ON THE DENSITY OF SUPERCUSPIDAL POINTS OF FIXEDREGULAR WEIGHT IN LOCAL DEFORMATION RINGS AND
GLOBAL HECKE ALGEBRASMATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Abstract. We study the Zariski closure in the deformation space of a local
Galois representation of the closed points corresponding to potentially semi-stable
representations with prescribed p-adic Hodge-theoretic properties. We show infavourable cases that the closure contains a union of irreducible components ofthe deformation space. We also study an analogous question for global Hecke
algebras.

1. Introduction
In this paper we study the Zariski closure in the deformation space of a local Galois
representation of the closed points corresponding to potentially semi-stable representations with prescribed properties. We show in a number of cases that the closure
contains a union of irreducible components of the deformation space. Our results
are most novel in the case when the family consists of potentially semi-stable representations with fixed Hodge–Tate weights and their Weil–Deligne representations
correspond to supercuspidal representations via the classical local Langlands correspondence; we call such points supercuspidal. We also obtain similar results in thesetting of global Hecke algebras, and thus, in certain circumstances, namely those towhich the results of [25] apply, in global Galois deformation rings as well.
Let us explain our results in more detail. Let F be a finite extension of Qp andlet GF be its absolute Galois group. Let L be a further finite extension of Qp withthe ring of integers O, a uniformizer $ and residue field k. Let ρ̄ : GF → GLn (k)be a continuous representation and let ρ : GF → GLn (R ) be the universal framed
deformation of ρ̄. In particular, R is a complete local noetherian O-algebra withresidue field k. If x is a maximal ideal of R [1/p] then its residue field κ(x) is a finite
extension of L; we denote its ring of integers by Oκ(x) . By specializing ρ at x we
obtain a representation ρx : GF → GLn (κ(x)). Moreover, its image is contained inGLn (Oκ(x) ) and by reducing the matrix entries modulo the maximal ideal of Oκ(x)we obtain ρ̄. If ρx is potentially semi-stable then we can associate to it p-adic Hodge
theoretic data: a multiset of integers wt(ρx ) called the Hodge–Tate weights and Weil–
Deligne representation WD(ρx ), which is a representation of WF (the Weil group ofF ) together with a nilpotent operator satisfying certain compatibilities. The following
theorem is a representative example of our results.
Theorem 1.1. Assume that p does not divide 2n and ρ̄ admits a potentially crystalline lift of regular weight, which is potentially diagonalisable. Let E be an extension of F of degree n, which is either unramified or totally ramified, and fix regularDate: February 7, 2020.
The first author is supported in part by the NSF grants DMS-1303450 and DMS-1601871. The
second author is supported in part by the DFG, SFB/TR45.
1

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MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Hodge–Tate weights k. Let Σ be the subset of maximal ideals of R [1/p] consisting ofx ∈ m-Spec R [1/p], such that ρx is potentially semi-stable, wt(ρx ) = k, WD(ρx ) is
irreducible as a representation of WF and is isomorphic to an induction from WE of a
1-dimensional representation. Then the closure of Σ in Spec R contains a non-empty
union of irreducible components of Spec R .
Remark 1.2. If HomGF (ρ̄, ρ̄(1)) = 0 then the deformation problem of ρ̄ is unobstructed and thus R is formally smooth over O, so the theorem implies the densityof Σ in the whole of Spec R .
Remark 1.3. The proof of the theorem uses a global input: the patched module M∞
constructed in [10]. In the process we identify GF with a decomposition group at aprime p of an absolute Galois group of a totally real field K, and ρ̄ as a restriction
to GF of a global automorphic representation of GK . In the general case, we obtain
precisely those components of Spec R which contain a closed point corresponding to
the restriction to GF of a global automorphic representation of the kind consideredin [10, §2.4] which is crystalline at p.
Remark 1.4. The assumption p - 2n comes from our use of the patched module
M∞ constructed in [10] and the assumptions made there. We also have a versionof Theorem 1.1, when the field extension E/F is wildly ramified. In that case, thecondition on WD(ρx ) is more complicated to describe; it can be phrased in terms ofBushnell–Kutzko classification of supercuspidal representations of GLn (F ) in terms oftypes. If we identify GLn (F ) = AutF (E), then WD(ρx ) is required to correspond toa supercuspidal representation πx under the classical local Langlands correspondence,such that πx contains a simple stratum [A, n0 , 0, α] with E = F [α]. Moreover, it isenough to consider those supercuspidals where α is minimal over F .
Remark 1.5. The assumption that ρ̄ admits a potentially crystalline lift of regularweight, which is potentially diagonalisable, also comes from our use of the patched
module M∞ . It has been conjectured in [20] and it will be shown in the forthcoming
joint work of Toby Gee and ME that such lift always exists. We refer the reader to[2, §1.4] for the notion of potential diagonalisability.
Remark 1.6. The first result of this flavour for arbitrary F and n was proved byHellmann and Schraen in [26], for the family of crystabelline representations with
fixed Hodge–Tate weights (i.e. those representations, which become crystalline after
a restriction to the Galois group of a finite abelian extension of F .) We also handle
this case; in fact our arguments allow us to mix both cases.
1.1. An idea of the proof. The proof of Theorem 1.1 uses in a crucial way thepatched module M∞ constructed in [10]. We will recall some properties of M∞ : it isan R∞ [G]-module, where R∞ is a complete local noetherian R -algebra with residue
field k, obtained as part of the patching construction, and G := GLn (F ). We let
K = GLn (OF ). As part of the construction of M∞ we know that the action of thegroup ring R∞ [K] ⊂ R∞ [G] on M∞ extends to the action of the completed groupring R∞ JKK, which makes M∞ into a finitely generated R∞ JKK-module. This allows
us to equip M∞ with a canonical topology with respect to which the action of R∞ JKK
is continuous. Moreover, M∞ is a projective OJKK-module in the category of linearlycompact OJKK-modules.
The proof then breaks up into two parts. In the first part we work in an abstract
setting, by axiomatising the properties of M∞ mentioned above, and develop a theory
which we call capture. This abstract setting applies both to M∞ and to the zero-thON THE DENSITY OF SUPERCUSPIDAL POINTS

3

completed homology of a symmetric space, when the group is compact at ∞, seesection 5.1. In fact the setting applies to the completed homology of more generalsymmetric spaces associated to a reductive group G defined over Q, whenever ρ̄ occurs
only in one homological degree. Conjecturally this holds whenever ρ̄ is irreducible andl0 = 0, see [19, Conj. 3.3]; the invariant l0 vanishes if for example G is semi-simple
and has discrete series.
The output of the theory says that for certain well chosen families {Vi }i∈I of finite
dimensional representations of K on L-vector spaces the R∞ -annihilator of the familyof R∞ -modules M∞ ⊗OJKK Vi , for i ∈ I, annihilates M∞ . This abstract theory is thenapplied with the family {Vi }i∈I consisting of Bushnell–Kutzko types for supercuspidal representations, tensored with a fixed irreducible algebraic representation W ofResF
Qp GLn /F evaluated at L and restricted to K. Properties of M∞ proved in [10]allow us to convert the automorphic information given by the Bushnell–Kutzko types
and the algebraic representation W into the Galois theoretic information defining theset Σ.
If f, g : N → N are functions, which tend to infinity as n goes to infinity, thenwe write f ∼ g, if log f (n) = log g(n) + O(1) as n tends to infinity. The following
proposition is a key tool in the abstract setting considered above; see section 2.4 forthe definition of the δ-dimension.
Proposition 1.7. Let K be a uniformly powerful pro-p group. Let M be a finitely
generated kJKK-module, let dim M be the δ-dimension of M as kJKK-module, and let
dim K be the dimension of K as a p-adic manifold. For each n ≥ 1 let Kn be theclosed subgroup of K generated by the pn -th powers of the elements of K. Thendim M

dimk MKn ∼ (K : Kn ) dim K ,where MKn denotes the space of Kn -coinvariants.
A weaker version of this Proposition has been proved by Harris in [23], see also[24].
In the second part of the proof carried out in section 6 we show:
Theorem 1.8. The support of M∞ in Spec R∞ is equal to a union of irreducible
components of Spec R∞ .
We follow the approach of Chenevier [11] and Nakamura [30], who proved that the
crystalline points are Zariski dense in the rigid analytic generic fibre (Spf R )rig of
Spf R . Indeed, we prove the following result, which implies the preceding theorem.
Theorem 1.9. The Zariski closure in (Spf R∞ )rig of the set of crystalline points lying
in the support of M∞ is equal to a union of irreducible components of (Spf R∞ )rig .
The argument of Chenevier and Nakamura uses “infinite fern”-style arguments. In
order to modify these arguments so as to be sensitive to the support of M∞ , we use
the locally analytic Jacquet module of [17], which provides a representation-theoretic
approach to the study of finite slope families. Breuil, Hellmann, and Schraen [3] havemade a careful study of the theory of the Jacquet module as it applies to M∞ , andwe use their results.
The argument outlined above proves the analog of Theorem 1.1 with R replaced
by R∞ . The statement for R is deduced from this by a commutative algebra argub O A, where A is a complete local noetherian O-algebra,
ment using that R∞ ∼= R ⊗
which is O-flat.

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MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

1.2. Historical remarks. Since we have been talking about the results of this paper
since the workshop on the p-adic Langlands correspondence in the Fields Institute in2012, some historical comments seem to be in order. At the beginning our focus wasto prove such density theorem for the global deformation ring of an absolutely irreducible representation ρ̄ : Gal(Q/Q) → GL2 (k), which we assume to be modular. The
strategy was to apply representation theoretical techniques in the spirit of Proposition
1.7 to the completed homology of modular curves localised at the ideal of the Hecke
algebra corresponding to ρ̄. So the first part of the proof was essentially the same as
it is now. For the second part of the proof one needed to show that the completed
homology of modular curves localised at the ideal of the Hecke algebra correspondingto ρ̄ is supported on a union of irreducible components of the global deformation ringof ρ̄, which amounted to producing a lower bound on the dimension of the Hecke
algebra, see the discussion of [19, §3.1.1]. The bound was obtained by showing that
the Krull dimension of the Hecke algebra is at least the δ-dimension of the completed
homology, which is equal to 5, minus the δ-dimension of the fibre, which we knew tohave dimension 1: the fibre is determined by results of [18] (see also [36, Lem. 5.8] fora different argument based on the results of [35]), and its δ-dimension is equal to 1;
our original argument is explained in the proof of [36, Lem. 5.14].
Such arguments have since appeared in the paper of Gee–Newton [21] in a moregeneral context; in particular they also prove a kind of converse theorem called ‘miracle
flatness’, which we did not know at the time, by working systematically with Cohen–
Macaulay modules.
We also proved density results for the local deformation rings of ρ̄ : Gal(Qp /Qp ) →GL2 (k) by using results of [35] and the p-adic Langlands correspondence for GL2 (Qp ).
Then a paper of Hellmann–Schraen appeared [26], proving the density result discussed in Remark 1.6. While studying their paper, we realised that the second partof our proof can be obtained using the ‘infinite fern’ arguments by constructing what
is now known as the patched eigenvariety. The construction of this patched eigenvariety appeared in the paper by Breuil–Hellmann–Schraen [3], although both ME andHellmann explained their ideas to VP independently at around the same time.
Finally a paper of Scholze [37] has appeared, in which he has a much simpler proofof one of the consequences of our theory, discussed in [19, 3.3.2]. We adapt and explain
his argument in section 4. And although our abstract theory is more general, the two
key arithmetically interesting examples are covered by this simpler argument due to
Scholze, which avoids counting dimensions, required by our criterion for capture.
In recent preprints [39], [40] Shen-Ning Tung has used our result on the support of
M∞ to obtain a new proof of the Fontaine–Mazur conjecture for GL2 /Q, obtaining
new cases of the conjecture when p = 3 and p = 2. This was one of the motivations
to finally finish this paper.

1.3. The structure of the paper. In section 2 we develop the theory of capture in
an abstract setting. Some of the results of section 2.1 have appeared in [12, §2.4], but
we recall the proofs for the convenience of the reader. The main result is Proposition
2.23, which establishes a numerical criterion for a family {Vi }i∈I of smooth irreducible
representations on finite dimensional L-vector spaces of a compact p-adic analytic
group K, which does not contain any torsion, to capture every projective linearlycompact OJKK-module in the sense of Definition 2.1. The key Proposition which
establishes the link between the growth of coinvariants and dimension is proved in
section 2.4.

