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 Shimura varieties 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 the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of ℚ_{p}, embeds into the reduction modulo p of a semistable 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 of certain unitary similitude groups. For example, we prove the following result. (See Corollary 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 varieties^{1} over a quadratic imaginary field E. Note that “sufficiently small level” means that the 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 \overline{𝔽}_{p}-vector spaces on X. Let 𝔪 be a maximal ideal of the Hecke algebra acting on the cohomology H^{•}(X, F), and suppose that there is a Galois representation ρ_{𝔪} : G_{E} → GL_{3}(\overline{𝔽}_{p}) associated to 𝔪. If we suppose further that we have SL_{3}(k) ⊂ ρ_{𝔪}(G_{E}) ⊂ \overline{𝔽}_{p}^{×}SL_{3}(k) for some finite extension k/𝔽_{p}, and that ρ_{𝔪}|_{G_{ℚp}} is 1-regular and irreducible, then the localisations H_{ét}^{i}(X\overline{ℚ}, F)_{𝔪} vanish in degrees i ≠ 2.

In fact, we prove a version of this result for U(n − 1, 1)-Shimura varieties under weaker assumptions on ρ_{𝔪}; however, in general we can only prove vanishing in degrees 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 a three-dimensional irreducible sub-G_{E}-representation of the étale cohomology group H_{ét}^{1}(X\overline{ℚ}, F), then either every irreducible subquotient of ρ_{|G_{ℚp}} is one-dimensional, or else ρ_{|G_{ℚp}} is not 1-regular, or else ρ(G_{E}) is not generated by its subset of regular elements.

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.

Note that in neither theorem do we make any assumption on the level of the Shimura variety at p.

A Galois representation ρ_{𝔪} 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 of this paper was written) implies that such a representation exists for any maximal ideal 𝔪.

It seems reasonable to believe that any irreducible sub-G_{E}-representation of any of the étale cohomology groups H_{ét}^{i}(X\overline{ℚ}, F) for any of the Shimura varieties under consideration should in fact be a constituent of ρ_{𝔪} for some maximal ideal 𝔪 of the Hecke algebra. However, this doesn’t seem to be known, and relating the “abstract” G_{E}-representations ρ_{𝔪} to the “physical” G_{E}-representations appearing on 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 results with 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 \overline{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 : G_{E} → GL_{3}(\overline{𝔽}_{p}) satisfies SL_{3}(k) ⊂ ρ(G_{E}) ⊂ \overline{𝔽}_{p}^{×}SL_{3}(k) for some finite extension k/𝔽_{p}, that ρ|_{G_{ℚp}} is irreducible and 1-regular, and that ρ is modular of some strongly generic weight. Then the set of generic weights 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 that appear in the L^{2}-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 sense of 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 be true for the mod p cohomology of Shimura varieties. Our Theorems A and B give further evidence in this direction.

p-adic Hodge theory. In order to prove these theorems, we establish some new results about the p-adic Hodge theoretic properties of the étale cohomology of varieties 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 ℚ_{p}, and write G_{K} 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 G_{K}-representation H_{ét}^{i}(X\overline{K}, \overline{𝔽}_{p}), then ρ also embeds as a subquotient of a G_{K}-representation over \overline{𝔽}_{p} which is the reduction modulo the maximal ideal of a G_{K}-invariant \overline{ℤ}_{p}-lattice in a G_{K}-representation over \overline{ℚ}_{p} which is semistable with Hodge–Tate weights contained in 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 the analysis 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 G_{ℚ_{p}} (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 of Section 3, while the results of Section 4 provide sufficient conditions for the various representation-theoretic hypotheses appearing in the results of Section 3 to be satisfied.

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 of applications 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, in the 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 ≠ p, by completely different methods to those of this paper. It seems plausible that via the mod p local Langlands correspondence for GL_{n}(ℚ_{l}), Shin’s hypothesis could be interpreted as a condition on the restriction to a 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 results appear 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 the subject of this article. We also thank an anonymous referee for their critical reading of the paper, which led to the filling of some gaps in the arguments of Subsection 1.5.

Conventions. For any field K we let G_{K} denote a choice of absolute Galois group of K.

If K is a finite field then by a Frobenius element in G_{K} we will always mean a 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 O_{K} denote the ring of integers of K, we let I_{K} denote the inertia subgroup of G_{K}, we let W_{K} denote the Weil group of K (the subgroup of G_{K} consisting of elements whose reduction modulo I_{K} is an integral power of Frobenius), and we let WD_{K} 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 K_{v} for the completion of K at v, and O_{K_{v}} for its ring of integers. We will write O_{K,(v)} for the localisation of O_{K} at the prime ideal v.

We will write \overline{ℤ}_{p} for the ring of integers in \overline{ℚ}_{p} (a fixed algebraic closure of ℚ_{p}), and 𝔪_{\overline{ℤ}_{p}} for the maximal ideal of \overline{ℤ}_{p}.

We let ω denote the mod p cyclotomic character. We will denote a Teichmüller lift with a tilde, so that for example ω̃ is the Teichmüller lift of ω.

We use the traditional normalisation of Hodge–Tate weights, with respect to which the cyclotomic character has Hodge–Tate weight 1.

By a closed geometric point \overline{x} of a scheme X, we mean a morphism of schemes \overline{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 \overline{x} is a closed geometric point of a Noetherian scheme X, then we let O_{\overline{x},X} denote the local ring of X at \overline{x}, i.e. the stalk, in the étale topology on X, of the structure sheaf of X at \overline{x}, we let { \atop O_{\overline{x},X}} denote the completion of O_{\overline{x},X}, and we write X̂\overline{x} := Spf { \atop O_{\overline{x},X}}, and refer to X̂\overline{x} as the formal completion of X along the closed geometric point \overline{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 dimensions of the varieties considered, and they also require that the field K be absolutely unramified and that the variety under consideration be of good reduction. More recently, 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 need a 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 as that 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 [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 of giving a p-adic Hodge-theoretic description of Galois-invariant lattices in crystalline or semistable Galois representations, and of the mod p Galois representations that appear 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 this direction, 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, the range 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 role in whatever analysis has to be undertaken, we also prove results in certain cases of potentially semistable reduction which allow us to gain some control over these descent 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 sequence

0 → H_{ét}^{i}(X\overline{K}, ℤ_{p})/pH_{ét}^{i}(X\overline{K}, ℤ_{p}) → H_{ét}^{i}(X\overline{K}, 𝔽_{p}) → H_{ét}^{i+1}(X\overline{K}, ℤ_{p})[p] → 0,

as well as an isomorphism

ℚ_{p} ⊗_{ℤ_{p}} H_{ét}^{i}(X\overline{K}, ℤ_{p}) ⟶^{∼} H_{ét}^{i}(X\overline{K}, ℚ_{p}).

Thus, if both H_{ét}^{i}(X\overline{K}, ℤ_{p}) and H_{ét}^{i+1}(X\overline{K}, ℤ_{p}) are torsion-free, then we see that H_{ét}^{i}(X\overline{K}, 𝔽_{p}) is the reduction mod p of H_{ét}^{i}(X\overline{K}, ℤ_{p}), which is a Galois-invariant lattice in H_{ét}^{i}(X\overline{K}, ℚ_{p}). 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 believe in general that H_{ét}^{i}(X\overline{K}, ℤ_{p}) and H_{ét}^{i+1}(X\overline{K}, ℤ_{p}) will be torsion-free. To get around this 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 ℤ_{p}-coefficients is certainly torsion-free, and so H_{ét}^{1}(X\overline{K}, 𝔽_{p}) is the reduction of a Galois-invariant lattice in H_{ét}^{1}(X\overline{K}, ℚ_{p}). Now a simple induction using the weak Lefschetz theorem shows that for any smooth projective variety X over K, there is an embedding

H_{ét}^{1}(X\overline{K}, 𝔽_{p}) ↪ H_{ét}^{1}(C\overline{K}, 𝔽_{p}),

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 the desired result in the case of H^{1} (ignoring for a moment the problem of obtaining a refinement 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 smooth and projective of dimension d, and if Y and Z are sufficiently generic hyperplane sections of X, then the cohomology of the pair ((X \ Y)\overline{K}, (Z \ Y)\overline{K}), with either ℤ_{p} or 𝔽_{p} coefficients, vanishes in degrees other than d (see Subsection A.3 of the appendix), so that H_{ét}^{d}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, ℤ_{p}) is torsion-free and is thus a Galoisinvariant lattice in H_{ét}^{d}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, ℚ_{p}), which is potentially semistable by [Yam11], and whose reduction is equal to H_{ét}^{d}((X \Y)\overline{K}, (Z\Y)\overline{K}, 𝔽_{p}). Such relative cohomology spaces are the essential ingredient of the basic lemma of Beǐlinson [Beǐ87], 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 Beǐlinson’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 a field of characteristic zero that is complete with respect to a discrete valuation with perfect residue field k_{K} 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 that a projective O_{K}-scheme X is semistable if it is regular and flat over Spec O_{K}, and if the special fibre X_{s} is reduced and is a normal crossings divisor; equivalently, a projective O_{K}-scheme X is semistable if at each closed geometric point \overline{x} of X_{s}, there is an isomorphism of complete local rings

{ \atop O_{\overline{x},X}} ≅ Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}),

where Ô_{K}^{sh} is the completion of the strict Henselisation of O_{K} (equivalently, the completion of the ring of integers in the maximal unramified algebraic extension of K), the element ϖ_{K} is a uniformiser of Ô_{K}^{sh}, and 1 ≤ m ≤ n. We say that X is strictly semistable if it is semistable, and if the special fibre X_{s} 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 O_{K}, and that it has (strictly) semistable reduction if it admits an extension to a projective O_{K}-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 O_{K,(v)}, and that it has (strictly) semistable reduction at v if it admits a (strictly) semistable projective model over O_{K,(v)}, i.e. a projective model over O_{K,(v)} whose base-change over O_{K_{v}} is (strictly) semistable in the sense of the preceding definition.

1.2.2. Proposition. Let X be a smooth projective variety over K with strictly semistable (respectively good) reduction (at v, in the global case). Then there are smooth 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 O_{K}- scheme that is projective and smooth (in the good reduction case) or strictly semistable (in the strictly semistable reduction case), and fix an embedding of X into some projective space over O_{K}. By Corollary 0 and Corollary 1 of [JS09] (or, perhaps more precisely, by their proofs) we can find a hypersurface section Y of X such that Y is again either smooth or strictly semistable over O_{K}. We take Y 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 X_{1}, …, X_{n} of smooth projective schemes in ℙ_{/k_{K}}^{N}, there is a common hypersurface section meeting each of them transversely. This is an immediate consequence of Theorem 1.3 of [Poo04], taking the set U_{P} there to be the subset of Ô_{P} consisting of the f such that f = 0 is transverse to each X_{i} 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 reduction case) or strictly semistable (in the strictly semistable reduction case) projective model of X over O_{K,(v)}. Let P_{d}^{∗} denote the projective space (over O_{K,(v)}) of degree d hypersurfaces in the ambient projective space containing X. Applying the argument in the local case to the base-change X_{/O_{Kv}}, we see that for some d ≥ 1, there is a K_{v}-valued point of P_{d}^{∗} corresponding to a hypersurface section of X_{/O_{Kv}} having either smooth or semistable intersection (depending on the case we are in) with X_{/O_{Kv}}. Furthermore, this point lies in an affinoid open subset of P_{d/K_{v}}^{∗} (the preimage of an open set in the special fibre of P_{d}^{∗}), all of whose points correspond to hyperplane sections of X_{/O_{Kv}} 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 same reference.) The intersection of the K_{v}-points of this affinoid open with P_{d}^{∗}(K_{v}) is a non-empty open subset of P_{d}^{∗}(K_{v}). Since K is dense in K_{v}, we see that this intersection also contains a K-point of P_{d}^{∗}, 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 this case is slightly stronger than our result in general degree, as we do not need to semisimplify the representation, so this result is not completely subsumed by our later 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 O_{K} and residue field k, assumed to be perfect of characteristic p. Let \overline{K} denote an algebraic closure of K, and set G_{K} := Gal(\overline{K}/K).

1.3.1. Theorem. If X is a smooth projective variety over K which has good (respectively strictly semistable) reduction, then H_{ét}^{1}(X\overline{K}, 𝔽_{p}) embeds G_{K}-equivariantly into the reduction modulo p of a G_{K}-invariant lattice in a crystalline (respectively semistable) p-adic representation of G_{K} whose Hodge–Tate weights are contained in [−1, 0].

Proof. We proceed by induction on the dimension d of X. If d ≤ 1 then H_{ét}^{1}(X\overline{K}, 𝔽_{p}) is isomorphic to the reduction modulo p of H_{ét}^{1}(X\overline{K}, ℤ_{p}), and the latter space (being torsion-free, by virtue of our assumption on d) is in turn a lattice in H_{ét}^{1}(X\overline{K}, ℚ_{p}), 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 applies to show that H_{ét}^{1}(Y\overline{K}, 𝔽_{p}) embeds as a subobject of a G_{K}-representation over 𝔽_{p} which is the reduction modulo p of a G_{K}-invariant lattice in a crystalline (respectively semistable) p-adic representation of G_{K} whose Hodge–Tate weights are contained in [−1, 0]. On the other hand, the weak Lefschetz Theorem with 𝔽_{p}-coefficients ([Gro73, XIV 3.3]) implies that the natural (restriction) map H_{ét}^{1}(X\overline{K}, 𝔽_{p}) → H_{ét}^{1}(Y\overline{K}, 𝔽_{p}) is an embedding (because 1 ≤ d − 1 by assumption). This 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 field of 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 this end we fix an algebraic extension E of ℚ_{p}, with ring of integers O_{E} and residue field k_{E}. (In applications, E will typically either be a finite extension of ℚ_{p}, or else will be \overline{ℚ}_{p}.)

We now recall some consequences of the weak Lefschetz Theorem. Among other notions, we will use the étale cohomology of a pair consisting of a variety and a closed subvariety; a precise definition of this cohomology, and a verification of its basic properties (such as those recalled in the next paragraph), is included in Appendix 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 also smooth. Let A denote either E, O_{E}, or k_{E}. In either the first or last case, the spaces H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, A) and H_{ét}^{2d−i}((X \ Z)\overline{K}, (Y \ Z)\overline{K}, A)(d) are naturally dual to one another, for each integer i. The weak Lefschetz Theorem implies that the 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 then follows that both spaces vanish unless i = d in the case when A is taken to be O_{E} as well, and hence that, when i = d, both spaces are torsion-free.

Let \overline{K} denote an algebraic closure of K, and set G_{K} := Gal(\overline{K}/K). Now let ρ : G_{K} → GL_{n}(k_{E}) be irreducible and continuous. In the global case, we fix a place v of K lying over p, and a decomposition group D_{v} ⊂ G_{K} for v.

1.4.1. Theorem. If X is a smooth projective variety over K which has strictly semistable (respectively good) reduction (at v, if we are in the global case), and if ρ embeds as a subquotient of H_{ét}^{i}(X\overline{K}, k_{E}), then ρ also embeds as a subquotient of a G_{K}-representation over k_{E} which is the reduction modulo the uniformiser of a G_{K}-invariant O_{E}-lattice in a G_{K}-representation which is semistable (respectively crystalline) (at v, in the global case) with Hodge–Tate weights contained in the interval [−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 and do choose smooth hypersurface sections Y and Z, having smooth intersection, and such 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))

⋯ → H_{Y\overline{K},ét}^{i}(X\overline{K}, A) → H_{ét}^{i}(X\overline{K}, A) → H_{ét}^{i}((X \ Y)\overline{K}, A) → ⋯,
⋯ → H_{(Y ∩Z)\overline{K},ét}^{i}(Z\overline{K}, A) → H_{ét}^{i}(Z\overline{K}, A) → H_{ét}^{i}((Z \ Y)\overline{K}, A) → ⋯,

and

⋯ → H_{ét}^{i}((X \Y)\overline{K}, (Z \Y)\overline{K}, A) → H_{ét}^{i}((X \Y)\overline{K}, A) → H_{ét}^{i}((Z \Y)\overline{K}, A) → ⋯,

with A taken to be either E or k_{E}. We also recall (cf. Section XIV.3 of [Gro73]) that there are canonical isomorphisms H_{ét}^{i−2}(Y\overline{K}, A)(−1) ⟶^{∼} H_{Y\overline{K},ét}^{i}(X\overline{K}, A) and

H_{ét}^{i−2}((Y ∩ Z)\overline{K}, A)(−1) ⟶^{∼} H_{(Y ∩Z)\overline{K},ét}^{i}(Z\overline{K}, A).

