally free of rank two in the neighbourhood of each point of non-critical slope and tame conductor M in E(1, M )par,cl .
∼Proof. The isomorphism (4.4.7) gives rise to a corresponding isomorphism M −→OW  M| Spec A0 , compatible with the isomorphisms (4.4.11) and (4.4.12). Thus it
suffices to check the statement of the corollary with E(1, M ) replaced by E(1, M )0 .
Proposition 4.4.18 shows that fibre of M over each classical point of non-criticalslope and tame conductor M in E(1, M )par,0 is two-dimensional. This already
implies the statement of the corollary in the neighbourhood of any such point thatis isolated in E(1, M )0 (although Corollary 4.4.16 shows that there are no such
points). Thus it suffices to check the statement of the corollary at classical points
of non-critical slope that lie in one-dimensional components of E(1, M )0 . As wasnoted above, the neighbourhood of any such point x contains an infinite number ofother such points. As x corresponds to a cuspidal newform of tame conductor M , aconsideration of associated Galois representations shows that it cannot be written
as the p-adic limit of Eisenstein series, or of newforms of tame conductor less than
M . Thus each neighbourhood of x contains infinitely many points of E(1, M )par,cl,0of non-critical slope and tame conductor M . Since E(1, M )0 is reduced, and since
M has fibre rank two at all such points, we conclude that indeed M is locally freeof rank two in the neighbourhood of x. (4.5) We close Section 4 by explaining how the results of the forthcoming paper[17] relating Jacquet functors and parabolic induction (which are summarized in[16, §5]) may be applied to construct a two-variable p-adic L-function over the
eigencurves E(1, M )par,0 considered in Subsection 4.4.
One-variable p-adic L-functions, following [26]. In [26] the authors define p-adic
L-functions attached to p-stabilized newforms of non-critical slope. We will briefly
recall their construction, using the notation and point of view adopted in this paper(which differs slightly from that of [26]).
If r ∈ P1 (Q), let {r, ∞} denote an arc in the upper half-plane X of C \ R joining
r to ∞. (If r = ∞ then we understand this arc to be trivial.) Let Kf be a compact
open subgroup of GL2 (Af ), and fix k ≥ 0. Given any element of Hc1 (Ỹ (Kf ), VW̌k ),we may pull this element back to X and then integrate it along {r, ∞} (for any
r ∈ P1 (Q)) and so obtain an element of W̌k . This may in turn be paired against
any element of Wk so as to obtain a scalar; thus each arc {r, ∞} induces an elementin the dual space to Hc1 (Ỹ (Kf ), VW̌k ) ⊗E Wk , which we again denote by {r, ∞}.
Let f be a p-stabilized newform f of weight k + 2, of p-stabilized conductor M pr ,of Up -eigenvalue α, and of non-critical slope (i.e. for which ordp (α) < k + 1). The
discussion of the preceding paragraph allows us in particular to regard {r, ∞} (forr ∈ P1 (Q)) as an element in the dual space to Hf ⊗E Wk . Equivalently, we obtaina map φ : Wk × P1 (Q) → Ȟf that is linear in the first variable.
Recall that W̌k is the kth symmetric power of the standard representation of GL2 .
Let (0, 1) be the indicated element of E 2 , and write w̌0 := (0, 1)k ∈ Symk E 2 = W̌k .
(If B denotes the Borel subgroup of GL2 consisting of lower triangular matrices,then w̌0 is a highest weight vector ofwith respect to B.) The map Qp ×
 W̌k 1 z
Wk → E defined by (z, w) 7→ hw̌0 ,wi identifies Wk with the space of
0 1
polynomial functions on Qp of degree ≤ k with coefficients in E. Having madethis identification, the map φ of the preceding paragraph is seen to be a variant of