ON THE DENSITY OF SUPERCUSPIDAL POINTS

5

In section 3 we exhibit two examples where our general theory applies, by taking K
to be a sufficiently small compact open subgroup of GLn (F ) and, in the supercuspidal
case in section 3.4, the family {Vi }i∈I inductions of simple characters in the sense ofBushnell–Kutzko theory. The main work is then to verify that our numerical criterion
holds, which amounts to counting types and their dimensions. We also handle theprincipal series case in section 3.3.
In section 4 we explain that both of these examples can be dealt with in an easier
way by adopting an argument of Scholze in [37]. This argument still uses Bushnell–
Kutzko theory, but avoids the counting arguments.
In section 5 we apply the theory developed above to the zero-th completed homology of a symmetric space, when the group is compact at infinity.
In section 6 we show that the support of M∞ is equal to a union of irreducible
components of R∞ .
1.4. Acknowledgments. This paper has its germ in conversations between the authors during the Workshop on Galois Representations and Automorphic Forms at the
IAS in March 2011. We would like to thank the organisers for the invitation. VP
would like to thank Michael Harris and Peter Schneider for their comments after histalk in January 2014, Représentations des groupes réductifs p-adiques et applications,
Luminy and Eugen Hellmann for explaining his work during his visit to Essen. The
authors thank Toby Gee for his detailed comments on an earlier version of the paper,Gaëtan Chenevier for answering our questions about his work, Gabriel Dospinescu forpointing out a simplification in the proof of Proposition 2.18, Yongquan Hu for pointing out a gap in the proof of the same proposition in an earlier draft, and Shen-Ning
Tung for his comments on section 6.
2. Generalities
2.1. Capture. Let K be a pro-finite group with an open pro-p subgroup. Let OJKK
be the completed group algebra, and let Modpro
K (O) be the category of compact lineartopological OJKK-modules. Let {Vi }i∈I be a family of continuous representations of
K on finite dimensional L-vector spaces, and let M ∈ Modpro
K (O). We will denote byVi∗ the L-linear dual of Vi .
Definition 2.1. We say that {Vi }i∈I captures M if the smallest quotient M  Q
cont
∗ ∼
∗
such that Homcont
OJKK (Q, Vi ) = HomOJKK (M, Vi ), for all i ∈ I, is equal to M .
The following equivalent characterizations of capture can be easily verified, see [12,§2.4] for details.
Lemma 2.2. Let
N=\

Ker φ,

φ∗
where the intersection is taken over all φ ∈ Homcont
OJKK (M, Vi ), for all i ∈ I. Then{Vi }i∈I captures M if and only if N = 0.
cont
Lemma 2.3. Let M ∈ Modpro
K (O) be O-torsion free, let Π(M ) := HomO (M, L)
be the associated L-Banach space equipped with its supremum norm. Then {Vi }i∈I
captures M if and only if the image of the evaluation map⊕i HomK Vi , Π(M ) ⊗L Vi −→ Π(M )is a dense subspace.
Our next lemmas describe various simple aspects of the notion of capture.

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MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Lemma 2.4. If {Vi }i∈I captures M , then M is O-torsion free.
Proof. Any O-torsion element is in the kernel of any morphism M → Vi∗ for any
i ∈ I; the lemma then follows from Lemma 2.2.
Lemma 2.5. If {Mj }j∈J isQa collection of objects of Modpro
K (O), then the family{Vi }i∈I captures the product j∈J Mj if and only if it captures each of the Mj .
Q
Proof. Noting that j Mj is O-torsion free if and only if each Mj is O-torsion free,
we see from Lemma 2.4 that it is no loss of generality, when proving the lemma, toassume further that each Mj is O-torsion free. We note also thatYY
b j HomcontbΠ( Mj ) := Homcont
Mj , L) = ⊕O (O (Mj , L) =: ⊕j Π(Mj ),jjwhere the completed direct sum is taken with respect to the sup norm on each summand, and where we are using the notationfromthe statementof Lemma 2.3. ThusQQ
the evaluation map ⊕i HomK Vi , Π( j Mj ) ⊗L Vi → Π( j Mj ) may be rewrittenb j HomK Vi , Π(Mj ) → ⊕b j Π(Mj ). This has dense image if and only
as the map ⊕i ⊕
if each of the evaluation maps ⊕i HomK Vi , Π(Mj ) → Π(Mj ) has dense image. The
present lemma thus follows from Lemma 2.3.
Lemma 2.6. Let W be a continuous representation of K on a finite dimensional Lvector space, and let Θ be a K-invariant O-lattice in W . If {Vi }i∈I captures M ⊗O Θ
then {Vi ⊗L W }i∈I captures M .
Proof. For each i ∈ I we have a natural isomorphism(1)
Homcont (M ⊗O Θ, V ∗ ) ∼= Homcont (M, (Vi ⊗L W )∗ ).
OJKK

i

OJKK

Thus if Q is a quotient of M , such thatcont∗∗
Homcont
OJKK (M, (Vi ⊗L W ) ) = HomOJKK (Q, (Vi ⊗L W ) ),for all i ∈ I, then we deduce from (1) thatcont∗∗
Homcont
OJKK (M ⊗O Θ, Vi ) = HomOJKK (Q ⊗O Θ, Vi ),for all i ∈ I. This implies that M ⊗O Θ = Q ⊗O Θ, as {Vi }i∈I captures M ⊗O Θ byassumption. Since Θ is a free O-module we deduce that M = Q.

Lemma 2.7. Assume that {Vi }i∈I captures M and let φ ∈ Endcont
OJKK (M ). If φ kills∗
Homcont
(M,
V
)
for
all
i
∈
I
then
φ
=
0.
OJKK
i
Proof. The assumption on φ implies that
Homcont (Coker φ, V ∗ ) ∼= Homcont (M, V ∗ ),
OJKK

i

OJKK

i

∀i ∈ I.

Since {Vi }i∈I captures M , we deduce that M = Coker φ and thus φ = 0.

Lemma 2.8. The following assertions are equivalent:(i) {Vi }i∈I captures every indecomposable projective in Modpro
K (O);(ii) {Vi }i∈I captures every projective in Modpro(O);K(iii) {Vi }i∈I captures OJKK.
Proof. This follows from Lemma 2.5 and the fact that every projective object inModproK (O) is isomorphic to a product of indecomposable projective objects, which
in turn are precisely the indecomposable direct summand of OJKK. A version of thisargument appears in [12, Lem. 2.11].
ON THE DENSITY OF SUPERCUSPIDAL POINTS

7

Lemma 2.9. Let W be a continuous representation of K on a finite dimensional Lvector space. If {Vi }i∈I captures every projective in Modpro
K (O) then the same holds
for {Vi ⊗L W }i∈I .
Proof. Let Θ be a K-invariant O-lattice in W . If Kn is an open normal subgroup ofK, which acts trivially on Θ/$n , then the map g ⊗ v 7→ g ⊗ g −1 v, where g ∈ K/Kn
and v ∈ Θ/$n induces an isomorphism of K-representations:
O/$n [K/Kn ] ⊗ Θ/$n ∼= O/$n [K/Kn ] ⊗ Θ/$n ,where K acts diagonally on the left hand side and only on the first factor on the righthand side of the isomorphism. By passing to a limit we deduce thatOJKK ⊗O Θ ∼= OJKK⊕das OJKK-module, where d = dimL W . In particular, OJKK ⊗O Θ is projective in
ModproK (O), and hence is captured by {Vi }i∈I by assumption. Lemma 2.6 implies
that {Vi ⊗L W }i∈I captures OJKK and the assertion follows from Lemma 2.8.

2.2. Density. Let K and M and {Vi }i∈I be as above, but we additionally assume
that K is p-adic analytic. In the applications K will be an open subgroup of GLn (F ),where F is a finite extension of Qp . Let R be a complete local noetherian O-algebrawith residue field k. We assume that we are given a faithful, continuous O-linear
action of R on M , which commutes with the action of K. In other words, we assume
that M is a compact RJKK-module, such that the action of R is faithful. For each∗i ∈ I let ai be the R-annihilator of Homcont
OJKK (M, Vi ).
T
Lemma 2.10. If {Vi }i∈I captures M then i∈I ai = 0.
Proof. This follows from Lemma 2.7.

Let m-Spec R[1/p] be the set of maximal ideals of R[1/p]. If x ∈ m-Spec R[1/p]then its residue field κ(x) is a finite extension of L, and we denote by Oκ(x) the ringof integers of κ(x). Then Oκ(x) ⊗R M is a compact OJKK-module and we defineΠx := Homcont
O (Oκ(x) ⊗R M, L).
Then Πx equipped with the supremum norm is a unitary L-Banach space representation of K.
Proposition 2.11. Assume that M is a finitely generated RJKK-module, that {Vi }i∈I√
captures M , and that there exists an integer e such that ( ai )e ⊂ ai for all i ∈ I.
Then the setΣ := {x ∈ m-Spec R[1/p] : HomK (Vi , Πx ) 6= 0 for some i ∈ I}
is dense in Spec R.
Proof. For each i ∈ I we choose a K-invariant bounded O-lattice Θi in Vi . Itd ddfollows from [34, Prop. 2.15] that M (Θi ) := (Homcont
OJKK (M, Θi )) , where (∗) :=cont
HomO (∗, O), is a finitely generated R-module. Moreover, it follows from [34,
Eqn. (11) and Rem. 2.18] that for a fixed i ∈ I the annihilator of M (Θi ) in R isequal to ai and it follows from [34, Prop. 2.22] that the set
Σi := {x ∈ m-Spec R[1/p] : HomK (Vi , Πx ) 6= 0}
is precisely m-Spec(R/ai )[1/p]. Since M (Θi ) is O-torsion free, R/ai is O-torsion free,√
and thus the intersection of all maximal ideals in Σi with R is equal to ai . Thus√
the intersection of all maximal ideals in Σ with R is equal to ∩i∈I ai . Note that R

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MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

is O-torsion free since by assumption it acts faithfully on an O-torsion free module√
M . Our assumption ( ai )e ⊂ ai for all i ∈ I implies that the e-th power of thisideal is contained in ∩i∈I ai , which is zero by Lemma 2.10. Thus Σ is Zariski dense inSpec R.

∗
Remark 2.12. If M is a finitely generated OJKK-module then Homcont
OJKK (M, Vi )is a finite dimensional L-vector space. If for each i ∈ I the action of R[1/p] on∗
Homcont
OJKK (M, Vi ) is semi-simple then we can conclude that R/ai is reduced, and√hence ai = ai , so that the conditions of Proposition 2.11 are satisfied.

Remark 2.13. The set-up of Proposition 2.11 can be relaxed to allow non-noetherianrings as follows. Let A be a projective limit of local artinian R-algebras with residue
field k with transition maps homomorphisms of local R-algebras. Then A is a local Ralgebra with residue field k, which need not be noetherian. Assume that the action of
RJKK on M extends to a continuous action of AJKK for the projective limit topology
on A. For each i ∈ I let Ai be a quotient of A, which acts faithfully on M (Θi ). Since
R is noetherian and M (Θi ) is a finitely generated R-module, we deduce that the rings
Ai are noetherian for all i ∈ I. Thus the proof of Proposition
2.11 shows that if Ai
S
are reduced for all i ∈ I and A acts faithfully on M then i∈I m-Spec Ai [1/p] is densein Spec A.
2.3. Rationality. From the representation theory point of view it is convenient towork over L, the algebraic closure of L. However this causes trouble from the padic functional analysis point of view, see for example the proof of Lemma 2.5. The
purpose of this section is to deal with this problem.
Lemma 2.14. If V is a finite dimensional L-vector space with a continuous Kaction then there is a finite extension L0 of L contained in L and a finite dimensional
K-invariant L0 -subspace W of V , such that the inclusion W ,→ V induces an isomor∼=
phism W ⊗L0 L −→ V .
Proof. We may rephrase the problem as follows: if ρ : K → GLn (L) is a continuous
group homomorphism then its image is contained in GLn (L0 ) for some finite extension
L0 of L contained in L. Such statement is proved in Lemma 2.2.1.1 in [4], with K
equal to the absolute Galois group of Qp . However, only compactness of the group is
used in the proof, so their argument applies in our setting.