When A = E, all the cohomology spaces that appear are potentially semistable [Yam11]. Since H_{ét}^{i}(X\overline{K}, E), H_{ét}^{i}(Y\overline{K}, E), and H_{ét}^{i}(Z\overline{K}, E) are semistable with Hodge–Tate weights lying in [−i, 0], we see that H_{ét}^{i}((X\Y)\overline{K}, E), H_{ét}^{i}((Z\Y)\overline{K}, E), and H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, E) are semistable, with Hodge–Tate weights lying in [−i, 0].

Now taking A = k_{E}, we see that since ρ is irreducible, it appears either as a subquotient of H_{ét}^{i−2}(Y\overline{K}, k_{E})(−1), as a subquotient of H_{ét}^{i}(Z\overline{K}, k_{E}), as a subquotient of H^{i−1}(Y ∩ Z)(−1), or else as a subquotient of H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, k_{E}). 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, H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, E) is the desired semistable representation, with invariant lattice H_{ét}^{i}((X \Y)\overline{K}, (Z \Y)\overline{K}, O_{E}), whose reduction H_{ét}^{i}((X \Y)\overline{K}, (Z \Y)\overline{K}, k_{E}) contains 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, having smooth 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 the dimension of X that it is enough to check that H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, E) is crystalline; but this follows immediately from Theorem 1.2 of [Yam11]. (Note that if in the notation of [Yam11] we take D^{1} = Z, and D^{2} = Y, then by (A.1.2) below we see that H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, E) appears on the left hand side of the Hyodo–Kato isomorphism in the statement of Theorem 1.2 of [Yam11], and since we have already shown that H_{ét}^{i}((X \ Y)\overline{K}, (Z \ Y)\overline{K}, E) is semistable, it is enough to show that the monodromy operator N vanishes on the right hand side of the Hyodo–Kato isomorphism. This follows easily from the definition of this operator 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 the preceding 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 to a discrete valuation with perfect residue field of characteristic p (“the local case”). We let \overline{K} denote an algebraic closure of K, and set G_{K} := Gal(\overline{K}/K). In the global case, we fix a place v of K lying over p, and a decomposition group D_{v} ⊂ G_{K} for v.

We now put ourselves in the following (somewhat elaborate) situation, which we call a tamely ramified semistable context, or a tame semistable context for short.

We suppose that X_{0} and X_{1} are smooth projective varieties over K, that G is a finite group which acts on X_{1}, and that π : X_{1} → X_{0} is a finite étale morphism which intertwines the given G-action on X_{1} with the trivial G-action on X_{0}, making X_{1} an étale G-torsor over X_{0}.

We suppose further that X_{0} admits a semistable projective model X_{0} over O_{K} (in the local case) or over O_{K,(v)} (in the global case). We also suppose that there is a finite extension L of K, and (in the global case) a prime w of L lying over v, such that (X_{1})_{/L} admits a semistable projective model X_{1} over O_{L} (in the local case) or over O_{L,(w)} (in the global case) to which the G-action extends, such that π extends to a morphism X_{1} → (X_{0})_{/O_{L}} which intertwines the G-action on its source with the trivial G-action on its target, and such that the action of I(L/K)^{op} (respectively I(L_{w}/K_{v})^{op} in the global case) on (X_{1})_{/L} extends to an action on X_{1}.^{2}

Finally (and most importantly) we assume that the composite morphism

(1.5.1)  X_{1} → (X_{0})_{/O_{L}} → X_{0}

(the first being the extension of π, and the second being the natural map) is tamely ramified along the special fibre (X_{0})_{s}, in the sense of [GM71, Def. 2.2.2].

In fact, in our applications we will consider the case that X_{0} 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 stipulation that (1.5.1) be tamely ramified along (X_{0})_{s}.

1.5.2. Lemma. The morphism (1.5.1) is tamely ramified along (X_{0})_{s} if and only if the following two conditions hold:

(1) L (respectively L_{w} in the global case) is tamely ramified over K (respectively K_{v} in the global case), of ramification degree e, say;

(2) For each closed geometric point \overline{x}_{1} of the special fibre (X_{1})_{s}, with image \overline{x}_{0} in (X_{0})_{s}, and for some choice of isomorphism

(1.5.3)  { \atop O_{\overline{x}_{0},X_{0}}} ≅ Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}),

where ϖ_{K} is a uniformiser of Ô_{K}^{sh} and 1 ≤ m ≤ n, there is a corresponding isomorphism

{ \atop O_{\overline{x}_{1},X_{1}}} ≅ Ô_{L}^{sh}[[y_{1}, …, y_{n}]]/(y_{1} ⋯ y_{m} − ϖ_{L}),

2Note that the tameness condition that we are going to require below ensures that L/K is in fact 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.

where ϖ_{L} is a uniformiser of Ô_{L}^{sh}, such that the induced morphism

{ \atop (X_{1})_{\overline{x}_{1}}} → { \atop (X_{0})_{\overline{x}_{0}}}

is defined by the formula x_{j} = y_{j}^{e} for 1 ≤ j ≤ m, and x_{j} = y_{j} 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 K_{v} as K and L_{w} as L, we may reduce ourselves to proving the lemma in the local case. Thus we assume that we in the local case from now on.

If conditions (1) and (2) (for some choice of isomorphism (1.5.3)) hold, then the morphism (1.5.1) is certainly tamely ramified along (X_{0})_{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 (X_{0})_{s}. Since this morphism factors through the natural morphism (X_{0})_{/O_{L}} → X_{0}, 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 the general case of faithfully flat descent for tamely ramified covers considered in that proposition) that Spec O_{L} → Spec O_{K} 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 \overline{x}_{1} of (X_{1})_{s} lying over the closed geometric point \overline{x}_{0} of (X_{0})_{s}, and fix an isomorphism of the form (1.5.3). Since X_{1} → X_{0} is tamely ramified along the divisor ϖ_{K} = 0 of X_{0}, Abhyankar’s Lemma [Gro71, XIII.V] (see also [GM71, Thm. 2.3.2] for a concise statement) implies that we may find regular elements {a_{j}}_{j=1,...,k} of Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} −ϖ_{K}) so that a_{1} ⋯ a_{k} generates the ideal (ϖ_{K}) of Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}), exponents e_{1}, …, e_{k} all coprime to p, and a subgroup H ⊂ μ_{e_{1}} × ⋯ × μ_{e_{k}}, such that the Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K})-algebra { \atop O_{\overline{x}_{1},X_{1}}} is isomorphic to

(Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{k}]/(x_{1} ⋯ x_{m} − ϖ_{K}, T_{1}^{e_{1}} − a_{1}, …, T_{k}^{e_{k}} − a_{k}))^{H}.

(Here μ_{e_{1}} ×⋯×μ_{e_{k}}, and hence H, acts on Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{k}]/(x_{1} ⋯ x_{m}− ϖ_{K}, T_{1}^{e_{1}} − a_{1}, …, T_{k}^{e_{k}} − a_{k}) in the obvious manner: namely, an element (ζ_{1}, …, ζ_{k}) acts on T_{j} via multiplication by ζ_{j}.)

Since each e_{j} is prime to p and Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}) is strictly Henselian, any unit in this ring has an e_{j}th root, and thus we are free to multiply any of the a_{j} by a unit. Consequently, we may assume that in fact a_{1} ⋯ a_{k} = ϖ_{K} = x_{1} ⋯ x_{m}, and hence (again taking advantage of our freedom to modify the a_{j} by units) that {1, …, m} is partitioned into k sets J_{1}, …, J_{k}, such that a_{j} = ∏_{i∈J_{j}} x_{i}. Now if we extract the e_{j}th roots of each x_{i} for i ∈ J_{j}, the resulting extension of Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}) contains

Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{k}]/(x_{1} ⋯ x_{m} − ϖ_{K}, T_{1}^{e_{1}} − a_{1}, …, T_{k}^{e_{k}} − a_{k});

thus it is no loss of generality to assume that k = m and that a_{j} = x_{j}, and so we conclude that { \atop O_{\overline{x}_{1},X_{1}}} is isomorphic, as an Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K})- algebra, to

(Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{m}]/(x_{1} ⋯ x_{m} − ϖ_{K}, T_{1}^{e_{1}} − x_{1}, …, T_{m}^{e_{m}} − x_{m}))^{H},

for some subgroup H ⊂ μ_{e_{1}} × ⋯ × μ_{e_{m}}.

Let I_{j} denote the subgroup 1 × ⋯ × μ_{e_{j}} × ⋯ × 1 of μ_{e_{1}} × ⋯ × μ_{e_{m}}; this is the inertia group of the divisor (x_{j}) with respect to the cover

Spec Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{m}]/(x_{1} ⋯ x_{m} − ϖ_{K}, T_{1}^{e_{1}} − x_{1}, …, T_{m}^{e_{m}} − x_{m})

→ Spec Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}).

If we write H_{j} = H ∩ I_{j}, then H′ := H_{1} × ⋯ × H_{m} is a subgroup of H, and the cover

Spec(Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{m}]/(x_{1} ⋯ x_{m} − ϖ_{K}, T_{1}^{e_{1}} − x_{1}, …, T_{m}^{e_{m}} − x_{m}))^{H′}

→ Spec(Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{m}]/(x_{1} ⋯ x_{m}−ϖ_{K}, T_{1}^{e_{1}}−x_{1}, …, T_{m}^{e_{m}}−x_{m}))^{H}

is unramified in codimension one. Since X_{1} is regular, being semistable over O_{L}, so is the target of this map (since we recall that this target is isomorphic to { \atop (X_{1})_{\overline{x}_{1}}}). 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′.

If we write

H_{j} = 1 × ⋯ × μ_{e′_{j}} × ⋯ × 1 ⊂ 1 × ⋯ × μ_{e_{j}} × ⋯ × 1 = I_{j},

and set d_{j} = e_{j}/e′_{j} and S_{j} = T_{j}^{e′_{j}}, then we conclude that

{ \atop O_{\overline{x}_{1},X_{1}}}

≅ (Ô_{K}^{sh}[[x_{1}, …, x_{n}]][T_{1}, …, T_{m}]/(x_{1} ⋯ x_{m} − ϖ_{K}, T_{1}^{e_{1}} − x_{1}, …, T_{m}^{e_{m}} − x_{m}))^{H′}

≅ Ô_{K}^{sh}[[x_{1}, …, x_{n}]][S_{1}, …, S_{m}]/(x_{1} ⋯ x_{m} − ϖ_{K}, S_{1}^{d_{1}} − x_{1}, …, S_{m}^{d_{m}} − x_{m}).

Now X_{1} is an O_{L}-scheme with reduced special fibre (again because it is semistable over O_{L}). Since { \atop O_{\overline{x}_{1},X_{1}}} is strictly Henselian, it contains Ô_{L}^{sh}, and we may choose a uniformiser ϖ_{L} of this ring such that ϖ_{L}^{e} = ϖ_{K}. Looking at the above description of { \atop O_{\overline{x}_{1},X_{1}}}, and taking into account that its reduction modulo ϖ_{L} must be reduced, we see that this special fibre must be the zero locus of the element S_{1} ⋯ S_{m}, hence that S_{1} ⋯ S_{m} = uϖ_{L} for some unit u, and thus that (S_{1} ⋯ S_{m})^{e} = u^{e}ϖ_{K}. We conclude that d_{1} = … = d_{m} = e and that u^{e} = 1, and hence, replacing ϖ_{L} by uϖ_{L}, we find that S_{1} ⋯ S_{m} = ϖ_{L}. This shows that (2) holds (for our given choice of isomorphism (1.5.3)). □

1.5.4. Remark. Note that we could avoid the appeal to the general theory of tame ramification (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 proof of Theorem 1.5.14 below, we will work directly with these conditions, and in our applications 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.

1.5.5. Lemma. In a tame strictly semistable context as above, the O_{L}-scheme X_{1} is also strictly semistable.

Proof. Since X_{1} is semistable by assumption, it is enough to show that the components of the special fibre (X_{1})_{s} are regular. Suppose that D is a non-regular component of (X_{1})_{s}, and let \overline{x}_{1} be a closed geometric point of D whose local ring on D is not regular. If we let \overline{x}_{0} denote the image of \overline{x}_{1} in (X_{0})_{s}, then Lemma 1.5.2 (2) shows that we may find isomorphisms { \atop O_{\overline{x}_{1},(X_{1})_{s}}} ≅ \overline{O_{K}/ϖ_{K}}[[y_{1}, …, y_{n}]]/(y_{1} ⋯ y_{m})

and { \atop O_{\overline{x}_{0},(X_{0})_{s}}} ≅ \overline{O_{K}/ϖ_{K}}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m}), with 1 ≤ m ≤ n, such that the

morphism { \atop O_{\overline{x}_{1},(X_{1})_{s}}} → { \atop O_{\overline{x}_{0},(X_{0})_{s}}} is given by x_{j} = y_{j}^{e} for 1 ≤ j ≤ m and x_{j} = y_{j}

for m < j ≤ n. Since by assumption \overline{x}_{1} is not a regular point of D, we find that necessarily m ≥ 2, and that (possibly after permuting indices) there is an

isomorphism { \atop O_{\overline{x}_{1},D}} ≅ \overline{O_{K}/ϖ_{K}}[[y_{1}, …, y_{n}]]/(y_{1} ⋯ y_{m′}), where 2 ≤ m′ ≤ m. If we

let D′ denote the image of D in (X_{0})_{s}, we conclude that there is an isomorphism { \atop O_{\overline{x}_{0},D′}} ≅ \overline{O_{K}/ϖ_{K}}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m′}), and thus that D′ is not regular. Hence X_{0} 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 action of I(L/K)^{op} × G on the special fibre (X_{1})_{s}. Let D be an irreducible component of the special fibre of (X_{0})_{s}, and let D̃ denote its preimage in (X_{1})_{s}, so that D̃ is an I(L/K)^{op} × G-invariant union of irreducible components of (X_{1})_{s}.

1.5.6. Lemma. If G is abelian, then there is a homomorphism ψ : I(L/K) → G such that the action of I(L/K)^{op} on D̃ is given by composing the action of G with ψ; i.e. if i ∈ I(L/K), then the action of i on D̃ coincides with the action of ψ(i).

We first prove a general lemma.

1.5.7. Lemma. Let S be a connected Noetherian scheme, let G and I be finite groups, 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 by composing 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 is required.

If we fix a geometric point \overline{s} of S, then the theory of the étale fundamental group [Gro71, Théorème V.4.1] shows that passing to the fibre over \overline{s} gives an equivalence of categories between the category of finite étale covers of S and the category of (discrete) finite sets with a continuous action of π_{1}(S, \overline{s}). In this way, T is classified by an object P of this latter category equipped with an action of I^{op} × G, with respect 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 as a principal homogeneous G-set via left multiplication. As the automorphisms of G as a principal homogeneous G-set are naturally identified with G^{op} acting by right multiplication, we obtain a homomorphism ψ_{p} : I^{op} → G^{op}, or equivalently a homomorphism ψ_{p} : 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 ψ_{gp} = gψ_{p}g^{−1}. Thus, if we now assume furthermore that G is abelian, then ψ_{p} = ψ_{gp}, and so it is reasonable in this case to write simply ψ for this homomorphism, which is well-defined independently of the choice of base-point for P. Furthermore, when G is abelian, left and right multiplication by an element g ∈ G coincide, and so the action of I^{op} on P is given by 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 X_{1} → (X_{0})_{/O_{L}} is étale on generic fibres, and the explicit local formulas for this morphism provided by Lemma 1.5.2 show that it is in fact étale over an open subset U_{0} of (X_{0})_{/O_{L}} whose intersection with the special fibre ((X_{0})_{/O_{L}})_{s} is Zariski dense. Replacing U_{0} with the intersection of all of its I(L/K)^{op}-translates, we may furthermore assume that U_{0} is invariant under the action of I(L/K)^{op} on (X_{0})_{/O_{L}}.

