p-ADIC HODGE-THEORETIC PROPERTIES OF ÉTALE
COHOMOLOGY WITH MOD p COEFFICIENTS, AND THE
COHOMOLOGY OF SHIMURA VARIETIES
MATTHEW EMERTON AND TOBY GEE

Abstract. We prove vanishing results for the cohomology of unitary Shimuravarieties with integral coefficients at arbitrary level, and deduce applications
to the weight part of Serre’s conjecture. In order to do this, we show that themod p cohomology of a smooth projective variety with semistable reduction
over K, a finite extension of Qp , embeds into the reduction modulo p of asemistable Galois representation with Hodge–Tate weights in the expected
range (at least after semisimplifying, in the case of the cohomological degree
greater than 1).

The aim of this paper is to establish vanishing results for the cohomology ofcertain unitary similitude groups. For example, we prove the following result. (SeeCorollary 3.5.1 and Lemma 4.1.9 below, and see Sections 2 and 3 for the precise
definitions that we are using; for simplicity we work with U (2, 1)-Shimura varieties1
over a quadratic imaginary field E. Note that “sufficiently small level” means thatthe compact open subgroup defining the level is sufficiently small.)A. Theorem. Let X be a projective U (2, 1)-Shimura variety of some sufficiently
small level, and let F be a canonical local system of Fp -vector spaces on X. Let m
be a maximal ideal of the Hecke algebra acting on the cohomology H • (X, F), andsuppose that there is a Galois representation ρm : GE → GL3 (Fp ) associated to m.
×
If we suppose further that we have SL3 (k) ⊂ ρm (GE ) ⊂ Fp SL3 (k) for some finite
extension k/Fp , and that ρm |GQp is 1-regular and irreducible, then the localisationsiHét(XQ , F)m vanish in degrees i 6= 2.
In fact, we prove a version of this result for U (n − 1, 1)-Shimura varieties under
weaker assumptions on ρm ; however, in general we can only prove vanishing indegrees outside of the range [n/2, (3n − 4)/2].
We also prove the following result, which makes no explicit reference to a maximal ideal in the Hecke algebra.
B. Theorem. Let X and F be as in the statement of Theorem A. If ρ is athree-dimensional irreducible sub-GE -representation of the étale cohomology group
1
Hét(XQ , F), then either every irreducible subquotient of ρ|GQp is one-dimensional,
or else ρ|GQp is not 1-regular, or else ρ(GE ) is not generated by its subset of regularelements.
The first author was supported in part by NSF grant DMS-1003339, and the second author
was supported in part by NSF grant DMS-0841491 and EPSRC Mathematics Platform grant
EP/I019111/1.
1These Shimura varieties might more properly be called GU (2, 1)-Shimura varieties; see Section
3 for their definition.
1

2

MATTHEW EMERTON AND TOBY GEENote that in neither theorem do we make any assumption on the level of the
Shimura variety at p.
A Galois representation ρm as in the statement of Theorem A is known to exist
if m corresponds to a system of Hecke eigenvalues arising from the reduction mod
p of the Hecke eigenvalues attached to some automorphic Hecke eigenform. Furthermore, recent work of Scholze [Sch13] (which appeared after the first version ofthis paper was written) implies that such a representation exists for any maximal
ideal m.
It seems reasonable to believe that any irreducible sub-GE -representation of any
iof the étale cohomology groups Hét
(XQ , F) for any of the Shimura varieties underconsideration should in fact be a constituent of ρm for some maximal ideal m ofthe Hecke algebra. However, this doesn’t seem to be known, and relating the“abstract” GE -representations ρm to the “physical” GE -representations appearingon étale cohomology is one of the problems we have to deal with in proving our
results.
Application to Serre-type conjectures. We are able to combine our resultswith those of [EGH11] so as to establish cases of the weight part of the Serretype conjecture of [Her09] for U (2, 1). More precisely, we have the following result(where the assertion that ρ is modular means that the corresponding system of
Hecke eigenvalues occurs in the mod p cohomology of some U (2, 1)-Shimura variety;see Theorem 3.5.6 and Lemma 4.1.9.)
C. Theorem. Suppose that ρ : GE → GL3 (Fp ) satisfies SL3 (k) ⊂ ρ(GE ) ⊂×
Fp SL3 (k) for some finite extension k/Fp , that ρ|GQp is irreducible and 1-regular,
and that ρ is modular of some strongly generic weight. Then the set of genericweights for which ρ is modular is exactly the set predicted by the recipe of [Her09].
Relationship with a mod p analogue of Arthur’s conjecture. Arthur has
made a quite precise conjecture regarding the systems of Hecke eigenvalues thatappear in the L2 -automorphic spectrum of any reductive group over a number field[Art89], which has consequences for the nature of the Hecke eigenvalues appearing
in the cohomology of Shimura varieties [Art89, §9]. For our purposes it suffices to
describe a qualitative version of these consequences: namely, Arthur’s conjecture
implies that if λ is a system of Hecke eigenvalues appearing in the degree i cohomology, where i is less than the middle dimension, then λ is attached (in the senseof e.g. [BG10], [Joh12]) to a reducible Galois representation (i.e. one which factors
through a parabolic subgroup of the L-group).
The fragmentary evidence available suggests that a similar statement will betrue for the mod p cohomology of Shimura varieties. Our Theorems A and B givefurther evidence in this direction.
p-adic Hodge theory. In order to prove these theorems, we establish somenew results about the p-adic Hodge theoretic properties of the étale cohomology ofvarieties over a number field or p-adic field with coefficients in a field of characteristic p. In the first section we establish results about the mod p étale cohomology
of varieties over number fields or p-adic fields which, although weaker in their conclusions, are substantially broader in the scope of their application than previously
known mod p comparison theorems. For example, we prove the following result(see Theorem 1.4.1 below).

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

3

D. Theorem. Let K be a finite extension of Qp , and write GK for the absolute Galois group of K. If X is a smooth projective variety over K which has
semistable reduction, and if ρ is an irreducible subquotient of the GK -representationi
Hét(XK , Fp ), then ρ also embeds as a subquotient of a GK -representation over Fp
which is the reduction modulo the maximal ideal of a GK -invariant Zp -lattice in a
GK -representation over Qp which is semistable with Hodge–Tate weights containedin the interval [−i, 0].
Both the hypotheses and the conclusions of our theorems are rather precisely
tailored to maximise (as far as we are able) their utility in applications to theanalysis of Galois representations occurring in the cohomology of Shimura varieties,
which we give in the third section.
The remaining two sections of the paper are devoted respectively to using integral p-adic Hodge theory (Breuil modules with descent data) to establish a result
related to the reductions of tamely potentially semistable p-adic representations of
GQp (Section 2) and to proving some technical results about group representations(Section 4). The result of Section 2 is an essential ingredient in the arguments ofSection 3, while the results of Section 4 provide sufficient conditions for the various representation-theoretic hypotheses appearing in the results of Section 3 to besatisfied.
Remark on related papers. Very general vanishing theorems for the mod p
cohomology of Shimura varieties have been proved by Lan and Suh (cf. [LS11]);
however, their results apply only in situations of good reduction and for coefficients
corresponding to small Serre weights, which makes them unsuitable for the kinds ofapplications we have in mind, e.g. to the weight part of Serre-type conjectures. In
the ordinary case there is the work of Mokrane–Tilouine in the Siegel case [MT02,§9] and Dimitrov in the case of Hilbert modular varieties [Dim05, §6.4]. Finally, inthe recent preprint [Shi11], Shin proves a general vanishing result for cohomology
outside of middle degree for the part of the mod p cohomology which is supercuspidal at some prime l 6= p, by completely different methods to those of thispaper. It seems plausible that via the mod p local Langlands correspondence forGLn (Ql ), Shin’s hypothesis could be interpreted as a condition on the restriction toa decomposition group at l of the relevant mod p Galois representations, whereas
our conditions involve the restriction to a decomposition group at p, so our resultsappear to be complementary.
Acknowledgements. We would like to thank Brian Conrad, Florian Herzig,
Mark Kisin, Kai-Wen Lan, Madhav Nori, Bjorn Poonen, Michael Rapoport, Junecue
Suh, and Teruyoshi Yoshida for helpful correspondence and conversations on thesubject of this article. We also thank an anonymous referee for their critical readingof the paper, which led to the filling of some gaps in the arguments of Subsection 1.5.
Conventions. For any field K we let GK denote a choice of absolute Galois
group of K.
If K is a finite field then by a Frobenius element in GK we will always meana geometric Frobenius element. We extend this convention in an evident way to
Frobenius elements at primes in Galois groups of number fields, and to Frobenius
elements in Galois groups of local fields.
If K is a local field, then we let OK denote the ring of integers of K, we let IK
denote the inertia subgroup of GK , we let WK denote the Weil group of K (the4

MATTHEW EMERTON AND TOBY GEE

subgroup of GK consisting of elements whose reduction modulo IK is an integral
power of Frobenius), and we let WDK denote the Weil–Deligne group of K.
If K is a number field, and v is a finite place of K, then we will write Kv for thecompletion of K at v, and OKv for its ring of integers. We will write OK,(v) for the
localisation of OK at the prime ideal v.
We will write Zp for the ring of integers in Qp (a fixed algebraic closure of Qp ),and mZp for the maximal ideal of Zp .
We let ω denote the mod p cyclotomic character. We will denote a Teichmüller
lift with a tilde, so that for example ωe is the Teichmüller lift of ω.
We use the traditional normalisation of Hodge–Tate weights, with respect towhich the cyclotomic character has Hodge–Tate weight 1.
By a closed geometric point x of a scheme X, we mean a morphism of schemes
x : Spec Ω → X for a separably closed field Ω, whose image is a closed point x of X,
and such that the induced embedding κ(x) ,→ Ω (where κ(x) denotes the residue
field of x) identifies Ω with a separable closure of κ(x). If x is a closed geometricpoint of a Noetherian scheme X, then we let Ox,X denote the local ring of X at x,
i.e. the stalk, in the étale topology on X, of the structure sheaf of X at x, we letc[[O
x,X denote the completion of Ox,X , and we write Xx := Spf Ox,X , and refer tocx as the formal completion of X along the closed geometric point x.
X
1. p-adic Hodge theoretic properties of mod p cohomology
1.1. Introduction. We now describe in more detail our results on the integral padic Hodge theory of the étale cohomology of projective varieties, which are perhaps
the most novel part of this paper.
It is well-known that integral p-adic Hodge theory is less robust than the corresponding theory with rational coefficients; for example, the comparison theorems for
integral and mod p étale cohomology due to Fontaine–Messing [FM87] and Faltings[Fal89] involve restrictions both on the degrees of cohomology and the dimensionsof the varieties considered, and they also require that the field K be absolutely
unramified and that the variety under consideration be of good reduction. Morerecently, Caruso [Car08] has proved an integral comparison theorem in the case of
semistable reduction for possibly ramified fields K, but there are still restrictions:his result requires that ei < p − 1, where e is the absolute ramification index of K,
and i is the degree of cohomology under consideration.
These restrictions are unfortunate, since mod p and integral p-adic Hodge theory
are among the most powerful local tools available for the analysis of Galois representations occurring in the mod p étale cohomology of varieties. The premise that
underlies the present work is that frequently in such applications, one does not needa precise comparison theorem relating the mod p étale cohomology to an analogous
structure involving mod p de Rham or crystalline cohomology. Rather, one often
uses the comparison theorem merely to draw much less specific conclusions, such asthat the Galois representations occurring in certain mod p étale cohomology spaces
are in the essential image of the Fontaine–Laffaille functor, applied to Fontaine–
Laffaille modules whose Fontaine–Laffaille numbers lie in some prescribed range.
Our aim is to establish results of the latter type in more general contexts than they
have previously been proved.
The precise direction of our work is informed to a significant extent by the
fairly recent development of a rich internal integral p-adic Hodge theory, by Breuil

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

5[Bre00], Kisin [Kis06], Liu [Liu08] and others. What we mean here by the word
“internal” is that these developments have been directed not so much at applications to comparison theorems, but rather at the purely Galois-theoretic problem ofgiving a p-adic Hodge-theoretic description of Galois-invariant lattices in crystalline
or semistable Galois representations, and of the mod p Galois representations thatappear in the reductions of such lattices. These tools, especially the theory of Breuil
modules [Bre00], which provides the desired description of the mod p representations arising as reductions of such lattices, have proved very useful in arithmetic
applications. Because of the availability of these tools, it has become both possible
and worthwhile to move beyond the Fontaine–Laffaille context in integral p-adic
Hodge theory. While Caruso’s work mentioned above is a significant step in thisdirection, an important aspect of the present work will be the consideration of situations in which the bound ei < p − 1, required for the validity of the comparison
theorem of [Car08], does not hold.
Our goal, then, is to establish in various situations that a Galois representation
appearing in the mod p étale cohomology of a variety can be embedded in the reduction of a Galois-invariant lattice contained in a crystalline or semistable Galois
representation, with Hodge–Tate weights lying in some specified range (namely, therange that one would expect given the degree of the cohomology space under consideration). Since, in arithmetic situations, one frequently has to make a ramified
base-change in order to obtain good or semistable reduction, and since the resulting
descent data on the associated Breuil module typically then play an important rolein whatever analysis has to be undertaken, we also prove results in certain casesof potentially semistable reduction which allow us to gain some control over thesedescent data.
The idea underlying our approach is very simple. Suppose that X is a variety
over a p-adic field K. If i is some degree of cohomology, then we have a short exact
sequencei+1iii
0 → Hét(XK , Zp )/pHét(XK , Zp ) → Hét
(XK , Fp ) → Hét
(XK , Zp )[p] → 0,

as well as an isomorphism∼ii
Qp ⊗Zp Hét
(XK , Zp ) −→ Hét
(XK , Qp ).
i+1i(XK , Zp ) are torsion-free, then we see that
Thus, if both Hét
(XK , Zp ) and Hétii
Hét
(XK , Fp ) is the reduction mod p of Hét
(XK , Zp ), which is a Galois-invariantilattice in Hét (XK , Qp ). Furthermore, the usual comparison theorems of rational
p-adic Hodge theory [Fal89, Tsu99] can be applied to conclude that this latter
representation is e.g. crystalline (if X is proper with good reduction) or semistable(if X is proper with semistable reduction).
The obstruction to implementing this idea is that we have no reason to believei+1
iin general that Hét
(XK , Zp ) and Hét
(XK , Zp ) will be torsion-free. To get aroundthis difficulty, we engage in various dévissages using the weak Lefschetz theorem.
To explain these, first consider the case when X is a projective curve and i = 1.
In this case all the cohomology with Zp -coefficients is certainly torsion-free, and1
1
so Hét(XK , Fp ) is the reduction of a Galois-invariant lattice in Hét
(XK , Qp ). Nowa simple induction using the weak Lefschetz theorem shows that for any smoothprojective variety X over K, there is an embedding
1
1(CK , Fp ),Hét(XK , Fp ) ,→ Hét

6

MATTHEW EMERTON AND TOBY GEE

where C is a smooth projective curve. Furthermore, if X has good (respectively
semistable) reduction, we can ensure that the same is true of C. This gives thedesired result in the case of H 1 (ignoring for a moment the problem of obtaining arefinement dealing with descent data in the potentially semistable case).
For higher degrees of cohomology, a more elaborate dévissage is required. The
key point, again established via the weak Lefschetz theorem, is that if X is smoothand projective of dimension d, and if Y and Z are sufficiently generic hyperplanesections of X, then the cohomology of the pair (X \ Y )K , (Z \ Y )K , with either
Zp or Fp coefficients, vanishes in degrees other than d (see Subsection A.3 of thedappendix), so that Hét
(X \ Y )K , (Z \ Y )K , Zp ) is torsion-free and is thus a Galoisd
invariant lattice in Hét
(X \ Y )K , (Z \ Y )K , Qp ), which is potentially semistable byd[Yam11], and whose reduction is equal to Hét
(X \Y )K , (Z \Y )K , Fp ). Such relative
cohomology spaces are the essential ingredient of the basic lemma of Beı̌linson[Beı̌87], and we learned the idea of using them as building blocks for the cohomology
of varieties from Nori [Nor02], who has used the basic lemma as the foundation of his
approach to the construction of motives. Indeed, our present approach to integral
p-adic Hodge theory was inspired by Beı̌linson’s and Nori’s work.
1.2. Bertini-type theorems. We begin by giving a straightforward generalisation of some of the results of [JS09], which build on the results of [Poo04] to prove
Bertini-type theorems for varieties with semistable reduction over a discrete valuation ring. It will be convenient to allow K to denote either a number field, or afield of characteristic zero that is complete with respect to a discrete valuation withperfect residue field kK of characteristic p. We abbreviate these two situations as“the global case” and “the local case” respectively, and in the former case we will
let v denote a place of K dividing p.
We recall the following definition:
1.2.1. Definition. Suppose firstly that we are in the local case. We then say thata projective OK -scheme X is semistable if it is regular and flat over Spec OK , andif the special fibre Xs is reduced and is a normal crossings divisor; equivalently, a
projective OK -scheme X is semistable if at each closed geometric point x of Xs ,there is an isomorphism of complete local ringsdsh [[x , . . . , x ]]/(x · · · x − $ ),
[ ∼
O
=O
x,X