Lemma 2.15. Let L0 be a finite extension of L contained in L. Let W be a finite
dimensional L0 -vector space with a continuous K-action. Then we have a natural
isomorphism
(2)

∼
=

cont
Homcont
OJKK (M, W ) ⊗L0 L −→ HomOJKK (M, W ⊗L0 L).

Proof. Since Homcont
OJKK (M, ∗) commutes with finite direct sums, the isomorphism in(2) holds if we replace L by a finite extension L00 of L0 contained in L. We thus obtain
∼
=

cont
cont
00
00
Homcont
OJKK (M, W ) ⊗L0 L → HomOJKK (M, W ⊗L0 L ) ,→ HomOJKK (M, W ⊗L0 L).

By passing to a direct limit we obtain an injectioncont
Homcont
OJKK (M, W ) ⊗L0 L ,→ HomOJKK (M, W ⊗L0 L).

In order to prove surjectivity we have to show that if ϕ : M → W ⊗L0 L is a continuous
map of O-modules then its image is contained in W ⊗L0 L00 for some finite extension
L00 of L0 . This can be proved in the same way as Lemma 2.2.1.1 in [4] using thecompactness of M .
ON THE DENSITY OF SUPERCUSPIDAL POINTS

9Let {Vi }i∈I be a family of continuous representations of K on finite dimensional
L-vector spaces, and let M ∈ Modpro
K (O) as in the previous section. We make theanalogous definition to Definition 2.1.
Definition 2.16. We say that {Vi }i∈I captures M if the smallest quotient M  Q,
cont
∗
∗ ∼
such that Homcont
OJKK (Q, Vi ) = HomOJKK (M, Vi ) for all i ∈ I is equal to M .
Lemma 2.17. Let {Vi }i∈I be a family of continuous representations of K on finite
dimensional L-vector spaces. For each i ∈ I let Li be a finite extension of L containedin L and let Wi be a K-invariant Li -subspace of Vi given by Lemma 2.14, so that
Wi ⊗Li L ∼= Vi is an isomorphism of K-representations. Then {Vi }i∈I captures M
in the sense of Definition 2.16 if and only if {Wi }i∈I captures M in the sense of
Definition 2.1.
Proof. The assertion follows from Lemma 2.15.



2.4. Growth of coinvariants. In this subsection we let K be a uniformly powerfulpro-p group; such groups are discussed in great detail in [13, §4]. We note thatuniformly powerful pro-p groups are p-adic analytic and every compact p-analytic
group contains an open uniformly powerful subgroup, so the assumption is harmless.
We denote by dim K the dimension of K as a p-adic manifold. Finitely generated
modules over kJKK have a good dimension theory, generalizing the Krull dimension,see [1], [41]. The purpose of this section is to relate this dimension to the growth of
coinvariants by certain open subgroups of K.
Let M be a finitely generated kJKK-module. We let
codimkJKK (M ) := min{i : ExtikJKK (M, kJKK) 6= 0},
dimkJKK (M ) := dim K − codimkJKK (M ).
n

We will refer to dimkJKK (M ) as the δ-dimension of M . For n ≥ 0 let Kn := K p
be the closed subgroup generated by the pn -th powers of elements of K. Since K is
uniformly powerful, Kn is open in K and we have(3)

(K : Kn ) = pn(dim K) ,∀n ≥ 0.

If f, g : N → N are functions, which tend to infinity as n goes to infinity, then we
write f ∼ g, if log f (n) = log g(n) + O(1) as n tends to infinity.
For example, if F is a finite extension of Qp , UF is the group of units of its ring ofintegers, and UnF is the n-th congruence subgroup of UF then (UF : UnF ) ∼ qFn , where
qF is the number of elements in the residue field of F .
1
Proposition 2.18. Let d = dimkJKK M . Then there are real numbers a ≥ b ≥ d!
such that
bpdn + O(p(d−1)n ) ≤ dimk MKn ≤ apdn + O(p(d−1)n ).
dim(M )

In particular, dimk MKn ∼ (K : Kn ) dim(K) .
Proof. Let g1 , . . . , gm be a minimal set of topological generators of K, let m be themaximal ideal of kJKK. Then there is an isomorphism of graded rings∼ k[x1 , . . . , xm ],gr• (kJKK) =mwhich sends gi − 1 + m2 to xi , see [1, Thm. 3.4]. Since K is uniformly powerfulnnpn
g1p , . . . , gm
is a minimal set of generators of K p [13, Thm. 3.6 (iii)]. Thus if we letnpn
In be the closed two-sided ideal of kJKK generated by g1p − 1, . . . , gm
− 1, thennp∼ M/In M  M/m M.
(4)
MK =
n

10

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

f be the localizations of gr•m (kJKK) and gr•m (M ) at the ideal gr>0
Let A and M
m (kJKK)
respectively, and let n be the maximal ideal of A. Thenn(5)dimk M/m M =n−1
Xf/nn Mf).
dimk (mi M/mi+1 M ) = `(M

i=0f,
Proposition 5.4 of [1] implies that the Krull dimension of gr•m (M ), and hence of Mnf/n Mf) is a polynomial in n of degree dis equal to d. We know that for large n, `(Mf, see [28,and leading coefficient e/d!, where e is the Hilbert–Samuel multiplicity of M
§14]. In particular,
f/npn Mf) = e pdn + O(p(d−1)n ).
(6)
`(M
d!
The surjection M  MKn induces a surjection of graded modules
gr•m (M )  gr•m (MKn ).
n

Since gip act trivially on MKn , the surjection factors throughnngr•m (M )/(xp1 , . . . , xpm ) gr•m (M )  gr•m (MKn ).

(7)
n

n

Let n[p ] be the ideal of A generated by xp for all x ∈ n. If a1 , . . . , am is a systemnnn
of parameters of A then n[p ] = (ap1 , . . . , apm ) as A is a regular ring. Hence, thennff [pn ] M
localization of gr•m (M )/(xp1 , . . . , xpm ) gr•m (M ) at gr>0
m (kJKK) is equal to M /nand we obtain
(8)

n

n

n

f/n[p ] Mf).
dimk (gr•m (M )/(xp1 , . . . , xpm ) gr•m (M )) = `(M

The main result of [29] says that there is a real number e0 , called the Hilbert–Kunz
multiplicity, such that(9)nf/n[p ] Mf) = e0 pdn + O(p(d−1)n ).
`(M

Actually, as G. Dospinescu pointed out to us, we don’t require the full strength of [29].
nnIndeed, since m is generated by m generators, we have nmp ⊆ n[p ] , and so (just using
the theory of the Hilbert–Samuel multiplicity), we find thatf) = e md pdn + O(p(d−1)n ),f/n[pn ] Mf) ≤ `(M
f/nmpn M
(10)
`(M
d!
which suffices for our purposes.
In any event, we conclude from (4), (5), (7), and (8) thatnnf/np Mf) ≤ dimk MK ≤ `(M
f/n[p ] Mf).
`(M
n
The assertion then follows from (6) and (9) (or its variant (10)).

Remark 2.19. Proposition 2.18 asserts thatdim(M )

(11)

dimk MKn ∼ (K : Kn ) dim(K) .

However, we caution the reader that the analogous statement might not hold for an
arbitrary filtration of K by open normal subgroups. For example, let K = Zp ⊕ Zp
and let M = kJZp K, on which K acts via the projection onto the second factor, letnnnKn = pn Zp ⊕ p2 Zp . Then dimk MKn = p2 and (K : Kn ) = pn+2 , dim M = 1
and dim K = 2, so that (11) does not hold. One should modify the statement of [9,Thm. 1.18] accordingly. The authors of [9] claim that a stronger version of Proposition
2.18 is proved in [1, §5], however we have not been able to verify this.

ON THE DENSITY OF SUPERCUSPIDAL POINTS

11

Remark 2.20. A weaker version of this statement follows from Theorem 1.10 in [23],see also [24].
2.5. Numerical criterion for capture. From now on we assume that K is a p-adic
analytic group, which doesn’t have any torsion. This allows for more general groups
K than considered in the previous section. As already noted every such K contains
an open subgroup K 0 , which is a uniformly powerful pro-p group, see [13, Thm. 8.23].
For a finitely generated OJKK-module M we let dimOJKK (M ) := dimOJK 0 K (M ). Thisdefinition does not depend on the choice of K 0 , see [1, §5.4]. Since the identity inK has a basis of open neighbourhoods consisting of open uniformly powerful pro-p
subgroups, this implies that dimOJKK (M ) = dimOJK 00 K (M ) for any open subgroup K 00
of K.
Our assumption on K implies that K is pro-p and the completed group algebras
kJKK and OJKK are integral domains, see [1, Thm. 4.3].
Lemma 2.21. Let M be a cyclic kJKK-module. If dimkJKK (M ) = dim K then thesurjection kJKK  M is an isomorphism.
Proof. Let Λ = kJKK. Then dimΛ M = dim K if and only if HomΛ (M, Λ) 6= 0. Since
M is cyclic, it is enough to prove that HomΛ (Λ/Λy, Λ) = 0 for all non-zero y in the
maximal ideal of Λ. This follows by applying the functor HomΛ (∗, Λ) to the exact·ysequence Λ → Λ → Λ/Λy → 0, and using the fact that our assumptions on K imply
that Λ is an integral domain, so that multiplication with y induces an injective map
from HomΛ (Λ, Λ) to itself.
Let {Kn }n≥1 be a basis of open neighbourhoods of 1 in K consisting of normal
subgroups. We assume that there is n0 ≥ 0, such that Kn0 is a uniform pro-p group,mand for each m ≥ 1 there is an n(m), such that Knp0 = Kn(m) . Let C(K, L) be thespace of continuous L-valued functions on K. Since K is compact the supremum norm
defines a Banach space structure on C(K, L); it is an admissible unitary L-Banach
space representation of K for the K-action given by right translations.
Lemma 2.22. Let Π be a closed K-invariant subspace of C(K, L). If dimL ΠKn ∼(K : Kn ) then Π = C(K, L).
Proof. Let Θ be the unit ball in Π with respect to the supremum norm on C(K, L).
The map K → OJKK, g 7→ g induces an isomorphism C(K, L) ∼= Homcont
O (OJKK, L)
d[38, Cor.2.2]. Thus, if we let Θ := HomO (Θ, O) then Schikhof’s duality induces asurjection of OJKK-modules, OJKK  Θd [38, Thm. 3.5]. We claim that the surjection
is an isomorphism. By dualizing back, the claim implies the Lemma. Since Θd is Otorsion free, it is enough to show that kJKK  Θd /$ is an isomorphism. Lemma
2.21 implies that it is enough to show that dimkJKK (Θd /$) = dim K. Since thedimension does not change if we pass to an open subgroup, it is enough to showthat dimkJKn0 K (Θd /$) = dim Kn0 . Proposition 2.18 implies that it is enough toshow that dimk ((Θd /$)Kn ) ∼ (Kn0 : Kn ). Since Θd /$ ∼= (Θ/$)∨ , we get thatKn ∨(Θd /$)Kn ∼((Θ/$)).
Thus=
dimk ((Θd /$)Kn ) = dimk ((Θ/$)Kn ) ≥ rankO ΘKn = dimL ΠKn ∼ (K : Kn ).
On the other handdimk ((Θd /$)Kn ) ≤ dimk kJKKKn = (K : Kn ).
Since (K : Kn ) ∼ (Kn0 : Kn ) we obtain the claim.

12

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Let {Vi }i∈I be a family of pairwise non-isomorphic, smooth irreducible L-representations of K. LetXd(n) :=(dimL ViKn )2 .
i∈I

We will show in the course of the proof of the next proposition that only finitely many
terms in the sum are non-zero, so that d(n) is finite.
Proposition 2.23. If d(n) ∼ (K : Kn ) then {Vi }i∈I captures OJKK and hence every
projective in Modpro
K (O).
Proof. Let OJKK  M be the smallest quotient such that
(12)

cont
∗
∗
Homcont
OJKK (OJKK, Vi ) = HomOJKK (M, Vi ),∀i ≥ 0.