If we let U_{1} denote the preimage of U_{0} in X_{1}, then U_{1} is invariant under the I(L/K) × G-action on X_{1}, and the morphism U_{1} → U_{0} is a finite étale cover, for which the corresponding map U_{1} → U_{0} on generic fibres realizes U_{1} as a G-torsor over U_{0}. It follows that the G-action on U_{1} realizes U_{1} as an étale G-torsor over U_{0}, and hence, passing to special fibres, that (U_{1})_{s} is an étale G-torsor over (U_{0})_{s}.

Now the induced I(L/K)^{op}-action on (U_{0})_{s} is trivial, and so I(L/K)^{op} acts on (U_{1})_{s} as a group of automorphisms of the G-torsor (U_{1})_{s} over (U_{0})_{s}. If D′ := D ∩ (U_{0})_{s}, then D′ is an irreducible component of (U_{0})_{s}, and D̃′ := D̃ ∩ (U_{1})_{s} is the restriction of (U_{1})_{s} to D′. Thus D̃′ → D̃ is again an étale G-torsor, with an action of I(L/K)^{op} via automorphisms. Lemma 1.5.7 then shows that there is a homomorphism ψ : I(L/K)^{op} → G^{op}, or equivalently a homomorphism ψ : I(L/K) → G, such that the action of I(L/K)^{op} on the points of D̃′ is given by composing the action of G with the homomorphism ψ. Since D̃ is equal to the Zariski closure of D̃′ in (X_{1})_{s}, the claim of the lemma follows. □

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 (X_{1})_{s}, then D and gD either coincide or are disjoint.

Proof. The images of D and gD in (X_{0})_{s} coincide, and it follows from Lemma 1.5.2 that two distinct components of (X_{1})_{s} that have non-empty intersection must have distinct images in (X_{0})_{s}. □

If we now suppose that we are in the global case, then the discussion applies with L/K everywhere replaced by L_{w}/K_{v}, and in particular for each component D we may define a character ψ : I(L_{w}/K_{v}) → G describing the action of I(L_{w}/K_{v}) on D.

Our next result describes how our tame semistable context behaves upon passage to a semistable hypersurface section of X_{0}. 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 Y_{0} be a regular hypersurface section of X_{0} such that the union of Y_{0} and (X_{0})_{s} forms a divisor with normal crossings on X_{0}. Let Y_{0} denote the generic fibre of Y_{0}, let Y_{1} denote the preimage of Y_{0} under the morphism π : X_{1} → X_{0}, and let Y_{1} be the preimage of Y_{0} under (1.5.1) (so that (Y_{1})_{/L} is the generic fibre of the O_{L}-scheme Y_{1}).

(1) The complement of Y_{1} in X_{1} is affine.

(2) The generic fibre Y_{0} of Y_{0} is smooth over K, the morphism Y_{1} → Y_{0} is an étale G-torsor (so in particular Y_{1} is also smooth over K), Y_{0} is a semistable model of Y_{0} over O_{K}, Y_{1} is a semistable model for (Y_{1})_{/L} over O_{L}, and the morphism Y_{1} → Y_{0} is tamely ramified; consequently Y_{1} → Y_{0} is again a tamely ramified semistable context.

(3) Suppose that G is abelian. If D′ is an irreducible component of (Y_{1})_{s}, contained in an irreducible component D of (X_{1})_{s}, then the homomorphism ψ : I(L/K) → G, which describes the action of I(L/K) on D, also describes the action of I(L/K) on D′.

Proof. Since Y_{0} is a hypersurface section of X_{0}, its generic fibre Y_{0} is a hypersurface section of X_{0}. Thus its complement is affine. Since π is a finite morphism by assumption, the complement of Y_{1} in X_{1} is again affine. Since Y_{0} is a regular projective K-scheme (being the generic fibre of Y_{0}, which is regular by assumption), it is in fact smooth over K. By definition Y_{1} is the preimage of Y_{0} under the morphism X_{1} → X_{0}, which is an étale G-torsor by assumption. Thus Y_{1} → Y_{0} is indeed an étale G-torsor (and so Y_{1} is also smooth over K).

Let \overline{x}_{0} be a closed geometric point of the special fibre (Y_{0})_{s}. Since ({ \atop (X_{0})_{\overline{x}_{0}}})_{s} ∪ { \atop (Y_{0})_{\overline{x}_{0}}} forms a divisor with normal crossings, since each component of ({ \atop (X_{0})_{\overline{x}_{0}}})_{s} is regular, and since Y_{0} is regular by assumption, it follows from [GM71, Lem. 1.8.4] that ({ \atop (X_{0})_{\overline{x}_{0}}})_{s} ∪ { \atop (Y_{0})_{\overline{x}_{0}}} is in fact a divisor with strictly normal crossings in { \atop (X_{0})_{\overline{x}_{0}}}, and hence the local equation ℓ of { \atop (Y_{0})_{\overline{x}_{0}}}, together with the elements x_{1}, …, x_{m} that cut out the irreducible components of ({ \atop (X_{0})_{\overline{x}_{0}}})_{s}, form part of a regular system of parameters for { \atop O_{\overline{x}_{0},X_{0}}}. Thus we may choose a model of the form (1.5.3) for { \atop (X_{0})_{\overline{x}_{0}}} for which m < n and in which ℓ is equal to the element x_{n}; i.e. in which { \atop (Y_{0})_{\overline{x}_{0}}} is the zero locus of the element x_{n}.

We now choose a closed geometric point \overline{x}_{1} of (X_{1})_{s} lying over \overline{x}_{0}, as well as a model for the tamely ramified morphism { \atop (X_{1})_{\overline{x}_{1}}} → { \atop (X_{0})_{\overline{x}_{0}}} as in part (2) of Lemma 1.5.2. Thus this morphism has the form

Spec Ô_{L}^{sh}[[y_{1}, …, y_{n}]]/(y_{1} ⋯ y_{m} − ϖ_{L}) → Spec Ô_{K}^{sh}[[x_{1}, …, x_{n}]]/(x_{1} ⋯ x_{m} − ϖ_{K}),

with x_{j} = y_{j}^{e} for 1 ≤ j ≤ m, and x_{j} = y_{j} for m < j ≤ n. In particular, we see that x_{n} = y_{n}, and thus we see that the induced morphism

(1.5.10)  { \atop (Y_{1})_{\overline{x}_{1}}} → { \atop (Y_{0})_{\overline{x}_{0}}}

can be written as

(1.5.11) Spec Ô_{L}^{sh}[[y_{1}, …, y_{n−1}]]/(y_{1} … y_{m} − ϖ_{L})

→ Spec Ô_{L}^{sh}[[x_{1}, …, x_{n−1}]]/(x_{1} … x_{m} − ϖ_{K}).

Thus we see that Y_{0} and Y_{1} are indeed semistable models of their generic fibres (over O_{K} and O_{L} respectively), and that the morphism Y_{1} → Y_{0} is tamely ramified. This completes the verification of (2). The claim of (3) follows from the fact that the action of I(L/K)^{op} × G on (Y_{1})_{s} is the restriction of the corresponding action on (X_{1})_{s}, together with the fact that any component of (Y_{1})_{s} is contained in a component of (X_{1})_{s}. □

We now suppose that E is an algebraic extension of ℚ_{p} containing K_{0}. Recall that if ρ : G_{K} → GL_{n}(E) is a potentially semistable representation, then we may attach a Weil–Deligne representation WD(ρ) to ρ by first passing to the potentially semistable Dieudonné module D_{pst}(ρ) of ρ, which is a module over E ⊗_{ℚ_{p}} K_{0}, then fixing an embedding K_{0} ↪ E, and hence a projection pr : E ⊗_{ℚ_{p}} K_{0} → E, and, finally, forming WD(ρ) := E ⊗_{E⊗_{ℚp} K_{0},pr} D_{pst}. (See for example Appendix B of [CDT99] for a detailed discussion of this construction.) Although WD(ρ) depends on the choice of the embedding K_{0} ↪ 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 X_{1}, or of a pair (X_{1}, Y_{1}) that arises in the context of the preceding proposition. Before stating the result we introduce some additional notation, and an additional assumption.

Assume that G is abelian, and let J denote the set of I(L/K) × G orbits on the set of irreducible components of (X_{1})_{s}, and let D_{j} (for j ∈ J) denote the union of the components lying in the orbit labelled by j. Let ψ_{j} : I(L/K) → G be the homomorphism provided by Lemma 1.5.6, describing the action of I(L/K) on the points of D_{j}.

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 potentially semistable G_{K}-representation H_{ét}^{i}((X_{1})_{/\overline{K}}, E), or else suppose that we are in the context of Proposition 1.5.9, and let W denote the Weil–Deligne representation associated to the potentially semistable G_{K}-representation H_{ét}^{i}((X_{1})_{/\overline{K}}, (Y_{1})_{/\overline{K}}), E) (here i is some given degree of cohomology); in either case W is a representation of the product WD_{K} × G. Assume furthermore that G is abelian.

Then, if, as in the above discussion, J̃ denotes the set of I(L/K) × G-orbits of irreducible components of (X_{1})_{s}, we may decompose W as a direct sum W = ⨁_{j̃∈J̃}Wj̃, such that on Wj̃, the action of the inertia group in W_{K} is obtained by

composing the G-action on Wj̃ with the homomorphism I_{K} → I(L/K) ⟶^{ψj̃} G.

Proof. Since the action of the inertia subgroup of W_{K} on W factors through a finite 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 of subobjects, quotients, and extensions (in the category of W_{K} × G-representations). A consideration of the long exact sequence of cohomology associated to the pair (X_{1}, Y_{1}) (cf. Appendix A) then reduces the claim for the cohomology of the pair to the claim for the cohomology of X_{1} and Y_{1} individually. Since Proposition 1.5.9 shows that the strictly semistable model Y_{1} of (Y_{1})_{/L} behaves in an identical manner to the strictly semistable model of X_{1} of (X_{1})_{/L}, it in fact suffices to consider the case of X_{1}.

Thus we now restrict our attention to the W_{K}-representation W underlying the potentially semistable Dieudonné module associated to H_{ét}^{i}((X_{1})_{/\overline{K}}, E). By [Tsu99] this Dieudonné module is naturally identified with the log-crystalline cohomology H^{i}((X_{1})_{s}^{×}/W (k)^{×}) ⊗_{W (k)} E of the special fibre (X_{1})_{s} with its natural log-structure. Lemma 1.5.8 shows that if an intersection of distinct components of the special fibre (X_{1})_{s} is non-empty, then the various components appearing must 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 {D_{j}}_{j∈J} of I(L/K) × G -orbits of components of (X_{1})_{s}, this log-crystalline cohomology may be computed by the following spectral sequence of [Mok93]:

E_{1}^{−m,i+m} = ⨁_{{l≥max\{0,−m\} \atop \{j_{1},...,j_{2l+m+1}\}⊂J}} H_{cris}^{i−2l−m}(D_{j_{1}}∩⋯∩D_{j_{2l+m+1}}/W (k))⊗_{W (k)}E(−l−m)

⟹ H^{i}((X_{1})_{s}^{×}/W (k)^{×}) ⊗_{W (k)} E.

Each of the summands in the E_{1}-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 ψj̃. Thus the E_{1}-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 E_{1}-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 K_{v}).

1.5.13. Theorem. Suppose that we are in the tame strictly semistable context described above. Then the G_{K} × G-representation H_{ét}^{1}((X_{1})\overline{K}, 𝔽_{p}) embeds G_{K} × Gequivariantly into the reduction modulo the uniformiser of a G_{K} × G-invariant O_{E}-lattice in a representation V of G_{K} × G over E, having the following properties:

(1) the restriction of V to G_{L} is semistable, with Hodge–Tate weights contained in the interval [−1, 0];

(2) The Weil–Deligne representation associated to V, which is naturally a representation of WD_{K} × G, when restricted to a representation of I_{K} × G, can be written as a direct sum ⨁_{j̃∈J̃}Wj̃ of I_{K} × G-representations, where j̃ runs over the same index set that labels the set of I(L/K) × G-orbits of irreducible components of (X_{1})_{s}, such that on Wj̃, the action of the inertia group in W_{K} is obtained by composing the G-action on Wj̃ with the

homomorphism I_{K} → I(L/K) ⟶^{ψj̃} G.

Proof. We follow the proof of Theorem 1.3.1, proceeding by descending induction on the dimension of X_{0} and X_{1}, and passing to appropriately chosen hypersurface sections Y_{0} of X_{0}, and their corresponding preimages Y_{1} in X_{1} and Y_{1} in (X_{1})_{/L}. Taking into account Proposition 1.5.9, we thus reduce to the case when X_{0} and X_{1} are curves, so that H_{ét}^{1}((X_{1})_{/\overline{K}}, 𝔽_{p}) is the reduction mod p of H_{ét}^{1}((X_{1})_{/\overline{K}}, ℤ_{p}), which is in turn a lattice in H_{ét}^{1}((X_{1})_{/\overline{K}}, ℚ_{p}). 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 : G_{K} × G → GL_{n}(k_{E}) be an irreducible and continuous representation that embeds as a subquotient of H_{ét}^{i}((X_{1})\overline{K}, k_{E}). Then ρ also embeds as a subquotient of a G_{K} × G-representation over k_{E} which is the reduction modulo the uniformiser of a G_{K} × G-invariant O_{E}-lattice in a representation V of G_{K} × G over E, having the following properties:

(1) the representation V becomes semistable when restricted to G_{L} (respectively to the decomposition group D_{w} ⊂ G_{L} in the global case), with Hodge–Tate weights contained in the interval [−i, 0];

(2) The Weil–Deligne representation associated to V, which is naturally a representation of WD_{K} × G (respectively of WD_{K_{v}} × G in the global case), when restricted to a representation of I_{K} × G (respectively of I_{K_{v}} × G in the global case), can be written as a direct sum ⨁j̃ Wj̃ of I_{K} × Grepresentations (respectively of I_{K_{v}} ×G-representations), where j̃ runs over the same index set that labels the set of I(L/K) × G-orbits (respectively of I_{K_{v}} ×G-orbits) of irreducible components of (X_{1})_{s}, such that on Wj̃, the action of the inertia group is obtained by composing the G-action on Wj̃ with

the homomorphism I_{K} → I(L/K) ⟶^{ψj̃} G (respectively the homomorphism

I_{K_{v}} → I(L_{w}/K_{v}) ⟶^{ψj̃} 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 some constraints on the reductions of certain tamely potentially semistable p-adic representations of G_{ℚ_{p}}.

2.1. Preliminaries. We begin by recalling a variety of results from Section 3 of [EGH11]. To this end, let p be an odd prime, let \overline{ℚ}_{p} be a fixed algebraic closure of ℚ_{p}, and let E and K be finite extensions of ℚ_{p} inside \overline{ℚ}_{p}. Assume that E contains the images of all embeddings K ↪ \overline{ℚ}_{p}. Let K_{0} be the maximal absolutely unramified subfield of K, so that K_{0} = W (k)[1/p], where k is the residue field of K. Let K/K′ be a Galois extension, with K′ a field lying between ℚ_{p} and K. Assume further that K/K′ is tamely ramified with ramification index e, and fix a uniformiser π ∈ K with π^{e} ∈ K′. Let E(u) ∈ W (k)[u] be the minimal polynomial of π over K_{0}.

Let k_{E} be the residue field of E, and let 0 ≤ r ≤ p − 2 be an integer. Recall that the category k_{E}-BrMod_{dd}^{r} of Breuil modules of weight r with descent data from K to K′ and coefficients k_{E} consists of quintuples (M, M_{r}, φ_{r}, ĝ, N) where:

• M is a finitely generated (k ⊗_{𝔽_{p}} k_{E})[u]/u^{ep}-module, free over k[u]/u^{ep}.