K

1

n

1

m

K

d
sh
where OK is the completion of the strict Henselisation of OK (equivalently, thecompletion of the ring of integers in the maximal unramified algebraic extensiondshof K), the element $K is a uniformiser of O
K , and 1 ≤ m ≤ n. We say that X isstrictly semistable if it is semistable, and if the special fibre Xs is a strict normal
crossings divisor.
Again in the local case, we say that a smooth projective K-scheme has good
reduction if it admits a smooth projective model over OK , and that it has (strictly)
semistable reduction if it admits an extension to a projective OK -scheme which is(strictly) semistable in the sense of the preceding definition.
In the global case, we say that a smooth projective K-scheme has good reduction
at v if it admits a smooth projective model over OK,(v) , and that it has (strictly)
semistable reduction at v if it admits a (strictly) semistable projective model overOK,(v) , i.e. a projective model over OK,(v) whose base-change over OKv is (strictly)
semistable in the sense of the preceding definition.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

7

1.2.2. Proposition. Let X be a smooth projective variety over K with strictlysemistable (respectively good) reduction (at v, in the global case). Then there aresmooth hypersurface sections Y and Z of X (with respect to an appropriately chosen embedding of X into some projective space) such that Y and Z intersect transversely, and all of Y , Z, Y ∩ Z have strictly semistable (respectively good) reduction(at v, in the global case).
Proof. We firstly handle the local case. Choose an extension X of X to an OK scheme that is projective and smooth (in the good reduction case) or strictlysemistable (in the strictly semistable reduction case), and fix an embedding of
X into some projective space over OK . By Corollary 0 and Corollary 1 of [JS09](or, perhaps more precisely, by their proofs) we can find a hypersurface section Yof X such that Y is again either smooth or strictly semistable over OK . We takeY to be the generic fibre of Y. By Remark 0 (ii) of [JS09], together with Lemma 1
and the remark immediately before Corollary 1 of op. cit., we see that in order
to find Z it is enough to check that given a finite collection X1 , . . . , Xn of smoothprojective schemes in PN/kK , there is a common hypersurface section meeting eachof them transversely. This is an immediate consequence of Theorem 1.3 of [Poo04],
taking the set UP there to be the subset of ÔP consisting of the f such that f = 0
is transverse to each Xi at P . (Since this set contains all the f which do not vanish
at P , it has positive Haar measure.)
We now pass to the global case. Let X be a smooth (in the good reductioncase) or strictly semistable (in the strictly semistable reduction case) projectivemodel of X over OK,(v) . Let Pd∗ denote the projective space (over OK,(v) ) ofdegree d hypersurfaces in the ambient projective space containing X . Applying theargument in the local case to the base-change X/OKv , we see that for some d ≥ 1,
there is a Kv -valued point of Pd∗ corresponding to a hypersurface section of X/OKv
having either smooth or semistable intersection (depending on the case we are in)∗with X/OKv . Furthermore, this point lies in an affinoid open subset of Pd/K
(thev∗
preimage of an open set in the special fibre of Pd ), all of whose points correspond to
hyperplane sections of X/OKv with either smooth or strictly semistable intersection.
(See Remark 0 (i) of [JS09], as well as the proofs of Theorems 0 and 1 of the samereference.) The intersection of the Kv -points of this affinoid open with Pd∗ (Kv )is a non-empty open subset of Pd∗ (Kv ). Since K is dense in Kv , we see that thisintersection also contains a K-point of Pd∗ , which gives the required hypersurface
section Y . We find the hypersurface section Z by applying the same argument. 
1.3. Cohomology in degree 1. Our arguments in degree 1 are rather simpler
than in general degree, so we warm up with this case. (In fact, our result in thiscase is slightly stronger than our result in general degree, as we do not need tosemisimplify the representation, so this result is not completely subsumed by ourlater results in general degree.) Fix a prime p. Let K denote a field of characteristic
zero, complete with respect to a discrete valuation, with ring of integers OK andresidue field k, assumed to be perfect of characteristic p. Let K denote an algebraic
closure of K, and set GK := Gal(K/K).
1.3.1. Theorem. If X is a smooth projective variety over K which has good (re1spectively strictly semistable) reduction, then Hét
(XK , Fp ) embeds GK -equivariantly
into the reduction modulo p of a GK -invariant lattice in a crystalline (respectively

8

MATTHEW EMERTON AND TOBY GEEsemistable) p-adic representation of GK whose Hodge–Tate weights are containedin [−1, 0].
1
Proof. We proceed by induction on the dimension d of X. If d ≤ 1 then Hét
(XK , Fp )1
is isomorphic to the reduction modulo p of Hét (XK , Zp ), and the latter space (being
1
torsion-free, by virtue of our assumption on d) is in turn a lattice in Hét
(XK , Qp ),which is crystalline (respectively semistable) with Hodge–Tate weights lying in[−1, 0], by the main result of [Tsu99].
Suppose now that d > 1. It follows from Corollary 0 (respectively Corollary 1)of [JS09] that if X has good reduction (respectively strictly semistable reduction)then we may choose a smooth hypersurface section Y of X defined over K which
has good (respectively strictly semistable) reduction. Our induction hypothesis
1applies to show that Hét
(YK , Fp ) embeds as a subobject of a GK -representation
over Fp which is the reduction modulo p of a GK -invariant lattice in a crystalline (respectively semistable) p-adic representation of GK whose Hodge–Tateweights are contained in [−1, 0]. On the other hand, the weak Lefschetz Theorem
with Fp -coefficients ([Gro73, XIV 3.3]) implies that the natural (restriction) map
1
1(XK , Fp ) → Hét
(YK , Fp ) is an embedding (because 1 ≤ d − 1 by assumption).
HétThis completes the proof.


1.4. Cohomology of arbitrary degree. As always we fix a prime p. The necessary dévissages in this subsection will be more elaborate than in the previous one,and so, to maximise the utility of our results for later applications, it will be convenient to again allow K to denote either a number field or a field of characteristic
zero that is complete with respect to a discrete valuation with perfect residue fieldof characteristic p. In applications it will also be useful to have flexibility in the
choice of coefficients in the various cohomology spaces that we consider, and to thisend we fix an algebraic extension E of Qp , with ring of integers OE and residue
field kE . (In applications, E will typically either be a finite extension of Qp , or else
will be Qp .)We now recall some consequences of the weak Lefschetz Theorem. Among othernotions, we will use the étale cohomology of a pair consisting of a variety anda closed subvariety; a precise definition of this cohomology, and a verification ofits basic properties (such as those recalled in the next paragraph), is included inAppendix A.
Let X be a smooth projective variety of dimension d over K, and suppose that
Y and Z are two smooth hypersurface sections of X, chosen so that Y ∩ Z is alsosmooth. Let A denote eitherE, OE , or kE . In either the first or last case, the spaces
2d−i
i
Hét
(X \ Y )K , (Z \ Y )K , A and Hét
(X \ Z)K , (Y \ Z)K , A (d) are naturally
dual to one another, for each integer i. The weak Lefschetz Theorem implies thatthe former space vanishes when i > d, and that the latter space vanishes when
2d − i > d, i.e. when i < d. Thus in fact both spaces vanish unless i = d. It thenfollows that both spaces vanish unless i = d in the case when A is taken to be OE
as well, and hence that, when i = d, both spaces are torsion-free.
Let K denote an algebraic closure of K, and set GK := Gal(K/K). Now let
ρ : GK → GLn (kE ) be irreducible and continuous. In the global case, we fix a placev of K lying over p, and a decomposition group Dv ⊂ GK for v.
1.4.1. Theorem. If X is a smooth projective variety over K which has strictlysemistable (respectively good) reduction (at v, if we are in the global case), and ifp-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

9iρ embeds as a subquotient of Hét
(XK , kE ), then ρ also embeds as a subquotient ofa GK -representation over kE which is the reduction modulo the uniformiser of a
GK -invariant OE -lattice in a GK -representation which is semistable (respectively
crystalline) (at v, in the global case) with Hodge–Tate weights contained in theinterval [−i, 0].

Proof. We proceed by induction on the dimension of X. Suppose initially that
we are in the strictly semistable reduction case. By Proposition 1.2.2 we can anddo choose smooth hypersurface sections Y and Z, having smooth intersection, andsuch that Y , Z, and Y ∩ Z all have strictly semistable reduction.
We then consider the long exact sequences (cf. [Mil80, III.1.25] for the first two,
which are local cohomology long exact sequences, and Appendix A for the third,
which is the long exact sequence of the pair (X \ Y, Z \ Y ))
ii
· · · → HYi K ,ét (XK , A) → Hét
(XK , A) → Hét
(X \ Y )K , A → · · · ,iii· · · → H(Y∩Z)K ,ét (ZK , A) → Hét (ZK , A) → Hét (Z \ Y )K , A → · · · ,
and



i
i
i
(X \Y )K , A → Hét
(Z \Y )K , A → · · · ,· · · → Hét
(X \Y )K , (Z \Y )K , A → Hét
with A taken to be either E or kE . We also recall (cf. Section XIV.3 of [Gro73])∼i−2that there are canonical isomorphisms Hét
(YK , A)(−1) −→ HYi ,ét (XK , A) and
K

∼
i−2
i
(Y ∩ Z)K , A (−1) −→ H(YHét∩Z)K ,ét (ZK , A).
When A = E, all the cohomology spaces that appear are potentially semistableiii(XK , E), Hét
(YK , E), and Hét
(ZK , E) arewith[Yam11]. Since Hét
 semistableii
Hodge–Tate weights lying in [−i,0], we see that Hét (X\Y )K , E , Hét (Z\Y )K , E ,i(X \ Y )K , (Z \ Y )K , E are semistable, with Hodge–Tate weights lying inand Hét[−i, 0].
Now taking A = kE , we see that since ρ is irreducible, it appears either as ai−2i(ZK , kE ), as a subquosubquotient of Hét
(YK , kE )(−1), as a subquotient of Héti−1
itient of H (Y ∩ Z)(−1), or else as a subquotient of Hét (X \ Y )K , (Z \ Y )K , kE .
In the first three cases the theorem follows by induction on the dimension. In the
final case, the conclusion follows
from the vanishing theorem noted above; namely,iHét(X \ Y )K , (Z \ Y )K , E is the desired
semistable representation, with invariantii(X \ Y )K , (Z \ Y )K , kE
lattice Hét
(X \ Y )K , (Z \ Y )K , OE , whose reduction Hétcontains ρ.
Finally, suppose that we are in the good reduction case. Again, by Proposition 1.2.2, we can and do choose smooth hypersurface sections Y and Z, havingsmooth intersection, and such that Y , Z, and Y ∩ Z all have good reduction. Applying the same argument as in the previous paragraph, we see by induction on
ithe dimension of X that it is enough to check that Hét
(X \ Y )K , (Z \ Y )K , E
is crystalline; but this follows immediately from Theorem 1.2 of [Yam11]. (Notethat if in the notation of [Yam11] we take D1 = Z, and D2 = Y , then by (A.1.2)
ibelow we see that Hét
(X \ Y )K , (Z \ Y )K , E appears on the left hand side of theHyodo–Kato isomorphism in the statement of Theorem 1.2 of [Yam11], and sincei
we have already shown that Hét
(X \ Y )K , (Z \ Y )K , E is semistable, it is enoughto show that the monodromy operator N vanishes on the right hand side of theHyodo–Kato isomorphism. This follows easily from the definition of this operator

10

MATTHEW EMERTON AND TOBY GEE

as a boundary map, as all objects concerned arise from base change from objects
with trivial log structures.)
1.5. Equivariant versions. In practice, we will need equivariant analogues of thepreceding results. As in the preceding section, we let K denote either a number field(“the global case”) or a field of characteristic zero that is complete with respect toa discrete valuation with perfect residue field of characteristic p (“the local case”).
We let K denote an algebraic closure of K, and set GK := Gal(K/K). In the global
case, we fix a place v of K lying over p, and a decomposition group Dv ⊂ GK for v.
We now put ourselves in the following (somewhat elaborate) situation, which wecall a tamely ramified semistable context, or a tame semistable context for short.
We suppose that X0 and X1 are smooth projective varieties over K, that G isa finite group which acts on X1 , and that π : X1 → X0 is a finite étale morphism
which intertwines the given G-action on X1 with the trivial G-action on X0 , making
X1 an étale G-torsor over X0 .
We suppose further that X0 admits a semistable projective model X0 over OK(in the local case) or over OK,(v) (in the global case). We also suppose that there isa finite extension L of K, and (in the global case) a prime w of L lying over v, suchthat (X1 )/L admits a semistable projective model X1 over OL (in the local case) orover OL,(w) (in the global case) to which the G-action extends, such that π extends
to a morphism X1 → (X0 )/OL which intertwines the G-action on its source with thetrivial G-action on its target, and such that the action of I(L/K)op (respectively
I(Lw /Kv )op in the global case) on (X1 )/L extends to an action on X1 .2
Finally (and most importantly) we assume that the composite morphism
X1 → (X0 )/OL → X0

(1.5.1)(the first being the extension of π, and the second being the natural map) is tamely
ramified along the special fibre (X0 )s , in the sense of [GM71, Def. 2.2.2].
In fact, in our applications we will consider the case that X0 is furthermore
strictly semistable, in which case we will say that we are in a tame strictly semistable
context.
The following lemma gives a more concrete interpretation of the stipulationthat (1.5.1) be tamely ramified along (X0 )s .
1.5.2. Lemma. The morphism (1.5.1) is tamely ramified along (X0 )s if and only if
the following two conditions hold:(1) L (respectively Lw in the global case) is tamely ramified over K (respectively
Kv in the global case), of ramification degree e, say;
(2) For each closed geometric point x1 of the special fibre (X1 )s , with image x0
in (X0 )s , and for some choice of isomorphism(1.5.3)sh∼ d\
O
x0 ,X0 = OK [[x1 , . . . , xn ]]/(x1 · · · xm − $K ),dshwhere $K is a uniformiser of O
K and 1 ≤ m ≤ n, there is a corresponding
isomorphismdsh [[y , . . . , y ]]/(y · · · y − $ ),\ ∼O=O
x1 ,X1

L

1

n

1

m

L

2Note that the tameness condition that we are going to require below ensures that L/K is infact tamely ramified, and hence that I(L/K) is abelian. Thus passing to the opposite group is
not actually necessary here when passing from the action on rings to the action on their Specs,
but we will keep the superscript op in the notation for the sake of conceptual clarity.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

11

d
sh
where $L is a uniformiser of O
L , such that the induced morphism\\(X1 )x1 → (X0 )x0
is defined by the formula xj = yje for 1 ≤ j ≤ m, and xj = yj for m < j ≤ n.
Furthermore, if these equivalent conditions hold, then condition (2) holds for every
choice of isomorphism (1.5.3).
Proof. We first note that if we are in the global case, than relabelling Kv as K and
Lw as L, we may reduce ourselves to proving the lemma in the local case. Thus weassume that we in the local case from now on.
If conditions (1) and (2) (for some choice of isomorphism (1.5.3)) hold, thenthe morphism (1.5.1) is certainly tamely ramified along (X0 )s . (This amounts
to the claim that we can verify tame ramification by passing to formal completions of closed geometric points, which is indeed the case, as follows from [GM71,
Cor. 4.1.5].)
Suppose conversely that (1.5.1) is tamely ramified along (X0 )s . Since this morphism factors through the natural morphism (X0 )/OL → X0 , it follows from [GM71,
Lemma 2.2.5] that this latter morphism is tamely ramified, and hence (e.g. by[GM71, Prop. 2.2.9], although our particular situation is much simpler than thegeneral case of faithfully flat descent for tamely ramified covers considered in that
proposition) that Spec OL → Spec OK is tamely ramified, i.e. that L is tamely
ramified over K, of some ramification degree e. Thus condition (1) holds.
Now choose a closed geometric point x1 of (X1 )s lying over the closed geometricpoint x0 of (X0 )s , and fix an isomorphism of the form (1.5.3). Since X1 → X0
is tamely ramified along the divisor $K = 0 of X0 , Abhyankar’s Lemma [Gro71,
XIII.V] (see also [GM71, Thm. 2.3.2] for a concise statement) implies that wedsh
may find regular elements {aj }j=1,...,k of OK [[x1 , . . . , xn ]]/(x1 · · · xm − $K ) so thatdsha · · · a generates the ideal ($ ) of O [[x , . . . , x ]]/(x · · · x − $ ), exponents
1

k

K

K

1

n

1

m

K

e1 , . . . , ek all coprime to p, and a subgroup H ⊂ µe1 × · · · × µek , such that thedsh [[x , . . . , x ]]/(x · · · x − $ )-algebra O\ is isomorphic to
O
K

1

n

1

m

x1 ,X1

K

H
ek
e1
d
sh
.
O
K [[x1 , . . . , xn ]][T1 , . . . , Tk ]/(x1 · · · xm − $K , T1 − a1 , . . . , Tk − ak )dsh(Here µe1 ×· · ·×µek , and hence H, acts on O
K [[x1 , . . . , xn ]][T1 , . . . , Tk ]/(x1 · · · xm −eke1
$K , T1 − a1 , . . . , Tk − ak ) in the obvious manner: namely, an element (ζ1 , . . . , ζk )acts on Tj via multiplication by ζj .)dsh [[x , . . . , x ]]/(x · · · x
Since each e is prime to p and O− $ ) is strictly
j