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

65

the map φ(f, – , – ) defined in [26, §1]. More precisely, the latter map is obtainedby composing the map φ we have constructed with the map Ȟf → C obtainedby pairing elements of Ȟf against the element of Hf defined by the holomorphic
de Rham cohomology class associated to f as in the proof of Proposition 4.4.2(multiplied by 2πi), and replacing the polynomial P (z) by P (−z).
We may now use the recipe of [26] to define the p-adic L-function attached to x.
This p-adic L-function is defined by a locally analytic distribution on Z×p that wewill denote by µf . We first define the values of µf on locally polynomial functionsmbeof degrees ≤ k. Let P (z) be a polynomial of degree ≤ k, let a ∈ Z×
p , and let psome positive power of p. We defineZa(4.5.1)
P (z)dµf (z) := α−m φ(P (pm z + a), m ) ∈ Ȟf .
pma+p Zp(Here P (pm z + a) is the indicated polynomial, regarded as an element of Wk via
the above identification.) The integral of any locally polynomial function of degrees≤ k over Z×
p with respect to µf can obviously be written as a linear combination ofsuch integrals, and so by linearity µf is defined on the space of all such functions.
Since f is of non-critical slope, the Theorem of Vishik and Amice-Vélu [26, Thm.,p. 13] shows that µf extends to a locally analytic Ȟf -valued distribution on Z×p.
The p-adic L-function attached to f is the
R Ȟf -valued rigid analytic L-function
on weight space W defined via L(f, χ) := Z×χdµf , for χ ∈ W(Qp ). (The p-adicp
Fourier theory of Amice [1] shows that this formula does indeed yield a rigid analytic
function on W.)
Two-variable p-adic L-functions. We now explain our construction of twovariable p-adic L-functions. We will use an obvious variation of the constructionsof Subsection 4.4, in which we work with compactly supported cohomology rather
than cohomology. We let Mc denote the sheaf on Spec Ac obtained by localizing Ec .
The natural map Ĥc1 (K p )la → Ĥ 1 (K p )la induces a corresponding map of Jacquet
modules, and hence a map E → Ec of coherent sheaves on T̂ . This in turn induces
a map(4.5.2)

M|E(1,M )par ,0 → Mc|E(1,M )par,0 .

Lemma 4.5.3. The map (4.5.2 ) is an isomorphism in the neighbourhood of eachpoint of tame conductor M and non-critical slope.
Proof. Proposition 4.4.20, and its obvious analogue in the case of compactly supported cohomology, show that the source and target of (4.5.2) are each locally freeof rank two in the neighbourhood of points of tame conductor M and non-criticalslope in E(1, M )par,0 . Furthermore, this map induces an isomorphism on fibres
at each such point (as follows from Proposition 4.4.18, its analogue in the compactly supported case, and the observations preceding its statement). The lemma
follows. 
Definition 4.5.4. Let E 0 be a finite extension of E. We say that a point x =(χ, λ) ∈ E(1, M )par,0 (E 0 ) is bad if the character χ is locally algebraic, of the form
χ = θψk for some smooth character θ and some k ≥ 0, and if the point (r−k−1 χ, λ)also lies in Spec Ac ; here r denotes the positive simple root of GL2 with respect to
B.