• M_{r} is a (k ⊗_{𝔽_{p}} k_{E})[u]/u^{ep}-submodule of M containing u^{er}M.

• φ_{r} : M_{r} → M is k_{E}-linear and φ-semilinear (where φ : k[u]/u^{ep} → k[u]/u^{ep} is the p-th power map) with image generating M as a (k ⊗_{𝔽_{p}} k_{E})[u]/u^{ep}- module.

• N : M → M is k ⊗_{𝔽_{p}} k_{E}-linear and satisfies N(ux) = uN(x) − ux for all x ∈ M, u^{e}N(M_{r}) ⊂ M_{r}, and φ_{r}(u^{e}N(x)) = cN(φ_{r}(x)) for all x ∈ M_{r}. Here, c = F̄(u)^{p} ∈ (k[u]/u^{ep})^{×}, where E(u) = u^{e} + pF (u) in W (k)[u].

• ĝ : M → M are additive bijections for each g ∈ Gal(K/K′), preserving M_{r}, commuting with the φ_{r}- and N-actions, and satisfying ĝ_{1} ∘ĝ_{2} = { \atop g_{1} ∘ g_{2}} for all g_{1}, g_{2} ∈ Gal(K/K′). Furthermore, if a ∈ k ⊗_{𝔽_{p}} k_{E}, m ∈ M then ĝ(au^{i}m) = g(a)((g(π)/π)^{i} ⊗ 1)u^{i}ĝ(m).

There is a covariant functor T_{st}^{∗,r} from k_{E}-BrMod_{dd}^{r} to the category of k_{E}-repre-sentations of G_{K′}.

2.1.1. Lemma. Suppose that M ∈ k_{E}-BrMod_{dd}^{r}, and that T′ is a G_{K′}-subrepre-sentation of T_{st}^{∗,r}(M) (so that in particular T′ has the structure of a k_{E}-vector space). Then there is a unique subobject M′ of M such that if f : M′ → M is the inclusion map, then T_{st}^{∗,r}(f) is identified with the inclusion T′ ↪ T_{st}^{∗,r}(M). (Here M′ is a subobject of M in the naive sense that it is a sub-(k ⊗_{𝔽_{p}} k_{E})[u]/u^{ep}-module of M, which inherits the structure of an object of k_{E}-BrMod_{dd}^{r} from M in the obvious 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 K_{0} be the unique unramified extension of ℚ_{p} of degree d, we take K = K_{0}((−p)^{1/(p^{d}−1)}), and we set K′ = K_{0}, so that e = p^{d} − 1. Fix π = (−p)^{1/(p^{d}−1)}. We write ω̃_{d} : Gal(K/K_{0}) → K_{0}^{×} for the character g ↦ g(π)/π, and we let ω_{d} be the reduction of ω̃_{d} modulo π. (By inflation we can also think of ω̃_{d} and ω_{d} as characters of I_{K_{0}} = I_{ℚ_{p}}. Note that ω_{d} is a tame fundamental character of niveau d and that ω̃_{d} is the Teichmüller lift of ω_{d}.) Note that when d = 1, we have ω_{1} = ω, the mod p cyclotomic character.

Let φ be the arithmetic Frobenius on k, and let σ_{0} : k ↪ k_{E} be a fixed embedding. Inductively define σ_{1}, …, σ_{d−1} by σ_{i+1} = σ_{i} ∘ φ^{−1}; we will often consider the numbering to be cyclic, so that σ_{d} = σ_{0}. There are idempotents e_{i} ∈ k ⊗_{𝔽_{p}} k_{E} such that if M is any k ⊗_{𝔽_{p}} k_{E}-module, then M = ⨁_{i} e_{i}M, and e_{i}M is the subset of M consisting of elements m for which (x ⊗ 1)m = (1 ⊗ σ_{i}(x))m for all x ∈ k. Note that (φ ⊗ 1)(e_{i}) = e_{i+1} for all i.

If ρ : G_{K_{0}} → GL_{n}(E) is a potentially semistable representation which becomes semistable over K, then the associated inertial type (that is, the restriction to I_{K_{0}} of the Weil–Deligne representation associated to ρ) is a representation of I_{K_{0}} which becomes trivial when restricted to I_{K}, so we can and do think of it as a representation of Gal(K/K_{0}) ≅ I_{K_{0}}/I_{K}.

2.1.2. Proposition. Maintaining our current assumptions on K, suppose that ρ : G_{K_{0}} → GL_{n}(E) is a continuous representation whose restriction to G_{K} is semistable with Hodge–Tate weights contained in [0, r], with r ≤ p − 2, and let the inertial type of ρ be χ_{1} ⊕ ⋯ ⊕ χ_{n}, where each χ_{i} is a character of I_{K_{0}}/I_{K}. If \overline{ρ} denotes the reduction modulo 𝔪_{E} of a G_{K_{0}}-stable O_{E}-lattice in ρ, then there is an element M of k_{E}-BrMod_{dd}^{r}, admitting a (k ⊗_{𝔽_{p}} k_{E})[u]/u^{ep}-basis v_{1}, …, v_{n} such that ĝ(v_{i}) = (1 ⊗ \overline{χ}_{i}(g))v_{i} for all g ∈ Gal(K/K_{0}), and for which T_{st}^{∗,r}(M) ≅ \overline{ρ}.

Proof. This is Proposition 3.3.1 of [EGH11]. (Note that the conventions on the sign of the Hodge–Tate weights in [EGH11] are the opposite of the conventions in this paper.) □

2.1.3. Lemma. Maintain our current assumptions on K, so that in particular we have e = p^{d} − 1. Then every rank one object of k_{E}-BrMod_{dd}^{r} may be written in the form:

• M = ((k ⊗_{𝔽_{p}} k_{E})[u]/u^{ep}) · m,
• e_{i}M_{r} = u^{r_{i}}e_{i}M,
• φ_{r}(∑_{i=0}^{d−1} u^{r_{i}}e_{i}m) = λm for some λ ∈ (k ⊗_{𝔽_{p}} k_{E})^{×},
• ĝ(m) = (∑_{i=0}^{d−1}(ω_{d}(g)^{k_{i}} ⊗ 1)e_{i})m for all g ∈ Gal(K/K_{0}), and
• N(m) = 0.

Here the integers 0 ≤ r_{i} ≤ (p^{d} − 1)r and k_{i} satisfy k_{i} ≡ p(k_{i−1} + r_{i−1}) mod (p^{d} − 1) for all i. Conversely, any module M of this form is a rank one object of k_{E}-BrMod_{dd}^{r}. Furthermore,

T_{st}^{∗,r}(M)|_{I_{K_{0}}} ≅ σ_{0} ∘ ω_{d}^{κ_{0}},

where κ_{0} ≡ k_{0} + p(r_{0}p^{d−1} + r_{1}p^{d−2} + ⋯ + r_{d−1})/(p^{d} − 1) mod (p^{d} − 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 k_{i} = (1 + p + ⋯ + p^{d−1})x_{i} for some 0 ≤ x_{i} < p − 1. In this case, the condition that pr_{i} ≡ k_{i+1} − pk_{i} mod (p^{d} − 1) implies that r_{i} ≡ (1 + p + ⋯ + p^{d−1})(x_{i+1} − x_{i}) mod (p^{d} − 1), so the condition that 0 ≤ r_{i} ≤ (p^{d} − 1)r means that we can write r_{i} = (1 + p + ⋯ + p^{d−1})(x_{i+1} − x_{i}) + (p^{d} − 1)y_{i} with 0 ≤ y_{i} ≤ r. An elementary calculation shows that we then have

κ_{0} ≡ x_{0} + y_{0} + p^{d−1}(x_{1} + y_{1}) + ⋯ + p(x_{d−1} + y_{d−1}) mod (p^{d} − 1).

2.2. Regularity. Let ℚ_{p^{n}} denote the unique unramified extension of ℚ_{p} of degree n, with residue field 𝔽_{p^{n}}. Regarding 𝔽_{p^{n}} as a subfield of \overline{𝔽}_{p}, we may then regard ω_{n} as a character I_{ℚ_{p}} → \overline{𝔽}_{p}^{×}.

2.2.1. Definition. Let \overline{ρ} : G_{ℚ_{p}} → GL_{n}(\overline{𝔽}_{p}) be an irreducible representation, so that

\overline{ρ} ≅ Ind_{G_{ℚ_{p^{n}}}}^{G_{ℚp}} χ for some character χ : G_{ℚ_{p}} → \overline{𝔽}_{p}^{×}. If we write

χ|_{I_{ℚp}} = ω_{n}^{(a_{0}+pa_{1}+···+p^{n−1}a_{n−1})},

where each a_{i} ∈ [0, p − 1] and not all the a_{i} = p − 1, then the multiset of exponents of \overline{ρ} is defined to be the multiset of residues of the a_{i} in ℤ/pℤ.

2.2.2. Definition. Let \overline{ρ} : G_{ℚ_{p}} → GL_{n}(\overline{𝔽}_{p}) be a representation. Then the multiset of exponents of \overline{ρ} is the union of the multisets of exponents of each of the Jordan– Hölder factors of \overline{ρ}.

2.2.3. Definition. Let \overline{ρ} : G_{ℚ_{p}} → GL_{n}(\overline{𝔽}_{p}) be a representation, and let r be a nonnegative integer. Then we say that \overline{ρ} is r-regular if the exponents a_{1}, …, a_{n} of \overline{ρ} are such that the residues a_{i} + 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 : G_{ℚ_{p}} → GL_{m}(\overline{ℚ}_{p}) be a potentially semistable representation, with Hodge–Tate weights contained in the interval [0, r], and inertial type χ_{1}⊕⋯⊕χ_{m}. Suppose that there are (not necessarily distinct) integers 0 ≤ a_{1}, …, a_{n} < p − 1 such that each χ_{i} is equal to some ω^{a_{j}}.

Suppose that \overline{ρ} : G_{ℚ_{p}} → GL_{n}(\overline{𝔽}_{p}) is a subquotient of \overline{s}, the reduction mod 𝔪_{O_{\overline{ℚ}_{p}}}

of some G_{ℚ_{p}}-stable \overline{ℤ}_{p}-lattice in s. Suppose also that • det \overline{ρ}|_{I_{ℚp}} = ω^{a_{1}+···+a_{n}+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 \overline{ρ} has dimension greater than one. Then \overline{ρ} is not r-regular.

Proof. Modifying the choice of \overline{ℤ}_{p}-lattice if necessary, it suffices to treat the case that \overline{ρ} and \overline{s} are semisimple. Let \overline{ρ}′ ≅ Ind_{G_{ℚ_{p^{d}}}}^{G_{ℚp}} χ be an irreducible subrepresentation

of \overline{ρ}. Take K′ = K_{0} = ℚ_{p^{d}} in the above notation, so that e = p^{d}−1. Taking E to be sufficiently large so that s is defined over E and \overline{ρ} is defined over k_{E}, and applying Proposition 2.1.2 to s|_{G_{ℚ_{p^{d}}}}, we see that there is an element M of k_{E}-BrMod_{dd}^{r} with

T_{st}^{∗,r}(M) ≅ \overline{s}|_{G_{ℚ_{p^{d}}}} ,

such that M has a (k⊗_{𝔽_{p}} k_{E})[u]/u^{ep}-basis v_{1}, …, v_{m} such that ĝ(v_{i}) = (1⊗\overline{χ}_{i}(g))v_{i} for all g ∈ Gal(K/K_{0}). Since \overline{s}|_{G_{ℚ_{p^{d}}}} contains a subrepresentation isomorphic to χ, we see from Lemma 2.1.1 that there is a rank one subobject N of M for which T_{st}^{∗,r}(N) ≅ χ.

Since N /uN embeds into M/uM (as N is a free k[u]/u^{ep}-submodule of the free k[u]/u^{ep}-module M), we see from Lemma 2.1.3 and our assumption on the characters \overline{χ}_{i} that we may write N in the form

• N = ((k ⊗_{𝔽_{p}} k_{E})[u]/u^{(p^{d}−1)p}) · w,
• e_{i}N_{r} = u^{r_{i}}e_{i}N,
• φ_{r}(∑_{i=0}^{d−1} u^{r_{i}}e_{i}w) = λw for some λ ∈ (k ⊗_{𝔽_{p}} k_{E})^{×},
• ĝ(w) = (∑_{i=0}^{d−1}(ω_{d}(g)^{k_{i}} ⊗ 1)e_{i})w for all g ∈ Gal(K/K_{0}), and
• N(w) = 0.

Here the integers 0 ≤ r_{i} ≤ (p^{d} − 1)r and k_{i} satisfy k_{i} ≡ p(k_{i−1} + r_{i−1}) (mod p^{d} − 1) for all i, and each k_{i} is equal to some (1 + p + ⋯ + p^{d−1})a_{j}. (The conditions on the r_{i} come from Lemma 2.1.3, and the fact that each k_{i} is equal to some (1 + p + ⋯ + p^{d−1})a_{j} comes from the fact that N is a submodule of M, which has a basis v_{1}, …, v_{n} such that ĝ(v_{i}) = (1 ⊗ \overline{χ}_{i}(g))v_{i}, and the assumption that each \overline{χ}_{i} is equal to some ω^{a_{j}}.) Writing k_{i} = (1 + p + ⋯ + p^{d−1})x_{i}, 0 ≤ x_{i} < p − 1, we see as in Remark 2.1.4 that we can write r_{i} = (1 + p + ⋯ + p^{d−1})(x_{i+1} − x_{i}) + (p^{d} − 1)y_{i}, with 0 ≤ y_{i} ≤ r. By Lemma 2.1.3 and Remark 2.1.4, we have

χ|_{I_{ℚp}} = σ_{0} ∘ ω_{d}^{x_{0}+y_{0}+p^{d−1}(x_{1}+y_{1})+···+p(x_{d−1}+y_{d−1})}.

Since r ≤ p − 2 we have 0 ≤ y_{i} ≤ p − 2, and we conclude (after allowing for “carrying”) that each exponent of \overline{ρ} is of the form a_{j} + k, with 0 ≤ k ≤ r + 1.

Suppose that \overline{ρ} is r-regular. It must then be the case that the x_{i} as above are all distinct. Applying this analysis to each irreducible subrepresentation of \overline{ρ}, we conclude that the a_{i} are all distinct. Since we have det(\overline{ρ}′)|_{I_{ℚp}} = χ^{1+p+···+p^{d−1}}|_{I_{ℚp}} = ω^{(x_{0}+y_{0})+···+(x_{d−1}+y_{d−1})}, we conclude that det \overline{ρ}|_{I_{ℚp}} = ω^{a_{1}+···+a_{n}+y} for some 0 ≤ y ≤ nr ≤ n(n − 1)/2. The assumption on det \overline{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 (by the additional hypothesis that we have assumed in this case) that the representation \overline{ρ}′ above has dimension d > 1. By the above analysis we must have y = n(n − 1)/2, so that each y_{i} = r. Since we have r_{i} ≤ (p^{d} − 1)r, we must have x_{i+1} − x_{i} ≤ 0 for each i, so that in fact x_{0} = x_{1} = ⋯ = x_{d−1}, a contradiction (as we already showed that the x_{i} are distinct). □

3. The cohomology of Shimura varieties

3.1. The semistable reduction of certain U(n−1, 1)-Shimura varieties. Fix n ≥ 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) = 𝔭\overline{𝔭} for some choice of 𝔭, let x ↦ \overline{x} be the nontrivial automorphism of F, and regard F as a subfield of ℂ via a fixed embedding F ↪ ℂ.

Let D be a division algebra over F of dimension n^{2}, and let ∗ be an involution of D of the second kind (that is, ∗|_{F} is nontrivial). Assume that D splits at 𝔭 (and hence at \overline{𝔭}), and fix an isomorphism D_{𝔭} ≅ M_{n}(ℚ_{p}) with the property that under the induced isomorphism

D ⊗ ℚ_{p} ≅ M_{n}(ℚ_{p}) × M_{n}(ℚ_{p})^{op},

the involution ∗ corresponds to (X, Y) ↦ (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 D_{𝔭} ≅ M_{n}(ℚ_{p}) induces an isomorphism G ×_{ℚ} ℚ_{p} ≅ GL_{n} × 𝔾_{m}.