K

1

n

1

m

K

Henselian, any unit in this ring has an ej th root, and thus we are free to multiply
any of the aj by a unit. Consequently, we may assume that in fact a1 · · · ak = $K =x1 · · · xm , and hence (again taking advantage of our freedom to modify the
Q aj byunits) that {1, . . . , m} is partitioned into k sets J1 , . . . , Jk , such that aj = i∈Jj xi .
Now if we extract the ej th roots of each xi for i ∈ Jj , the resulting extension ofdsh [[x , . . . , x ]]/(x · · · x − $ ) contains
O
K

1

n

1

m

K

ek
e1
d
sh
O
K [[x1 , . . . , xn ]][T1 , . . . , Tk ]/(x1 · · · xm − $K , T1 − a1 , . . . , Tk − ak );12

MATTHEW EMERTON AND TOBY GEE

thus it is no loss of generality to assume that k = m and that aj = xj , and sodsh [[x , . . . , x ]]/(x · · · x − $ )\ is isomorphic, as an O
we conclude that O
x1 ,X1

K

1

n

1

m

K

algebra, to
H
e1
em
d
sh
O
,
K [[x1 , . . . , xn ]][T1 , . . . , Tm ]/(x1 · · · xm − $K , T1 − x1 , . . . , Tm − xm )for some subgroup H ⊂ µe1 × · · · × µem .
Let Ij denote the subgroup 1 × · · · × µej × · · · × 1 of µe1 × · · · × µem ; this is theinertia group of the divisor (xj ) with respect to the cover
e1
em
d
sh
Spec OK [[x1 , . . . , xn ]][T1 , . . . , Tm ]/(x1 · · · xm − $K , T1 − x1 , . . . , Tm − xm )dsh→ Spec O
K [[x1 , . . . , xn ]]/(x1 · · · xm − $K ).
If we write Hj = H ∩ Ij , then H 0 := H1 × · · · × Hm is a subgroup of H, and the
cover
H 0
e1
em
d
sh
Spec OK [[x1 , . . . , xn ]][T1 , . . . , Tm ]/(x1 · · · xm − $K , T1 − x1 , . . . , Tm − xm )
H
e1
em
d
sh
→ Spec O
K [[x1 , . . . , xn ]][T1 , . . . , Tm ]/(x1 · · · xm −$K , T1 −x1 , . . . , Tm −xm )is unramified in codimension one. Since X1 is regular, being semistable over OL , so\is the target of this map (since we recall that this target is isomorphic to (X1 )x1 ).
The purity of the branch locus then implies that this cover is étale, and hence is
an isomorphism (since its target is strictly Henselian). Consequently H = H 0 .
If we write
Hj = 1 × · · · × µe0j × · · · × 1 ⊂ 1 × · · · × µej × · · · × 1 = Ij ,e0and set dj = ej /e0j and Sj = Tj j , then we conclude that\O
x1 ,X1
H
e1
em
d
sh
∼
= O
K [[x1 , . . . , xn ]][T1 , . . . , Tm ]/(x1 · · · xm − $K , T1 − x1 , . . . , Tm − xm )

0

d1
dm
d
sh
∼
=O
K [[x1 , . . . , xn ]][S1 , . . . , Sm ]/(x1 · · · xm − $K , S1 − x1 , . . . , Sm − xm ).

Now X1 is an OL -scheme with reduced special fibre (again because it is semistabledsh\over OL ). Since O
x1 ,X1 is strictly Henselian, it contains OL , and we may choose aeuniformiser $L of this ring such that $L = $K . Looking at the above description\of O
x1 ,X1 , and taking into account that its reduction modulo $L must be reduced,
we see that this special fibre must be the zero locus of the element S1 · · · Sm , hence
that S1 · · · Sm = u$L for some unit u, and thus that (S1 · · · Sm )e = ue $K . We
conclude that d1 = . . . = dm = e and that ue = 1, and hence, replacing $L by
u$L , we find that S1 · · · Sm = $L . This shows that (2) holds (for our given choiceof isomorphism (1.5.3)).

1.5.4. Remark. Note that we could avoid the appeal to the general theory of tameramification (in particular, to Abhyankar’s Lemma) by just directly stipulating in
our context that conditions (1) and (2) of Lemma 1.5.2 hold; indeed, in the proofof Theorem 1.5.14 below, we will work directly with these conditions, and in ourapplications to Shimura varieties, we will also see directly that these conditions
hold. Nevertheless, we have included Lemma 1.5.2 as an assurance to ourselves(and perhaps to the reader) that these conditions are somewhat natural.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

13

1.5.5. Lemma. In a tame strictly semistable context as above, the OL -scheme X1
is also strictly semistable.
Proof. Since X1 is semistable by assumption, it is enough to show that the components of the special fibre (X1 )s are regular. Suppose that D is a non-regular component of (X1 )s , and let x1 be a closed geometric point of D whose local ring on D
is not regular. If we let x0 denote the image of x1 in (X0 )s , then Lemma 1.5.2 (2)∼shows that we may find isomorphisms O\
x1 ,(X1 )s = OK /$K [[y1 , . . . , yn ]]/(y1 · · · ym )∼and O\
x ,(X ) = OK /$K [[x1 , . . . , xn ]]/(x1 · · · xm ), with 1 ≤ m ≤ n, such that the
0

0 s

e
\
morphism O\
x1 ,(X1 )s → Ox0 ,(X0 )s is given by xj = yj for 1 ≤ j ≤ m and xj = yj
for m < j ≤ n. Since by assumption x1 is not a regular point of D, we findthat necessarily m ≥ 2, and that (possibly after permuting indices) there is an0∼\
isomorphism O
x1 ,D = OK /$K [[y1 , . . . , yn ]]/(y1 · · · ym0 ), where 2 ≤ m ≤ m. If we0
let D denote the image of D in (X0 )s , we conclude that there is an isomorphism
0∼\O
x0 ,D 0 = OK /$K [[x1 , . . . , xn ]]/(x1 · · · xm0 ), and thus that D is not regular. Hence
X0 is not strictly semistable, a contradiction.


We now suppose that we are in a tame semistable context, as described above,and suppose for the moment that we are in the local case. Then we have an actionof I(L/K)op × G on the special fibre (X1 )s . Let D be an irreducible component ofe denote its preimage in (X1 )s , so that D
e is anthe special fibre of (X0 )s , and let D
op
I(L/K) × G-invariant union of irreducible components of (X1 )s .
1.5.6. Lemma. If G is abelian, then there is a homomorphism ψ : I(L/K) → G
e is given by composing the action of G withsuch that the action of I(L/K)op on De coincides with the action of ψ(i).
ψ; i.e. if i ∈ I(L/K), then the action of i on D
We first prove a general lemma.
1.5.7. Lemma. Let S be a connected Noetherian scheme, let G and I be finitegroups, and let f : T → S be a finite étale morphism with the property that I op × G
acts on T over S in such a way that T becomes a G-torsor over S. If G is abelian,then there exists a morphism ψ : I → G such that the action of I on T is given bycomposing the action of G on T with the morphisms ψ.
Proof. For clarity, we will not impose the assumption that G is abelian until it isrequired.
If we fix a geometric point s of S, then the theory of the étale fundamental group[Gro71, Théorème V.4.1] shows that passing to the fibre over s gives an equivalenceof categories between the category of finite étale covers of S and the category of(discrete) finite sets with a continuous action of π1 (S, s). In this way, T is classified
by an object P of this latter category equipped with an action of I op × G, withrespect to which the G-action makes P a principal homogeneous G-set.
If we fix a base-point p ∈ P , then we may identify P with G, thought of asa principal homogeneous G-set via left multiplication. As the automorphisms ofG as a principal homogeneous G-set are naturally identified with Gop acting by
right multiplication, we obtain a homomorphism ψp : I op → Gop , or equivalently
a homomorphism ψp : I op → G, describing the action of I op on P . If we replace
p by g · p (for some g ∈ G), then one finds that ψgp = gψp g −1 . Thus, if we now
assume furthermore that G is abelian, then ψp = ψgp , and so it is reasonable in this14

MATTHEW EMERTON AND TOBY GEEcase to write simply ψ for this homomorphism, which is well-defined independentlyof the choice of base-point for P . Furthermore, when G is abelian, left and rightmultiplication by an element g ∈ G coincide, and so the action of I op on P is givenby the formula i · p = ψ(i) · p for all p ∈ P . Since the automorphisms of T over S
induced by i and ψ(i) coincide on P , they in fact coincide on all of T .
Proof of Lemma 1.5.6. The morphism X1 → (X0 )/OL is étale on generic fibres, andthe explicit local formulas for this morphism provided by Lemma 1.5.2 show that
it is in fact étale over an open subset U0 of (X0 )/OL whose intersection with thespecial fibre (X0 )/OL s is Zariski dense. Replacing U0 with the intersection of allof its I(L/K)op -translates, we may furthermore assume that U0 is invariant under
the action of I(L/K)op on (X0 )/OL .
If we let U1 denote the preimage of U0 in X1 , then U1 is invariant under the
I(L/K) × G-action on X1 , and the morphism U1 → U0 is a finite étale cover, forwhich the corresponding map U1 → U0 on generic fibres realizes U1 as a G-torsor
over U0 . It follows that the G-action on U1 realizes U1 as an étale G-torsor over U0 ,and hence, passing to special fibres, that (U1 )s is an étale G-torsor over (U0 )s .
Now the induced I(L/K)op -action on (U0 )s is trivial, and so I(L/K)op acts on(U1 )s as a group of automorphisms of the G-torsor (U1 )s over (U0 )s . If D0 :=e 0 := De ∩ (U1 )sD ∩ (U0 )s , then D0 is an irreducible component of (U0 )s , and D00ee
is the restriction of (U1 )s to D . Thus D → D is again an étale G-torsor, withan action of I(L/K)op via automorphisms. Lemma 1.5.7 then shows that there
is a homomorphism ψ : I(L/K)op → Gop , or equivalently a homomorphism ψ :e 0 is given by
I(L/K) → G, such that the action of I(L/K)op on the points of D
e is equal to thecomposing the action of G with the homomorphism ψ. Since D
e 0 in (X1 )s , the claim of the lemma follows.
Zariski closure of D

1.5.8. Lemma. Suppose that we are in a tame strictly semistable context. If g ∈I(L/K) × G and D is a component of (X1 )s , then D and gD either coincide or are
disjoint.
Proof. The images of D and gD in (X0 )s coincide, and it follows from Lemma 1.5.2
that two distinct components of (X1 )s that have non-empty intersection must havedistinct images in (X0 )s .
If we now suppose that we are in the global case, then the discussion applies
with L/K everywhere replaced by Lw /Kv , and in particular for each component D
we may define a character ψ : I(Lw /Kv ) → G describing the action of I(Lw /Kv )on D.
Our next result describes how our tame semistable context behaves upon passage
to a semistable hypersurface section of X0 . In its statement we assume for simplicity
that we are in the local case.
1.5.9. Proposition. Suppose that we are in the tamely ramified semistable context
described above, and let Y0 be a regular hypersurface section of X0 such that theunion of Y0 and (X0 )s forms a divisor with normal crossings on X0 . Let Y0 denotethe generic fibre of Y0 , let Y1 denote the preimage of Y0 under the morphism π :X1 → X0 , and let Y1 be the preimage of Y0 under (1.5.1) (so that (Y1 )/L is thegeneric fibre of the OL -scheme Y1 ).
(1) The complement of Y1 in X1 is affine.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

15(2) The generic fibre Y0 of Y0 is smooth over K, the morphism Y1 → Y0 is anétale G-torsor (so in particular Y1 is also smooth over K), Y0 is a semistablemodel of Y0 over OK , Y1 is a semistable model for (Y1 )/L over OL , andthe morphism Y1 → Y0 is tamely ramified; consequently Y1 → Y0 is again
a tamely ramified semistable context.
(3) Suppose that G is abelian. If D0 is an irreducible component of (Y1 )s ,contained in an irreducible component D of (X1 )s , then the homomorphismψ : I(L/K) → G, which describes the action of I(L/K) on D, also describes
the action of I(L/K) on D0 .
Proof. Since Y0 is a hypersurface section of X0 , its generic fibre Y0 is a hypersurface
section of X0 . Thus its complement is affine. Since π is a finite morphism byassumption, the complement of Y1 in X1 is again affine. Since Y0 is a regularprojective K-scheme (being the generic fibre of Y0 , which is regular by assumption),
it is in fact smooth over K. By definition Y1 is the preimage of Y0 under themorphism X1 → X0 , which is an étale G-torsor by assumption. Thus Y1 → Y0 is
indeed an étale G-torsor (and so Y1 is also smooth over K).
\
Let x0 be a closed geometric point of the special fibre (Y0 )s . Since ((X0 )x0 )s ∪\\(Y0 )x0 forms a divisor with normal crossings, since each component of ((X0 )x0 )s isregular, and since Y0 is regular by assumption, it follows from [GM71, Lem. 1.8.4]\\\
that ((X0 )x0 )s ∪ (Y0 )x0 is in fact a divisor with strictly normal crossings in (X0 )x0 ,\and hence the local equation ` of (Y0 )x0 , together with the elements x1 , . . . , xm that\cut out the irreducible components of ((X0 )x0 )s , form part of a regular system of\\parameters for Ox0 ,X0 . Thus we may choose a model of the form (1.5.3) for (X0 )x0\
for which m < n and in which ` is equal to the element xn ; i.e. in which (Y)
0 x0 isthe zero locus of the element xn .
We now choose a closed geometric point x1 of (X1 )s lying over x0 , as well as\\a model for the tamely ramified morphism (X1 )x1 → (X0 )x0 as in part (2) ofLemma 1.5.2. Thus this morphism has the form
d
d
sh
sh
Spec OL [[y1 , . . . , yn ]]/(y1 · · · ym − $L ) → Spec OK [[x1 , . . . , xn ]]/(x1 · · · xm − $K ),with xj = yje for 1 ≤ j ≤ m, and xj = yj for m < j ≤ n. In particular, we see thatxn = yn , and thus we see that the induced morphism\\(Y1 )x1 → (Y0 )x0(1.5.10)can be written as(1.5.11)dshSpec OL [[y1 , . . . , yn−1 ]]/(y1 . . . ym − $L )dsh
→ Spec O
L [[x1 , . . . , xn−1 ]]/(x1 . . . xm − $K ).

Thus we see that Y0 and Y1 are indeed semistable models of their generic fibres(over OK and OL respectively), and that the morphism Y1 → Y0 is tamely ramified.
This completes the verification of (2). The claim of (3) follows from the fact thatthe action of I(L/K)op × G on (Y1 )s is the restriction of the corresponding action
on (X1 )s , together with the fact that any component of (Y1 )s is contained in acomponent of (X1 )s .


16

MATTHEW EMERTON AND TOBY GEE

We now suppose that E is an algebraic extension of Qp containing K0 . Recall
that if ρ : GK → GLn (E) is a potentially semistable representation, then we may
attach a Weil–Deligne representation WD(ρ) to ρ by first passing to the potentiallysemistable Dieudonné module Dpst (ρ) of ρ, which is a module over E ⊗Qp K0 , thenfixing an embedding K0 ,→ E, and hence a projection pr : E ⊗Qp K0 → E, and,
finally, forming WD(ρ) := E ⊗E⊗Qp K0 ,pr Dpst . (See for example Appendix B of
[CDT99] for a detailed discussion of this construction.) Although WD(ρ) depends
on the choice of the embedding K0 ,→ E, up to isomorphism it is independent of
this choice.
In the tame strictly semistable case, the following result will allow us to describe
the inertial part of the Weil–Deligne representation associated to the p-adic étale
cohomology of X1 , or of a pair (X1 , Y1 ) that arises in the context of the preceding
proposition. Before stating the result we introduce some additional notation, andan additional assumption.
Assume that G is abelian, and let J denote the set of I(L/K) × G orbits on theset of irreducible components of (X1 )s , and let Dj (for j ∈ J) denote the unionof the components lying in the orbit labelled by j. Let ψj : I(L/K) → G be thehomomorphism provided by Lemma 1.5.6, describing the action of I(L/K) on the
points of Dj .
1.5.12. Proposition. Suppose that we are in a tame strictly semistable context as
above. Either let W denote the Weil–Deligne representation
associated to the poten
itially semistable GK -representation Hét
(X1 )/K , E , or else suppose that we are in
the context of Proposition 1.5.9, and let W denote the Weil–Deligne representation
iassociated to the potentially semistable GK -representation Hét
(X1 )/K , (Y1 )/K ), E
(here i is some given degree of cohomology); in either case W is a representationof the product WDK × G. Assume furthermore that G is abelian.
Then, if, as in the above discussion, Je denotes the set of I(L/K) × G-orbitsof
L irreducible components of (X1 )s , we may decompose W as a direct sum W =ej , such that on We
j , the action of the inertia group in WK is obtained byj∈Je Weψjecomposing the G-action on Wej with the homomorphism IK → I(L/K) −→ G.
Proof. Since the action of the inertia subgroup of WK on W factors through afinite group, and representations of a finite group over a field of characteristic
zero are semisimple, the claimed property of W is stable under the formation ofsubobjects, quotients, and extensions (in the category of WK × G-representations).
A consideration of the long exact sequence of cohomology associated to the pair
(X1 , Y1 ) (cf. Appendix A) then reduces the claim for the cohomology of the pairto the claim for the cohomology of X1 and Y1 individually. Since Proposition 1.5.9
shows that the strictly semistable model Y1 of (Y1 )/L behaves in an identical mannerto the strictly semistable model of X1 of (X1 )/L , it in fact suffices to consider thecase of X1 .
Thus we now restrict our attention to the WK -representation W underlyingithe potentially semistable Dieudonné module associated to Hét
(X1 )/K , E . By
[Tsu99] this Dieudonné module is naturally identified with the log-crystalline co×homology H i (X1 )×/W(k)⊗W (k) E of the special fibre (X1 )s with its naturals
log-structure. Lemma 1.5.8 shows that if an intersection of distinct components ofthe special fibre (X1 )s is non-empty, then the various components appearing must

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY17

lie in mutually distinct orbits of I(L/K) × G acting on the set of components. Recalling that J denotes the indexing set for the collection {Dj }j∈J of I(L/K) × G-orbits of components of (X1 )s , this log-crystalline cohomology may be computedby the following spectral sequence of [Mok93]:
E1−m,i+m =M
i−2l−mHcris
Dj1 ∩· · ·∩Dj2l+m+1 /W (k) ⊗W (k) E(−l−m)l≥max{0,−m}{j1 ,...,j2l+m+1 }⊂J×⊗W (k) E.
=⇒ H i (X1 )×
s /W (k)Each of the summands in the E1 -term is naturally an I(L/K)×G-representation,and furthermore the action of I(L/K) is given by the composite of the action of G
with one of the characters ψej . Thus the E1 -terms of this spectral sequence satisfies
the claimed property of W . Thus W also satisfies this property, since it is obtained
as a successive extension of subquotients of these E1 -terms.