Then M is O-torsion free, and Π := Homcont
O (M, L) is a closed K-invariant subspaceof C(K, L). Lemma 2.22 and Schikhof duality imply that it is enough to show that
dimL ΠKn ∼ (K : Kn ). Let I(n) := {i ∈ I : ViKn 6= 0}. Since Vi are absolutely
irreducible and Kn are normal in K, ViKn = Vi for all i ∈ I(n). In particular, such
Vi are representations of a finite group K/Kn . Thus, I(n) is a finite set and d(n) is
finite. Thus for a fixed n there is a finite extension Ln of L, such that all Vi , i ∈ I(n)are defined over Ln . Alternatively, we could use Lemma 2.14.
Since (ΠLn )Kn ∼= (ΠKn )Ln , after extending the scalars to Ln we may assume that
Ln = L. Schikhof duality [38, Thm.3.5], together with (12), implies that(13)

HomK (Vi , Π) ∼= Vi∗ ,= HomK (Vi , C(K, L)) ∼∀i ∈ I(n)(the second isomorphism holding by Frobenius reciprocity). This implies that⊕i∈I(n) Vi ⊗ Vi∗
occurs as a subspace of ΠKn , and so dimL ΠKn ≥ d(n). Since ΠKn is a subspace ofC(K, L)Kn , dimL ΠKn ≤ (K : Kn ). Since d(n) ∼ (K : Kn ) by assumption, we deducethat dimL ΠKn ∼ (K : Kn ). It follows from Lemma 2.22 and Lemma 2.3 that {Vi }i∈I
captures OJKK. The last assertion follows from Lemma 2.8.

3. Examples of capture
3.1. Locally algebraic representations. Let G be an affine group scheme of finite
type over Zp such that GL is a split connected reductive group over L. Let Alg(G)be the set isomorphism classes of irreducible rational representations of GL , which weview as representations of G(Zp ) via the inclusion G(Zp ) ⊂ G(L).
Proposition 3.1. Alg(G) captures every projective object in Modpro
G(Zp ) (O).
Proof. [12, Prop. 2.12].



Corollary 3.2. Let W be any finite dimensional L-representation of G(Zp ), suchthat the action factors through a finite quotient. Then {V ⊗L W }V ∈Alg(G) captures
every projective object in Modpro
G(Zp ) (O).
Proof. This follows from Proposition 3.1 and Lemma 2.9.



ON THE DENSITY OF SUPERCUSPIDAL POINTS

13

3.2. Subgroups and filtrations. Let F be a finite extension of Qp with the ring ofintegers OF , the maximal ideal pF and a uniformizer $F . Let V be an N -dimensional
F -vector space. Let A = EndF (V ) and let G = AutF (V ). A lattice chain L is a setof OF -lattices in V , {Li : i ∈ Z}, such that Li+1 ⊂ Li for all i ∈ Z and there is an
integer e = e(L), called the period of L, such that $F Li = Li+e , for all i ∈ Z. To
a lattice chain we associate a hereditary order, A := {a ∈ A : aLi ⊂ Li , ∀i ∈ Z}. Ifn ∈ Z then let Pn := {a ∈ A : aLi ⊂ Li+n , ∀i ∈ Z}. Then P := P1 is the Jacobsonradical of A, and Pm Pn = Pn+m , for all n, m ∈ Z. Let
U(A) := U0 (A) := A× ,Un (A) := 1 + Pn ,∀n ≥ 1.

nThen U(A) is a compact open subgroup of G and {U (A)}n≥0 is a basis of open
neighborhoods of 1 by normal subgroups of U(A).
Example 3.3. If L = {$Fi L0 : ∀i ∈ Z}, then U(A) = AutOF (L0 ) and Un (A) is thekernel of the natural map AutOF (L0 )  AutOF (L0 /$Fn L0 ).
Lemma 3.4. Let e(L|Qp ) := e(L)e(F |Qp ) and if p > 2 then let c = e(L|Qp ), if p = 2
then let c = 2e(L|Qp ). Then Un (A)p = Un+e(L|Qp ) (A) for all n ≥ c.
Proof. If x ∈ Pn then (1 + x)p ∈ 1 + pPn = Un+e(L|Qp ) (A). This yields one inclusion.
For the other inclusion, we observe that the series defining exp converges on pA = Pc ,if p > 2, and on 4A = Pc , if p = 2, to give a homeomorhism exp : Pc → Uc (A), with
the inverse map given by the series for log. If y ∈ Pn+e(L|Qp ) and n ≥ c, thenlog(1 + y)/p ∈ Pn , and hence exp(log(1 + y)/p) ∈ Un (A) is a p-th root of 1 + y. Proposition 3.5. Let c be the constant defined in Lemma 3.4. If n ≥ c then Un (A)is a uniform pro-p group.
Proof. If 2n ≥ m then the map g 7→ g − 1 induces an isomorphism of groups
(14)

∼
=

Un (A)/Um (A) → Pn /Pm ,where the group operation on the right-hand-side is addition, and so both groups areabelian. Lemma 3.4 implies that the derived subgroup [Un (A), Un (A)] is containedin Un (A)p , if p > 2 and in Un (A)4 if p = 2. Hence, Un (A) is powerful. It follows
from (14) and Lemma 3.4 that Un (A)/Un (A)p is finite. Since Un (A) is a pro-p group,
this implies that Un (A) is finitely generated. Since exp(ma) = exp(a)m , for m ∈ Z,
and Pn is a torsion free Zp -module, the homeomorphism exp : Pn ∼= Un (A) implies
that Un (A) is torsion free. Theorem 4.5 of [13] implies that Un (A) is uniformly
powerful.

3.3. Principal series types for GL(N, F ). We keep the notation of the previous
subsection, but fix a basis of V and identify G = GL(N, F ). Let UF := U0F := OF×and UiF = 1 + piF for i ≥ 1. Let T the subgroup of diagonal matrices in G and let Ti
be the subgroup of T with all diagonal entries in UiF .
For n ≥ 1 let I n be the inverse image in GL(N, OF ) of unipotent upper-triangularmatrices in GL(N, OF /pnF ) under the reduction map. Then I n is a compact p-adic
analytic group.
Lemma 3.6. There is n0 such that I n is torsion free for all n ≥ n0 .
(N −i)m

Proof. Let t be a diagonal matrix with diagonal entries tii = $F
, 1 ≤ i ≤ N,with n ≥ N m. A matrix in g ∈ tI n t−1 is congruent to 1 modulo $Fm . It follows fromthe last part of the proof of Proposition 3.5 that if m is large enough then g cannot
be a torsion element unless it is equal to the identity.
14

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

We choose n0 , such that I n0 is torsion free and n0 > N . We say that a smooth×
i+1
character χ : Umand nontrivial on UiF .
F → L has conductor i, if χ is trivial on UF
For n ≥ n0 let X(n) be the set of ordered N -tuples (χ1 , . . . , χN ) of smooth charactersof UnF0 , such that χi has conductor n − i + 1. Then for n ≥ n0 + N we have
(15)

|X(n)| =

N
Y

n−i+1
nN
(|UnF0 /UF| − |UnF0 /Un−i
F |) ∼ qF .

i=1

Let

 n
pF
pnF

 .
Jn :=  ..

pn

OF
pn−1
F
..
.

...
OF

...
..
.

OF
OF
..
.

pn−1
...
OF
F
n−N +1
pn−1
.
.
.
p
F
F
If x, y ∈ Jn then x + y and xy lie in Jn . Hence,
F
pnF





.



J n := 1 + Jn
is an open subgroup of G normalized by T 0 . We may interpret an element of X(n) asa character of T n0 , which maps a diagonal matrix (d1 , . . . , dN ) to χ1 (d1 ) . . . χN (dN ).
Given such χ ∈ X(n) we may extend it to a character of T n0 J n+1 by letting χ(tk) =χ(t) for all t ∈ T n0 and k ∈ J n+1 . Letn0Vχ := IndIT n0 J n+1 χ.
Proposition 3.7. If π is a smooth irreducible L-representation of G thenHomI n0 (Vχ , π) 6= 0 ⇐⇒ π ∼= IndG ψ
Bwith ψ|T n0 = χ1 ⊗ . . . ⊗ χN .
×

Proof. Let χ̃ : T 0 → L be any character of T 0 , such that χ̃|T n0 = χ and letWχ̃ := IndK
T 0 J n+1 χ̃, where K = GLN (OF ). It follows from [8], see [26, Prop. 3.10],
that Wχ̃ is an irreducible representation of K and is a type for the inertial Bernstein
component [T 0 , χ̃]. This implies that HomK (Wχ̃ , π) 6= 0 if and only if π ∼= IndGBψ
with ψ|T 0 = χ̃. Since
M
KK
T 0 J n+1
∼
∼
∼
Wχ̃ ,
(16)
IndK
I n0 Vχ = IndT n0 J n+1 χ = IndT 0 J n+1 (IndT n0 J n+1 χ) =χ̃×where the sum is taken over all χ̃ : T 0 → L , such that χ̃|T n0 = χ, the assertion
follows from Frobenius reciprocity.

Definition 3.8. X := ∪n≥1 X(n).
Lemma 3.9. The representation Vχ is irreducible. If χ1 , χ2 ∈ X are distinct thenVχ1 6∼= Vχ2 .
Proof. Let χ1 ∈ X(n1 ) and χ2 ∈ X(n2 ) and let χ̃1 and χ̃2 be characters of T 0 extending χ1 , and χ2 , respectively. We use the notation introduced in the proof of
Proposition 3.7. If HomK (Wχ̃1 , Wχ̃2 ) 6= 0 then Wχ̃1 ∼= Wχ̃2 . Since these representations are types for the components [T 0 , χ̃1 ] and [T 0 , χ̃2 ] respectively, we conclude
that χ̃1 = χ̃2 . It follows from (16) and Frobenius reciprocity, that if χ1 6= χ2 thenHomI n0 (Vχ1 , Vχ2 ) = 0, and if χ1 = χ2 = χ then
K
K
K
dimL HomK (IndK
T n0 J n+1 χ, IndT n0 J n+1 χ) = dimL HomK (IndI n0 Vχ , IndI n0 Vχ )

ON THE DENSITY OF SUPERCUSPIDAL POINTS

15

×is equal to the number of characters χ̃ : T 0 → L , such that χ̃|T n0 = χ, which isequal to|T 0 /T n0 | = |T n0 J n+1 \T 0 J n+1 /T n0 J n+1 |.
Since T 0 J n+1 is contained in the K-intertwining of χ, we deduce that the K-intertwiningof χ is equal to T 0 J n+1 . Since I n0 ∩ T 0 J n+1 = T n0 J n+1 , we deduce that the I n0 intertwining of χ is equal to T n0 J n+1 and so Vχ is irreducible.

For n ≥ 1 let Kn be the n-th congruence subgroup in GL(N, OF ).
Lemma 3.10.
X(dim VχKn+1 )2 ∼ (In0 : Kn+1 ).

χ∈XProof. If χ ∈ X(n) then Kn+1 acts trivially on Vχ and Kn does not act trivially on
Vχ , and, since Vχ is irreducible, VχKn = 0. Hence, it is enough to check thatX
(17)(dim Vχ )2 ∼ (I n0 : Kn+1 ).
χ∈X(n)Let U−

be the subgroup of unipotent lower triangular matrices in G. Thendim Vχ = (I n0 : T n0 J n+1 ) = (I n0 ∩ U − : J n+1 ∩ U − )

(18)

=

N
−1
Y

(N −i)(n−n0 −i)qF

nN (N −1)/2

∼ qF

i=1
2

Since |X(n)| ∼ qFnN by (15) the right-hand-side of (17) grows as qFnN . Since(19)2(n+1−n0 )N (N −1)/2 (n+1−n0 )N (n+1)N (N −1)/2
qF
qF∼ qFnN ,(I n0 : Kn+1 ) = qF

we deduce that (17) holds.



Proposition 3.11. The set {Vχ }χ∈X captures every projective object in Modpro
I n0 (O).
Proof. The assertion follows from Proposition 2.23 applied with K = I n0 and Kn asin Example 3.3 with L0 = OF⊕N , using Lemma 3.10.