66

MATTHEW EMERTON

Proposition 4.5.5. The set of bad points (defined over some finite extension of E)
is a discrete subset of E(1, M )par,0 . It does not contain any point of E(1, M )par,cl,0that is of non-critical slope.
Proof. If x ∈ E(1, M )par,0 is a classical point of non-critical slope, then an argument like that in the proof of [14, Thm. 4.4.5] shows that x does not satisfy
Definition 4.5.4. This completes the proof of the second claim of the proposition.
Since “slope” is a locally constant function on E(1, M )par,0 , we see that any pointof E(1, M )par,0 that does satisfy Definition 4.5.4 has a neighbourhood all of whose
points of classical weight, other than x itself, are in fact of non-critical slope. This
proves the first claim of the proposition. 
Corollary 4.4.16 shows that E(1, M )par,0 is a union of irreducible componentsthat are each of dimension one. Since the fibre of E(1, M )par over any point of W
is discrete, it follows that each component of E(1, M )par,0 is flat over W.
Definition 4.5.6. We let E(1, M )∗par,0 denote the complement of the set of bad
points in E(1, M )par,0 , and let M∗c denote the quotient of the restriction of Mc toE(1, M )∗par,0 by its maximal OW -torsion submodule.
Let x ∈ E(1, M )par,cl,0 be a point of non-critical slope, corresponding via Proposition 4.4.2 to a p-stabilized newform f of tame conductor M . Proposition 4.4.20
and Lemma 4.5.3 imply that Mc is locally free of rank two in a neighbourhoodof x, and hence (by the flatness result discussed prior to Definition 4.5.6) that
Mc is OW -torsion free in the neighbourhood of x. Thus Mc and M∗c coincidein a neighbourhood of x. Proposition 4.4.18 (and Lemma 4.5.3) then imply thatthe fibre M∗c,x of M∗c over x is canonically dual to Hf ⊗E (Wk )N . (As in thestatement of Proposition 4.4.18, we let (Wk )N denote the highest weight space of
Wk with respect to B.) There is a unique basis element of (Wk )N that satisfies
hw̌0 , wi = 1 (where as above w̌0 denotes the vector (0, 1)k ∈ Symk E 2 = W̌k ). This∼basis element determines an isomorphism (Wk )N −→ E, and hence an isomorphism∼M∗c,x −→ Ȟf .
Our goal in the remainder of this subsection is to prove the following theorem.
Theorem 4.5.7. There is a rigid analytic section of the coherent sheaf OW M∗c on W × E(1, M )∗par,0 , which we denote by L, with the property that at eachpoint x ∈ E(1, M )par,cl,0 of non-critical slope, corresponding via Proposition 4.4.2
to a p-stabilized newform f of tame conductor M , the fibre Lx (which is a rigid∼
analytic section of OW ⊗ M∗c,x −→ OW ⊗ Ȟf , the isomorphism being provided by
the preceding discussion) is equal to the p-adic L-function Lf constructed above.
Proof. We let U denote the compact type space that is topologically dual to theFréchet space of global sections of M∗c . The natural locally closed immersion
E(1, M )∗par,0 → T̂ × Spec H(K p )sph allows us to regard the space of global sectionsof M∗c as a topological module over C an (T̂ , E) := Γ(T̂ , OT̂ ), with a commuting action of H(K p )sph . Thus U is a locally analytic T -representation, equipped with acommuting H(K p )sph -action. The locally analytic T -representation U is allowablein the sense of [16, Def. 5.18], since it is dual to the space of global sections of acoherent sheaf on a quasi-Stein rigid analytic variety.
We may restrict global sections of Mc to E(1, M )∗par,0 and so obtain global
sections of M∗c . The dual of this restriction map is a continuous T × H(K p )sph -ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

67

equivariant map of compact type spaces(4.5.8)U → JB (Ĥc1 (K p )la ).

Lemma 4.5.12 below shows that (4.5.8) is balanced, in the sense of [16, Def. 5.17].
As usual, let G = GL2 (Qp ), and let B := B(Qp ) denote the Borel subgroup ofGlower triangular matrices. Let IB(U (δ −1 )) be the closed subspace of the locallyG−1analytic induction IndB U (δ ) defined in [16, Def. 5.13]. Lemma 4.5.12 below
GG
U (δ −1 ). Thus [16, Thm. 5.19] shows thatshows that in fact IB(U (δ −1 )) = IndB
the map (4.5.8) induces a continuous G × H(K p )sph -equivariant map(4.5.9)IndG
U (δ −1 ) → Ĥc1 (K p )la .
B

As above, let N denote the unipotent radical of B (the Borel of upper triangular
1 zmatrices). It is naturally isomorphic to Qp (via z 7→), and embeds as an
0 1
open subset of G/B. (See the discussion at the beginning of [16, §5].) In particular,
we obtain an embedding Z×
p → G/B, and a corresponding closed embedding ofcompact type spaces(4.5.10)G−1
ˆ
C la (Z×).
p , E) ⊗E U → IndB U (δ(The source is a closed subspace of Ccla (N, U (δ)), which in turn is a closed subspaceof the target.) Recall from [1] that the dual to C la (Z×p , E) is isomorphic to the space
C an (W, E) := Γ(W, OW ). Thus composing (4.5.10) with (4.5.9), and then passing
to topological duals, we obtain a H(K p )sph -equivariant morphism(4.5.11)(Ĥc1 (K p )la )0 → Γ( W × E(1, M )∗par,0 , OW  M∗c ).