We are now ready to prove our equivariant versions of Theorems 1.3.1 and 1.4.1.
For the first result, we place ourselves in the local case (since the global case immediately reduces to the local case by passing from K to Kv ).
1.5.13. Theorem. Suppose that we are in the tame strictly semistable
context de1(X1 )K , Fp embeds GK × Gscribed above. Then the GK × G-representation Hétequivariantly into the reduction modulo the uniformiser of a GK × G-invariant
OE -lattice in a representation V of GK × G over E, having the following properties:(1) the restriction of V to GL is semistable, with Hodge–Tate weights containedin the interval [−1, 0];(2) The Weil–Deligne representation associated to V , which is naturally a representation of WDK × G, whenLrestricted to a representation of IK × G,
can be written as a direct sum ej∈Je Wej of IK × G-representations, whereej runs over the same index set that labels the set of I(L/K) × G-orbits ofirreducible components of (X1 )s , such that on Wej , the action of the inertia group in WK is obtained by composing the G-action on Wej with theψjehomomorphism IK → I(L/K) −→ G.
Proof. We follow the proof of Theorem 1.3.1, proceeding by descending induction
on the dimension of X0 and X1 , and passing to appropriately chosen hypersurface
sections Y0 of X0 , and their corresponding preimages Y1 in X1 and Y1 in (X1 )/L .
Taking into account Proposition 1.5.9,
 we thus reduce to the case1 when X0 and
1(X1 )/K , Zp ,X1 are curves, so that Hét
(X1 )/K , Fp is the reduction mod p of Hét1which is in turn a lattice in Hét
(X1 )/K , Qp . This latter representation is potentially semistable with Hodge–Tate weights in [−1, 0], by [Tsu99]; the claim regarding
Weil–Deligne representations follows from Proposition 1.5.12.
For our second result, we allow ourselves to be in either one of the local or global
contexts.
1.5.14. Theorem. Suppose that we are in the tame strictly semistable context described above, and let ρ : GK × G → GLn (kE ) be an irreducible and continuous

18

MATTHEW EMERTON AND TOBY GEEirepresentation that embeds as a subquotient of Hét
(X1 )K , kE . Then ρ also embeds as a subquotient of a GK × G-representation over kE which is the reduction
modulo the uniformiser of a GK × G-invariant OE -lattice in a representation V of
GK × G over E, having the following properties:(1) the representation V becomes semistable when restricted to GL (respectively
to the decomposition group Dw ⊂ GL in the global case), with Hodge–Tateweights contained in the interval [−i, 0];(2) The Weil–Deligne representation associated to V , which is naturally a representation of WDK × G (respectively of WDKv × G in the global case),
when restricted to a representation of IK × G (respectively
of IKv × G
Lin the global case), can be written as a direct sum
Weejj of IK × Grepresentations (respectively of IKv × G-representations), where ej runs overthe same index set that labels the set of I(L/K) × G-orbits (respectively of
IKv ×G-orbits) of irreducible components of (X1 )s , such that on Wej , the action of the inertia group is obtained by composing the G-action on Wej withψje

the homomorphism IK → I(L/K) −→ G (respectively the homomorphism
ψje

IKv → I(Lw /Kv ) −→ G in the global case).
Proof. We can be proved in exactly the same way as Theorem 1.4.1, taking into
account Propositions 1.5.9 and Proposition 1.5.12.

2. Breuil modules with descent data
In this section we establish a result (Theorem 2.2.4 below) which imposes someconstraints on the reductions of certain tamely potentially semistable p-adic representations of GQp .
2.1. Preliminaries. We begin by recalling a variety of results from Section 3of [EGH11]. To this end, let p be an odd prime, let Qp be a fixed algebraic closureof Qp , and let E and K be finite extensions of Qp inside Qp . Assume that E contains the images of all embeddings K ,→ Qp . Let K0 be the maximal absolutely
unramified subfield of K, so that K0 = W (k)[1/p], where k is the residue fieldof K. Let K/K 0 be a Galois extension, with K 0 a field lying between Qp and K.
Assume further that K/K 0 is tamely ramified with ramification index e, and fix auniformiser π ∈ K with π e ∈ K 0 . Let E(u) ∈ W (k)[u] be the minimal polynomialof π over K0 .
Let kE be the residue field of E, and let 0 ≤ r ≤ p − 2 be an integer. Recall thatthe category kE -BrModrdd of Breuil modules of weight r with descent data from K
to K 0 and coefficients kE consists of quintuples (M, Mr , ϕr , ĝ, N ) where:• M is a finitely generated (k ⊗Fp kE )[u]/uep -module, free over k[u]/uep .
• Mr is a (k ⊗Fp kE )[u]/uep -submodule of M containing uer M.
• ϕr : Mr → M is kE -linear and ϕ-semilinear (where ϕ : k[u]/uep → k[u]/uepis the p-th power map) with image generating M as a (k ⊗Fp kE )[u]/uep module.
• N : M → M is k ⊗Fp kE -linear and satisfies N (ux) = uN (x) − ux for allx ∈ M, ue N (Mr ) ⊂ Mr , and ϕr (ue N (x)) = cN (ϕr (x)) for all x ∈ Mr .
Here, c = F̄ (u)p ∈ (k[u]/uep )× , where E(u) = ue + pF (u) in W (k)[u].

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

19• ĝ : M → M are additive bijections for each g ∈ Gal(K/K 0 ), preserving
Mr , commuting with the ϕr - and N -actions, and satisfying ĝ1 ◦ ĝ2 = g\
1 ◦ g2for all g1 , g2 ∈ Gal(K/K 0 ). Furthermore, if a ∈ k ⊗Fp kE , m ∈ M thenĝ(aui m) = g(a)((g(π)/π)i ⊗ 1)ui ĝ(m).
rThere is a covariant functor T∗,r
st from kE -BrModdd to the category of kE -representations of GK 0 .
2.1.1. Lemma. Suppose that M ∈ kE -BrModrdd , and that T 0 is a GK 0 -subrepre0
sentation of T∗,rst (M) (so that in particular T has the structure of a kE -vector
0space). Then there is a unique subobject M of M such that if f : M0 → M is the∗,r0
inclusion map, then T∗,rst (f ) is identified with the inclusion T ,→ Tst (M). (HereM0 is a subobject of M in the naive sense that it is a sub-(k ⊗Fp kE )[u]/uep -moduleof M, which inherits the structure of an object of kE -BrModrdd from M in theobvious way.)
Proof. This is Corollary 3.2.9 of [EGH11].



We now specialise to the particular situation of interest to us in this paper,
namely we let K0 be the unique unramified extension of Qp of degree d, we take K =ddK0 ((−p)1/(p −1) ), and we set K 0 = K0 , so that e = pd − 1. Fix π = (−p)1/(p −1) .
We write ω
ed : Gal(K/K0 ) → K0× for the character g 7→ g(π)/π, and we let ωd be
the reduction of ωed modulo π. (By inflation we can also think of ω
ed and ωd as
characters of IK0 = IQp . Note that ωd is a tame fundamental character of niveau dand that ω
ed is the Teichmüller lift of ωd .) Note that when d = 1, we have ω1 = ω,the mod p cyclotomic character.
Let ϕ be the arithmetic Frobenius on k, and let σ0 : k ,→ kE be a fixed embedding. Inductively define σ1 , . . . , σd−1 by σi+1 = σi ◦ ϕ−1 ; we will often consider thenumbering to be cyclic, so that σd = σ0 . There are
L idempotents ei ∈ k ⊗Fp kE suchthat if M is any k ⊗Fp kE -module, then M = i ei M , and ei M is the subset of
M consisting of elements m for which (x ⊗ 1)m = (1 ⊗ σi (x))m for all x ∈ k. Notethat (ϕ ⊗ 1)(ei ) = ei+1 for all i.
If ρ : GK0 → GLn (E) is a potentially semistable representation which becomes
semistable over K, then the associated inertial type (that is, the restriction to
IK0 of the Weil–Deligne representation associated to ρ) is a representation of IK0
which becomes trivial when restricted to IK , so we can and do think of it as arepresentation of Gal(K/K0 ) ∼= IK0 /IK .
2.1.2. Proposition. Maintaining our current assumptions on K, suppose that
ρ : GK0 → GLn (E) is a continuous representation whose restriction to GK issemistable with Hodge–Tate weights contained in [0, r], with r ≤ p − 2, and let theinertial type of ρ be χ1 ⊕ · · · ⊕ χn , where each χi is a character of IK0 /IK . Ifρ denotes the reduction modulo mE of a GK0 -stable OE -lattice in ρ, then there isan element M of kE -BrModrdd , admitting a (k ⊗Fp kE )[u]/uep -basis v1 , . . . , vn such∼that ĝ(vi ) = (1 ⊗ χi (g))vi for all g ∈ Gal(K/K0 ), and for which T∗,rst (M) = ρ.
Proof. This is Proposition 3.3.1 of [EGH11]. (Note that the conventions on the signof the Hodge–Tate weights in [EGH11] are the opposite of the conventions in thispaper.)

2.1.3. Lemma. Maintain our current assumptions on K, so that in particular wehave e = pd − 1. Then every rank one object of kE -BrModrdd may be written in the
form:

20

MATTHEW EMERTON AND TOBY GEE• M = ((k ⊗Fp kE )[u]/uep ) · m,
• ei Mr = uri ei M,
Pd−1• ϕr ( i=0 uri ei m) = λm for some λ ∈ (k ⊗Fp kE )× ,Pd−1
• ĝ(m) = ( i=0 (ωd (g)ki ⊗ 1)ei )m for all g ∈ Gal(K/K0 ), and• N (m) = 0.
Here the integers 0 ≤ ri ≤ (pd − 1)r and ki satisfy ki ≡ p(ki−1 + ri−1 ) mod(pd − 1) for all i. Conversely, any module M of this form is a rank one object of
kE -BrModrdd . Furthermore,∼ σ0 ◦ ω κ0 ,T∗,r (M)|I =stwhere κ0 ≡ k0 + p(r0 pd−1+ r1 pd−2

dK0+ · · · + rd−1 )/(pd − 1) mod (pd − 1).

Proof. This is Lemma 3.3.2 of [EGH11].



2.1.4. Remark. In the sequel, we will only be interested in the case that for each
i we have ki = (1 + p + · · · + pd−1 )xi for some 0 ≤ xi < p − 1. In this case,the condition that pri ≡ ki+1 − pki mod (pd − 1) implies that ri ≡ (1 + p + · · · +pd−1 )(xi+1 − xi ) mod (pd − 1), so the condition that 0 ≤ ri ≤ (pd − 1)r means that
we can write ri = (1 + p + · · · + pd−1 )(xi+1 − xi ) + (pd − 1)yi with 0 ≤ yi ≤ r. An
elementary calculation shows that we then have
κ0 ≡ x0 + y0 + pd−1 (x1 + y1 ) + · · · + p(xd−1 + yd−1 ) mod (pd − 1).
2.2. Regularity. Let Qpn denote the unique unramified extension of Qp of degree n, with residue field Fpn . Regarding Fpn as a subfield of Fp , we may then×
regard ωn as a character IQp → Fp .
2.2.1. Definition. Let ρ : GQp → GLn (Fp ) be an irreducible representation, so that
GQ

×

ρ∼
= IndGQpn χ for some character χ : GQp → Fp . If we writepn−1

χ|IQp = ωn(a0 +pa1 +···+pan−1 ),

where each ai ∈ [0, p − 1] and not all the ai = p − 1, then the multiset of exponentsof ρ is defined to be the multiset of residues of the ai in Z/pZ.
2.2.2. Definition. Let ρ : GQp → GLn (Fp ) be a representation. Then the multisetof exponents of ρ is the union of the multisets of exponents of each of the Jordan–Hölder factors of ρ.
2.2.3. Definition. Let ρ : GQp → GLn (Fp ) be a representation, and let r be a nonnegative integer. Then we say that ρ is r-regular if the exponents a1 , . . . , an of ρ
are such that the residues ai + k ∈ Z/pZ, 1 ≤ i ≤ n, 0 ≤ k ≤ r + 1, are pairwise
distinct.
The following theorem is the main result we will need from explicit p-adic Hodge
theory.
2.2.4. Theorem. Let r be a non-negative integer, and let s : GQp → GLm (Qp ) bea potentially semistable representation, with Hodge–Tate weights contained in theinterval [0, r], and inertial type χ1 ⊕· · ·⊕χm . Suppose that there are (not necessarily
distinct) integers 0 ≤ a1 , . . . , an < p − 1 such that each χi is equal to some ωe aj .
Suppose that ρ : GQp → GLn (Fp ) is a subquotient of s, the reduction mod mOQpof some GQp -stable Zp -lattice in s. Suppose also that

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

21• det ρ|IQp = ω a1 +···+an +n(n−1)/2 ,• r ≤ (n − 1)/2, and• p > n(n − 1)/2 + 1.
If r = (n − 1)/2 then assume further that some irreducible subquotient of ρ hasdimension greater than one. Then ρ is not r-regular.
Proof. Modifying the choice of Zp -lattice if necessary, it suffices to treat the case∼ IndGQp χ be an irreducible subrepresentationthat ρ and s are semisimple. Let ρ0 =
GQ d
p

0

of ρ. Take K = K0 = Qpd in the above notation, so that e = pd −1. Taking E to be
sufficiently large so that s is defined over E and ρ is defined over kE , and applying
Proposition 2.1.2 to s|GQ d , we see that there is an element M of kE -BrModrdd with
p

∼
T∗,r
st (M) = s|GQ d ,pepsuch that M has a (k ⊗Fp kE )[u]/u -basis v1 , . . . , vm such that ĝ(vi ) = (1⊗χi (g))vi
for all g ∈ Gal(K/K0 ). Since s|GQ d contains a subrepresentation isomorphic to χ,p
we see from Lemma 2.1.1 that there is a rank one subobject N of M for which∼T∗,rst (N ) = χ.
Since N /uN embeds into M/uM (as N is a free k[u]/uep -submodule of thefree k[u]/uep -module M), we see from Lemma 2.1.3 and our assumption on thecharacters χi that we may write N in the formd

• N = ((k ⊗Fp kE )[u]/u(p −1)p ) · w,
• ei Nr = uri ei N ,Pd−1• ϕr ( i=0 uri ei w) = λw for some λ ∈ (k ⊗Fp kE )× ,Pd−1
• ĝ(w) = ( i=0 (ωd (g)ki ⊗ 1)ei )w for all g ∈ Gal(K/K0 ), and• N (w) = 0.
Here the integers 0 ≤ ri ≤ (pd − 1)r and ki satisfy ki ≡ p(ki−1 + ri−1 ) (mod pd − 1)for all i, and each ki is equal to some (1 + p + · · · + pd−1 )aj . (The conditions
on the ri come from Lemma 2.1.3, and the fact that each ki is equal to some
(1 + p + · · · + pd−1 )aj comes from the fact that N is a submodule of M, which has
a basis v1 , . . . , vn such that ĝ(vi ) = (1 ⊗ χi (g))vi , and the assumption that each χiis equal to some ω aj .) Writing ki = (1 + p + · · · + pd−1 )xi , 0 ≤ xi < p − 1, we see as
in Remark 2.1.4 that we can write ri = (1 + p + · · · + pd−1 )(xi+1 − xi ) + (pd − 1)yi ,with 0 ≤ yi ≤ r. By Lemma 2.1.3 and Remark 2.1.4, we havex +y0 +pd−1 (x1 +y1 )+···+p(xd−1 +yd−1 )χ|IQp = σ0 ◦ ωd 0.