3.4. Supercuspidal types for GL(N, F ). Let F be a finite extension of Qp and let
E be a field extension of F of degree N . Let f = f (E|F ) and e = e(E|F ) denote theinertial degree and the ramification index of E over F respectively.
Definition 3.12 ([7, (1.4.14)]). An element α ∈ E is minimal over F if e is prime to−v (α)vE (α) and $F E αe + pE generates the field extension kE /kF .
Remark 3.13. If α is minimal then E = F [α], since the fields have the same ramification indices and inertial degrees over F .
Remark 3.14. If α is minimal over F then α + a is also minimal over F for all a ∈ Ewith vE (a) > vE (α).
In general, not every extension E will contain an element which is minimal over F ,as the following example shows.
Example 3.15. Let f be an even integer, e = pf /2 + 1, F = Qp and let E be the× ecompositum of Qp (p1/e ) and the unramified extension of Qp of degree f . If x ∈ (kE)
pf /2
then x= x, and hence x cannot generate kE over kF . Any α ∈ E will be of the×form pn/e ξ for some ξ ∈ OE
and the image of $−n αe in kE will be equal to the imageeof ξ , and hence cannot generate kE over kF .

16

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

However, there are plenty extensions that do. For example, let F 0 be a finite
unramified extension of F , and let f (x) = xe + . . . + $F ξ be an Eisenstein polynomial
in F 0 [x], such that ξ + pF 0 generates kF 0 /kF then E = F 0 [x]/(f (x)) is a field and x +(f (x)) is minimal over F . In particular, all unramified or totally ramified extensions
will contain a minimal element.
Lemma 3.16. If α ∈ E is minimal over F then αm is also minimal over F for allm prime to e(qE − 1).
Proof. If x ∈ kE then x generates kE over kF if and only if it is not contained in any
×proper subfield, which is equivalent to the order of x in the group kE
not dividing
qFd − 1 for all divisors d of f with d < f . If m is prime to qFf − 1 then x and xm havethe same order, which implies the assertion.

From now on we assume that E contains an element that is minimal over F . It
follows from Lemma 3.16 that for any n ≥ 0 there exists α ∈ E minimal over F , suchthat vE (α) < −n.
Let A = EndF (E) and G = AutF (E), so that G ∼= GL(N, F ), E ⊂ A and E × ⊂ G.
Let A be the hereditary OF -order in A associated to the lattice chain {piE : i ∈ Z}
and let P be its Jacobson radical. Then Pn = {a ∈ A : apiE ⊂ pi+n
E , ∀i ∈ Z} for alln ∈ Z. We let U(A) = U0 (A) = A× and Un (A) = 1 + Pn for n ≥ 1. Then for alln ≥ 0, Un (A) is a normal subgroup of U(A) normalized by E × .
Let α ∈ E be minimal with respect to E/F with vE (α) = −n < 0. Then [A, n, 0, α]
is a simple stratum in the sense of [7, (1.5.5)]. For m ≥ 0, we let[n/2]+1
H m (α) := Um(A),
EU[(n+1)/2]
J m (α) := Um(A).
EUSince α is minimal and [E : F ] = N , these groups coincide with those defined in [7,(3.1.14)]. If it is clear that vE (α) is fixed and is equal to −n then we will write H m
and J m .
×
We fix an additive character ψF : F → L , which is trivial on pF and non-trivial on×
OF . If b ∈ A then ψb : A → L is the function a 7→ ψF (trA (b(a−1))). The restrictionof ψα to U[n/2]+1 (A) defines a character, which is trivial on Un+1 (A) and non-trivial
×on Un (A). Let C(m, α) be the set of characters θ : H m+1 (α) → L , such that the
restriction of θ to U[n/2]+1 (A) is equal to ψα . Since α is minimal and [E : F ] = N ,C(m, α) is the set of simple characters defined in [7, (3.2.1)]. If θ ∈ C(m, α) thenthe G-intertwining of θ is equal to E × J m+1 (α). It follows from [5, (8.3.3)], using [7,(3.4.1)], that there is a unique irreducible representation η(θ) of J m+1 (α), such that
HomH m+1 (α) (θ, η(θ)) 6= 0. Thus,(20)

J m+1 (α)

IndH m+1 (α) θ ∼= η(θ)⊕a ,where a is an integer such that a dim η(θ) = (J m+1 (α) : H m+1 (α)). It is shown in [5,(8.3.3)] that(dim η(θ))2 = (J m+1 (α) : H m+1 (α)) = (U[(n+1)/2] (A) : U[n/2]+1 (A)).
p
In particular, a = dim η(θ) and is equal to 1 if n is odd, and is equal to (A : P) if
n is even.
(21)

Um+1 (A)Definition 3.17. V (α, θ) := IndJ m+1 (α) η(θ).
Lemma 3.18. V (α, θ) is an irreducible representation of Um+1 (A).

ON THE DENSITY OF SUPERCUSPIDAL POINTS

17

Proof. It follows from (20) that the G-intertwining of η(θ) is equal to the G-intertwiningof θ. Hence, the Um+1 (A)-intertwining of η(θ) is equal to Um+1 (A) ∩ E × J m+1 (α) =J m+1 (α). Hence, V (α, θ) is irreducible.
Proposition 3.19. If π is a smooth irreducible L-representation of G then
HomUm+1 (A) (V (α, θ), π) 6= 0 ⇐⇒ π ∼= c-IndG
E × J 0 (α) Λ,(22)where Λ is an irreducible representation, such that Λ|H m+1 (α) is isomorphic to a direct
sum of copies of θ. In particular, π is supercuspidal.
Moreover, if E is a tame extension of F and ρ is the N -dimensional representationof WF corresponding to π via the local Langlands correspondence, then (22) is equiv×F
is a character, such that if we identify
alent to ρ ∼= IndW
WE ψ, where ψ : WE → L
WEab ∼= E × then ψ|Um+1 = θ.
E

Proof. Since α is minimal over F and [E : F ] = N , [7, (3.2.5), (3.3.18)] imply that1
∼ L θ̃, where the sum is taken over all θ̃ ∈ C(0, α), such that θ̃|H m+1 = θ.
IndH
H m+1 θ =1
It follows from (20) that IndJH 1 θ̃ ∼= η(θ̃)⊕a . Theorem (5.2.2) of [7] implies thatL
00
IndJJ 1 η(θ̃) ∼=σ κ ⊗ σ, where κ is an irreducible representation of J , such that∼
κ|J 1 = η(θ̃), the so called β-extension, and the sum is taken over all characters σ of
UE /U1E , which we inflate to J 0 via the isomorphism J 0 /J 1 ∼= UE /U1E . Thus
M
0
(23)
IndJH m+1 θ ∼λ)⊕a=(where the sum is taken over all simple types λ, [7, (5.5.10)], associated to a simple
stratum [A, n, 0, α], such that λ|H m+1 is a direct sum of copies of θ. If HomJ 0 (λ, π) 6= 0
then π is supercuspidal and there is a unique irreducible representation Λ of E × J 0 ,such that Λ|J 0 ∼= λ and π ∼= c-IndG
E × J 0 Λ, [7, (6.2.2)]. The first part of the Proposition
now follows from (20) and Frobenius reciprocity. The second part follows from [6]. 
Lemma 3.20. Un+1 (A) acts trivially on V (α, θ). The space of Un (A)-invariants in
V (α, θ) is zero.
Proof. This follows from the fact that the restriction of V (α, θ) to U[n/2]+1 (A) is
isomorphic to a direct sum of copies of ψα .
Lemma 3.21.
dim V (α, θ) =(Um+1 (A) : U[n/2]+1 (A))[n/2]+1

(Um+1
: UE
E

−n/2

) dim η(θ)∼ (Um+1 (A) : Un+1 (A))1/2 qE.

Proof. It follows from the definition of V (α, θ) as induced representation that its
dimension is equal to (Um+1 (A) : J m+1 (α)) dim η(θ). The Lemma follows from thisusing (21).
×

We fix an additive character ψE : E → L , which is trivial on pE and nontrivial
on OE . There exists a unique map s : A → E such that(24)

ψA (ab) = ψE (s(a)b),∀a ∈ A,∀b ∈ E.

Moreover, s is an E × E-bimodule homomorphism, such that s(Pm ) = pm
E for allm ∈ Z, [7, (1.3.4)]. Since E is maximal in A and E is separable over F , trA/F (E) =trE/F (E) = F . It follows from (24) that s(1) 6= 0. We let δ := vE (s(1)).

18

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Lemma 3.22. Let n, r be natural numbers with n − δ > r. Let α1 , α2 ∈ E be−n+δ+1minimal over F with vE (α1 ) = vE (α2 ) = −n. If α1 − α2 ∈ pE
and α1 + P−r =
−r−δ
−1
−r
xα2 x + P for some x ∈ U(A) then α1 − α2 ∈ pE .
Proof. Since p−n+δ+1⊂ P−n+δ+1 we haveE
α2 + P−n+δ+1 = α1 + P−n+δ+1 = xα2 x−1 + P−n+δ+1 .
Since α2 is minimal it follows from [27, Thm. 2.4] that x = uy with u ∈ E × andy ∈ Uδ+1 (A). Then xα2 x−1 = yα2 y −1 and so α1 + P−r = yα2 y −1 + P−r . Thus
α1 y − yα2 ∈ P−r . By applying s, we get s(y)(α1 − α2 ) ∈ s(P−r ) = p−r
E . Sincey ∈ Uδ+1 (A), s(y) ∈ s(1) + s(Pδ+1 ) = s(1) + pδ+1
,
hence
v
(s(y))=
δandα1 − α2 ∈
E
E
p−r−δ
.

E
Lemma 3.23. For i = 1, 2 let αi ∈ E be minimal over F with vE (αi ) = −n and−[n/2]−δ
n > 2δ, and let θi ∈ C(m, αi ). If α1 − α2 ∈ p−n+δ+1
and α1 − α2 6∈ pE
then θ1Eand θ2 do not intertwine in G.
Proof. If g ∈ G intertwines θ1 and θ2 then there exists x ∈ U(A) normalizing H m+1
such that θ1x = θ2 , [7, (3.5.11)]. By considering the restrictions of θ1 and θ2 toU[n/2]+1 (A) we obtain that ψα1 = ψαx 2 = ψxα2 x−1 as characters of U[n/2]+1 (A). Hence,ψA ((α1 − xα2 x−1 )a) = 1 for all a ∈ P[n/2]+1 , which implies α1 − xα2 x−1 ∈ P−[n/2] ,−[n/2]−δ
[7, p.22]. Lemma 3.22 implies that α1 −α2 ∈ pE
, leading to a contradiction. 
Lemma 3.24. Let α ∈ E be minimal with vE (α) = −n, where n ≥ 2m ≥ 0. Thenn/2[n/2]+1) ∼ qE . Moreover, distinct θ1 , θ2 ∈ C(m, α) do not|C(m, α)| = (Um+1: UE
E
intertwine.
Proof. The set C(m, α) is non-empty, [7, (3.3.18)]. If θ ∈ C(m, α) then all the other[n/2]+1
characters in C(m, α) are of the form θµ, where µ is any character of Um+1
/UEE
inflated to H m+1 . This implies the first assertion. The second assertion follows fromthe proof of [33, Prop. 7.3]. Namely, if g intertwines θ and θµ then it will intertwine
their restrictions to U[n/2]+1 (A), which are both equal to ψα . It follows from [7,(3.3.2)] that g ∈ E × U[(n+1)/2] (A). Any such g will normalize H m+1 , and so theintertwining maybe rewritten as(25)θµ(h) = θ(ghg −1 ),∀h ∈ H m+1 .

By applying [7, (3.3.2)] again, we get that g will also intertwine θ with itself. It then
follows from (25) that µ(h) = 1 for all h ∈ H m+1 .
We fix m ≥ 0. For n ≥ 2m, let I(n) be the set of isomorphism classes of therepresentations of the form V (α, θ), where θ ∈ C(m, α) with α ∈ E minimal over F
and vE (α) = −n.
nProposition 3.25. |I(n)| ∼ qE.