Now let h_{0} : ℂ → D_{ℝ} be an ℝ-algebra homomorphism with the properties that h_{0}(z)^{∗} = h_{0}(\overline{z}) and the involution x ↦ h_{0}(i)^{−1}x^{∗}h_{0}(i) is positive. Let B = D^{op} and let V = D, which we consider as a free left B-module of rank one by multiplication on the right. Then End_{B}(V) = D, and one can find an element ξ ∈ D^{×} with the properties that ξ^{∗} = −ξ and such that the involution ι of B defined by x^{ι} = ξx^{∗}ξ^{−1} is positive (that is, tr_{B/ℚ}(xx^{ι}) > 0 for all nonzero x) (see Section I.7 of [HT01] or Section 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, b^{ι}y), and that ψ(·, h_{0}(i)·) is either positive- or negative-definite. After possibly replacing ξ by −ξ, we can and 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 in fact {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(𝔸^{p,∞})) is sufficiently small if for some prime q (respectively some prime q ≠ p) the projection of K (respectively K^{p}) to G(ℚ_{q}) contains no element of finite order other than 1. If K is sufficiently small, we may consider the Shimura variety Sh(G, h_{0}|_{ℂ^{×}}^{−1}, K). It has a canonical model over F, which we denote 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).

3The 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.

We say that a compact open subgroup K of 𝔾(A) is of level dividing N, for some integer N ≥ 1, if for all primes l ∤ N we can write K = K_{l}K^{l}, where K_{l} is a hyperspecial maximal compact subgroup of G(ℚ_{l}) and K^{l} is a compact open subgroup of G(𝔸^{∞,l}). (Note then that in fact K = K_{N} ×∏_{l∤N} K_{l}, for some compact open subgroup K_{N} of ∏_{l|N} G(ℚ_{l}).) If K is of level dividing N, then we similarly refer 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 specific kinds of level structure. We begin by introducing notation related to the level structures in question.

We write I_{0} for the Iwahori subgroup of GL_{n}(ℚ_{p}) × ℚ_{p}^{×}; namely, I_{0} is the subgroup of GL_{n}(ℤ_{p}) × ℤ_{p}^{×} consisting of elements whose first factor lies in the usual Iwahori subgroup of matrices which are upper triangular mod p. We write I_{1} to denote the pro-p-Iwahori subgroup of GL_{n}(ℚ_{p}) × ℚ_{p}^{×}; namely, I_{1} is the (unique) pro-p Sylow subgroup of I_{0}, and consists of those elements of GL_{n}(ℤ_{p}) × ℤ_{p}^{×} whose first factor is upper triangular unipotent mod p, and whose second factor is congruent to 1 mod p. We write I_{1}^{∗} to denote the subgroup of I_{0} consisting of matrices in GL_{n}(ℤ_{p}) × ℤ_{p}^{×} whose first factor is upper triangular unipotent mod p.

There is a natural isomorphism ℤ_{p}^{×} = 𝔽_{p}^{×} × (1 + pℤ_{p}), and this induces a natural isomorphism I_{1}^{∗} = 𝔽_{p}^{×} ×I_{1}. If we let T denote the diagonal torus in GL_{n}, then there is also a natural isomorphism I_{0} = T (𝔽_{p}) × I_{1}^{∗}.

We will define integral models for X(I_{0}K^{p}) and X(I_{1}^{∗}K^{p}) over the local rings O_{F,(𝔭)} and O_{F (ζ_{p−1}),(v)} respectively, where ζ_{p−1} denotes a primitive (p − 1)st root of unity, and v is some fixed place in F (ζ_{p−1}) above 𝔭; here K^{p} is a sufficiently small compact open subgroup of G(𝔸^{∞,p}), and we consider I_{0} and I_{1}^{∗} as subgroups of G(ℚ_{p}) ≅ GL_{n}(ℚ_{p}) × ℚ_{p}^{×}. We will typically not include K^{p} in the notation, and we will write X_{0}(p), X_{1}(p) for our integral models of X_{0}(p) := X(I_{0}K^{p}) and X_{1}(p) := X(I_{1}^{∗}K^{p}) respectively.

3.1.1. Remark. Note that in [HR10], [HT02], and [HT01], the authors work over ℤ_{p}, but we follow [Kot92] in working over O_{F,(𝔭)}, so as to satisfy the hypothesis required to be in the global case of Section 1. The appearance of ζ_{p−1} in the ring of definition of X_{1}(p) is a consequence of our use of Oort–Tate theory in the definition of the integral model in this case.

In order to define these integral models, we firstly recall a certain category of abelian schemes (up to isogeny) with polarisations and endomorphisms. If F is a set-valued functor on the category of connected, locally noetherian O_{F,(p)}-schemes, we will also consider it to be a functor on the category of all locally noetherian O_{F_{(}p)}-schemes by setting

F(∐ S_{i}) := ∏ F(S_{i}).

Let O_{B} be the unique maximal ℤ_{(p)}-order in B which under our fixed identification B ⊗_{ℚ} ℚ_{p} = M_{n}(ℚ_{p}) × M_{n}(ℚ_{p})^{op} is identified with M_{n}(ℤ_{p}) × M_{n}(ℤ_{p})^{op}. Let S be a connected, locally noetherian O_{F,(𝔭)}-scheme, and let AV_{S} be the category whose objects are pairs (A, i), where A is an abelian scheme over S of dimension n^{2}, and i : O_{B} → End_{S}(A) ⊗ ℤ_{(p)} is a homomorphism. We define homomorphisms in AV_{S} by

Hom((A_{1}, i_{1}), (A_{2}, i_{2})) = Hom_{O_{B}}((A_{1}, i_{1}), (A_{2}, i_{2})) ⊗ ℤ_{(p)}

(that is, the elements of Hom_{S}(A_{1}, A_{2}) ⊗ ℤ_{(p)} which commute with the action of O_{B}). The dual of an object (A, i) of AV_{S} is (A^{∨}, i^{∨}), where A^{∨} is the dual abelian scheme of A, and i^{∨}(b) = (i(b^{ι}))^{∨}. 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 equivalence class of homomorphisms (A, i) → (A^{∨}, i^{∨}) which contains a polarisation, under the equivalence relation “differing by a ℚ^{×}-scalar locally on S”. We define a ℚ-class of principal polarisations in the same way.

Fix K^{p} a sufficiently small open compact subgroup of G(𝔸^{∞,p}). Let A_{0} be the set-valued functor on the category of locally noetherian schemes over O_{F,(𝔭)} which sends a connected, locally noetherian scheme S over O_{F,(𝔭)} to the set of isomorphism classes of the following data.

• A commutative diagram of morphisms in the category AV_{S} of the form

\xymatrix{A_{0} \ar[r]^{α_{0}} \ar[d]^{λ_{0}} & A_{1} \ar[r]^{α_{1}} \ar[d]^{λ_{1}} & ⋯ \ar[r]^{α_{n−2}} & A_{n−1} \ar[r]^{α_{n−1}} \ar[d]^{λ_{n−1}} & A_{0} \ar[d]^{λ_{0}} \\ Â_{0} \ar[r]^{α̂_{0}} & Â_{1} \ar[r]^{α̂_{1}} & ⋯ \ar[r]^{α̂_{n−2}} & Â_{n−1} \ar[r]^{α̂_{n−1}} & Â_{0}}

where each α_{i} is an isogeny of degree p^{2n}, and their composite is just multiplication by p. In addition, λ_{0} is a ℚ-class of principal polarisations. Furthermore, we require that each A_{i} satisfies the usual compatibility condition that the two actions of O_{F} on the Lie algebra of A_{i} coincide (one action coming from the structure morphism O_{F,(𝔭)} → O_{S}, and the other from the O_{B}-action; see [HT01, §III.4] for a discussion of this condition).

• A geometric point s of S, and a π_{1}(S, s)-invariant K^{p}-orbit of isomorphisms

\overline{η} : V ⊗_{ℚ} 𝔸^{p,∞} ⟶^{∼} H_{1}((A_{0})_{s}, 𝔸^{p,∞})

which are O_{B}-linear, and up to a constant in (𝔸^{p,∞})^{×} take the ψ-pairing on the left hand side to the λ_{0}-Weil pairing on the right hand side. (This data 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 AV_{S} which preserve the λ_{i} up to an overall ℚ^{×}-scalar.

The functor A_{0} is represented by a projective scheme X_{0}(p) over O_{F,(𝔭)}, which is an integral model for X(I_{0}K^{p}). (See the proof of Lemma 3.2 of [TY07], which shows that X_{0}(p) is projective over the usual integral model at hyperspecial level. At hyperspecial level, quasi-projectivity is proved on page 391 of [Kot92], and projectivity 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 unchanged in our setting) shows that the special fibre of X_{0}(p) is a strict normal crossings divisor.

Our next goal is to describe an integral model X_{1}(p), over O_{F (ζ_{p−1}),(v)}, for X(I_{1}^{∗}K^{p}). Recalling that T denotes the diagonal torus in GL_{n}, we let ı_{i} : 𝔾_{m} → T denote the embedding of tori identifying 𝔾_{m} with the subgroup of T consisting of elements which are 1 away from the ith diagonal entry. We use the same notation ı_{i} to denote the map 𝔽_{p}^{×} → T (𝔽_{p}) induced by the map of tori.

The quotient I_{0}/I_{1}^{∗} is naturally identified with T (𝔽_{p}), and so T (𝔽_{p}) acts on X_{1}(p), with quotient isomorphic to X_{0}(p).

The scheme X_{0}(p) over Spec O_{F,(𝔭)} classifies chains C of p-isogenies of abelian varieties (with additional structure as specified above)

C : A_{0} ⟶^{α_{0}} A_{1} ⟶^{α_{1}} ⋯ ⟶^{α_{n−2}} A_{n−1} ⟶^{α_{n−1}} A_{n} := A_{0},

whose composite is equal to multiplication by p.

We let OT denote the Artin stack over O_{F (ζ_{p−1}),(v)} given by

OT := [Spec O_{F (ζ_{p−1}),(v)}[X, Y]/(XY − w_{p})/𝔾_{m}],

where 𝔾_{m} acts via λ · (X, Y) = (λ^{p−1}X, λ^{1−p}Y), and w_{p} is some explicit element of O_{F (ζ_{p−1}),(v)} of valuation one. Oort–Tate theory shows that OT classifies finite flat group schemes of order p over O_{F (ζ_{p−1}),(v)}-schemes. The universal group scheme over OT is the stack

G := [Spec O_{F (ζ_{p−1}),(v)}[X, Y, Z]/(XY − w_{p}, Z^{p} − XZ)/𝔾_{m}],

where 𝔾_{m} 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 the language of stacks.)

We define X_{1}(p) via the Cartesian diagram

(3.1.2)  \xymatrix{X_{1}(p) \ar[d] \ar[r] & G^{×} ×_{O_{F (ζ_{p−1}),(v)}} ⋯ ×_{O_{F (ζ_{p−1}),(v)}} G^{×} \ar[d] \\ X_{0}(p)_{/O_{F (ζ_{p−1}),(v)}} \ar[r] & OT ×_{O_{F (ζ_{p−1}),(v)}} ⋯ ×_{O_{F (ζ_{p−1}),(v)}} OT,}

where the bottom horizontal arrow is given by

C ↦ (ker(α_{0}), …, ker(α_{n−1})).

Note that the right-hand vertical arrow is finite and relatively representable by construction, and so the left-hand vertical arrow is a finite morphism of schemes. The action of T (𝔽_{p}) on X_{1}(p) extends to an action on X_{1}(p), namely the action pulled back from the action of T (𝔽_{p}) on G^{×} ×_{O_{F (ζ_{p−1}),(v)}} ⋯ ×_{O_{F (ζ_{p−1}),(v)}} G^{×} over OT ×_{O_{F (ζ_{p−1}),(v)}} ⋯ ×_{O_{F (ζ_{p−1}),(v)}} OT .

Let π := (−p)^{1/(p−1)}, and let w be the unique finite place of L := F (ζ_{p−1}, π) lying over our fixed place v of F (ζ_{p−1}). We let O denote the localization of O_{L} at w, and we let X_{1}(p)_{O} denote the normalisation of the base-change X_{1}(p)_{/O} of X_{1}(p) over O. We write I := Gal(L/F (ζ_{p−1})); this is also the inertia group at w in Gal(L/F). The group I acts naturally on X_{1}(p)_{O}. The mod p cyclotomic character ω induces an isomorphism (which we continue to denote by ω)

ω : I ⟶^{∼} 𝔽_{p}^{×}.

For each i = 0, …, n − 1, we let α_{i} : I → T denote the composite ı_{i} ∘ ω^{−1}.

3.1.3. Lemma. The scheme X_{1}(p)_{O} is a semistable projective model for X_{1}(p)_{/L} over O, the natural morphism

(3.1.4)  X_{1}(p)_{O} → X_{0}(p)

is tamely ramified, and the action of I × T on X_{1}(p)_{/L} extends to an action on X_{1}(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 α_{i}.

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 X_{0}(p) is there denoted X_{U} (and the integral model there is considered over ℤ_{p} rather than O_{F,(𝔭)}, but this is immaterial for our present purposes), while the scheme there denoted X_{U_{0}} is an integral model of the Shimura variety (in the notation of the present paper) X(K_{p}K^{p}), where K_{p} = GL_{n}(ℤ_{p}) × ℤ_{p}^{×}. We let X_{0}(p)_{s}^{(h)} (for 0 ≤ h ≤ n − 1) denote the locally closed subset of the special fibre X_{0}(p)_{s}, obtained by pulling back the locally closed subset \overline{X}_{U_{0}}^{(h)} defined in section 3 of [TY07] under the natural projection X_{0}(p)_{s} = \overline{X}_{U} → \overline{X}_{U_{0}}.

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 X_{0}(p)_{s}, and hence (by Lemma 1.5.2) that it is tamely ramified. We first note that the morphism X_{0}(p)_{/O_{F (ζ_{p−1}),(v)}} → X_{0}(p) induces an isomorphism on special fibres, and so we are free to replace X_{0}(p) by X_{0}(p)_{/O_{F (ζ_{p−1}),(v)}} in our considerations. We next note that the completion of the bottom arrow of (3.1.2) at a closed geometric point \overline{x}_{0} of X_{0}(p)_{s} depends up to isomorphism only on the value of h for which \overline{x}_{0} ∈ X_{0}(p)_{s}^{(h)} (since the p-divisible group attached to the point \overline{x}_{0} depends only on the value of h), and hence that the restriction of (3.1.4) to a formal neighbourhood of \overline{x}_{0} depends only on the value of h. Then by [TY07, Lem. 3.1], the closure of X_{0}(p)_{s}^{(h)} contains X_{0}(p)_{s}^{(0)}. Since being tamely ramified is an open condition, we conclude from these two conditions that in order to prove the lemma, it suffices to show that the restriction of (3.1.4) to a formal neighbourhood of \overline{x}_{0} is tamely ramified at closed geometric points \overline{x}_{0} of X_{0}(p)_{s}^{(0)}. (As already indicated, this argument is a variation on Deligne’s homogeneity principle.)

Thus, consider a closed geometric supersingular point \overline{x}_{0} of X_{0}(p)_{s}^{(0)}, so that \overline{x}_{0} admits a formal neighbourhood of the form

Spec W (\overline{𝔽}_{p})[[T_{1}, …, T_{n}]]/(T_{1} ⋯ T_{n} − w_{p}).

The proof of [TY07, Prop. 3.4] shows that the T_{i} may be taken to be the matrix of α_{i−1} on tangent spaces, so that the map X_{0}(p) → OT ×_{O_{F (ζ_{p−1}),(v)}} ⋯ ×_{O_{F (ζ_{p−1}),(v)}} OT may be defined in the formal neighbourhood of \overline{x}_{0} by the map

(T_{1}, …, T_{n}) ↦ ((T_{1}, U_{1}), …, (T_{n}, U_{n})),

where U_{i} = T_{1} ⋯ T̂_{i} ⋯ T_{n} (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 \overline{x}_{0} in (X_{1}(p)_{/O})_{s} is isomorphic to

Spec W (\overline{𝔽}_{p})[π][[V_{1}, …, V_{n}]]/((V_{1} ⋯ V_{n})^{p−1} − w_{p}).