Since r ≤ p − 2 we have 0 ≤ yi ≤ p − 2, and we conclude (after allowing for“carrying”) that each exponent of ρ is of the form aj + k, with 0 ≤ k ≤ r + 1.
Suppose that ρ is r-regular. It must then be the case that the xi as above are
all distinct. Applying this analysis to each irreducible subrepresentation of ρ, wed−1
conclude that the ai are all distinct. Since we have det(ρ0 )|IQp = χ1+p+···+p |IQp =ω (x0 +y0 )+···+(xd−1 +yd−1 ) , we conclude that det ρ|IQp = ω a1 +···+an +y for some 0 ≤y ≤ nr ≤ n(n − 1)/2. The assumption on det ρ then implies that y ≡ n(n − 1)/2
(mod p − 1). If in fact r < (n − 1)/2 then we have 0 ≤ y < n(n − 1)/2, which
contradicts the assumption that p > n(n − 1)/2 + 1.
It remains to treat the case that r = (n − 1)/2, in which case we may assume (bythe additional hypothesis that we have assumed in this case) that the representation

22

MATTHEW EMERTON AND TOBY GEEρ0 above has dimension d > 1. By the above analysis we must have y = n(n − 1)/2,
so that each yi = r. Since we have ri ≤ (pd − 1)r, we must have xi+1 − xi ≤ 0 foreach i, so that in fact x0 = x1 = · · · = xd−1 , a contradiction (as we already showedthat the xi are distinct).

3. The cohomology of Shimura varieties
3.1. The semistable reduction of certain U (n − 1, 1)-Shimura varieties. Fixn ≥ 2, and fix an odd prime p.3 We now recall the definitions of the U (n − 1, 1)Shimura varieties with which we will work, and some associated integral models.
For simplicity we work over Q (or rather an imaginary quadratic extension of Q)
rather than over a general totally real field.
For the most part we will follow Section 3 of [HR10] (which uses a similar approach to [HT02]), with the occasional reference to [HT01]. Fix an imaginary
quadratic field F in which the prime p splits, say (p) = pp for some choice of p, let
x 7→ x be the nontrivial automorphism of F , and regard F as a subfield of C via afixed embedding F ,→ C.
Let D be a division algebra over F of dimension n2 , and let ∗ be an involutionof D of the second kind (that is, ∗|F is nontrivial). Assume that D splits at p (andhence at p), and fix an isomorphism Dp ∼= Mn (Qp ) with the property that under
the induced isomorphism
D ⊗ Qp ∼= Mn (Qp ) × Mn (Qp )op ,the involution ∗ corresponds to (X, Y ) 7→ (Y t , X t ).
Let G/Q be the algebraic group whose R-points are
G(R) = {x ∈ (D ⊗Q R)× |x · x∗ ∈ R× }for any Q-algebra R. Thus our fixed isomorphism Dp ∼= Mn (Qp ) induces an isomorphism G ×Q Qp ∼= GLn × Gm .
Now let h0 : C → DR be an R-algebra homomorphism with the properties thath0 (z)∗ = h0 (z) and the involution x 7→ h0 (i)−1 x∗ h0 (i) is positive. Let B = Dop andlet V = D, which we consider as a free left B-module of rank one by multiplicationon the right. Then EndB (V ) = D, and one can find an element ξ ∈ D× with theproperties that ξ ∗ = −ξ and such that the involution ι of B defined by xι = ξx∗ ξ −1
is positive (that is, tr B/Q (xxι ) > 0 for all nonzero x) (see Section I.7 of [HT01] orSection 5.2 of [Hai05] for the existence of such a ξ).
We have an alternating pairing ψ(·, ·) : D × D → Q defined by ψ(x, y) =tr D/Q (xξy ∗ ), and one sees easily that ψ(bx, y) = ψ(x, bι y), and that ψ(·, h0 (i)·)
is either positive- or negative-definite. After possibly replacing ξ by −ξ, we canand do assume that it is positive-definite.
It is easy to see that one has
G(R) ∼= GU(p, q)
for some p, q with p + q = n. We impose the additional assumption that infact {p, q} = {n − 1, 1}. Note that by Lemma I.7.1 of [HT01] one can find division
algebras D for which this holds. We say that a compact open subgroup K ⊂ G(A∞ )(respectively K p ⊂ G(Ap,∞ )) is sufficiently small if for some prime q (respectively
some prime q 6= p) the projection of K (respectively K p ) to G(Qq ) contains no3The reason for assuming that the prime p is odd is that below we will want to apply the

discussion and results of Subsection 2, in which this assumption was made.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

23

element of finite order other than 1. If K is sufficiently small, we may consider
the Shimura variety Sh(G, h0 |−1
C× , K). It has a canonical model over F , which wedenote by X(K) (note that if n > 2 the reflex field is F , while if n = 2 the reflex
field is Q, and we let X(K) denote the base-change of the canonical model from Q
to F ).
We say that a compact open subgroup K of G(A) is of level dividing N , forsome integer N ≥ 1, if for all primes l - N we can write K = Kl K l , where Kl
is a hyperspecial maximal compact subgroup of G(Ql ) Q
and K l is a compact open∞,lsubgroup of G(A ). (Note then that in fact K = KN × l-N Kl , for some compact
Q
open subgroup KN of l|N G(Ql ).) If K is of level dividing N , then we similarlyrefer to X(K) as a U (n − 1, 1)-Shimura variety of level dividing N .
We will now define integral models of these Shimura varieties for two specifickinds of level structure. We begin by introducing notation related to the levelstructures in question.
We write I0 for the Iwahori subgroup of GLn (Qp ) × Q×
p ; namely, I0 is the sub×group of GLn (Zp ) × Zp consisting of elements whose first factor lies in the usual
Iwahori subgroup of matrices which are upper triangular mod p. We write I1 todenote the pro-p-Iwahori subgroup of GLn (Qp ) × Q×
p ; namely, I1 is the (unique)pro-p Sylow subgroup of I0 , and consists of those elements of GLn (Zp ) × Z×
p whosefirst factor is upper triangular unipotent mod p, and whose second factor is congruent to 1 mod p. We write I1∗ to denote the subgroup of I0 consisting of matrices
in GLn (Zp ) × Z×
p whose first factor is upper triangular unipotent mod p.
×
There is a natural isomorphism Z×
p = Fp × (1 + pZp ), and this induces a natural∗×
isomorphism I1 = Fp × I1 . If we let T denote the diagonal torus in GLn , then there
is also a natural isomorphism I0 = T (Fp ) × I1∗ .
We will define integral models for X(I0 K p ) and X(I1∗ K p ) over the local rings
OF,(p) and OF (ζp−1 ),(v) respectively, where ζp−1 denotes a primitive (p − 1)st rootof unity, and v is some fixed place in F (ζp−1 ) above p; here K p is a sufficiently
small compact open subgroup of G(A∞,p ), and we consider I0 and I1∗ as subgroupspof G(Qp ) ∼= GLn (Qp ) × Q×
p . We will typically not include K in the notation,
and we will write X0 (p), X1 (p) for our integral models of X0 (p) := X(I0 K p ) andX1 (p) := X(I1∗ K p ) respectively.
3.1.1. Remark. Note that in [HR10], [HT02], and [HT01], the authors work over Zp ,but we follow [Kot92] in working over OF,(p) , so as to satisfy the hypothesis required
to be in the global case of Section 1. The appearance of ζp−1 in the ring of definitionof X1 (p) is a consequence of our use of Oort–Tate theory in the definition of theintegral model in this case.
In order to define these integral models, we firstly recall a certain category ofabelian schemes (up to isogeny) with polarisations and endomorphisms. If F is a
set-valued functor on the category of connected, locally noetherian OF,(p) -schemes,
we will also consider it to be a functor on the category of all locally noetherianOF( p) -schemes by settingYaF( Si ) :=F(Si ).
Let OB be the unique maximal Z(p) -order in B which under our fixed identification
B ⊗Q Qp = Mn (Qp ) × Mn (Qp )op is identified with Mn (Zp ) × Mn (Zp )op . Let S bea connected, locally noetherian OF,(p) -scheme, and let AVS be the category whose

24

MATTHEW EMERTON AND TOBY GEE

objects are pairs (A, i), where A is an abelian scheme over S of dimension n2 , andi : OB → EndS (A) ⊗ Z(p) is a homomorphism. We define homomorphisms in AVSby
Hom (A1 , i1 ), (A2 , i2 ) = HomOB (A1 , i1 ), (A2 , i2 ) ⊗ Z(p)(that is, the elements of HomS (A1 , A2 ) ⊗ Z(p) which commute with the actionof OB ). The dual of an object (A, i) of AVS is (A∨ , i∨ ), where A∨ is the dual abelianscheme of A, and i∨ (b) = (i(bι ))∨ . A polarisation of (A, i) is a homomorphism(A, i) → (A∨ , i∨ ) which is a Z×(p) -multiple of a polarisation of A, and a principal
polarisation is defined in the same way. A Q-class of polarisations is an equivalenceclass of homomorphisms (A, i) → (A∨ , i∨ ) which contains a polarisation, under theequivalence relation “differing by a Q× -scalar locally on S”. We define a Q-class ofprincipal polarisations in the same way.
Fix K p a sufficiently small open compact subgroup of G(A∞,p ). Let A0 be theset-valued functor on the category of locally noetherian schemes over OF,(p) which
sends a connected, locally noetherian scheme S over OF,(p) to the set of isomorphismclasses of the following data.
• A commutative diagram of morphisms in the category AVS of the form
A0

α0

/ A1

α̂0


/ Â
1

λ0


Â0

α1

λ1
α̂1

/ ···

αn−2

/ An−1 αn−1 / A0


λn−1

λ0


α̂n−1
/ Â
/ · · · α̂n−2/ Ân−10where each αi is an isogeny of degree p2n , and their composite is just multiplication by p. In addition, λ0 is a Q-class of principal polarisations.
Furthermore, we require that each Ai satisfies the usual compatibility condition that the two actions of OF on the Lie algebra of Ai coincide (oneaction coming from the structure morphism OF,(p) → OS , and the otherfrom the OB -action; see [HT01, §III.4] for a discussion of this condition).
• A geometric point s of S, and a π1 (S, s)-invariant K p -orbit of isomorphisms∼

η : V ⊗Q Ap,∞ −→ H1 ((A0 )s , Ap,∞ )which are OB -linear, and up to a constant in (Ap,∞ )× take the ψ-pairingon the left hand side to the λ0 -Weil pairing on the right hand side. (Thisdata is canonically independent of the choice of s; see the discussion on
pages 390-391 of [Kot92].)
An isomorphism of this data is one induced by isomorphisms in AVS which preservethe λi up to an overall Q× -scalar.
The functor A0 is represented by a projective scheme X0 (p) over OF,(p) , which
is an integral model for X(I0 K p ). (See the proof of Lemma 3.2 of [TY07], whichshows that X0 (p) is projective over the usual integral model at hyperspecial level.
At hyperspecial level, quasi-projectivity is proved on page 391 of [Kot92], andprojectivity can be checked via the valuative criterion for properness as on page 392
of [Kot92].) The proof of Proposition 3.4(3) of [TY07] (which goes over unchangedin our setting) shows that the special fibre of X0 (p) is a strict normal crossings
divisor.
Our next goal is to describe an integral model X1 (p), over OF (ζp−1 ),(v) , for
X(I1∗ K p ). Recalling that T denotes the diagonal torus in GLn , we let ıi : Gm → T

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

25

denote the embedding of tori identifying Gm with the subgroup of T consisting ofelements which are 1 away from the ith diagonal entry. We use the same notation
ıi to denote the map F×
p → T (Fp ) induced by the map of tori.
The quotient I0 /I1∗ is naturally identified with T (Fp ), and so T (Fp ) acts on
X1 (p), with quotient isomorphic to X0 (p).
The scheme X0 (p) over Spec OF,(p) classifies chains C of p-isogenies of abelianvarieties (with additional structure as specified above)
α

α

αn−2

αn−1

0
1
C : A0 −→A1 −→· · · −→ An−1 −→ An := A0 ,whose composite is equal to multiplication by p.
We let OT denote the Artin stack over OF (ζp−1 ),(v) given by
OT := Spec OF (ζp−1 ),(v) [X, Y ]/(XY − wp )/Gm ,where Gm acts via λ · (X, Y ) = (λp−1 X, λ1−p Y ), and wp is some explicit element ofOF (ζp−1 ),(v) of valuation one. Oort–Tate theory shows that OT classifies finite flatgroup schemes of order p over OF (ζp−1 ),(v) -schemes. The universal group scheme
over OT is the stack
G := Spec OF (ζp−1 ),(v) [X, Y, Z]/(XY − wp , Z p − XZ)/Gm ,where Gm acts on X and Y as above, and on Z via λ · Z = λZ. The morphism
G → OT is the evident one, the zero section of G is cut out by the equation
Z = 0, and we let G × denote the closed subscheme of G cut out by the equation
Z p−1 − X = 0; it is the so-called scheme of generators of G. (See Theorem 6.5.1
of [GT05] for these facts, which are a restatement of the results of [TO70] in thelanguage of stacks.)
We define X1 (p) via the Cartesian diagram(3.1.2)X1 (p)/ G × ×OF (ζp−1 ),(v)· · · ×OF (ζp−1 ),(v) G ×


X0 (p)/OF (ζp−1 ),(v)/ OT ×OF (ζp−1 ),(v)· · · ×OF (ζp−1 ),(v) OT,

where the bottom horizontal arrow is given byC 7→ ker(α0 ), . . . , ker(αn−1 ) .
Note that the right-hand vertical arrow is finite and relatively representable byconstruction, and so the left-hand vertical arrow is a finite morphism of schemes.
The action of T (Fp ) on X1 (p) extends to an action on X1 (p), namely the action
pulled back from the action of T (Fp ) on G × ×OF (ζp−1 ),(v) · · · ×OF (ζp−1 ),(v) G × over
OT ×OF (ζp−1 ),(v) · · · ×OF (ζp−1 ),(v) OT .
Let π := (−p)1/(p−1) , and let w be the unique finite place of L := F (ζp−1 , π)lying over our fixed place v of F (ζp−1 ). We let O denote the localization of OL
at w, and we let X1 (p)O denote the normalisationof the base-change X1 (p)/O of
X1 (p) over O. We write I := Gal L/F (ζp−1 ) ; this is also the inertia group at w in
Gal(L/F ). The group I acts naturally on X1 (p)O . The mod p cyclotomic characterω induces an isomorphism (which we continue to denote by ω)∼ω : I −→ F×p.
For each i = 0, . . . , n − 1, we let αi : I → T denote the composite ıi ◦ ω −1 .

26

MATTHEW EMERTON AND TOBY GEE

3.1.3. Lemma. The scheme X1 (p)O is a semistable projective model for X1 (p)/Lover O, the natural morphism(3.1.4)

X1 (p)O → X0 (p)is tamely ramified, and the action of I × T on X1 (p)/L extends to an action on
X1 (p)O . Furthermore, on each irreducible component of its special fibre, the inertia
group I acts through the composite of the T -action with one of the characters αi .
Proof. We will apply a form of Deligne’s homogeneity principle, as described in the
proof of [TY07, Prop. 3.4], to the morphism (3.1.4). The scheme here denoted X0 (p)is there denoted XU (and the integral model there is considered over Zp rather
than OF,(p) , but this is immaterial for our present purposes), while the scheme
there denoted XU0 is an integral model of the Shimura variety (in the notation of(h)the present paper) X(Kp K p ), where Kp = GLn (Zp ) × Z×
(forp . We let X0 (p)s0 ≤ h ≤ n − 1) denote the locally closed subset of the special fibre X0 (p)s , obtained(h)by pulling back the locally closed subset X U0 defined in section 3 of [TY07] under
the natural projection X0 (p)s = X U → X U0 .
We will show that the morphism (3.1.4) is tamely ramified in the formal neighbourhood of any closed geometric point of the special fibre X0 (p)s , and hence(by Lemma 1.5.2) that it is tamely ramified. We first note that the morphism
X0 (p)/OF (ζp−1 ),(v) → X0 (p) induces an isomorphism on special fibres, and so we arefree to replace X0 (p) by X0 (p)/OF (ζp−1 ),(v) in our considerations. We next note thatthe completion of the bottom arrow of (3.1.2) at a closed geometric point x0 of(h)
X0 (p)s depends up to isomorphism only on the value of h for which x0 ∈ X0 (p)s(since the p-divisible group attached to the point x0 depends only on the valueof h), and hence that the restriction of (3.1.4) to a formal neighbourhood of x0(h)depends only on the value of h. Then by [TY07, Lem. 3.1], the closure of X0 (p)s(0)
contains X0 (p)s . Since being tamely ramified is an open condition, we conclude
from these two conditions that in order to prove the lemma, it suffices to showthat the restriction of (3.1.4) to a formal neighbourhood of x0 is tamely ramified(0)
at closed geometric points x0 of X0 (p)s . (As already indicated, this argument isa variation on Deligne’s homogeneity principle.)(0)Thus, consider a closed geometric supersingular point x0 of X0 (p)s , so that x0
admits a formal neighbourhood of the form
Spec W (Fp )[[T1 , . . . , Tn ]]/(T1 · · · Tn − wp ).
The proof of [TY07, Prop. 3.4] shows that the Ti may be taken to be the matrix of
αi−1 on tangent spaces, so that the map X0 (p) → OT ×OF (ζp−1 ),(v) · · · ×OF (ζp−1 ),(v)OT may be defined in the formal neighbourhood of x0 by the map(T1 , . . . , Tn ) 7→ (T1 , U1 ), . . . , (Tn , Un ) ,where Ui = T1 · · · T̂i · · · Tn (and, as is usual in these situations, a hat on a variable
denotes that that variable is omitted in the expression). Thus a formal neighbourhood of a closed geometric point lying over x0 in (X1 (p)/O )s is isomorphic toSpec W (Fp )[π][[V1 , . . . , Vn ]]/ (V1 · · · Vn )p−1 − wp .
If we write u := V1 · · · Vn /π, then we see that up−1 = −wp /p, and hence thatu lies in the normalisation of this formal neighbourhood. Furthermore, on eachp-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

27

component of this normalisation, u is equal to one of the (p − 1)st roots of −wp /plying in W (Fp ). Thus the normalisation of this formal neighbourhood is a union ofcomponents, each isomorphic toSpec O[[V1 , . . . , Vn ]]/(V1 · · · Vn − uπ),with the morphism (3.1.4) being given by Ti = Vip−1 . Thus this morphism is indeed
tamely ramified in the formal neighbourhood of x0 .
It is clear that the I × T -action on X1 (p)L extends to an action on X1 (p)/O , andhence to its normalisation X1 (p)O . As for the final statement, note that I acts on
π via ω, and hence on u via ω −1 , while I fixes each Vi . Also T acts on Vi throughmultiplication by the ith diagonal entry, and so acts on u via multiplication by thedeterminant. Combining these facts, we see that I acts on the components of thespecial fibre on which Vi = 0 via ıi ◦ ω −1 .