Proof. Let us note that it follows from Lemma 3.16 that there exists α ∈ E minimal
over F with n := −vE (α) arbitrarily large. It is enough to show that g(n) ≤ |I(n)| ≤nf (n), with f (n) ∼ g(n) ∼ qE
. For the upper bound we observe that[−[n/2]n|I(n)| ≤ | C(m, α)| ≤ |C(m, α)||p−n| ∼ qE,E /pE
αwhere the union is taken over all minimal α ∈ E with vE (α) = −n. In the above−[n/2]
estimates we use Lemma 3.24 and the fact that if α1 − α2 ∈ pE
then ψα1 = ψα2

ON THE DENSITY OF SUPERCUSPIDAL POINTS

19

on U[n/2]+1 (A) and hence C(m, α1 ) = C(m, α2 ). For the lower bound we choose a−[n/2]−δ
set of representatives {ai }i of p−n+δ+1/pE
and let αi := α + ai . Then each αiE−[n/2]−δ
−n+δ+1is minimal over F , αi − αj ∈ pE
and αi − αj =if i 6= j.
S ai − aj 6∈ pE
Lemmas 3.24 and 3.23 imply that distinct θ1 , θ2 ∈ i C(m, αi ) do not intertwine.
Um+1 (A)Um+1 (A)This implies that HomUm+1 (A) (IndH m+1 θ1 , IndH m+1 θ2 ) = 0, and (20) implies
that V (α, θ1 ) ∼6 V (α0 , θ2 ). Hence=[−[n/2]−δn|I(n)| ≥ | C(m, αi )| = |C(m, α)||p−n+δ+1/pE| ∼ qE,E
iwhere we use Lemma 3.24 again.

We let I = ∪n≥2m I(n) and for each i ∈ I we fix a representative Vi of the isomorphism class i.
Proposition 3.26. Let m be large enough, so that Um+1 (A) is torsion free. Thenthe set {Vi }i∈I captures every projective object in Modpro
Um+1 (A) (O).
Proof. We will deduce the assertion from Proposition 2.23 with K = Um+1 (A) andKn = Un+1 (A). Proposition 3.5 and Lemma 3.4 imply that {Kn }n≥m satisfy theconditions of Proposition 2.23. Lemma 3.20 implies that i ∈ I(n) if and only if
Un+1 (A) acts trivially on Vi and Un (A) acts nontrivially on Vi . Thus it is enough toshow thatX(dim Vi )2 ∼ (Um+1 (A) : Un+1 (A)).
i∈I(n)

This follows from Lemma 3.21 and Proposition 3.25.



4. Capture a la Scholze
The following section is motivated by the proof of Corollary 7.3 in [37]. Let K be a
pro-finite group and let {Kn }n≥1 be a basis of open neighbourhood of 1 consisting of∗
normal subgroups of K. We assume that Kn is pro-p for all n ≥ 1. Let ψn : Kn → L
be a smooth character for all n ≥ 1.
Let R be a complete noetherian local O-algebra with residue field k. Let M be a
finitely generated RJKK-module.
Proposition 4.1. Assume M is projective as OJKK-module. If φ ∈ Endcont
RJKK (M )cont∗kills HomOJKn K (M, ψn ) for all n ≥ 1 then φ kills M .
∗Proof. Since ψn is smooth, ψn (Kn ) is a finite subgroup of L . Since Kn is pro-p, theorder of ψn (Kn ) is a power of p. Hence, we may assume that ψn takes values in On× ,where On is the ring of integers of a finite extension Ln of L obtained by adding somep-power roots of unity to L. Let On (ψn∗ ) be a free On -module of rank 1, on which Kn
acts by ψn∗ . Since On (ψn∗ ) is an OJKn K-submodule of ψn∗ we have an injectioncont∗∗
Homcont
OJKn K (M, On (ψn )) ,→ HomOJKn K (M, ψn ).

Moreover, this is an injection of Endcont
OJKK (M )-modules. Thus φ annihilates∗
Homcont
OJKn K (M, On (ψn ))for all n ≥ 1. Let $n be a uniformizer in On . Then the residue field is equal to k
and all the p-power roots of unity in On are mapped to 1 in k. We have an exact
sequence of Kn -representations
(26)

$

n
0 → On (ψn∗ ) −→On (ψn∗ ) → k → 0

20

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

with the trivial action of Kn on the quotient. Since M is projective as an OJKKmodule, it is also projective as an OJKn K-module and by applying Homcont
OJKn K (M, ∗)cont∗
to (26) we see that Homcont
(M,
k)
is
a
quotient
of
Hom
(M,
O
(ψ
n
n )) and thus
OJKn KOJKn Kis killed by φ. This module is isomorphic to the continuous k-linear dual of Kn coinvariants of M/$M , which we denote by (M/$M )Kn . Using Pontryagin duality
we conclude that φ acts trivially on (M/$M )Kn .
Let m be the maximal ideal of R and let Im,n = mm RJKK + I(Kn )RJKK, where
I(Kn ) is the augmentation ideal of OJKn K. Since M is finitely generated as RJKKmodule, its topology coincides with the topology defined by the ideals Im,n , see [32,(5.2.17)]. Thus
∼ lim(M/$M )/In,m (M/$M ).
M/$M =←−
n,m
Since (M/$M )/I(Kn )(M/$M ) = (M/$M )Kn , by the previous part we deducethat φ kills all the terms in this projective limit. Thus φ kills M/$M and so φ(M )is contained in $M .
If φ is non-zero then there is a largest integer n such that φ(M ) is contained in$n M and not contained in $n+1 M . We may replace φ by $1n φ in the argument
above to obtain a contradiction.

Corollary 4.2. Let M be as in Proposition 4.1 and assume that R acts faithfully on
M . Let {Vi }i∈I be the family of irreducible subquotients of IndK
Kn ψn for all n ≥ 1.
cont∗
For each i ∈ I let ai be the R-annihilator of HomOJKK (M, Vi ). Then ∩i∈I ai = 0.
Proof. This follows from Proposition 4.1 by observing that IndK
Kn ψn is a finite direct
sum of some Vi ’s with some multiplicity and then using Frobenius reciprocity to obtainan isomorphism of R-modules
K
cont
∗ ∼
∗
Homcont
OJKn K (M, ψn ) = HomOJKK (M, (IndKn ψn ) ).


Remark 4.3. Although the theory outlined in section 2 is more general, all thesmooth examples in section 3 are covered by Corollary 4.2, which avoids all the
intricate counting arguments. For example in the supercuspidal case, considered in
section 3.4, it is enough to apply Corollary 4.2 with K = U(A) and consider families(U[n/2]+1 (A), ψα ), for all n ≥ 1, such that [A, n, 0, α] is a simple stratum, E = F [α]and α is minimal over F .
5. Applications
5.1. Global applications. Let G be a reductive group over Q with the property
that the maximal Q-split torus in the centre of G is a maximal R-split torus in G.
This means that G(R) is compact modulo its centre, and that if the level Kf is small◦enough, then G(Q) acts with trivial stabilizers on G(A)/A◦∞ K∞
Kf . We let◦
Y (Kf ) := G(Q)\G(A)/A◦∞ K∞
Kf .

Here we use the usual notation that A∞ is the group of real points of the maximal Qsplit torus in the centre of G, and K∞ denotes a choice of maximal compact subgroupof G(R); and, of course, the superscript ◦ denotes the connected component of theidentity.

ON THE DENSITY OF SUPERCUSPIDAL POINTS

21

We are in the situation of [14, (3.2)]. In particular, we may consider the completed
e 0 (K p ) with O-coefficients, for some choice of tame level K p ; recall that
cohomology H
this is defined as
e 0 (K p ) := lim lim H 0 (Y (Kp K p ), O/$s ),H←s− −→
Kp

where Kp runs over all compact open subgroups of G(Qp ). We may also consider thecompleted homology, with O-coefficients, which we denote by M ; it is defined as
M := lim H0 (Y (Kp K p ), O).
←−
Kp

We then have that M is finitely generated and (if K p is small enough) free over OJKK,
and
(27)

e 0 (K p ) = Homcont
H
O (M, O),

where K is a compact open subgroup of G(Qp ).
For each Kp , let T(Kp ) denote the commutative algebra of endomorphisms of
H0 (Y (Kp K p ), O) generated by the spherical Hecke operators at the primes ` 6= p at
which K p is unramified (i.e. at which G is unramified and K p is of the form K p,` K` ,where K` is maximal compact hyperspecial). We then define
T := lim T(Kp );←−Kpthis is an algebra of G-equivariant endomorphisms of M . We endow each T(Kp ) with
its $-adic topology, and endow T with its projective limit topology. Then T is a proArtinian O-algebra (indeed, since each T(Kp ) is finitely generated as an O-modules,
we may describe T as the projective limit of (literally) finite O-algebras), and is the
product (as a topological ring) of finitely many local pro-Artinian O-algebras, these
local factors being in bijection with the maximal ideals of T; we let Tm denote thefactor corresponding to a given maximal ideal m. The local factor Tm is naturally
identified with the m-adic completion of T as an abstract ring; note, though, that the
topology on Tm coincides with the m-adic topology if and only if Tm is Noetherian. In
fact, because of the anticipated relationship with Galois deformation rings, we expectthat T, and hence each Tm , is Noetherian, but as far as we know, this is not provedin the generality that we consider here. If V is any locally algebraic representationof K, then it follows from (27) and Schikhof duality [38, Thm. 3.5] that∗e0 p
Homcont
OJKK (M, V ) = HomK V, H (K ) ,and the target is isomorphic to the space of algebraic automorphic forms on G withvalues in V ∗ [14, Prop. 3.2.4]. In particular, the action of T ⊗O L on this space issemi-simple.
Now, let m be a maximal ideal of T. We may form the localization Mm ; this is adirect summand of M , and so projective as an OJKK-module. It has a commuting∗
action of the local ring Tm . The space Homcont
OJKK (Mm , V ) is a finite-dimensional
L-vector space. The action of Tm on this space factors through an order in thefinite-dimensional L-algebra generated by the Hecke operators.
Remark 2.12 applies, and so Proposition 2.11 together with Remark 2.13 showsthat if the set {Vi }i∈I captures M , then the systems of Hecke eigenvalues arising∗in the spaces of algebraic automorphic forms Homcont
OJKK (M, Vi ) are Zariski dense inSpec Tm .

22

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

5.2. Hecke algebras and functoriality. Let us suppose that we have two groups
G1 and G2 as in the previous subsection. We keep the notation of the previous
section, except that we add the index 1 or 2 to indicate with respect to which groupthe objects are defined. Assume that both Hecke algebras T1 and T2 are completionsof the quotients of the same universal Hecke algebra Tuniv , which is a polynomial
ring over O with variables corresponding to the Hecke operators at the unramified
places for both groups. Let m be a maximal ideal of Tuniv and assume that m is a
maximal ideal of both T1 and T2 . Then both T1,m and T2,m are quotients of theb univ .
m-adic completion of Tuniv , which we denote by Tm
Let {Vi }i∈I be a family of irreducible locally algebraic representations of K1 , whichb univ [1/p] in thecaptures M1 . As explained above, a maximal ideal x ∈ m-Spec T
m

p
0
e
support of HomK1 Vi , H (K1 )m corresponds to an algebraic automorphic form on
G1 . Let us further assume that functoriality between G1 and G2 is available andwe may transfer such automorphic forms on G1 to automorphic forms on G2 . Moreb univ [1/p], the map Tb univ → κ(x),precisely, we assume that for such x ∈ m-Spec Tmmwhere κ(x) is the residue field of x, fits into a commutative diagram:b univTm/ / T1,m


T2,m


/ κ(x).

b univ [1/p] consisting of maximal ideals contained in the
Let Σ be the subset of m-Spec Tm
pe 0 (K )m for some i ∈ I. Since {Vi }i∈I captures M1 it will
support of HomK1 Vi , H
1
b univ → Q
also capture its direct summand M1,m . Hence, the image of Tmx∈Σ κ(x) is
equal to T1,m . The commutativity of the above diagram implies that this map factors
through T2,m . Thus we obtain a surjection T2,m  T1,m and so the Hecke algebrae 0 (K p ).
T2,m acts on the completed cohomology H
1
The meta-argument outlined above can be modified to give a different proof ofCorollary 7.3 in [37], where the functoriality is given by the (classical) Jacquet–
Langlands correspondence, by taking {Vi }i∈I to be a family of supercuspidal types
considered in subsection 3.4 and using Proposition 3.26 together with Proposition
3.19. A similar example has been discussed in [19, §3.3.2].
5.3. The case of definite unitary groups. In the case when G = GU (n), we let
T0m denote the subring of Tm topologically generated by spherical Hecke operators atunramified primes ` which are furthermore split in the appropriate quadratic imaginary field. The ring T0m is a subring of Tm , and (assuming that m corresponds toan irreducible ρ) is a quotient of a global deformation ring Rρ . (See [10, §2.4] for anindication of what exact kinds of Galois representations Rρ parameterizes.)
Thus we get the Zariski density in Spec T0m of the collection of Galois representa∗
tions arising from the various spaces Homcont
OJKK (M, Vi ).
We formulate this discussion into a theorem to make it more concrete. We freelySp
use the notation of [10]. Let R be the Hecke algebra denoted by Tξ,τ(U p , O)m in [10]pdbefore Corollary 2.11 and let M = S̃ξ,τ (U , O)m . It follows from the proof of [10,Prop. 2.10] that S̃ξ,τ (U p , O)dm is a finitely generated projective OJGLn (OF )K-module,
where p is a fixed place above p of totally real field denoted by Fe+ in [10] and F is a
finite extension of Qp such that F is the p-adic completion of Fe+ .