(Here the source is the topological dual to the space Ĥc1 (K p )la , and the right handside denotes the Fréchet space of global sections of the coherent sheaf OW  M∗con W × E(1, M )∗par,0 .)One way to construct elements in the source of (4.5.11) is via the arcs {r, ∞}for r ∈ P1 (Q). Indeed, integrating along such an arc gives rise to a homomorphism
Hc1 (Ỹ (Kf ), A) → A for any level Kf and any ring A, compatibly with change oflevel and change of ring. Consequently {r, ∞} induces an element of the topologicaldual to Ĥc1 (K p ), and so also (simply by composing with the continuous injectionĤc1 (K p )la → Ĥc1 (K p )) an element of (Ĥc1 (K p )la )0 . Applying (4.5.11) to the element
in its source corresponding to {0, ∞}, we obtain a global section of OW × M∗ overW × E(1, M )∗par,0 , which we denote by L.
If x ∈ E(1, M )∗par,0 (Qp ), then let Lx denote the restriction of L to the closed
subspace W × x of W × E(1, M )par,0 ; it is a section of OW × M∗c,x . Let µx denotethe Ȟf -valued locally analytic distribution on Z×p that corresponds via the p-adic
Fourier theory of [1] to the rigid analytic function Lx on W.
If x is a classical point of non-critical slope, corresponding to a p-stabilized newform f of tame conductor M , then by its construction the distribution µx coincides
with the distribution µg appearing in the statement of Proposition 4.9 of [15] (taking the p-stabilized newform g of that proposition to be f ). That proposition showsthat Lx coincides with the p-adic L-function of f , and so completes the proof ofthe present theorem. 

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MATTHEW EMERTON

Lemma 4.5.12. (i ) The map (4.5.8 ) is balanced.
G(ii ) The closed embedding IB(U (δ −1 )) → IndG
U (δ −1 ) is an isomorphism.
B
Proof. By [14, Prop. 3.5.6], the map (4.5.8) induces a B-equivariant map
Ccsm (Qp , U (δ −1 )) → Ĥc1 (K p )la ,and thus a (gl2 , B)-equivariant map(4.5.13)

U (gl2 ) ⊗U (b) Ccsm (Qp , U (δ −1 )) → Ĥc1 (K p )la .

(As in the proof of Theorem 4.5.7, we are identifying Qp with the unipotent radical
N of B.)Let Cclp (Qp , U (δ −1 )) denote the space of compactly supported locally polynomial∼
U (δ −1 )-valued functions on Qp . The inclusion Qp −→ N → G/B allows us toregard Cclp (Qp , U (δ −1 )) as a (gl2 , B)-invariant subspace of IndG
U (δ −1 ). The inB
−1
−1
lp
sm
clusion of Cc (Qp , U (δ )) in Cc (Qp , U (δ )) thus induces a (gl2 , B)-equivariant
map(4.5.14)

U (gl2 ) ⊗U (b) Ccsm (Qp , U (δ −1 )) → Cclp (Qp , U (δ −1 )).

There are isomorphisms∼

Ccsm (Qp , U (δ −1 )) −→ Ccsm (Qp , E) ⊗ U (δ −1 )and∼Cclp (Qp , U (δ −1 )) −→ Ccsm (Qp , E) ⊗ U (δ −1 )[z](where U (δ −1 )[z] denotes the space of polynomials in z with coefficients in U (δ −1 )),and (4.5.14) may be factored as the tensor product of the identity automorphismof Ccsm (Qp , E) and a map(4.5.15)

U (gl2 ) ⊗U (b) U → U [z](we drop the twists by δ from our notation, since we will consider only Lie algebra
actions from now on, and δ is smooth),definedby the following
explicit formula: Let



0 0
1 0
n− and h denote the elements
andof gl2 . Then U (gl2 ) ⊗U (b)
1 0
0 −1
L
U = i (n− )i U, and (4.5.15) is defined by the formula(4.5.16)(n− )i u 7→ ((h − i + 1) . . . (h − 1)hu) z i .