3.1.5. Remark. This lemma (and the fact that X0 (p) is strictly semistable) showsthat the map X1 (p)O → X0 (p) provides a tame strictly semistable context, in thesense of Subsection 1.5. In particular, we can apply Theorem 1.5.14 in this setting(of course taking the group G to be the abelian group T ), and we see that eachcharacter ψj as in the statement of that Theorem is equal to one of the characters αi .
3.2. Canonical local systems. If K is a sufficiently small compact open subgroupof G(A∞ ), and V is a continuous representation of K on an Fp -vector space (thisvector space being equipped with its discrete topology), then we may associate to Van étale local system FV of Fp -vector spaces on X(K) as follows: Choose an open
normal subgroup K 0 ⊂ K lying in the kernel of V , and regard V as a representationof the quotient K/K 0 . Since X(K 0 ) is naturally an étale K/K 0 -torsor over X(K),
we may form the étale local system of Fp -vector spaces over X(K) associated to
the K/K 0 -representation V . This local system is independent of the choice of K 0 ,up to canonical isomorphism, and we define it to be FV .
3.2.1. Definition. We refer to the étale local systems FV that arise by the preceding
construction as the canonical local systems on X(K). If we may choose K 0 in
the kernel of V to be of level dividing N (so that in particular X(K) is of level
dividing N ), then we say that FV can be trivialised at level N .
3.3. The Eichler–Shimura relation. Let X be a U (n − 1, 1)-Shimura variety oflevel dividing N . Let w be a place of F such that l := w|Q splits in F and does notdivide N . There is a natural action via correspondences on X of Hecke operators(i)(i)
Tw , 0 ≤ i ≤ n, where Tw is the double coset operator corresponding to


l1i
0
× 1 ∈ GLn (Ql ) × Zl × ,0 1n−iwhere we use the assumption that l - N and identify a hyperspecial maximal compact subgroup of G(Ql ) with GLn (Zl ) × Zl × via an isomorphism Dw ∼= Mn (Ql ).
j
These correspondences then act on the cohomology Hét
(XQ , Fp ).
More generally if FV is a canonical local system on X that can be trivialised at(i)level N , then we obtain an action of the double coset operators Tw on FV , andjhence on the cohomology Hét (XQ , FV ).
The following theorem regarding this action is then an immediate consequenceof the main result of [Wed00] (which proves the Eichler–Shimura relation for PEL
Shimura varieties at places of good reduction at which the group is split).

28

MATTHEW EMERTON AND TOBY GEE

3.3.1. Theorem. Let X be a U (n − 1, 1)-Shimura variety of level dividing N , andFV a canonical local system on X. Let w be a place of F such that w|Q splits inPn(i)
F and does not divide N p. Then i=0 (−1)i (Norm w)i(i−1)/2 Tw Frobn−iacts as 0w
jon each Hét
(XQ , FV ).
3.4. Vanishing and torsion-freeness of cohomology for certain U (n − 1, 1)Shimura varieties. Let X denote a U (n − 1, 1)-Shimura variety as above, and let
FV denote a canonical local system on X. Choose N so that X has level dividing N ,so that FV can be trivialised at level N , and so that p divides N . Assume that the
projection of the corresponding level K to G(Ap,∞ ) is sufficiently small.
(i)(i)
Let T = Zp [Tw ] be the polynomial ring in the variables Tw , 1 ≤ i ≤ n, wherew runs over the places of F such that w|Q splits in F and does not divide N .
Let m be a maximal ideal in T with residue field Fp , and suppose that there
exists a continuous irreducible representation ρm : GF → GLn (Fp ) which is unramified at all finite places not dividing N , and which satisfies char ρm (Frobw ) ≡Pnii(i−1)/2 (i) n−i
Tw X
mod m for all w - N such that w|Q splits in F .
i=0 (−1) (Norm w)
Continue to fix a choice of a place p of F dividing p, and write GQp for GFp from now
on. Recall that the choice of p also gives us an isomorphism G(Qp ) ∼= GLn (Qp )×Q×p
as in Section 3.1.
We consider the following further hypothesis on ρm (this is Hypothesis 4.1.1
below):
3.4.1. Hypothesis. If θ : GQ → GLm (k) is any continuous, irreducible representation with the property that the characteristic polynomial of ρm (g) annihilates θ(g)for every g ∈ GQ , then θ is equivalent to ρm .
We will now prove our first main result, a vanishing theorem for the cohomology
of X with FV -coefficients.
3.4.2. Theorem. Suppose that ρm satisfies Hypothesis 3.4.1, that ρm |GQp is rregular for some r ≤ (n − 1)/2, that p > n(n − 1)/2 + 1 and, if r = (n − 1)/2,
suppose in addition that ρm |GQp contains an irreducible subquotient of dimensioni(XQ , FV )m vanish for i ≤ r and for
greater than one. Then the localisations Héti ≥ 2(n − 1) − r.
3.4.3. Remark. In Subsection 4.1 we will show that Hypothesis 3.4.1 is satisfied
if either ρm is induced from a character of GK for some degree n cyclic Galois
×
extension K/Q, or if p ≥ n and SLn (k) ⊆ ρm (G) ⊆ Fp GLn (k) for some subfieldk ⊂ Fp .
3.4.4. Remark. While we work here with étale local systems and étale cohomology,
by virtue of Artin’s comparison theorem [Gro73, XI 4.3] our vanishing results areequivalent to vanishing results for the cohomology of the complex U (n − 1, 1)Shimura varieties with coefficients in the corresponding canonical local systems forthe complex topology.
Proof of Theorem 3.4.2. Firstly, note that it suffices to prove vanishing in degree i ≤r for all V , as vanishing in degree i ≥ 2(n − 1) − r then follows by Poincaré duality.
(Note that the dual of the canonical local system FV attached to a representation
V is the canonical local system attached to the contragredient representation V ∨ .)

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

29

We prove the theorem for i ≤ r by induction on i, the case when i < 0 being
trivial. We begin by reducing to the case when FV is trivial. To this end, write
X = X(K), let K 0 be an open normal subgroup of K of level dividing N such that
V is representation of K/K 0 , and choose an embedding of K/K 0 -representations
V ,→ Fp [K/K 0 ]n for some n; denote the cokernel by W . Let π denote the projection
X(K 0 ) → X(K). Passing to the associated canonical local systems, we obtain anshort exact sequence 0 → FV → π∗ Fp → FW → 0, which gives rise to an exact
sequence of cohomology
n

i−1
i
i
i
(XQ , FV ) → Hét
(XQ , π∗ Fp ) = Hét
(X(K 0 )Q , Fp )n .
Hét
(XQ , FW ) → Hét

Localizing at m and applying our inductive hypothesis (with FW in place of FV ),we reduce to the case of constant coefficients (with X(K 0 ) in place of X). We
therefore turn to establishing the claimed vanishing in this case.
Suppose now that K 0 is any open normal subgroup of K of level dividing N .
Combining the Hochschild–Serre spectral sequence
 

m+n
m
E2m,n = Hét
K/K 0 , H n X(K 0 )Q , Fp m =⇒ Hét
X(K)Q , Fp mi
X(K 0 ), Fp m implies
with our inductive hypothesis, we find that vanishing of Hét
i
X(K)Q , Fp m . Thus, without loss of generality, we may andthe vanishing of Hét
do assume that K = Kp K p , where Kp is an open normal subgroup of I1 , and K p
is a sufficiently small compact open subgroup of G(A∞,p ).
We again consider a Hochschild–Serre spectral sequence, this time the one relating the cohomology of X(K) and X(I1 K p ), which takes the form
 

m+n
n
E2m,n = H m I1 /Kp , Hét
X(K)Q , Fp m =⇒ Hét
X(I1 K p )Q , Fp m .
Once more taking into account our inductive hypothesis, we obtain an isomorphismI1 /Kp∼ii
X(I1 K p )Q , Fp m −→ Hét
Hét
X(K)Q , Fp m
. Since I1 /Kp is a p-group, while
iHét
X(K)Q , Fp m is a vector space over a field of characteristic p, we see that thelatter space vanishes if and only if its space of I1 /Kp -invariants does. Thus we are
reduced to establishing the theorem in the case when K = I1 K p .
Now recall that I1∗ = F×p × I1 , and that the projection onto the first factor
arises from the similitude projection GU (n − 1, 1) → Gm . From this it follows that
X(I1 K p ) is isomorphic to the product X(I1∗ K p )×F Spec A, where A := F [x]/Φp (x)∗(where Φp (x) denotes the pth cyclotomic polynomial; the action of F×
p = I1 /I1 on
×a
X(I1 K p ) is induced by the action of F×p on A given by x 7→ x , for a ∈ Fp ).
Consequently there is an isomorphism of Galois representations
 ∼

i
∗ p
j
H i X(I1 K p )Q , Fp m −→ ⊕p−2
j=0 H X(I1 K )Q , Fp m ⊗ ω ,jwhere mj is the maximal ideal in T which corresponds to the twisted Galois representation ρmj := ρm ⊗ ω −j . Since each of the Galois representations ρmj satisfies
the hypotheses of the theorem (as these hypotheses are invariant under twisting),
we are reduced to proving the theorem in the case of X(I1∗ K p ).
iConsider an irreducible GF × I0 /I1∗ -subrepresentation of Hét
X(I1∗ K p )Q , Fp m ,which we may write in the form θ ⊗ β, where θ is an absolutely irreducible GF representation and β is a character of the abelian group T (Fp ) = I0 /I1∗ . Theorem 3.3.1 and Hypothesis 3.4.1 taken together imply that θ is equivalent to ρm .
We consider β as an n-tuple of characters β1 , . . . , βn of F×
p . Write β̃j for the Teichmüller lift of βj , and χj for the character of IQp given by βj . By Lemma 3.1.3,

30

MATTHEW EMERTON AND TOBY GEE

Remark 3.1.5 and Theorem 1.5.14, we see that θ can be embedded as the reduction mod p of a potentially semistable representation with Hodge–Tate weights in
the range [−i, 0], whose inertial type is a direct sum of characters belonging to thecollection {β̃j }. (To see this, note that as θ is in the β-part of the cohomology, it necessarily occurs in the reduction of the β̃-part of the GF × T -representation provided
by Theorem 1.5.14.) We claim that det θ|IQp = det ρm |IQp = χ1 · · · χn ω −n(n−1)/2 .
Admitting this for the moment, we may apply Theorem 2.2.4 to the representa∨tion ρm |∨GQp , and we deduce that ρm |GQp is not r-regular. Equivalently, we see thatρm |GQp is not r-regular, which contradicts our assumptions.
It remains to establish the equality det θ|IQp = det ρm |IQp = χ1 · · · χn ω −n(n−1)/2 .
To see this, note that Theorem 3.3.1 implies that for each place w - N of F such that(n)w|Q splits in F , we have det ρm (Frobw ) = (Norm w)n(n−1)/2 Tw . Let ψm be the×character of AF /F × corresponding to the character ω n(n−1)/2 det ρm by global class= β1 · · · βn . The centre of G(AQ )field theory; then we need to prove that ψm |Z×p
×
is AF , so by the definition of the Shimura variety X there is a natural action of×iA×
on Hét
X(I1∗ K p )Q , Fp m . By the definition of the Hecke operators, we see
F /F
that if $w is a uniformiser at a place w - N of F such that w|Q splits in F , then(n)$w acts as Tw . By the Chebotarev density theorem, we deduce that the action
××of AF /F on the underlying
vector space of θ (which by definition is a subspacei
X(I1∗ K p )Q , Fp m ) is via the character ψm . In order to compute ψm |Z×of Hét
, it
p

∗ p
×
i
is thus sufficient to compute the action of Zp on Hét X(I1 K )Q , Fp m , and inparticular sufficient to compute the action of the Iwahori subgroup I0 . Now, sinceθ is assumed to be in the β-part of the cohomology, I0 acts via the character β of
= β1 · · · βn , as required.

I0 /I1∗ , so that ψm |Z×p
3.4.5. Corollary. Suppose that ρm satisfies Hypothesis 3.4.1, that ρm |GQp is rregular for some r ≤ min{(n − 1)/2, p − 2}, and, if r = (n − 1)/2, suppose inaddition that ρm |GQp contains an irreducible subquotient of dimension greater thanr+1i
(XQ , Zp )m vanishes for i ≤ r, while Hétone. Then the localisation Hét
(XQ , Zp )mis torsion-free.
Proof. This follows at once from Theorem 3.4.2 and the short exact sequenceiii
0 → Hét(XQ , Zp )m /mZp Hét
(XQ , Zp )m → Hét
(XQ , Fp )mi+1→ Hét(XQ , Zp )m [mZp ] → 0. 

3.4.6. Remark. As already remarked in the introduction, we expect some kind of
mod p analogue of Arthur’s conjectures to hold, and so in particular, we expect thatstronger results than Theorem 3.4.2 and Corollary 3.4.5 should hold. In particular,
if m is any maximal ideal in the Hecke algebra attached to an irreducible continuous representation ρm : GF → GLn (Fp ), then we expect that the localizationsiiHét(XQ , FV )m and Hét
(XQ , Zp )m should vanish in degrees i < n − 1.
iOn the other hand, it need not be the case that (for example) Hét
(XQ , Fp )ivanishes in all degrees in which Hét (XQ , Qp ) vanishes. For example, in the casen = 3, for the unitary Shimura varieties that we consider here, namely those that1
are associated to division algebras, it is known that Hét
(XQ , Qp ) = 0 [Rog90,
Thm. 15.3.1]. (Under additional restrictions on the division algebra that is allowed,

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

31

an analogous result is known for all values of n [Clo93, Thm. 3.4].) On the other1
hand, one can construct examples for which Hét
(XQ , Fp ) 6= 0, and hence for which2
Hét (XQ , Zp ) is not torsion-free, via congruence cohomology. (See e.g. the proof of[Suh08, Thm. 3.4].) The existence of such classes does not contradict our theorems
or expectations, since congruence cohomology is necessarily Eisenstein (i.e. gives
rise to Eisenstein systems of Hecke eigenvalues, in the sense that the associatedGalois representation ρm is completely reducible).
3.5. On the mod p cohomology of certain U (2, 1)-Shimura varieties. Let
X := X(K) denote a U (2, 1)-Shimura variety, with K of level dividing N , for somenatural number N divisible by p, and such that the projection of K to G(Ap,∞ )is sufficiently small. Let FV be a canonical local system on X, which may betrivialized at level N . The results of Section 3.4 are particularly powerful in thiscase, as we now demonstrate.
3.5.1. Corollary. Suppose that ρm satisfies Hypothesis 3.4.1, that ρm |GQp is 1regular, and that ρm |GQp contains an irreducible subquotient of dimension greateri(XQ , FV )m vanish for i 6= 2.
than one. Then the localisations Hét
Proof. This follows immediately from Theorem 3.4.2, noting that the hypothesisthat ρm |GQp is 1-regular implies that p > 4 (indeed, that p ≥ 11).
We now prove a result which does not require the existence of a Galois representation ρm . We begin with a lemma.
3.5.2. Lemma. If m is a maximal ideal of T with residue field Fp , such that0
(XQ , FV )m 6= 0, then there is an abelian representation ρ0 : GF → GL3 (Fp )
Hét
 P3
(i)
such that char ρ0 (Frobw ) = i=0 (−1)i (Norm w)i(i−1)/2 Tw X n−i (mod m) for all
split places w of E for which w - N p.
Proof. This is standard and follows for example from Section 2.1 of [Del79].



3.5.3. Theorem. If ρ is a three-dimensional irreducible sub-GF -representation of
1(XQ , FV ), then either every irreducible subquotientthe étale cohomology group Hétof ρ|GQp is one-dimensional, or else ρ|GQp is not 1-regular, or else ρ(GF ) is not
generated by its subset of regular elements.
3.5.4. Remark. Recall that a square matrix is said to be regular if its minimal andcharacteristic polynomials coincide. In Subsection 4.2 we will show that ρ(GF ) is
generated by its subset of regular elements if either ρm is induced from a character of
GK for some cubic Galois extension K/Q, or if ρ(GF ) contains a regular unipotent
element.
3.5.5. Remark. In the proof of the theorem we use some of the results of Section 4
below.
Proof of Theorem 3.5.3. The argument follows similar lines to the proof of Theorem 3.4.2, although it is slightly more involved, since we are not giving ourselves
the existence of the Galois representation ρm .4 The key point will be that in the
Hochschild–Serre spectral sequences that appear, the only other cohomology to
4In fact, as noted in the introduction, recent work of Scholze [Sch13] implies that ρ exists form
any maximal ideal m in the Hecke algebra. We have left our argument as originally written.