If we write u := V_{1} ⋯ V_{n}/π, then we see that u^{p−1} = −w_{p}/p, and hence that u lies in the normalisation of this formal neighbourhood. Furthermore, on each component of this normalisation, u is equal to one of the (p − 1)st roots of −w_{p}/p lying in W (\overline{𝔽}_{p}). Thus the normalisation of this formal neighbourhood is a union of components, each isomorphic to

Spec O[[V_{1}, …, V_{n}]]/(V_{1} ⋯ V_{n} − uπ),

with the morphism (3.1.4) being given by T_{i} = V_{i}^{p−1}. Thus this morphism is indeed tamely ramified in the formal neighbourhood of \overline{x}_{0}.

It is clear that the I × T -action on X_{1}(p)_{L} extends to an action on X_{1}(p)_{/O}, and hence to its normalisation X_{1}(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 V_{i}. Also T acts on V_{i} through multiplication by the ith diagonal entry, and so acts on u via multiplication by the determinant. Combining these facts, we see that I acts on the components of the special fibre on which V_{i} = 0 via ı_{i} ∘ ω^{−1}. □

3.1.5. Remark. This lemma (and the fact that X_{0}(p) is strictly semistable) shows that the map X_{1}(p)_{O} → X_{0}(p) provides a tame strictly semistable context, in the sense 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 each character ψ_{j} as in the statement of that Theorem is equal to one of the characters α_{i}.

3.2. Canonical local systems. If K is a sufficiently small compact open subgroup of G(𝔸^{∞}), and V is a continuous representation of K on an \overline{𝔽}_{p}-vector space (this vector space being equipped with its discrete topology), then we may associate to V an étale local system F_{V} of \overline{𝔽}_{p}-vector spaces on X(K) as follows: Choose an open normal subgroup K′ ⊂ K lying in the kernel of V, and regard V as a representation of the quotient K/K′. Since X(K′) is naturally an étale K/K′-torsor over X(K), we may form the étale local system of \overline{𝔽}_{p}-vector spaces over X(K) associated to the K/K′-representation V. This local system is independent of the choice of K′, up to canonical isomorphism, and we define it to be F_{V}.

3.2.1. Definition. We refer to the étale local systems F_{V} that arise by the preceding construction as the canonical local systems on X(K). If we may choose K′ 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 F_{V} can be trivialised at level N.

3.3. The Eichler–Shimura relation. Let X be a U(n − 1, 1)-Shimura variety of level dividing N. Let w be a place of F such that l := w|_{ℚ} splits in F and does not divide N. There is a natural action via correspondences on X of Hecke operators T_{w}^{(i)}, 0 ≤ i ≤ n, where T_{w}^{(i)} is the double coset operator corresponding to

(\begin{matrix}l1_{i} & 0 \\ 0 & 1_{n−i} \end{matrix}) × 1 ∈ GL_{n}(ℚ_{l}) × ℤ_{l}^{×},

where we use the assumption that l ∤ N and identify a hyperspecial maximal compact subgroup of G(ℚ_{l}) with GL_{n}(ℤ_{l}) × ℤ_{l}^{×} via an isomorphism D_{w} ≅ M_{n}(ℚ_{l}). These correspondences then act on the cohomology H_{ét}^{j}(X\overline{ℚ}, \overline{𝔽}_{p}).

More generally if F_{V} is a canonical local system on X that can be trivialised at level N, then we obtain an action of the double coset operators T_{w}^{(i)} on F_{V}, and hence on the cohomology H_{ét}^{j}(X\overline{ℚ}, F_{V}).

The following theorem regarding this action is then an immediate consequence of 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).

3.3.1. Theorem. Let X be a U(n − 1, 1)-Shimura variety of level dividing N, and F_{V} a canonical local system on X. Let w be a place of F such that w|_{ℚ} splits in F and does not divide Np. Then ∑_{i=0}^{n}(−1)^{i}(Norm w)^{i(i−1)/2}T_{w}^{(i)} Frob_{w}^{n−i} acts as 0 on each H_{ét}^{j}(X\overline{ℚ}, F_{V}).

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 F_{V} denote a canonical local system on X. Choose N so that X has level dividing N, so that F_{V} can be trivialised at level N, and so that p divides N. Assume that the projection of the corresponding level K to G(𝔸^{p,∞}) is sufficiently small.

Let 𝕋 = \overline{ℤ}_{p}[T_{w}^{(i)}] be the polynomial ring in the variables T_{w}^{(i)}, 1 ≤ i ≤ n, where w 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 \overline{𝔽}_{p}, and suppose that there exists a continuous irreducible representation ρ_{𝔪} : G_{F} → GL_{n}(\overline{𝔽}_{p}) which is unramified at all finite places not dividing N, and which satisfies char (ρ_{𝔪}(Frob_{w})) ≡ ∑_{i=0}^{n}(−1)^{i}(Norm w)^{i(i−1)/2}T_{w}^{(i)}X^{n−i} mod 𝔪 for all w ∤ N such that w|_{ℚ} splits in F. Continue to fix a choice of a place 𝔭 of F dividing p, and write G_{ℚ_{p}} for G_{F_{𝔭}} from now on. Recall that the choice of 𝔭 also gives us an isomorphism G(ℚ_{p}) ≅ GL_{n}(ℚ_{p})×ℚ_{p}^{×} as in Section 3.1.

We consider the following further hypothesis on ρ_{𝔪} (this is Hypothesis 4.1.1 below):

3.4.1. Hypothesis. If θ : G_{ℚ} → GL_{m}(k) is any continuous, irreducible representation with the property that the characteristic polynomial of ρ_{𝔪}(g) annihilates θ(g) for every g ∈ G_{ℚ}, then θ is equivalent to ρ_{𝔪}.

We will now prove our first main result, a vanishing theorem for the cohomology of X with F_{V} -coefficients.

3.4.2. Theorem. Suppose that ρ_{𝔪} satisfies Hypothesis 3.4.1, that ρ_{𝔪}|_{G_{ℚp}} 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 ρ_{𝔪 |G_{ℚp}} contains an irreducible subquotient of dimension greater than one. Then the localisations H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} vanish for i ≤ r and for i ≥ 2(n − 1) − r.

3.4.3. Remark. In Subsection 4.1 we will show that Hypothesis 3.4.1 is satisfied if either ρ_{𝔪} is induced from a character of G_{K} for some degree n cyclic Galois extension K/ℚ, or if p ≥ n and SL_{n}(k) ⊆ ρ_{𝔪}(G) ⊆ \overline{𝔽}_{p}^{×}GL_{n}(k) for some subfield k ⊂ \overline{𝔽}_{p}.

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 are equivalent to vanishing results for the cohomology of the complex U(n − 1, 1)- Shimura varieties with coefficients in the corresponding canonical local systems for the 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 F_{V} attached to a representation V is the canonical local system attached to the contragredient representation V^{∨}.)

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 F_{V} is trivial. To this end, write X = X(K), let K′ be an open normal subgroup of K of level dividing N such that V is representation of K/K′, and choose an embedding of K/K′-representations V ↪ \overline{𝔽}_{p}[K/K′]^{n} for some n; denote the cokernel by W. Let π denote the projection X(K′) → X(K). Passing to the associated canonical local systems, we obtain a short exact sequence 0 → F_{V} → π_{∗}\overline{𝔽}_{p}^{n} → F_{W} → 0, which gives rise to an exact sequence of cohomology

H_{ét}^{i−1}(X\overline{ℚ}, F_{W}) → H_{ét}^{i}(X\overline{ℚ}, F_{V}) → H_{ét}^{i}(X\overline{ℚ}, π_{∗}\overline{𝔽}_{p}^{n}) = H_{ét}^{i}(X(K′)\overline{ℚ}, \overline{𝔽}_{p})^{n}.

Localizing at 𝔪 and applying our inductive hypothesis (with F_{W} in place of F_{V}), we reduce to the case of constant coefficients (with X(K′) in place of X). We therefore turn to establishing the claimed vanishing in this case.

Suppose now that K′ is any open normal subgroup of K of level dividing N. Combining the Hochschild–Serre spectral sequence

E_{2}^{m,n} = H_{ét}^{m}(K/K′, H^{n}(X(K′)\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}) ⟹ H_{ét}^{m+n}(X(K)\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}

with our inductive hypothesis, we find that vanishing of H_{ét}^{i}(X(K′), \overline{𝔽}_{p})_{𝔪} implies the vanishing of H_{ét}^{i}(X(K)\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}. Thus, without loss of generality, we may and do assume that K = K_{p}K^{p}, where K_{p} is an open normal subgroup of I_{1}, 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(I_{1}K^{p}), which takes the form

E_{2}^{m,n} = H^{m}(I_{1}/K_{p}, H_{ét}^{n}(X(K)\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}) ⟹ H_{ét}^{m+n}(X(I_{1}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}.

Once more taking into account our inductive hypothesis, we obtain an isomorphism

H_{ét}^{i}(X(I_{1}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪} ⟶^{∼} H_{ét}^{i}(X(K)\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}^{I_{1}/K_{p}}. Since I_{1}/K_{p} is a p-group, while

H_{ét}^{i}(X(K)\overline{ℚ}, \overline{𝔽}_{p})_{𝔪} is a vector space over a field of characteristic p, we see that the latter space vanishes if and only if its space of I_{1}/K_{p}-invariants does. Thus we are reduced to establishing the theorem in the case when K = I_{1}K^{p}.

Now recall that I_{1}^{∗} = 𝔽_{p}^{×} × I_{1}, and that the projection onto the first factor arises from the similitude projection GU(n − 1, 1) → 𝔾_{m}. From this it follows that X(I_{1}K^{p}) is isomorphic to the product X(I_{1}^{∗}K^{p})×_{F} Spec A, where A := F [x]/Φ_{p}(x) (where Φ_{p}(x) denotes the pth cyclotomic polynomial; the action of 𝔽_{p}^{×} = I_{1}^{∗}/I_{1} on X(I_{1}K^{p}) is induced by the action of 𝔽_{p}^{×} on A given by x ↦ x^{a}, for a ∈ 𝔽_{p}^{×}). Consequently there is an isomorphism of Galois representations

H^{i}(X(I_{1}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪} ⟶^{∼} ⊕_{j=0}^{p−2}H^{i}(X(I_{1}^{∗}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪_{j}} ⊗ ω^{j},

where 𝔪_{j} is the maximal ideal in 𝕋 which corresponds to the twisted Galois representation ρ_{𝔪_{j}} := ρ_{𝔪} ⊗ ω^{−j}. Since each of the Galois representations ρ_{𝔪_{j}} 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(I_{1}^{∗}K^{p}).

Consider an irreducible G_{F} × I_{0}/I_{1}^{∗}-subrepresentation of H_{ét}^{i}(X(I_{1}^{∗}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}, which we may write in the form θ ⊗ β, where θ is an absolutely irreducible G_{F} - representation and β is a character of the abelian group T (𝔽_{p}) = I_{0}/I_{1}^{∗}. Theorem 3.3.1 and Hypothesis 3.4.1 taken together imply that θ is equivalent to ρ_{𝔪}. We consider β as an n-tuple of characters β_{1}, …, β_{n} of 𝔽_{p}^{×}. Write β̃_{j} for the Teichmüller lift of β_{j}, and χ_{j} for the character of I_{ℚ_{p}} given by β_{j}. By Lemma 3.1.3, 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 the collection {β̃_{j}}. (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 G_{F} ×T -representation provided by Theorem 1.5.14.) We claim that det θ|_{I_{ℚp}} = det ρ_{𝔪}|_{I_{ℚp}} = \overline{χ}_{1} ⋯ \overline{χ}_{n}ω^{−n(n−1)/2}. Admitting this for the moment, we may apply Theorem 2.2.4 to the representation ρ_{𝔪}|_{G_{ℚp}}^{∨}, and we deduce that ρ_{𝔪}|_{G_{ℚp}}^{∨} is not r-regular. Equivalently, we see that ρ_{𝔪}|_{G_{ℚp}} is not r-regular, which contradicts our assumptions.

It remains to establish the equality det θ|_{I_{ℚp}} = det ρ_{𝔪}|_{I_{ℚp}} = \overline{χ}_{1} ⋯ \overline{χ}_{n}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 w|_{ℚ} splits in F, we have det ρ_{𝔪}(Frob_{w}) = (Norm w)^{n(n−1)/2}T_{w}^{(n)}. Let ψ_{𝔪} be the character of 𝔸_{F}^{×}/F^{×} corresponding to the character ω^{n(n−1)/2} det ρ_{𝔪} by global class field theory; then we need to prove that ψ_{𝔪}|_{ℤ_{p}^{×}} = β_{1} ⋯ β_{n}. The centre of G(𝔸_{ℚ}) is 𝔸_{F}^{×}, so by the definition of the Shimura variety X there is a natural action of 𝔸_{F}^{×}/F^{×} on H_{ét}^{i}(X(I_{1}^{∗}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}. By the definition of the Hecke operators, we see that if ϖ_{w} is a uniformiser at a place w ∤ N of F such that w|_{ℚ} splits in F, then ϖ_{w} acts as T_{w}^{(n)}. By the Chebotarev density theorem, we deduce that the action of 𝔸_{F}^{×}/F^{×} on the underlying vector space of θ (which by definition is a subspace of H_{ét}^{i}(X(I_{1}^{∗}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}) is via the character ψ_{𝔪}. In order to compute ψ_{𝔪}|_{ℤ_{p}^{×}}, it is thus sufficient to compute the action of ℤ_{p}^{×} on H_{ét}^{i}(X(I_{1}^{∗}K^{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}, and in particular sufficient to compute the action of the Iwahori subgroup I_{0}. Now, since θ is assumed to be in the β-part of the cohomology, I_{0} acts via the character β of I_{0}/I_{1}^{∗}, so that ψ_{𝔪}|_{ℤ_{p}^{×}} = β_{1} ⋯ β_{n}, as required. □

3.4.5. Corollary. Suppose that ρ_{𝔪} satisfies Hypothesis 3.4.1, that ρ_{𝔪 |G_{ℚp}} is rregular for some r ≤ min{(n − 1)/2, p − 2}, and, if r = (n − 1)/2, suppose in addition that ρ_{𝔪 |G_{ℚp}} contains an irreducible subquotient of dimension greater than one. Then the localisation H_{ét}^{i}(X\overline{ℚ}, \overline{ℤ}_{p})_{𝔪} vanishes for i ≤ r, while H_{ét}^{r+1}(X\overline{ℚ}, \overline{ℤ}_{p})_{𝔪} is torsion-free.

Proof. This follows at once from Theorem 3.4.2 and the short exact sequence

0 → H_{ét}^{i}(X\overline{ℚ}, \overline{ℤ}_{p})_{𝔪}/𝔪_{\overline{ℤ}_{p}}H_{ét}^{i}(X\overline{ℚ}, \overline{ℤ}_{p})_{𝔪} → H_{ét}^{i}(X\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}

→ H_{ét}^{i+1}(X\overline{ℚ}, \overline{ℤ}_{p})_{𝔪}[𝔪_{\overline{ℤ}_{p}}] → 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 that stronger 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 ρ_{𝔪} : G_{F} → GL_{n}(\overline{𝔽}_{p}), then we expect that the localizations H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} and H_{ét}^{i}(X\overline{ℚ}, \overline{ℤ}_{p})_{𝔪} should vanish in degrees i < n − 1.