Referring to the definitions of [16, §5], we find that part (i) of the lemma isequivalent to showing that the kernel of (4.5.13) contains the kernel of (4.5.14).
Consider the map(4.5.17)U → (Ĥc1 (K p )la )N0

obtained as the composition of (4.5.8) with the canonical lift JB (Ĥc1 (K p )la ) →Ĥc1 (K p )la . From the explicit formula (4.5.16), we see that part (i) holds provided

ON THE INTERPOLATION OF SYSTEMS OF HECKE EIGENVALUES

69

that (4.5.17) takes any element u of U on which h acts via an integer k ≥ 0 to a
locally Wk -algebraic element of Ĥc1 (K p )la . Note that since the T0 -action on U is
fixed to be trivial, the t-action on any such point u under (4.5.17) is completely
determined. The fibre of Spec Ac over a given character of ť is discrete, and so we
may assume (after extending scalars if necessary) that such a point u is a generalized
eigenvector for the action of T , with respect to some character χ ∈ T̂ (E), as well asa generalized eigenvector for the action of H(K p )sph , with respect to some characterλ ∈ (Spec H(K p )sph )(E). If v denotes the image of u under (4.5.17), then v is a
generalized χ-eigenvector for the action of the Hecke operators πN0 ,t (for t ∈ T + ),as well as a generalized λ-eigenvector for the action of H(K p )sph . A minor variant of[14, Prop. 4.4.4] shows that (n− )k+1 v lies in the generalized (r−k−1 χ, λ)-eigenspace
for the action of the operators πN0 ,t (t ∈ T + ) and the Hecke algebra H(K p )sph .
From the construction of U , we see that the point (χ, λ) must not be a bad pointof Spec Ac , and thus that (r−k−1 χ, λ) must not lie on Spec Ac . Consequently
the generalized (r−k−1 χ, λ)-eigenspace of (Ĥc1 (K p )la )N0 vanishes, and so (n− )k+1
annihilates v. Hence v is Wk -locally algebraic, as we wanted to show.
Again referring to the definitions of [16, §5], we see that part (ii) of the lemma
will follow if we show that (4.5.15) (and hence (4.5.14)) is surjective. Since U is dual
to the space of global sections of a flat OW -sheaf, it is divisible as a module under
the action of the polynomial ring E[h]. Formula (4.5.16) thus implies that (4.5.15)
is indeed surjective, as required. 
The global section L of OW  M∗c may be regarded as a two-variable p-adic
L-function: one variable is provided by the factor OW (with an appropriate normalization this becomes the classical “s”-variable), the other by the punctured
eigencurve E(1, M )∗par,0 .
If we decompose the coherent sheaf M∗c into its ± eigenspaces under the actionof π0 , then the section L decomposes into the sum of two sections L± , whose fibres
at classical points of non-critical slope correspond to the usual + and − p-adic Lfunctions (which vanish when evaluated on odd and even characters respectively).
Let us remark that Stevens [33] has previously constructed two-variable p-adic
L-functions along the eigencurve, via a method quite different to ours. He uses
his theory of rigid analytic modular symbols to construct a coherent sheaf (which
we will denote by S) along the eigencurve E(1, M )par,0 , whose sections are certain
modular symbols with values in the space of locally analytic distributions on Z×p.
The fibre of S at a classical point of non-critical slope maps isomorphically, via a
specialization map, to the dual of the fibre of Mc at such a point. If U is an open
subset of E(1, M )par,0 for which S|U is free of rank two, then by choosing a basisof each of the ±-parts of S|U , evaluating these modular symbols on the arc {0, ∞},and passing via the p-adic Fourier theory of [1] from distributions on Z×p to rigid±
analytic functions on W, Stevens obtains p-adic L-functions L as rigid analytic
functions on W × U.
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Northwestern University
Department of Mathematics
2033 Sheridan Rd.
Evanston, IL 60208-2730, USA
E-mail address: emerton@math.northwestern.edu