32

MATTHEW EMERTON AND TOBY GEEcontribute besides H 1 will be H 0 , and for maximal ideals of T in the support ofH 0 , we do have associated Galois representations,by Lemma 3.5.2.
0
We first show that if Hét
XQ , FW m 6= 0 for some canonical local system FW 1on X and some maximal ideal m of T, then HomGF ρ, Hét
XQ , FV [m] = 0. To


 
1
0
XQ , FV [m] 6= 0,
see this, note that if Hét
XQ , FW m 6= 0 and HomGF ρ, Hétthen Lemma 3.5.2 and Theorem 3.3.1 together imply that there exists an abelian
representation ρ0 : GF → GL3 (Fp ) such that for all g ∈ GF , the characteristic
polynomial of ρ0 (g) annihilates ρ(g). By Lemma 4.1.3 this implies that ρ is abelian,
which is impossible as ρ is irreducible.
1
Now Hét(XQ , FV ) is the direct sum of its localisations at the various maximal1
ideals m of T, and hence, since HomGF ρ, Hét
(X, FV ) 6= 0 by hypothesis, we
1
see that HomGF ρ, Hét
(X, FV )m 6= 0 for some maximal ideal m of T. Since this1(X, FV )[m]is a finite length Tm -module, we see that its Tm -socle HomGF ρ, Hét
must also be non-zero, and hence, by the preceding paragraph, we conclude that
0
(X, FW )m = 0 for any canonical local system FW on X.
Hét
As in the proof of Theorem 3.4.2, choose a short exact sequence of canonicalnlocal systems 0 → FV → π∗ Fp → FW → 0, for some π : X 0 → X. Passing to thethe long exact sequence
10
1
1(X 0 , Fp )n ,Hét
(X, FW )m → Hét
(X, FV )m → Hét
(X, π∗ Fp )n = Hét0
and using the result of the preceding paragraph, namely that Hét
(X, FW )m = 0,01
we conclude that ρ embeds into Hét (X , Fp ). Thus, replacing X by X 0 , we reduceto the case when FV is constant, which we assume from now on.
1(XQ , Fp )m . We will prove
We now suppose that ρ is a subrepresentation of Hét1that ρ is then necessarily a subrepresentation of Hét X(K p I1 ), Fp m , for some sufficiently small open subgroup K p of G(A∞,p ). The result will then follow from
Lemmas 3.1.3 and 4.2.2, and Theorems 1.5.14, 3.3.1, and 2.2.4.
As in the proof of Theorem 3.4.2, we write X = X(K), and choose a normal
open subgroup K 0 := K p Kp of K, with Kp ⊂ I1 . The Hochschild–Serre spectral
sequence associated to the cover X(K 0 ) → X(K) gives rise to an exact sequence K/K 01
1
0 → H 1 K/K 0 , H 0 X(K 0 ), Fp m → Hét
X(K), Fp m → Hét
X(K 0 ), Fp m 
→ H 2 K/K 0 , H 0 X(K 0 ), Fp m .
The same argument as above, using Lemma 3.5.2 (applied now with X(K 0 ) in placeof X), Theorem 3.3.1, and Lemma 4.1.3 below, shows that  
H 1 K/K 0 , H 0 X(K 0 ), Fp m = H 2 K/K 0 , H 0 X(K 0 ), Fp m = 0.
 ∼
K/K 01
1
Thus in fact we have an isomorphism Hét
X(K), Fp −→ Hét
X(K 0 ), Fp m
, andhence an isomorphism ∼K/K 0 
1
1
HomGF ρ, Hét
X(K), Fp −→ HomGF ρ, Hét
X(K 0 ), Fp m K/K 0
1.
= HomGF ρ, Hét
X(K 0 ), Fp m
1
In particular, if ρ appears as a subrepresentation of Hét
X(K), Fp )m , then it ap0
1
pears as a subrepresentation of Hét X(K ), Fp )m .

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY33

Now considering the Hochschild–Serre spectral
sequence for thecover X(K 0 ) →
1X(K p I1 ), and using the fact that if HomGF ρ, Hét
X(K 0 ), Fp m 6= 0, then also I1 /Kp
1
HomGF ρ, Hét
X(K 0 ), Fp m
6= 0 (since I1 /Kp is a p-group), we conclude
1that if ρ appears as a subrepresentation of Hét
X(K 0 ), Fp )m , then it appears as a1
subrepresentation of Hét
X(K p I1 ), Fp )m .
Arguing exactly as in the proof of Theorem 3.4.2, we then deduce that some1
X(K p I1∗ ), Fp ), and so, replacing ρ by this twist, it
twist of ρ appears in Hétsuffices to prove that if ρ is an irreducible three-dimensional representation ρ of1
X(K p I1∗ ), Fp ) that is generated by its regular elements, then either every irHét
reducible subquotient of ρ|GQp is one-dimensional, or else ρ|GQp is not 1-regular.
This follows from Lemma 3.1.3 and Theorems 1.5.14, 3.3.1, and 2.2.4 exactly as in
the proof of Theorem 3.4.2, replacing the appeal to Hypothesis 3.4.1 with one toLemma 4.2.2 below.


Our other main theorem concerns the weight part of the Serre-type conjectureof [Her09] for U (2, 1). It is proved by combining our techniques with those of[EGH11], where a similar theorem is proved for U (3) (which is simpler, because
one has vanishing of cohomology outside of degree 0). We begin by recalling someterminology from [EGH11]. We will call an irreducible Fp -representation of GL3 (Fp )a Serre weight. Fix an irreducible representation ρ : GF → GL3 (Fp ). Let X :=X(K) be a U (2, 1)-Shimura variety such that K is of level dividing N and has
sufficiently small projection to G(Ap,∞ ), where now we assume that (N, p) = 1.
Assume furthermore that ρ is unramified at all places not dividing N p, and define
a maximal ideal m of T with residue field Fp by demanding that for each placew - N p of F such that w|Q splits in F , the characteristic polynomial of ρ(Frobw ) isP3
(i)equal to the reduction modulo m of i=0 (−1)i (Norm w)i(i−1)/2 Tw X n−i .
Let V be a Serre weight; since (N, p)=1, we may write K = Kp K p where Kp ⊂×
G(Qp ) ∼= GL3 (Qp ) × Q×
p is conjugate to GL3 (Zp ) × Zp , and we may regard V as arepresentation of Kp via the projection GL3 (Zp )  GL3 (Fp ). As usual, write FV
for the canonical local system associated to V . We say that ρ is modular of weight
V if for some N , X as above and for some 0 ≤ i ≤ 4 we haveiHét
(XQ , FV )m 6= 0.

Assume now that ρ|GQp is irreducible. Definition 6.2.2 of [EGH11] defines what
it means for a Serre weight to be (strongly) generic, and Section 5.1 of [EGH11](using the recipe of [Her09]) defines a set W ? (ρ) of Serre weights in which it ispredicted that ρ is modular. Let Wgen (ρ) be the set of generic weights for which ρis modular.
3.5.6. Theorem. Suppose that ρ satisfies Hypothesis 4.1.1 below, and that ρ|GQp
is irreducible and 1-regular. Suppose that ρ is modular of some strongly genericweight. Then Wgen (ρ) = W ? (ρ). In fact, for each V ∈ Wgen (ρ), we haveiHét
(XQ , FV )m 6= 0

if and only if i = 2 and V ∈ W ? (ρ).
Proof. By the definition of m, the representation ρ satisfies the defining propertiesof the representation ρm considered in Section 3.4. Applying Corollary 3.5.1, we

34

MATTHEW EMERTON AND TOBY GEEisee that for any Serre weight V , we have Hét
(XQ , FV )m = 0 if i 6= 2. We will now
deduce the result from Theorem 6.2.3 of [EGH11] (taking r̄ there to be our ρ). By
Theorem 4.3.3 of [EGH11], we see that it suffices to show that we can define S
and Se as in Section 4 of [EGH11] so that Axioms Ã1–Ã3 of Section 4.3 of [EGH11]
are satisfied. Following [EGH11], we define S and Se using completed cohomology
in the sense of [Eme06] (in [EGH11] the use of completed cohomology was somewhat disguised, but the constructions with algebraic modular forms in [EGH11] areequivalent to the use of completed cohomology of U (3) in degree 0). In fact, given
our vanishing results the verification of the axioms of [EGH11] is very similar tothat carried out for U (3) in [EGH11], and we content ourselves with sketching thearguments.
From now on we regard the prime-to-p level structure K p of X as fixed, and wewill vary Kp in our arguments. We will write Kp (0) for GL3 (Zp ) × Z×p ⊂ G(Qp ).
We fix a sufficiently large extension E/Qp with ring of integers OE , residue field kE ,and uniformiser $E , and we define2
S := lim Hét
(X(K p Kp )Q , Fp )m ,
−→
Kp

2
s
Se := ((lim lim Hét
(X(K p Kp )Q , OE /$E)m ) ⊗OE Zp )l.alg
←− −→
s

Kp

(that is, the locally algebraic vectors in the localisation at m of the completed
cohomology of degree 2). Using the Hochschild–Serre spectral sequence and thei(XQ , FV )m = 0 if i 6= 2, it is straightforward to verify Axioms
vanishing of HétÃ1–Ã3 of Section 4.3 of [EGH11], as we now explain.
Firstly, we need to check that our definition of “modular” is consistent withthat of Definition 4.2.2 of [EGH11]. This amounts to showing that for any Serreweight V ,2(X(Kp (0)K p )Q , FV )m .
(S ⊗Fp V )Kp (0) = Hét
To see this, note that since any sufficiently small Kp acts trivially on V , we have
2
(X(K p Kp )Q , Fp )m ⊗ V
S ⊗Fp V = lim Hét
−→
Kp

∼

2
−→ lim Hét
(X(K p Kp )Q , FV )m ,−→Kp

so it is enough to check that for all compact open subgroups Kp ⊂ Kp (0) we have
K (0)

2
Hét
(X(K p Kp )Q , FV )m p

2
= Hét(X(K p Kp (0))Q , FV )m ,which is an easy consequence of the Hochschild–Serre spectral sequence and ourvanishing result. We also need an embedding S ,→ Se ⊗Zp Fp which is compatible
with the actions of GL3 (Qp ) and the Hecke algebra. In fact, it is easy to see that
we have Se ⊗Zp Fp = S. For example, there is a natural isomorphism
3
3
3
Hét
(XQ , OE )m /$E Hét
(XQ , OE )m = Hét
(XQ , kE )m ,3
and hence the vanishing of Hét
(XQ , kE )m implies that of the finitely generated OE 3
module Hét (XQ , OE )m . One then sees that for all s we have a natural isomorphism
2
s
2
2
s
Hét
(XQ , OE )m /$E
Hét(XQ , OE )m = Hét
(XQ , OE /$E)m ,from which the claim follows easily.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY35

We now examine Axiom Ã1. We must show that if Ve is a finite free Zp -modulewith a locally algebraic action of Kp (0) (acting through GL3 (Zp )), then (Se ⊗ZpVe )Kp (0) is a finite free Zp -module, and for A = Qp , Fp we have(Se ⊗Zp Ve )Kp (0) ⊗Zp A = (Se ⊗Zp Ve ⊗Zp A)Kp (0) .
This is straightforward, the key point being that if FVe denotes the lisse étale sheaf
attached to Ve , then a straightforward argument with Hochschild–Serre as abovegives
2(X(K p Kp (0))Q , FVe )m ,(Se ⊗Zp Ve )Kp (0) = Hétwhich is certainly a finite free Zp -module (it is torsion-free by the proof of Corollary 3.4.5).
The verification of Axioms Ã2 and Ã3 is now exactly the same as in Proposition
7.4.4 of [EGH11], as the Galois representations occurring in the localised cohomol2(X(K p Kp (0))Q , FVe )m are associated to automorphic forms exactly
ogy module Hét
as in [EGH11].5 (In fact, at least for Axiom Ã2 this is a rather roundabout way
of proceeding, as the Galois representations in question are constructed in [HT01]
2
by using Hét
(X(K p Kp (0))Q , FVe ), and one can read off the required properties directly from the comparison theorems of p-adic Hodge theory. For Axiom Ã3 we are
not aware of any comparison theorems in sufficient generality, so it is necessary atpresent to take a lengthier route through the theory of automorphic forms.)

4. Group theory lemmas
The theorems of Section 3 contain certain hypotheses on the Galois representations involved. Our goal in this section is to establish some group-theoretic lemmas
which give sufficient criteria for these hypotheses to be satisfied. Throughout thesection G is a finite group, and k is an algebraically closed field of characteristic p. For any square matrix A with entries in k, we write char (A) to denote thecharacteristic polynomial of A.
4.1. Characterising representations by their characteristic polynomials.
Let ρ : G → GLn (k) be an irreducible representation. In this subsection we establish
some criteria for ρ to satisfy the following hypothesis:

4.1.1. Hypothesis. If θ : G → GLm (k) is irreducible, and if char ρ(g) annihilatesθ(g) for every g ∈ G, then θ is equivalent to ρ.
4.1.2. Remark. Any irreducible ρ of dimension 2 satisfies Hypothesis 4.1.1, as wasproved by Mazur (see the proof of Proposition 14.2 of [Maz77]). However, it is
not satisfied in general if the dimension n of ρ is greater than 2 (for instance, thisalready fails if ρ is the irreducible 3-dimensional representation of A4 , cf. Section 5of [BLR91] as well as Remark 4.1.7 below).
4.1.3. Lemma. Let ρ : G → GLn (k)and θ : G → GLm (k) be two representations.
If θ is irreducible, and if char ρ(g) annihilates θ(g) for every g ∈ G, then the
kernel of θ contains the kernel of ρ.
5In the interests of full disclosure, we are not aware of a reference in the literature giving the
precise base change result from U (2, 1) to GL3 that we need, but it seems to be well known tothe experts, and will follow from the much more general work in progress of Mok and Kaletha–
Minguez–Shin–White.

36

MATTHEW EMERTON AND TOBY GEE

Proof. If ρ(g) is trivial, then the assumption implies that every eigenvalue of θ(g)is equal to 1, and hence that θ(g) is unipotent, and so of order a power of p. Thus
the image of ker(ρ) under θ is a normal subgroup H of θ(G) of p-power order, andwe see that the space of invariants (k m )H is a non-trivial subspace of k m . Since
H is normal in θ(G), we see that θ(G) leaves (k m )H invariant, and hence, since θ
is assumed to be irreducible, we see that in fact (k m )H = k m . Thus H is trivial,which is to say that ker(ρ) ⊂ ker(θ), as claimed.

4.1.4. Lemma. If ρ : G → GLn (k) is a direct sum of one-dimensional charactersof G, and if θ(g) : G → GLm (k) is an irreducible representation of G such thatchar ρ(g) annihilates θ(g) for every g ∈ G, then m = 1, so that θ is a character,and every element of G lies in the kernel of at least one of the summands of ρ⊗θ−1 .
Proof. Since ρ is a direct sum of characters, it factors through Gab . Lemma 4.1.3
then shows that θ also factors through Gab . Since θ is also assumed to be irreducible,
we find that θ must be a character. Twisting by ρ by θ−1 , we may in fact assume
that θ is trivial, and writing ρ = χ1 ⊕ · · · ⊕ χn , we find that for each g ∈ G,
the value
χ
(g)
is
equal
to
1
for
at
least
one
value
of
i
(since
char
ρ(g)
= X−
i


χ1 (g) · · · X − χn (g) annihilates θ(g) = 1). Thus G is equal to the union of its
subgroups ker(χi ).

4.1.5. Remark. In the context of the preceding proposition, we can’t conclude
in general that θ coincides with one the summands of ρ. E.g. if G denotes the
Klein four group, if p is odd, and if ρ denotes the three-dimensional representation
obtained by taking the direct sum of the three non-trivial characters of G, thentaking θ to be the trivial representation, the hypotheses of the proposition aresatisfied, but θ is certainly not one of the summands of ρ.
4.1.6. Lemma. If ρ : G → GLn (k) is irreducible, and is isomorphic to an induction
×
IndG
is
H ψ, where H is a cyclic normal subgroup of G of index n and ψ : H → ka character, then ρ satisfies Hypothesis 4.1.1.
Proof. The restriction ρ|H is isomorphic to the direct sum ⊕g∈G/H ψ g . If we let θ0 be
a Jordan–Hölder constituent of the restriction θ|H , then Lemma 4.1.4 (applied to therepresentations ρ|H and θ0 of H) implies that θ0 is a character of H and (because H iscyclic) that θ0 = ψ g for some g ∈ G/H. The H-equivariant inclusion ψ g = θ0 → θ|HG gthen induces a non-zero G-equivariant map ρ = IndG
H ψ = IndH ψ → θ, which must
be an isomorphism, since both its source and target are irreducible by assumption.
This proves the lemma.

4.1.7. Remark. If we take G = A4 and H to be the normal subgroup of G oforder four (so that H is a Klein four group), then the induction of any non-trivial
character of H gives an irreducible representation ρ : G → SO3 (k). For everyg ∈ G, the characteristic polynomial of ρ(g) thus has 1 as an eigenvalue, and so ifθ denotes the trivial character of G, the element θ(g) is annihilated by char ρ(g)for every g ∈ G. Thus the analogue of Lemma 4.1.6 does not hold in general if H
is not cyclic.
We thank Florian Herzig for providing the proof of the following lemma.
4.1.8. Lemma. Suppose that G is a finite subgroup of GLn (k), which contains
SLn (k 0 ) for some subfield k 0 of k, and is contained in k × GLn (k 0 ).