On the other hand, it need not be the case that (for example) H_{ét}^{i}(X\overline{ℚ}, \overline{𝔽}_{p}) vanishes in all degrees in which H_{ét}^{i}(X\overline{ℚ}, \overline{ℚ}_{p}) vanishes. For example, in the case n = 3, for the unitary Shimura varieties that we consider here, namely those that are associated to division algebras, it is known that H_{ét}^{1}(X\overline{ℚ}, \overline{ℚ}_{p}) = 0 [Rog90, Thm. 15.3.1]. (Under additional restrictions on the division algebra that is allowed, an analogous result is known for all values of n [Clo93, Thm. 3.4].) On the other hand, one can construct examples for which H_{ét}^{1}(X\overline{ℚ}, \overline{𝔽}_{p}) ≠ 0, and hence for which H_{ét}^{2}(X\overline{ℚ}, \overline{ℤ}_{p}) 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 associated Galois representation ρ_{𝔪} 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 some natural number N divisible by p, and such that the projection of K to 𝔾(𝔸^{p,∞}) is sufficiently small. Let F_{V} be a canonical local system on X, which may be trivialized at level N. The results of Section 3.4 are particularly powerful in this case, as we now demonstrate.

3.5.1. Corollary. Suppose that ρ_{𝔪} satisfies Hypothesis 3.4.1, that ρ_{𝔪}|_{G_{ℚp}} is 1regular, and that ρ_{𝔪 |G_{ℚp}} contains an irreducible subquotient of dimension greater than one. Then the localisations H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} vanish for i ≠ 2.

Proof. This follows immediately from Theorem 3.4.2, noting that the hypothesis that ρ_{𝔪}|_{G_{ℚp}} 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 ρ_{𝔪}. We begin with a lemma.

3.5.2. Lemma. If 𝔪 is a maximal ideal of 𝕋 with residue field \overline{𝔽}_{p}, such that H_{ét}^{0}(X\overline{ℚ}, F_{V})_{𝔪} ≠ 0, then there is an abelian representation \overline{ρ}_{0} : G_{F} → GL_{3}(\overline{𝔽}_{p})

such that char (\overline{ρ}_{0}(Frob_{w})) = ∑_{i=0}^{3}(−1)^{i}(Norm w)^{i(i−1)/2}T_{w}^{(i)}X^{n−i} (mod 𝔪) for all split places w of E for which w ∤ Np.

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-G_{F} -representation of the étale cohomology group H_{ét}^{1}(X\overline{ℚ}, F_{V}), then either every irreducible subquotient of ρ_{|G_{ℚp}} is one-dimensional, or else ρ_{|G_{ℚp}} is not 1-regular, or else ρ(G_{F}) 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 and characteristic polynomials coincide. In Subsection 4.2 we will show that ρ(G_{F}) is generated by its subset of regular elements if either ρ_{𝔪} is induced from a character of G_{K} for some cubic Galois extension K/ℚ, or if ρ(G_{F}) 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 ρ_{𝔪}.^{4} The key point will be that in the Hochschild–Serre spectral sequences that appear, the only other cohomology to contribute besides H^{1} will be H^{0}, and for maximal ideals of 𝕋 in the support of H^{0}, we do have associated Galois representations, by Lemma 3.5.2.

4In fact, as noted in the introduction, recent work of Scholze [Sch13] implies that ρ_{𝔪} exists for any maximal ideal 𝔪 in the Hecke algebra. We have left our argument as originally written.

We first show that if H_{ét}^{0}(X\overline{ℚ}, F_{W} )_{𝔪} ≠ 0 for some canonical local system F_{W}

on X and some maximal ideal 𝔪 of 𝕋, then Hom_{G_{F}} (ρ, H_{ét}^{1}(X\overline{ℚ}, F_{V} )[𝔪]) = 0. To

see this, note that if H_{ét}^{0}(X\overline{ℚ}, F_{W} )_{𝔪} ≠ 0 and Hom_{G_{F}} (ρ, H_{ét}^{1}(X\overline{ℚ}, F_{V} )[𝔪]) ≠ 0, then Lemma 3.5.2 and Theorem 3.3.1 together imply that there exists an abelian representation \overline{ρ}_{0} : G_{F} → GL_{3}(\overline{𝔽}_{p}) such that for all g ∈ G_{F}, the characteristic polynomial of \overline{ρ}_{0}(g) annihilates ρ(g). By Lemma 4.1.3 this implies that ρ is abelian, which is impossible as ρ is irreducible.

Now H_{ét}^{1}(X\overline{ℚ}, F_{V}) is the direct sum of its localisations at the various maximal ideals 𝔪 of 𝕋, and hence, since Hom_{G_{F}} (ρ, H_{ét}^{1}(X, F_{V})) ≠ 0 by hypothesis, we see that Hom_{G_{F}} (ρ, H_{ét}^{1}(X, F_{V})_{𝔪}) ≠ 0 for some maximal ideal 𝔪 of 𝕋. Since this is a finite length 𝕋_{𝔪}-module, we see that its 𝕋_{𝔪}-socle Hom_{G_{F}} (ρ, H_{ét}^{1}(X, F_{V})[𝔪]) must also be non-zero, and hence, by the preceding paragraph, we conclude that H_{ét}^{0}(X, F_{W})_{𝔪} = 0 for any canonical local system F_{W} on X.

As in the proof of Theorem 3.4.2, choose a short exact sequence of canonical local systems 0 → F_{V} → π_{∗}\overline{𝔽}_{p}^{n} → F_{W} → 0, for some π : X′ → X. Passing to the the long exact sequence

H_{ét}^{0}(X, F_{W})_{𝔪} → H_{ét}^{1}(X, F_{V})_{𝔪} → H_{ét}^{1}(X, π_{∗}\overline{𝔽}_{p})^{n} = H_{ét}^{1}(X′, \overline{𝔽}_{p})^{n},

and using the result of the preceding paragraph, namely that H_{ét}^{0}(X, F_{W})_{𝔪} = 0, we conclude that ρ embeds into H_{ét}^{1}(X′, \overline{𝔽}_{p}). Thus, replacing X by X′, we reduce to the case when F_{V} is constant, which we assume from now on.

We now suppose that ρ is a subrepresentation of H_{ét}^{1}(X\overline{ℚ}, \overline{𝔽}_{p})_{𝔪}. We will prove that ρ is then necessarily a subrepresentation of H_{ét}^{1}(X(K^{p}I_{1}), \overline{𝔽}_{p})_{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′ := K^{p}K_{p} of K, with K_{p} ⊂ I_{1}. The Hochschild–Serre spectral sequence associated to the cover X(K′) → X(K) gives rise to an exact sequence

0 → H^{1}(K/K′, H^{0}(X(K′), \overline{𝔽}_{p})_{𝔪}) → H_{ét}^{1}(X(K), \overline{𝔽}_{p})_{𝔪} → H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪}^{K/K′}

→ H^{2}(K/K′, H^{0}(X(K′), \overline{𝔽}_{p})_{𝔪}).

The same argument as above, using Lemma 3.5.2 (applied now with X(K′) in place of X), Theorem 3.3.1, and Lemma 4.1.3 below, shows that

H^{1}(K/K′, H^{0}(X(K′), \overline{𝔽}_{p})_{𝔪}) = H^{2}(K/K′, H^{0}(X(K′), \overline{𝔽}_{p})_{𝔪}) = 0.

Thus in fact we have an isomorphism H_{ét}^{1}(X(K), \overline{𝔽}_{p}) ⟶^{∼} H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪}^{K/K′}, and hence an isomorphism

Hom_{G_{F}} (ρ, H_{ét}^{1}(X(K), \overline{𝔽}_{p})) ⟶^{∼} Hom_{G_{F}} (ρ, H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪}^{K/K′})

= Hom_{G_{F}} (ρ, H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪})^{K/K′}.

In particular, if ρ appears as a subrepresentation of H_{ét}^{1}(X(K), \overline{𝔽}_{p})_{𝔪}, then it appears as a subrepresentation of H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪}.

Now considering the Hochschild–Serre spectral sequence for the cover X(K′) → X(K^{p}I_{1}), and using the fact that if Hom_{G_{F}} (ρ, H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪}) ≠ 0, then also

Hom_{G_{F}} (ρ, H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪})^{I_{1}/K_{p}} ≠ 0 (since I_{1}/K_{p} is a p-group), we conclude

that if ρ appears as a subrepresentation of H_{ét}^{1}(X(K′), \overline{𝔽}_{p})_{𝔪}, then it appears as a subrepresentation of H_{ét}^{1}(X(K^{p}I_{1}), \overline{𝔽}_{p})_{𝔪}.

Arguing exactly as in the proof of Theorem 3.4.2, we then deduce that some twist of ρ appears in H_{ét}^{1}(X(K^{p}I_{1}^{∗}), \overline{𝔽}_{p}), and so, replacing ρ by this twist, it suffices to prove that if ρ is an irreducible three-dimensional representation ρ of H_{ét}^{1}(X(K^{p}I_{1}^{∗}), \overline{𝔽}_{p}) that is generated by its regular elements, then either every irreducible subquotient of ρ_{|G_{ℚp}} is one-dimensional, or else ρ_{|G_{ℚp}} 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 to Lemma 4.2.2 below. □

Our other main theorem concerns the weight part of the Serre-type conjecture of [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 some terminology from [EGH11]. We will call an irreducible \overline{𝔽}_{p}-representation of GL_{3}(𝔽_{p}) a Serre weight. Fix an irreducible representation \overline{ρ} : G_{F} → GL_{3}(\overline{𝔽}_{p}). 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(𝔸^{p,∞}), where now we assume that (N, p) = 1. Assume furthermore that \overline{ρ} is unramified at all places not dividing Np, and define a maximal ideal 𝔪 of 𝕋 with residue field \overline{𝔽}_{p} by demanding that for each place w ∤ Np of F such that w|_{ℚ} splits in F, the characteristic polynomial of \overline{ρ}(Frob_{w}) is equal to the reduction modulo 𝔪 of ∑_{i=0}^{3}(−1)^{i}(Norm w)^{i(i−1)/2}T_{w}^{(i)}X^{n−i}.

Let V be a Serre weight; since (N, p)=1, we may write K = K_{p}K^{p} where K_{p} ⊂ G(ℚ_{p}) ≅ GL_{3}(ℚ_{p}) × ℚ_{p}^{×} is conjugate to GL_{3}(ℤ_{p}) × ℤ_{p}^{×}, and we may regard V as a representation of K_{p} via the projection GL_{3}(ℤ_{p}) ↠ GL_{3}(𝔽_{p}). As usual, write F_{V} for the canonical local system associated to V. We say that \overline{ρ} is modular of weight V if for some N, X as above and for some 0 ≤ i ≤ 4 we have

H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} ≠ 0.

Assume now that \overline{ρ}|_{G_{ℚp}} 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^{?}(\overline{ρ}) of Serre weights in which it is predicted that \overline{ρ} is modular. Let W_{gen}(\overline{ρ}) be the set of generic weights for which \overline{ρ} is modular.

3.5.6. Theorem. Suppose that \overline{ρ} satisfies Hypothesis 4.1.1 below, and that \overline{ρ}|_{G_{ℚp}} is irreducible and 1-regular. Suppose that \overline{ρ} is modular of some strongly generic weight. Then W_{gen}(\overline{ρ}) = W^{?}(\overline{ρ}). In fact, for each V ∈ W_{gen}(\overline{ρ}), we have

H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} ≠ 0

if and only if i = 2 and V ∈ W^{?}(\overline{ρ}).

Proof. By the definition of 𝔪, the representation \overline{ρ} satisfies the defining properties of the representation ρ_{𝔪} considered in Section 3.4. Applying Corollary 3.5.1, we see that for any Serre weight V, we have H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} = 0 if i ≠ 2. We will now deduce the result from Theorem 6.2.3 of [EGH11] (taking r̄ there to be our \overline{ρ}). By Theorem 4.3.3 of [EGH11], we see that it suffices to show that we can define S and S̃ 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 S̃ 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] are equivalent 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 to that carried out for U(3) in [EGH11], and we content ourselves with sketching the arguments.

From now on we regard the prime-to-p level structure K^{p} of X as fixed, and we will vary K_{p} in our arguments. We will write K_{p}(0) for GL_{3}(ℤ_{p}) × ℤ_{p}^{×} ⊂ G(ℚ_{p}). We fix a sufficiently large extension E/ℚ_{p} with ring of integers O_{E}, residue field k_{E}, and uniformiser ϖ_{E}, and we define

S := \underrightarrow{lim}_{K_{p}} H_{ét}^{2}(X(K^{p}K_{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪},

S̃ := ((\underleftarrow{lim}_{s} \underrightarrow{lim}_{K_{p}} H_{ét}^{2}(X(K^{p}K_{p})\overline{ℚ}, O_{E}/ϖ_{E}^{s})_{𝔪}) ⊗_{O_{E}} \overline{ℤ}_{p})^{l.alg}

(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 the vanishing of H_{ét}^{i}(X\overline{ℚ}, F_{V})_{𝔪} = 0 if i ≠ 2, it is straightforward to verify Axioms Ã1–Ã3 of Section 4.3 of [EGH11], as we now explain.

Firstly, we need to check that our definition of “modular” is consistent with that of Definition 4.2.2 of [EGH11]. This amounts to showing that for any Serre weight V,

(S ⊗_{\overline{𝔽}_{p}} V)^{K_{p}(0)} = H_{ét}^{2}(X(K_{p}(0)K^{p})\overline{ℚ}, F_{V})_{𝔪}.

To see this, note that since any sufficiently small K_{p} acts trivially on V, we have

S ⊗_{\overline{𝔽}_{p}} V = \underrightarrow{lim}_{K_{p}} H_{ét}^{2}(X(K^{p}K_{p})\overline{ℚ}, \overline{𝔽}_{p})_{𝔪} ⊗ V

⟶^{∼} \underrightarrow{lim}_{K_{p}} H_{ét}^{2}(X(K^{p}K_{p})\overline{ℚ}, F_{V})_{𝔪},

so it is enough to check that for all compact open subgroups K_{p} ⊂ K_{p}(0) we have

H_{ét}^{2}(X(K^{p}K_{p})\overline{ℚ}, F_{V})_{𝔪}^{K_{p}(0)} = H_{ét}^{2}(X(K^{p}K_{p}(0))\overline{ℚ}, F_{V})_{𝔪},

which is an easy consequence of the Hochschild–Serre spectral sequence and our vanishing result. We also need an embedding S ↪ S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} which is compatible with the actions of GL_{3}(ℚ_{p}) and the Hecke algebra. In fact, it is easy to see that we have S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} = S. For example, there is a natural isomorphism

H_{ét}^{3}(X\overline{ℚ}, O_{E})_{𝔪}/ϖ_{E}H_{ét}^{3}(X\overline{ℚ}, O_{E})_{𝔪} = H_{ét}^{3}(X\overline{ℚ}, k_{E})_{𝔪},

and hence the vanishing of H_{ét}^{3}(X\overline{ℚ}, k_{E})_{𝔪} implies that of the finitely generated O_{E}- module H_{ét}^{3}(X\overline{ℚ}, O_{E})_{𝔪}. One then sees that for all s we have a natural isomorphism

H_{ét}^{2}(X\overline{ℚ}, O_{E})_{𝔪}/ϖ_{E}^{s}H_{ét}^{2}(X\overline{ℚ}, O_{E})_{𝔪} = H_{ét}^{2}(X\overline{ℚ}, O_{E}/ϖ_{E}^{s})_{𝔪},

from which the claim follows easily.

We now examine Axiom Ã1. We must show that if Ṽ is a finite free \overline{ℤ}_{p}-module with a locally algebraic action of K_{p}(0) (acting through GL_{3}(ℤ_{p})), then (S̃ ⊗_{\overline{ℤ}_{p}}

Ṽ)^{K_{p}(0)} is a finite free \overline{ℤ}_{p}-module, and for A = \overline{ℚ}_{p}, \overline{𝔽}_{p} we have

(S̃ ⊗_{\overline{ℤ}_{p}} Ṽ)^{K_{p}(0)} ⊗_{\overline{ℤ}_{p}} A = (S̃ ⊗_{\overline{ℤ}_{p}} Ṽ ⊗_{\overline{ℤ}_{p}} A)^{K_{p}(0)}.