ON THE DENSITY OF SUPERCUSPIDAL POINTS

23

Let Σcl be the subset of m-Spec R[1/p] corresponding to the Hecke eigenvalues
corresponding to the classical (algebraic) automorphic forms on the unitary groupconsidered in [10, §2.3]. To each x ∈ Σcl one may associate a Galois representation,
which is potentially semi-stable at p. For each embedding κ : F → Qp we fix an
n-tuple of distinct integers kκ := (kκ,1 > kκ,2 > . . . > kκ,n ) and let k be the multiset{kκ }κ:F ,→Qp . Let Σ(k) be the subset of Σcl corresponding to the automorphic forms
such that the restriction of the associated Galois representation at the decompositiongroup at p has Hodge–Tate weights equal to k.
To each classical automorphic form we may associate an automorphic representation. Let Σps be the subset of Σcl such that the local factor of the automorphic
representation at p is a principal series representation of GLn (F ). In terms of Galois
representations the subset Σps correspond to those automorphic forms such that the
restriction of the associated Galois representation at the decomposition group at p
becomes crystalline after a restriction to the Galois group of a finite abelian extensionof F .
We fix either a totally ramified or unramified extension E of F of degree n andidentify GLn (F ) with AutF (E). More generally we could take any extension E of F ofdegree n, which is generated by a minimal element in the sense of section 3.4. Let Σsc
be the subset of Σcl such that the local factor of the automorphic representation at p
is a supercuspidal representation of GLn (F ) containing a minimal stratum [A, m, 0, α]with E = F [α]. If E over F is tamely ramified and ρ is a Galois representation associated to an automorphic form in Σsc then the Weil–Deligne representation associatedto the restriction of ρ to the decomposition group at p is isomorphic to the induction
from WE to WF of a 1-dimensional representation; the requirement for [A, m, 0, α] tobe minimal imposes further restrictions on the 1-dimensional representation of WE .
Theorem 5.1. The sets Σ(k) ∩ Σps and Σ(k) ∩ Σsc are both Zariski dense inside
Sp
Spec Tξ,τ
(U p , O)m .
Proof. We will carry out the proof in the supercuspidal case. The proof in the principal series case is the same using the results of section 3.3. Out of the multiset k one
may manufacture a highest weight and hence a corresponding irreducible algebraic
representation of G0 := ResF
Qp GLn /F , see section [10, §1.8]. Let W be its evaluation
at L and restriction to GLn (F ) via GLn (F ) = G0 (Qp ) ,→ G0 (L).
Sp
We will now apply the theory developed in section 3.4 with R = Tξ,τ(U p , O)m andpdm+1
M = S(U , O)m . Let K := U(A). Since M is projective as OJGLn (OF )K-module
it is also projective as OJKK-module. It follows from Lemma 2.6 and Proposition 3.26
that the family {W ⊗ V (α, θ)}α,θ , where α runs over the elements of E, such that
E = F [α] such that [A, n0 , 0, α] is a simple stratum with n0 ≥ 2m and α is minimal
over F and θ runs over all simple characters θ ∈ C(m, α), captures M . Since the Hecke
algebra acts faithfully on the module by construction we only have to understand the∗
R-annihilator of Homcont
OJKK (M, (W ⊗ V (α, θ)) ) and then the assertion follows from
Proposition 2.11. We first note that since M is finitely generated over OJKK and both∗
W and V (α, θ) are finite dimensional L-vector space Homcont
OJKK (M, (W ⊗ V (α, θ)) ) isa finite dimensional L-vector space. By Schikhof duality we get that
Homcont (M, (W ⊗ V (α, θ))∗ ) ∼= HomK (W ⊗ V (α, θ), S̃(U p , O)m ⊗O L),OJKK∼ Homcont (M, L) is an admissible unitary L-Banach space
where S̃(U p , O)m ⊗O L =Orepresentation of GLn (F ). Since W ⊗ V (α, θ) is a locally algebraic representation ofK, the image of a K-invariant homomorphism will be contained in the subspace of

24

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

locally algebraic vectors (S̃(U p , O)m ⊗O L)alg . It follows from [14, Prop. 3.2.4] that the
representation (S̃(U p , O)m ⊗O L)alg can be computed in terms of classical automorphic
forms. In particular, it follows from [22, Prop. 6.11] that the action of R[1/p] on(S̃(U p , O)m ⊗O L)alg is semi-simple, and (essentially by definition) the support isequal to Σcl . Moreover, if x ∈ Σcl and mx is the corresponding maximal ideal ofR[1/p] then the subspace of (S̃(U p , O)m ⊗O L)alg annihilated by mx is isomorphic to
πx,p ⊗ Wx , where πx,p is the local component at p of the automorphic representation
associated to x, and Wx is the restriction to GLn (F ) of the algebraic representationof G0 encoding the Hodge–Tate weights of the Galois representation attached to x.
Thus the action of R[1/p] on HomK (W ⊗ V (α, θ), (S̃(U p , O)m ⊗O L)alg ) is semisimple
and the support is equal to a finite subset of Σcl consisting of x such that W ∼= Wx
and HomK (V (α, θ), πx,p ) 6= 0. The first condition implies that x ∈ Σ(k), the second
condition implies that x ∈ Σsc via Proposition 3.19.

Remark 5.2. In the recent paper [25], Hellman, Margerin and Schraen have proved
a “big R equals big T theorem” in the context of definite unitary groups. We refer totheir paper for the precise details, but note here that, when their result applies, we
may rephrase Theorem 5.1 as a statement about the Zariski density of certain sets ofpoints in the Spec of a global Galois deformation ring.
5.4. Local application. By applying the theory of capture to M∞ of the six author
paper [10], we will get density in the support of M∞ . This support will equal the full
local deformation space, provided that this latter space is irreducible. The proof ofthis is the subject of the next section.
Let R be the framed deformation ring of ρ̄ : GF → GLn (k). If R is a local noetherian R -algebra with residue field k then to x ∈ m-Spec R[1/p] one may associate
Galois representation ρx : GF → GLn (κ(x)) by specialising the universal framed deformation along the map R → R → κ(x). We will define the local analogs of thesets Σcl , Σ(k), Σps and Σsc defined in the previous section.
Let Σpst be the subset of m-Spec R[1/p] consisting of those x for which the representation ρx is potentially semi-stable. Let Σ(k) be the subset of Σpst such that the
Hodge–Tate weights of ρx are equal to k. Let Σps be the subset of Σpst such that ρx
become crystalline after a restriction to the Galois group of an abelian extension of F .
Alternatively Σps can be described as those x ∈ Σpst such that the Weil–Deligne representation associated to ρx corresponds to a principal series representation via theclassical local Langlands correspondence. Let E be either a totally ramified or anunramified extension of F of degree n as in the previous section. We let Σsc be thesubset of Σpst consisting of those x such that the Weil–Deligne representation associated to ρx corresponds to a supercuspidal representation πx via the classical local
Langlands correspondence, such that πx contains a simple stratum [A, n0 , 0, α], suchthat E = F [α] and α is minimal over F .
Recall that in [10] we have constructed a local noetherian R -algebra R∞ withresidue field k and an arithmetically interesting R∞ [GLn (F )]-module M∞ . We assume that p does not divide 2n and that ρ̄ admits a potentially crystalline lift ofregular weight, which is potentially diagonalisable, since this is assumed in [10]. As
noted in Remark 1.5, this last assumption will become redundant in the future. We
recall that M∞ is projective as an OJKK-module. (This follows from [10, Prop. 2.10],
which shows that M∞ is projective over S∞ JKK, where S∞ is a certain formal power
series ring in a finite number of variables over O.)

ON THE DENSITY OF SUPERCUSPIDAL POINTS

25

Theorem 5.3. If a denotes the R∞ -annihilator of M∞ then Σ(k) ∩ Σps ∩ V (a) andΣ(k) ∩ Σsc ∩ V (a) are both Zariski dense subsets of V (a).
Proof. We will treat the supercuspidal case, since the principal series is similar (andeasier). The proof is essentially the same as the proof of Theorem 5.1. We only needto verify that R∞ /aα,θ is reduced and all the maximal ideals of (R∞ /aα,θ )[1/p] lie∗in Σ(k) ∩ Σsc , where aα,θ is the R∞ -annihilator of Homcont
OJKK (M∞ , (W ⊗ V (α, θ)) ).
This is proved in the same way as part 1 of [10, Lem. 4.17] by using Proposition 3.19
together with local–global compatibility.

Lemma 5.4. Let A and B be complete local noetherian O-algebras with residue field
k. Assume that B is a flat A-algebra. Let Σ be a subset of m-Spec A[1/p] and let Σ0be its preimage in m-Spec B[1/p]. If the closure of Σ0 in Spec B is a union of irreducible components of Spec B then the closure of Σ in Spec A is a union of irreducible
components of Spec A.
Proof. Let Btf be the maximal p-torsion free quotient of B. The map p 7→ p ∩ Btf
induces a bijection between the minimal primes of B[1/p] and Btf and we will identify
both sets. Moreover, every minimal prime of Btf is also a minimal prime of B: if p
is a minimal prime of Btf then the quotient Btf /p is p-torsion free, as p is regular on
Btf . If q is a minimal prime of B then either B/q is p-torsion free in which case q isa minimal prime of Btf or p is zero in B/q in which case B/q cannot have Btf /p as aquotient. Thus it is enough to study the question after inverting p.
Let b be an ideal of B[1/p] such that V (b) is the closure of Σ0 and let a = b∩A[1/p].
Then V (a) is equal to the closure of Σ. By assumption we may write V (b) = V (q1 ) ∪. . . ∪ V (qr ), where qi are minimal prime ideals of B[1/p]. We may assume thatb = q1 ∩ . . . ∩ qr . Since B is A-flat the going down theorem implies that pi :=qi ∩ A[1/p] is a minimal prime of A[1/p]. Since a = b ∩ A[1/p] = p1 ∩ . . . ∩ pr we have
V (a) = V (p1 ) ∪ . . . ∪ V (pr ).