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

37(1) Any irreducible representation of G over k remains irreducible upon restriction to SLn (k 0 ).
(2) Any two irreducible representations of G which become isomorphic upon
restriction to SLn (k 0 ) can be obtained one from the other via twisting by acharacter of G that is trivial on SLn (k 0 ).
Proof. Let G act via θ on the k-vector space V , and let (θ, W ) be an irreducible
subrepresentation of θ|SLn (k0 ) . Then W is obtained by restriction from a representation of the algebraic group SLn /k 0 (cf. Section 1 of [Jan87]), so the actionof SLn (k 0 ) on W may be extended to an action of GLn (k) and thus of G. By
Frobenius reciprocity we obtain a surjective map (IndG
SLn (k0 ) 1) ⊗ W → V of Grepresentations. Since G/SLn (k 0 ) is a finite abelian group of prime to p order, we
see that (IndG
SLn (k0 ) 1) is a direct sum of one-dimensional representations, so that V
is a twist of W by some character which is trivial on SLn (k 0 ). Thus the restrictionof θ to SLn (k 0 ) is just W , which is irreducible, proving (1).
This same argument also serves to establish (2).

4.1.9. Lemma. Assume that p ≥ n. If ρ : G → GLn (k) is irreducible, and if
SLn (k 0 ) ⊆ ρ(G) ⊆ k × GLn (k 0 ) for some subfield k 0 of k, then ρ satisfies Hypothesis 4.1.1.
Proof. The case n = 1 follows from Lemma 4.1.4, and as remarked above the casen = 2 is proved in the course of the proof of Proposition 14.2 of [Maz77], so we
may assume that n ≥ 3. By Lemma 4.1.3, we may assume that ρ is faithful, sothat we can identify G with ρ(G). In particular, SLn (k 0 ) is a subgroup of G, andby Lemma 4.1.8, the restriction of θ to SLn (k 0 ) remains irreducible.
Since G is finite, our assumption that SLn (k 0 ) ⊂ G implies that k 0 is finite;
suppose that k 0 has cardinality q. We recall some basic facts about the representation theory of SLn (k 0 ); see for example Section 1 of [Jan87]. The irreducible k-representations of SLn (k 0 ) are obtained by restriction from the algebraic
group SLn /k 0 , and are precisely those representations whose highest weights are
q-restricted. (With the usual choice of maximal torus T of SLn , if we identify
the weight lattice with Zn modulo the diagonally embedded copy of Z, a weight(a1 , . . . , an ) is q-restricted if 0 ≤ ai − ai+1 ≤ q − 1 for all 1 ≤ i ≤ n − 1.) Supposethat θ has highest weight (a1 , . . . , an ). Let g ∈ SLn (k 0 ) be a semisimple elementwith eigenvalues α1 , . . . , αn . Then, since g is conjugate to an element of T (k), by
considering the formal character of the corresponding representation of SLn /k 0 we
see that among the eigenvalues of θ(g) are each of the quantitiesnY

αixi ,

i=1

where the xi are a permutation of a1 , . . . , an .
Qn
Our assumption on θ and ρ implies that for each such permutation, i=1 αixi
must equal one of α1 , . . . , αn . In particular, if we let α be a primitive (q n −1)/(q − 1)-st root of unity, we may consider a semisimple element g with eigenn−1
values α, αq , . . . , αq . Then for any x1 , . . . , xn as above, there must be an integer0 ≤ β ≤ n − 1 such thatq n−1 x1 + q n−2 x2 + · · · + xn ≡ q β(mod (q n − 1)/(q − 1)).

38

MATTHEW EMERTON AND TOBY GEE

Fix some 1 ≤ i ≤ n−1, and consider two permutations x1 , . . . , xn and x01 , . . . , x0n
as above, which satisfy xi = x0i for 1 ≤ i ≤ n−2, xn−1 = ai , xn = ai+1 , x0n−1 = ai+1
and x0n = ai . Taking the difference of the two expressions q n−1 x1 +q n−2 x2 +· · ·+xn
and q n−1 x01 + q n−2 x02 + · · ·+ x0n , we conclude that there are integers 0 ≤ β, γ ≤ n − 1such that(q − 1)(ai − ai+1 ) ≡ q β − q γ (mod (q n − 1)/(q − 1)).
Since n ≥ 3, we have 0 ≤ (q − 1)(ai − ai+1 ) ≤ (q − 1)2 < (q n − 1)/(q − 1),and we conclude that either β ≥ γ and (q − 1)(ai − ai+1 ) = q β − q γ , or β < γand (q − 1)(ai − ai+1 ) = (q n − 1)/(q − 1) + q β − q γ . In the second case, we have(q−1)2 ≥ (q−1)(ai −ai+1 ) = (q n −1)/(q−1)+q β −q γ ≥ (q n −1)/(q−1)+1−q n−1 =q n−2 + · · · + q + 2. This is a contradiction if n ≥ 4. If n = 3, (q 3 − 1)/(q − 1) ≡ 3(mod q − 1), so that (q − 1)|3, which is a contradiction as p > 2.
Thus it must be the case that β ≥ γ and (q − 1)(ai − ai+1 ) = q β − q γ , so thatai − ai+1 is congruent to 0 or 1 (mod q), and thus ai − ai+1 = 0 or 1. In particular,
for each i we have 0 ≤ ai − an ≤ n − 1 ≤ p − 1.
We now repeat the above analysis. Fix some 1 ≤ i ≤ n − 1, and consider
two permutations x1 , . . . , xn and x01 , . . . , x0n as above, which satisfy xi = x0i for1 ≤ i ≤ n − 2, xn−1 = ai , xn = an , x0n−1 = an and x0n = ai . Taking the differenceof the two expressions q n−1 x1 +q n−2 x2 +· · ·+xn and q n−1 x01 +q n−2 x02 +· · ·+x0n , andusing that 0 ≤ ai − an ≤ p − 1 ≤ q − 1, we conclude as before that each ai − an = 0
or 1. Thus there must be an integer 1 ≤ r ≤ n with a1 = · · · = ar = an + 1,
ar+1 = · · · = an .
Returning to the original congruences
q n−1 x1 + q n−2 x2 + · · · + xn ≡ q β(mod (q n − 1)/(q − 1)),

we see that the left hand side is congruent to a sum of precisely r distinct values
q j , 1 ≤ j ≤ n − 1. Thus r = 1, and θ|SLn (k0 ) is the standard representation ofSLn (k 0 ), i.e. θ|SLn (k0 ) ∼= ρ|SLn (k0 ) . By part (2) of Lemma 4.1.8, we see that there isa character χ : G → k × with χ|SLn (k0 ) = 1 such that θ ∼= ρ ⊗ χ.
To complete the proof, we must show that χ is trivial. Take g ∈ G; we willshow that χ(g) = 1. If g 0 ∈ G has det g 0 = det g, then g(g 0 )−1 ∈ SLn (k 0 ), so
χ(g 0 ) = χ(g). Firstly, note that by assumption we can write g = λh, with λ ∈ k × ,h ∈ GLn (k 0 ). Choose h0 ∈ GLn (k 0 ) to have eigenvalues {1, . . . , 1, det(h)}. Thenh(h0 )−1 ∈ SLn (k 0 ) ⊂ G, so g 0 := λh0 is an element of G. Then the hypothesis onθ and ρ shows that the eigenvalues of χ(g 0 )g 0 = χ(g)g 0 are contained in the set of
eigenvalues of g 0 , so that if χ(g) 6= 1 we must have χ(g) = det(h) = −1. Assume forthe sake of contradiction that this is the case. If n is odd, then we now choose h0
to have eigenvalues {−1, . . . , −1}, and we immediately obtain a contradiction fromthe same argument. If n is even then since p ≥ n we have p ≥ 5 (recall that we areassuming n ≥ 3), and we may choose a ∈ (k 0 )× , a 6= ±1. Then choosing h0 to haveeigenvalues {1, . . . , 1, a, −1/a} gives a contradiction.

4.2. Representations whose image is generated by regular elements. Recall that a square matrix with entries in k is said to be regular if its minimal
polynomial and characteristic polynomial coincide.
4.2.1. Lemma. If ρ : G → GLn (k) and θ : G → GLn (k) are representations suchthat the image θ(G) is generated by its subset of regular elements, and for everyg ∈ G the characteristic polynomial of ρ(g) annihilates θ(g), then det ρ = det θ.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

39

Proof. Let g ∈ G be an element such that θ(g) is regular. Then the characteristic
polynomials of ρ(g) and θ(g) must be equal (since the minimal and characteristic
polynomials of θ(g) coincide, and the characteristic polynomial of ρ(g) annihilates
θ(g)), so det ρ(g) = det θ(g). Since θ(G) is generated by its subset of regularelements, the result follows.
In fact, we actually need a slight generalisation of this result, where we simply
have a collection of characteristic polynomials, rather than a representation ρ. Suppose that for each g ∈ G we have a monic polynomial ρg (X) = X n − a1 (g)X n−1 +· · · + (−1)n an (g) ∈ k[X] of degree n, with the property that for all g, h ∈ G, wehave an (gh) = an (g)an (h).
4.2.2. Lemma. Suppose that θ : G → GLn (k) is a representation with the property
that θ(G) is generated by its subset of regular elements, and that for each g ∈ G wehave ρg (θ(g)) = 0. Then for each g ∈ G we have det θ(g) = an (g).
Proof. This may be proved in exactly the same way as Lemma 4.2.1.

Let ρ : G → GL3 (k) be irreducible. Our goal is to give criteria for ρ to satisfy
the following hypothesis, in order to apply the previous lemmas:
4.2.3. Hypothesis. The image ρ(G) is generated by its subset of regular elements.
4.2.4. Lemma. If ρ : G → GL3 (k) is irreducible, and if either:(1) ρ is isomorphic to an induction IndG
H ψ, where H is a normal subgroup ofindex 3 in G and ψ : H → k × is a character, or(2) ρ(G) contains a regular unipotent element,
then ρ satisfies Hypothesis 4.2.3.
Proof. Suppose firstly that ρ is isomorphic to an induction IndG
H ψ. Since H is aproper subgroup of G, the set of elements G − H generates G. If g ∈ G − H thenthe characteristic polynomial of ρ(g) is of the form X 3 − α. If p 6= 3 then this has
distinct roots, so ρ(g) is regular, and if p = 3 then it is easy to check that ρ(g) is
the product of a scalar matrix and a unipotent matrix, and is regular.
Suppose now that ρ(G) contains a regular unipotent element. For ease of notation, we will refer to ρ(G) as G from now on. Let H be the subgroup of G generated
by the regular elements, and assume for the sake of contradiction that H is a proper
subgroup of G. We claim that H contains every scalar matrix in G; this is true
because the product of a scalar matrix and a regular matrix is again a regular matrix. Consider an element g ∈ G − H; since it is not regular, and not scalar, it acts
as a scalar on some unique plane in k 3 . We write `g for the corresponding line in
P2 (k).
Let h be the given regular unipotent element in H. Then h stabilises a uniqueline in k 3 , so a unique point P ∈ P2 (k). As g ∈ G − H, we also have gh ∈ G − H.
Then `g ∩ `gh is non-empty, so there is a point Q ∈ P2 (k) which is fixed by g
and gh. It is thus also fixed by h, so in fact Q = P . Since g was an arbitrary
element of G − H, we see that every element of G − H fixes P , and since G isgenerated by G − H, this implies that every element of G fixes P . This contradicts
the assumption that ρ is irreducible.
40

MATTHEW EMERTON AND TOBY GEEAppendix A. Cohomology of pairs
A.1. Étale cohomology of a pair. Let X be a scheme, finite-type and separated
over a field, let Z be a closed subscheme, and write j : U ,→ X for the open
immersion of the complement U := X \ Z into X. As in Section 1, we let E be
an algebraic extension of Qp , where p is invertible on X, let kE denote the residue
field of E, and we let A be one of E or kE . We define the étale cohomology of thepair (X, Z) with coefficients in A to be the étale cohomology of the sheaf j! A on X,
i.e. we write••Hét(X, Z, A) := Hét
(X, j! A).
If i : Z ,→ X is the closed immersion of Z, then the short exact sequence
0 → j! A → A → i∗ A → 0
gives rise to a long exact sequence
i+1
i+1
i+1
i
· · · → Hét
(Z, A) → Hét
(X, Z, A) → Hét
(X, A) → Hét
(Z, A) → · · · ,which is the long exact cohomology sequence of the pair (X, Z).
We are particularly interested in the case of a pair (X \ Y, Z \ Y ), where X isa smooth projective variety over a separably closed field, and Y and Z are smoothdivisors on X which meet transversely. In this case we have a Cartesian diagramof open immersions(A.1.1)

X \ (Y ∪ Z)j

k0

/ X \Y
k


X \Z

j0


/ X.

According to the above definition, the cohomology of the pair (X \ Y, Z \ Y ) is computed as the cohomology of the sheaf j! A on X \ Y , which is canonically isomorphic
to the cohomology of the complex Rk∗ j! A on X. An important point is that there
is a canonical isomorphism(A.1.2)∼

j!0 Rk∗0 A −→ Rk∗ j! A.

(See the discussion of §III (b) on p. 44 of [Fal89].)
A.2. Verdier duality. Verdier duality ([Gro73, XVIII], [Ver67]) states that if f :X → S is a morphism of finite-type and separated schemes over a separably closed
field k, then for any constructible étale A-sheaves F on X and G on S, there is a
canonical isomorphism (in the derived category) of complexes of étale sheaves on S
RHom(Rf! F, G) ∼= Rf∗ RHom(F, f ! G).
We recall some standard special cases of this isomorphism, in the context of the
diagram (A.1.1).
Taking f to be k 0 (and recalling that k 0 is an open immersion), we obtain an
isomorphism
RHom(k!0 A, A) ∼= Rk∗0 RHom(A, A) = Rk∗0 A,
and hence by double duality, an isomorphism(A.2.1)

RHom(Rk∗0 A, A) ∼= k!0 A.

p-ADIC HODGE-THEORETIC PROPERTIES OF MOD p ÉTALE COHOMOLOGY

41

Next, taking f to be j 0 , and taking into account (A.1.2) and (A.2.1), we obtain
isomorphisms
RHom(Rk∗ j! A, A) ∼= Rj 0 k 0 A.
= Rj 0 RHom(Rk 0 A, A) ∼= RHom(j 0 Rk 0 A, A) ∼
∗

!

∗

∗ !

∗

Finally, taking f to the natural map X → Spec k, and recalling that in this casef ! A = A[2d](d) (where d is the dimension of X; cf. [Gro73, XVIII 3.2.5]) and thatRf∗ = Rf! (since f is proper, the variety X being projective by assumption), we
find that
RHom(Rf∗ Rk∗ j! A, A) ∼= Rf∗ Rj 0 k 0 A[2d](d).
= Rf∗ RHom Rk∗ j! A, A[2d](d) ∼∗ !

iPassing to cohomology, we find that Hét
(X \ Y, Z \ Y, A) is in natural duality with
2d−i
H
(X \ Z, Y \ Z, A)(d).

A.3. Vanishing outside of, and torsion-freeness in, the middle degree. We
continue to assume that X is a smooth projective variety of dimension d over the
separably closed field k, and that Y and Z are smooth divisors on X which meettransversely, but in addition, we now assume that the complements X \ Y and
X \ Z are affine (and hence also that Z \ Y and Y \ Z are affine). This latter
ii
Z \ Y, Aassumption implies that Hét
X \ Y, A vanishes if i > d, and that Hétvanishes if i ≥ d ([Gro73] XIV 3.3). By the long exact
cohomology
sequenceof theipair (X \ Y, Y \ Z), we see that Hét
X \ Y, Z \ Y, A vanishes if i > d. Similarly, we
see that H 2d−i (X \ Z, Y \ Z, A)(d) vanishes if i < d. Hence, by the duality betweeniHét(X \ Y, Z \ Y, A) and H 2d−i (X \ Z, Y \ Z, A)(d), we find that both vanish unlessi = d.
Let OE denote the ring of integers in E. Suppose momentarily that E/Qp isfinite, and let $ be a uniformiser of OE . From a consideration of the cohomology
long exact sequence arising from the short exact sequence of sheaves$·

0 → OE /$n −→ OE /$n+1 −→ kE → 0,iand arguing inductively on n, we find that Hét
(X \ Y, Z \ Y, OE /$n ) vanishes indegrees other than i = d for all n. Passing to the projective limit over n, we see thatithe same is true of Hét
(X \Y, Z \Y, OE ). Finally, a consideration of the cohomology
long exact sequence arising from the short exact sequence$·

0 → OE −→ OE −→ kE → 0dshows that Hét
(X \ Y, Z \ Y, OE ) is torsion-free.
By passage to the direct limit over subfields of E which are finite over Qp , we
see that these properties continue to hold for arbitrary algebraic extensions E/Qp .

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(Matthew Emerton) Mathematics Department, University of Chicago(Toby Gee) Mathematics Department, Imperial College London
E-mail address, Matthew Emerton: emerton@math.uchicago.edu
E-mail address, Toby Gee: toby.gee@imperial.ac.uk