This is straightforward, the key point being that if FṼ denotes the lisse étale sheaf attached to Ṽ, then a straightforward argument with Hochschild–Serre as above gives

(S̃ ⊗_{\overline{ℤ}_{p}} Ṽ)^{K_{p}(0)} = H_{ét}^{2}(X(K^{p}K_{p}(0))\overline{ℚ}, FṼ)_{𝔪},

which is certainly a finite free \overline{ℤ}_{p}-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 cohomology module H_{ét}^{2}(X(K^{p}K_{p}(0))\overline{ℚ}, FṼ)_{𝔪} are associated to automorphic forms exactly 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] by using H_{ét}^{2}(X(K^{p}K_{p}(0))\overline{ℚ}, FṼ), 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 at present 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 the section 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 the characteristic polynomial of A.

4.1. Characterising representations by their characteristic polynomials. Let ρ : G → GL_{n}(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 → GL_{m}(k) is irreducible, and if char (ρ(g)) annihilates θ(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 was proved 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, this already fails if ρ is the irreducible 3-dimensional representation of A_{4}, cf. Section 5 of [BLR91] as well as Remark 4.1.7 below).

4.1.3. Lemma. Let ρ : G → GL_{n}(k) and θ : G → GL_{m}(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 GL_{3} that we need, but it seems to be well known to the experts, and will follow from the much more general work in progress of Mok and Kaletha– Minguez–Shin–White.

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, and we 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 → GL_{n}(k) is a direct sum of one-dimensional characters of G, and if θ(g) : G → GL_{m}(k) is an irreducible representation of G such that char (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 G^{ab}. Lemma 4.1.3 then shows that θ also factors through G^{ab}. 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 = χ_{1} ⊕ ⋯ ⊕ χ_{n}, we find that for each g ∈ G, the value χ_{i}(g) is equal to 1 for at least one value of i (since char (ρ(g)) = (X − χ_{1}(g)) ⋯ (X − χ_{n}(g)) annihilates θ(g) = 1). Thus G is equal to the union of its subgroups ker(χ_{i}). □

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, then taking θ to be the trivial representation, the hypotheses of the proposition are satisfied, but θ is certainly not one of the summands of ρ.

4.1.6. Lemma. If ρ : G → GL_{n}(k) is irreducible, and is isomorphic to an induction Ind_{H}^{G} ψ, where H is a cyclic normal subgroup of G of index n and ψ : H → k^{×} is a 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 θ′ be a Jordan–Hölder constituent of the restriction θ_{|H}, then Lemma 4.1.4 (applied to the representations ρ_{|H} and θ′ of H) implies that θ′ is a character of H and (because H is cyclic) that θ′ = ψ^{g} for some g ∈ G/H. The H-equivariant inclusion ψ^{g} = θ′ → θ_{|H} then induces a non-zero G-equivariant map ρ = Ind_{H}^{G} ψ = Ind_{H}^{G} ψ^{g} → 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 = A_{4} and H to be the normal subgroup of G of order 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 → SO_{3}(k). For every g ∈ G, the characteristic polynomial of ρ(g) thus has 1 as an eigenvalue, and so if θ 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 GL_{n}(k), which contains SL_{n}(k′) for some subfield k′ of k, and is contained in k^{×}GL_{n}(k′).

(1) Any irreducible representation of G over k remains irreducible upon restriction to SL_{n}(k′).

(2) Any two irreducible representations of G which become isomorphic upon restriction to SL_{n}(k′) can be obtained one from the other via twisting by a character of G that is trivial on SL_{n}(k′).

Proof. Let G act via θ on the k-vector space V, and let (θ, W) be an irreducible subrepresentation of θ|_{SL_{n}(k′)}. Then W is obtained by restriction from a representation of the algebraic group SL_{n}/k′ (cf. Section 1 of [Jan87]), so the action of SL_{n}(k′) on W may be extended to an action of GL_{n}(k) and thus of G. By Frobenius reciprocity we obtain a surjective map (Ind_{SL_{n}(k′)}^{G} 1) ⊗ W → V of Grepresentations. Since G/SL_{n}(k′) is a finite abelian group of prime to p order, we see that (Ind_{SL_{n}(k′)}^{G} 1) is a direct sum of one-dimensional representations, so that V is a twist of W by some character which is trivial on SL_{n}(k′). Thus the restriction of θ to SL_{n}(k′) 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 → GL_{n}(k) is irreducible, and if SL_{n}(k′) ⊆ ρ(G) ⊆ k^{×}GL_{n}(k′) for some subfield k′ 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 case n = 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, so that we can identify G with ρ(G). In particular, SL_{n}(k′) is a subgroup of G, and by Lemma 4.1.8, the restriction of θ to SL_{n}(k′) remains irreducible.

Since G is finite, our assumption that SL_{n}(k′) ⊂ G implies that k′ is finite; suppose that k′ has cardinality q. We recall some basic facts about the representation theory of SL_{n}(k′); see for example Section 1 of [Jan87]. The irreducible k-representations of SL_{n}(k′) are obtained by restriction from the algebraic group SL_{n}/k′, and are precisely those representations whose highest weights are q-restricted. (With the usual choice of maximal torus T of SL_{n}, if we identify the weight lattice with ℤ^{n} modulo the diagonally embedded copy of ℤ, a weight (a_{1}, …, a_{n}) is q-restricted if 0 ≤ a_{i} − a_{i+1} ≤ q − 1 for all 1 ≤ i ≤ n − 1.) Suppose that θ has highest weight (a_{1}, …, a_{n}). Let g ∈ SL_{n}(k′) be a semisimple element with eigenvalues α_{1}, …, α_{n}. Then, since g is conjugate to an element of T (k), by considering the formal character of the corresponding representation of SL_{n}/k′ we see that among the eigenvalues of θ(g) are each of the quantities

∏_{i=1}^{n} α_{i}^{x_{i}},

where the x_{i} are a permutation of a_{1}, …, a_{n}.

Our assumption on θ and ρ implies that for each such permutation, ∏_{i=1}^{n} α_{i}^{x_{i}} must equal one of α_{1}, …, α_{n}. 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 eigenvalues α, α^{q}, …, α^{q^{n−1}}. Then for any x_{1}, …, x_{n} as above, there must be an integer 0 ≤ β ≤ n − 1 such that

q^{n−1}x_{1} + q^{n−2}x_{2} + ⋯ + x_{n} ≡ q^{β} (mod (q^{n} − 1)/(q − 1)).

Fix some 1 ≤ i ≤ n−1, and consider two permutations x_{1}, …, x_{n} and x′_{1}, …, x′_{n} as above, which satisfy x_{i} = x′_{i} for 1 ≤ i ≤ n−2, x_{n−1} = a_{i}, x_{n} = a_{i+1}, x′_{n−1} = a_{i+1} and x′_{n} = a_{i}. Taking the difference of the two expressions q^{n−1}x_{1}+q^{n−2}x_{2}+⋯+x_{n} and q^{n−1}x′_{1} +q^{n−2}x′_{2} +⋯+x′_{n}, we conclude that there are integers 0 ≤ β, γ ≤ n−1 such that

(q − 1)(a_{i} − a_{i+1}) ≡ q^{β} − q^{γ} (mod (q^{n} − 1)/(q − 1)).

Since n ≥ 3, we have 0 ≤ (q − 1)(a_{i} − a_{i+1}) ≤ (q − 1)^{2} < (q^{n} − 1)/(q − 1), and we conclude that either β ≥ γ and (q − 1)(a_{i} − a_{i+1}) = q^{β} − q^{γ}, or β < γ and (q − 1)(a_{i} − a_{i+1}) = (q^{n} − 1)/(q − 1) + q^{β} − q^{γ}. In the second case, we have (q−1)^{2} ≥ (q−1)(a_{i}−a_{i+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)(a_{i} − a_{i+1}) = q^{β} − q^{γ}, so that a_{i} − a_{i+1} is congruent to 0 or 1 (mod q), and thus a_{i} − a_{i+1} = 0 or 1. In particular, for each i we have 0 ≤ a_{i} − a_{n} ≤ n − 1 ≤ p − 1.

We now repeat the above analysis. Fix some 1 ≤ i ≤ n − 1, and consider two permutations x_{1}, …, x_{n} and x′_{1}, …, x′_{n} as above, which satisfy x_{i} = x′_{i} for 1 ≤ i ≤ n − 2, x_{n−1} = a_{i}, x_{n} = a_{n}, x′_{n−1} = a_{n} and x′_{n} = a_{i}. Taking the difference of the two expressions q^{n−1}x_{1}+q^{n−2}x_{2}+⋯+x_{n} and q^{n−1}x′_{1}+q^{n−2}x′_{2}+⋯+x′_{n}, and using that 0 ≤ a_{i} − a_{n} ≤ p − 1 ≤ q − 1, we conclude as before that each a_{i} − a_{n} = 0 or 1. Thus there must be an integer 1 ≤ r ≤ n with a_{1} = ⋯ = a_{r} = a_{n} + 1, a_{r+1} = ⋯ = a_{n}.

Returning to the original congruences

q^{n−1}x_{1} + q^{n−2}x_{2} + ⋯ + x_{n} ≡ 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|_{SL_{n}(k′)} is the standard representation of SL_{n}(k′), i.e. θ|_{SL_{n}(k′)} ≅ ρ|_{SL_{n}(k′)}. By part (2) of Lemma 4.1.8, we see that there is a character χ : G → k^{×} with χ|_{SL_{n}(k′)} = 1 such that θ ≅ ρ ⊗ χ.

To complete the proof, we must show that χ is trivial. Take g ∈ G; we will show that χ(g) = 1. If g′ ∈ G has det g′ = det g, then g(g′)^{−1} ∈ SL_{n}(k′), so χ(g′) = χ(g). Firstly, note that by assumption we can write g = λh, with λ ∈ k^{×}, h ∈ GL_{n}(k′). Choose h′ ∈ GL_{n}(k′) to have eigenvalues {1, …, 1, det(h)}. Then h(h′)^{−1} ∈ SL_{n}(k′) ⊂ G, so g′ := λh′ is an element of G. Then the hypothesis on θ and ρ shows that the eigenvalues of χ(g′)g′ = χ(g)g′ are contained in the set of eigenvalues of g′, so that if χ(g) ≠ 1 we must have χ(g) = det(h) = −1. Assume for the sake of contradiction that this is the case. If n is odd, then we now choose h′ to have eigenvalues {−1, …, −1}, and we immediately obtain a contradiction from the same argument. If n is even then since p ≥ n we have p ≥ 5 (recall that we are assuming n ≥ 3), and we may choose a ∈ (k′)^{×}, a ≠ ±1. Then choosing h′ to have eigenvalues {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 → GL_{n}(k) and θ : G → GL_{n}(k) are representations such that the image θ(G) is generated by its subset of regular elements, and for every g ∈ G the characteristic polynomial of ρ(g) annihilates θ(g), then det ρ = det θ.

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 regular elements, 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 ρ_{g}(X) = X^{n} − a_{1}(g)X^{n−1} + ⋯ + (−1)^{n}a_{n}(g) ∈ k[X] of degree n, with the property that for all g, h ∈ G, we have a_{n}(gh) = a_{n}(g)a_{n}(h).

4.2.2. Lemma. Suppose that θ : G → GL_{n}(k) is a representation with the property that θ(G) is generated by its subset of regular elements, and that for each g ∈ G we have ρ_{g}(θ(g)) = 0. Then for each g ∈ G we have det θ(g) = a_{n}(g).

Proof. This may be proved in exactly the same way as Lemma 4.2.1. □

Let ρ : G → GL_{3}(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 → GL_{3}(k) is irreducible, and if either:

(1) ρ is isomorphic to an induction Ind_{H}^{G} ψ, where H is a normal subgroup of index 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 Ind_{H}^{G} ψ. Since H is a proper subgroup of G, the set of elements G − H generates G. If g ∈ G − H then the characteristic polynomial of ρ(g) is of the form X^{3} − α. If p ≠ 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 ℓ_{g} for the corresponding line in ℙ^{2}(k).

Let h be the given regular unipotent element in H. Then h stabilises a unique line in k^{3}, so a unique point P ∈ ℙ^{2}(k). As g ∈ G − H, we also have gh ∈ G − H. Then ℓ_{g} ∩ ℓ_{gh} is non-empty, so there is a point Q ∈ ℙ^{2}(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 is generated by G − H, this implies that every element of G fixes P. This contradicts the assumption that ρ is irreducible. □

Appendix 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 ℚ_{p}, where p is invertible on X, let k_{E} denote the residue field of E, and we let A be one of E or k_{E}. We define the étale cohomology of the pair (X, Z) with coefficients in A to be the étale cohomology of the sheaf j_{!}A on X, i.e. we write

H_{ét}^{•}(X, Z, A) := H_{ét}^{•}(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

⋯ → H_{ét}^{i}(Z, A) → H_{ét}^{i+1}(X, Z, A) → H_{ét}^{i+1}(X, A) → H_{ét}^{i+1}(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 is a smooth projective variety over a separably closed field, and Y and Z are smooth divisors on X which meet transversely. In this case we have a Cartesian diagram of open immersions

(A.1.1)  \xymatrix{X \ (Y ∪ Z) \ar[r]^{j} \ar[d]^{k′} & X \ Y \ar[d]^{k} \\ X \ Z \ar[r]^{j′} & 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′_{!}Rk′_{∗}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′ (and recalling that k′ is an open immersion), we obtain an isomorphism

RHom(k′_{!}A, A) ≅ Rk′_{∗}RHom(A, A) = Rk′_{∗}A,

and hence by double duality, an isomorphism

(A.2.1)  RHom(Rk′_{∗}A, A) ≅ k′_{!}A.

Next, taking f to be j′, and taking into account (A.1.2) and (A.2.1), we obtain isomorphisms

RHom(Rk_{∗}j_{!}A, A) ≅ RHom(j′_{!}Rk′_{∗}A, A) ≅ Rj′_{∗}RHom(Rk′_{∗}A, A) ≅ Rj′_{∗}k′_{!}A.

Finally, taking f to the natural map X → Spec k, and recalling that in this case f^{!}A = A[2d](d) (where d is the dimension of X; cf. [Gro73, XVIII 3.2.5]) and that Rf_{∗} = Rf_{!} (since f is proper, the variety X being projective by assumption), we find that

RHom(Rf_{∗}Rk_{∗}j_{!}A, A) ≅ Rf_{∗}RHom(Rk_{∗}j_{!}A, A[2d](d)) ≅ Rf_{∗}Rj′_{∗}k′_{!}A[2d](d).

Passing to cohomology, we find that H_{ét}^{i}(X \ Y, Z \ Y, A) is in natural duality with H^{2d−i}(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 meet transversely, 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 assumption implies that H_{ét}^{i}(X \ Y, A) vanishes if i > d, and that H_{ét}^{i}(Z \ Y, A) vanishes if i ≥ d ([Gro73] XIV 3.3). By the long exact cohomology sequence of the pair (X \ Y, Y \ Z), we see that H_{ét}^{i}(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 between H_{ét}^{i}(X \ Y, Z \ Y, A) and H^{2d−i}(X \ Z, Y \ Z, A)(d), we find that both vanish unless i = d.

Let O_{E} denote the ring of integers in E. Suppose momentarily that E/ℚ_{p} is finite, and let ϖ be a uniformiser of O_{E}. From a consideration of the cohomology long exact sequence arising from the short exact sequence of sheaves

0 → O_{E}/ϖ^{n} ⟶^{ϖ·} O_{E}/ϖ^{n+1} ⟶ k_{E} → 0,

and arguing inductively on n, we find that H_{ét}^{i}(X \ Y, Z \ Y, O_{E}/ϖ^{n}) vanishes in degrees other than i = d for all n. Passing to the projective limit over n, we see that the same is true of H_{ét}^{i}(X \Y, Z \Y, O_{E}). Finally, a consideration of the cohomology long exact sequence arising from the short exact sequence

0 → O_{E} ⟶^{ϖ·} O_{E} ⟶ k_{E} → 0

shows that H_{ét}^{d}(X \ Y, Z \ Y, O_{E}) is torsion-free.

By passage to the direct limit over subfields of E which are finite over ℚ_{p}, we see that these properties continue to hold for arbitrary algebraic extensions E/ℚ_{p}.

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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