Corollary 5.5. Let a be the R∞ -annihilator of M∞ . If V (a) is equal to a unionof irreducible components of Spec R∞ , then the Zariski closure of Σ(k) ∩ Σsc (resp.
Σ(k) ∩ Σps ) in Spec R contains a non-empty union of irreducible components ofSpec R .
Proof. Given Lemma 5.4 and Theorem 5.3, it is enough to verify that R∞ is flat overR . Now R∞ is formally smooth over the ring denoted by Rloc in [10, §2.6], andb O A, where A is a completed tensor product over O of some deformation
Rloc = R ⊗rings at a finite number of places away from p. All them are O-torsion free, see [10,§2.4], thus A is O-torsion free and hence O-flat, which implies that Rloc is flat overR .
The condition on V (a) in Corollary 5.5 will be verified in the next section.
6. The support of M∞
We begin with a definition.
Definition 6.1. If A is a complete Noetherian local O-algebra, and M is a finitely
generated AJKK-module, then we define the support of M in Spec A, denoted SuppA M ,as follows: SuppA M := V (a), where a is the A-annihilator of M .
Since a finitely generated AJKK-module is typically not finitely generated as an
A-module, it is not immediately clear that defining its support in the above manner

26

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

is sensible. However, the following lemma gives us assurance that the preceding
definition does in fact give a reasonable notion of support.
Lemma 6.2. If m1 , . . . , mr are generators of M as an AJKK-module, and M 0 is the
A-submodule of M generated by m1 , . . . , mr , then Supp M 0 = Supp M .
Proof. Let a be the A-annihilator of M and let a0 be the A-annihilator of M 0 . Since
M 0 ⊂ M we have a ⊂ a0 . On the other hand, since the actions of A and K on M
commute, we see that a0 annihilates AJKKM 0 = M.
The main goal of this section is to prove the following theorem.
Theorem 6.3. The support of M∞ in Spec R∞ is equal to a union of irreducible
components of Spec R∞ .
Proof. It follows from Proposition 3.1 that, if ξ runs over all the algebraic representations of G, then the support of M∞ is equal to the Zariski closure of the unions of the
supports of the various modules M∞ (ξ). (See [10] for a discussion of these patched
modules.) The support of each M∞ (ξ) is reduced [10, Lem. 4.17], and so it follows
from Theorem 6.5 below that the support of M∞ [1/p] in Spec R∞ [1/p] is equal to aunion of its irreducible components. Since M∞ is p-torsion free by construction, thesupport of M∞ is equal to a union of irreducible components of Spec R∞,tf , where
Spec R∞,tf denotes the maximal p-torsion free quotient of R∞ . As explained in the
proof of Lemma 5.4 these are also irreducible components of Spec R∞ .
Remark 6.4. Typically R is a formally smooth O-algebra and thus irreducible. In
such cases, our theorem together with Corollary 5.5 implies that M∞ is supported on
all of Spf R . We expect that M∞ is in fact always supported on all of Spf R , but
it seems a priori possible that this is not the case, when Spf R has more than one
irreducible component.
Theorem 6.5. The Zariski closure in (Spf R∞ )rig of the set of crystalline points lying
in the support of M∞ is equal to a union of irreducible components of (Spf R∞ )rig .
We follow the approach of Chenevier [11] and Nakamura [30], who proved (undervarious technical hypotheses) that the crystalline points are Zariski dense in the rigid
analytic generic fibre (Spf R )rig of Spf R .
The arguments of Chenevier and Nakamura use “infinite fern”-style techniques. In
order to modify these arguments so as to be sensitive to the support of M∞ , we use
the locally analytic Jacquet module of [17], which provides a representation-theoretic
approach to the study of finite slope families. Breuil, Hellmann, and Schraen [3] havemade a careful study of the theory of the Jacquet module as it applies to M∞ , andwe will use their results.1
Following the notation of [3], we write Π∞ := Homcont
O (M∞ , L); then Π∞ is an
R∞ -admissible Banach space representation of G. We may pass to its R∞ -locallyanalytic vectors, see [3, §3.1], and then form the locally analytic Jacquet module∞ −an∞ −an 0∞ −an). The continuous dual JB (ΠR) of JB (ΠR) may naturally be
JB (ΠR∞∞∞
regarded as the space of global sections of a coherent sheaf M∞ on the rigid analyticb We let X∞ denote the support of M∞ ; then X∞ is a Zariski
space (Spf R∞ )rig × T.
b If x is a closed point of Spec R∞ [1/p]closed rigid analytic subspace of (Spf R∞ )rig × T.
1In fact, they work with a slightly different version of M to us (which is literally the construction∞of [10]), but all their arguments and results apply equally well in our setting, and we will cite andapply them without further discussion of this issue.

ON THE DENSITY OF SUPERCUSPIDAL POINTS

27

and mx is the corresponding maximal ideal, then Π(x) is a closed subspace of Π∞
consisting of elements in Π∞ , killed by mx .
We recall again from [3], see their Definition 2.4, that there is another important
Zariski closed rigid analytic subspacetribX∞,→ (Spf R∞ )rig × T,

which by definition is the Zariski closure of the set of pointsb(x, δ) ∈ (Spf R∞ )rig × T

reg,where δ is the parameter of a triangulation of the Galois representation rx , see [3,§2.2].
We then have the following key result of [3].
triTheorem 6.6 ([3, Thm. 3.21]). There is an inclusion X∞ ⊂ X∞
, and X∞ is furtri
thermore a union of irreducible components of X∞ .

We also need to recall the key point of the infinite fern argument of [11] and [30],in a form that is well-suited to our argument.
kDefinition 6.7. If k is a system of labelled Hodge–Tate weights, then we let R∞
denote the reduced and O-flat quotient of R∞ characterized by the property that akpoint x ∈ (Spf R∞ )rig lies in (Spf R∞ )rig if and only if rx is crystalline with labelled
Hodge–Tate weights given by k.
Let f := [F0 : Qp ] be the degree of the maximal unramified subextension of F .
Definition 6.8. We say that a crystalline representation r of GF with regular Hodge–
Tate weights k is benign if the following hold:(i) the eigenvalues of the linearization of the crystalline Frobenius ϕf of r are
pairwise distinct;(ii) all the refinements of r are non-critical;(iii) the ratio of any two eigenvalues in (i) is not equal to p±f .
We refer the reader to [3, §2.3] for the terminology used in (i) and (ii). Our definition coincides with the definition of benign in [30, Def. 2.8]. It is more restrictive than[3, Def. 2.8], since we additionally impose the condition (iii). Let WD(r) be the WeilDeligne representation associated to r. The condition (iii) implies that the smooth
representation πp of GLn (F ) associated to WD(r) via the classical local Langlands
correspondence is an irreducible unramified principal series representation.
Proposition 6.9. If k is any regular system of labelled Hodge–Tate weights, and Ckis any irreducible component of (Spf R∞ )rig , then the set of points x ∈ C for whichkrx is benign forms a non-empty Zariski open subset of (Spf R∞ )rig .
Proof. This is a result of Chenevier [11, Lem. 4.4] in the case when K = Qp andNakamura [30, Lem. 4.2] in the general case.

Theorem 6.10. Let z be a benign crystalline point of Spec R∞ .
tri(1) If (z, δ) is a point of X∞
which lies over z, then (z, δ) is a smooth point oftri
X∞ .
(2) The tangent space T(Spf R∞ )rig ,z is spanned by the images of the tangent spacestri
tri ,(z,δ) , as (z, δ) runs over all points of X
TX∞
∞ lying over z.

28

MATTHEW EMERTON AND VYTAUTAS PAŠKŪNAS

Proof. The first claim follows by combining Corollary 2.12 and Theorem 2.6 (iii) of [3].
The second claim follows from [11] in the case that K = Qp and [31, Thm. 2.62] intri ,(z,δ) in T(Spf R )rig ,z coincides with thegeneral once we note that the image of TX∞∞
closed subspace of T(Spf R∞ )rig ,z classifing trianguline deformations of (z, δ). See e.g.
the discussion of this in the proof of [3, Thm. 2.6].
With the preceding results recalled, we are now in a position to study the supportof M∞ .
If ξ is an irreducible algebraic representation of G, and if k is the system of labelled
Hodge–Tate weights corresponding to ξ, then (as already recalled in the proof of
Theorem 6.3), we may form the R∞ [1/p]-module M∞ (ξ) as in [10], whose support iska union of some number of irreducible components of Spec R∞ [1/p], or, equivalently,ka union of some number of irreducible components of (Spf R∞ )rig . In particular, if
M∞ (ξ) is non-zero, then Proposition 6.9 implies that its support contains a Zariski
dense open subset of points x for which rx is benign.
Lemma 6.11. If ξ is an algebraic representation of G, if x is a point lying in thetrilying over x,
support of M∞ (ξ) for which rx is benign, and if (x, δ) is a point of X∞
then (x, δ) ∈ X∞ .
Proof. We see from [3, Cor. 2.12] that the δ is necessarily a triangulation of rx . The
T =δ
fibre of M∞ over (x, δ) is dual to the eigenspace JB Π(x)anin the Jacquet
module of Π(x), where an denotes the subspace of locally analytic vectors in the
Banach space Π(x), see [3, Prop. 3.7]. Thus in order to show that (x, δ) lies in X∞ ,it suffices to show that for every parameter δ describing a triangulation of rx , thecorresponding eigenspace in the Jacquet module of Π(x)an is non-zero.
To see this, note that local-global compatibility shows that Π(x) contains a locallyalgebraic representation πp ⊗ ξ, with πp the parabolic induction of an unramified
character, see [10, Thm. 4.35]. Note that the condition (iii) in Definition 6.8 ensuresthat πp is irreducible and hence generic, so that the conditions of [10, Thm. 4.35] aresatisfied. The representation πp ⊗ ξ equipped with the finest locally convex topology
is a locally analytic subrepresentation of Π(x)an . Since
 the Jacquet-functor is left
exact we have an inclusion JB (πp ⊗ ξ) ⊂ JB Π(x)an . Moreover, [17, Prop. 4.3.6]
implies that JB (πp ⊗ ξ) ∼= (πp )N ⊗ ξ N , where N is the unipotent radical of B. Hence,
JB (πp ⊗ξ) consists of n! characters which fill out all the necessary eigenspaces. (These
correspond to the n! possible orderings of the Frobenius eigenvalues, which in turnmatch with all the possible triangulations.)
Theorem 6.12. The Zariski closure, in (Spf R∞ )rig , of the union of the supports ofthe modules M∞ (ξ), as ξ ranges over all the irreducible algebraic representations of
G, is equal to a non-empty union of components of (Spf R∞ )rig .
Proof. To ease notation, write Y := (Spf R∞ )rig , and let Z ,→ Y the Zariski closure
under consideration in the statement of the theorem. We note that M∞ (ξ) is nonzero for at least one choice of ξ, so that Z is non-empty. Since Z is furthermore
a reduced rigid analytic space over a field of characteristic zero, its smooth locus
Z sm is a non-empty and open subset. Thus the correspondence V 7→ U := V ∩ Z sm
induces a bijection between the irreducible components V of Z and the irreducible(or, equivalently, connected) components U of Z sm ; one recovers V from U by taking
the Zariski closure of U in Z (or, equivalently, in Y ).
We next note that the points of X∞ of the form (z, δ) with z lying in the supportof one of M∞ (ξ) are actually Zariski dense in X∞ . Indeed, this follows from [3,ON THE DENSITY OF SUPERCUSPIDAL POINTS

29

Thm. 3.19], since any (z, δ) ∈ X∞ , which is strongly classical in the sense of [3,Def. 3.15], is classical in the sense of [3, Def. 3.17] and hence will have z lying in the
support of M∞ (ξ), where the highest weight of ξ corresponds to the algebraic part of δ.
b →YThis implies that the image of X∞ in Y under the composition π : X∞ ⊂ Y × T
is contained in Z, as π −1 (Z) is a closed subset of X∞ containing a dense set of points.
As noted above, if k is the system of labelled Hodge–Tate weights corresponding toan algebraic representation ξ, then the support of M∞ (ξ) is a union of some numberkof components of (Spf R∞ )rig , each of which contains a Zariski dense subset of points
z for which rz is benign. Thus Z is equal to the Zariski closure of the set B of such
points, i.e. the set B of points z lying in the support of some M∞ (ξ), and for whichrz is benign. Thus if V is an irreducible component of Z, and if U := V ∩ Z sm (which,as we noted above, is Zariski dense in V ), then we find that the B ∩ U is Zariski densein U ; in particular, it is non-empty.
triChoose a point z ∈ B ∩ U . If (z, δ) is any point of X∞
lying over z, the point (z, δ)trilies in the smooth locus of X∞ , by Theorem 6.10 (1). Lemma 6.11 shows that any
such point (z, δ) in fact lies in X∞ . We then conclude from Theorem 6.6 that (z, δ) istri
is an isomorphism
a smooth point of X∞ , and that the closed immersion X∞ ,→ X∞
at any such point (z, δ).
Combining this last conclusion with the inclusion π(X∞ ) ⊆ Z that was proved
tri ,(z,δ) ,above, we thus find that TZ,z = TU,z contains the image of the tangent space TX∞
for each choice of (z, δ) lying over z. Since Theorem 6.10 (2) shows that the images oftri ,(z,δ) span TY,z , we find that TU,z = TY,z , and hence
the various tangent spaces TX∞
that U and Y coincide locally at z. This implies that the Zariski closure V of U ,which is an irreducible component of Z, is also an irreducible component of Y , asrequired.


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