Deformations of automorphic Galois representations Fourth draft, January 2003 Michael Harris

UFR de Mathématique, Université Paris 7, 2 Pl. Jussieu 75251 Paris cedex 05, FRANCE

Richard Taylor

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

The present manuscript is the fourth draft of a project begun in the summer of 1996, to apply the methods of [TW] to Galois representations of dimension > 2. We show that a (mod ℓ) representation \overline{ρ} of dimension n that ”comes from” automorphic forms on certain unitary groups has the property that every lifting of \overline{ρ} to characteristic zero also comes from automorphic forms. In the language of [W] and [TW], we are constructing isomorphisms between ℓ-adic Hecke algebras and ℓ-adic deformation rings. The precise statements are Theorems V.1.6 and V.1.7.

Let E be a totally real field, K_{0} an imaginary quadratic field, and K = E · K_{0}. We consider automorphic representations π of the unitary similitude group G of a division algebra D over K of dimension n^{2} with involutions “of the second kind”; i.e., which restrict to complex conjugation on K. To such data we can attach Shimura varieties Sh which are uniformized by the unit ball in ℂ^{n−1}, and which admit canonical models over K. The automorphic representations of interest to us are those that contribute to middle-dimensional (singular or ℓ-adic) cohomology with coefficients in group-theoretic local systems over Sh; in the present article only the trivial local system is considered, but this is mainly for convenience. The Galois representation on the piece of the ℓ-adic cohomology cut out by π is rather well understood, both at (most) unramified places [K1, C1] and at places of K split over E and where π is supercuspidal [H1, H2]. Via base change, π can be associated to an automorphic representation Π of GL(n, K), and, up to normalization, the L-function of the corresponding Galois representation, denoted r_{ρ}(π), coincides with that of Π. In particular, r_{ρ}(π) is n-dimensional and enjoys a number of arithmetically interesting properties; e.g., it is crystalline at primes dividing ℓ. Let O be an ℓ-adic integer ring over which r_{ρ}(π) can be realized, and let k be its residue field.

We assume that the only ramification of π comes at a finite set SC of places that split in K/E, and that π_{v} is full induced from supercuspidals for v ∈ SC. When D is not split at v we assume the representation of GL(n) corresponding to π is supercuspidal. We also assume ℓ is prime to the order of GL(n, k(v)) for all v such that π_{v} is ramified; here k(v) is the residue field. In Vignéras’ terminology, ℓ is banal for such v. This implies that ℓ > n. It thus follows from ℓ-adic Hodge theory [FM, F1] that the residual representation \overline{r}_{ρ}(π) over k is also crystalline, i.e. it can be obtained by the construction of Fontaine-Laffaille [FL]. It can be shown that \overline{r}_{ρ}(π) restricts to an absolutely irreducible representation of the decomposition group Z_{v} at any v ∈ SC. Using these facts, we can define a universal ring R(π) classifying deformations of \overline{r}_{ρ}(π) to crystalline representations of Gal(\overline{K}/K), via the theory developed by Mazur and Ramakrishna (cf. [DDT]). For technical reasons,

Typeset by AMS-TEX

\overline{r}_{ρ}(π) is replaced in the deformation theory by the representation \overline{ρ}(π)of Gal(\overline{E}/E) induced from \overline{r}_{ρ}(π), which we view as a homomorphism to a disconnected group with identity component GL(n).

The goal is to construct a natural isomorphism between R(π) and an appropriate Hecke algebra. The Hecke algebra acts on the space of functions on the zero-dimensional Shimura variety Sh′ attached to an inner form J of G. The correspondence between functions on Sh′ and cohomology of Sh, analogous to the Jacquet-Langlands correspondence for inner forms of GL(2), is established in most cases in [H1], and in general in [HL]. It is relatively easy to study congruences for Hecke algebras acting on zero-dimensional Shimura varieties. Thus we can hope to apply the techniques of [TW]. The possibility of extending these techniques to groups other than GL(2, ℚ) is based on the crucial fact, observed simultaneously and independently by Diamond and Fujiwara [D, Fu], that one can work directly with Hecke algebra modules, rather than with the Hecke algebras themselves.

Few general techniques are available for studying the images of Galois representations of dimension > 3. In order to carry out the sorts of calculations of Galois cohomology familiar to readers of [W], [TW], and [DDT], we need to impose several hypotheses to guarantee that the image of \overline{ρ} is not too small (Hypotheses IV.5.2), in addition to the conditions on ℓ already mentioned. Our hypotheses on \overline{ρ} are certainly not optimal, but they have the advantage of economy. We hope to relax some of them before completing the definitive version. For this reason, the present draft develops techniques in greater generality than is immediately necessary.

The article is divided into five sections. Section I introduces the unitary groups and the associated Shimura varieties, and summarizes results on the spaces of automorphic forms on these groups that are derived from the trace formula. Section II is mainly concerned with applications of Vignéras’ theory of modular representations of p-adic reductive groups, especially GL(n). Vignéras’ results replace some of the theory of the U-operator, familiar from work on GL(2). Section III describes the Shimura varieties attached to unitary groups as moduli spaces of abelian varieties of PEL type, and develops the properties of the Galois representations realized in the ℓ-adic cohomology of these Shimura varieties. Section IV is concerned with deformations of Galois representations, and extends the Galois-cohomological techniques of [W] and (especially) [TW] to dimension > 2. Finally, section V generalizes the arguments and the main result of [TW]. At the end of V we indicate how the main theorems can be applied to construct automorphic tensor products.

The authors would like to thank the University of Durham (Harris and Taylor), Harvard University (Harris), the Institut Henri Poincaré (Taylor), and RIMS and Kyoto University (Harris) for providing a hospitable working environment at different stages of the project. We also thank the many colleagues who have answered questions during the last 18 months. They include: Blasius, Carayol, Clozel, Colmez, de Jong, Diamond, Fontaine, Fujiwara, Gabber, Henniart, Kottwitz, Labesse, Laumon, Niziol, Rapoport, Rogawski, Vignéras, Zink. This list is incomplete and will be amended in future versions of the manuscript.

I. Automorphic forms on unitary groups

I.1. Cohomological automorphic forms.

Let E be a totally real number field of degree d over ℚ and let K be a totally imaginary quadratic extension of E; let c ∈ Gal(K/E) denote the non-trivial automorphism. We assume K contains an imaginary quadratic field K_{0}, so that K = K_{0} · E. We denote by Σ_{E} and Σ_{K} the sets of complex embeddings of E and K. Let D be a central simple algebra of dimension n^{2} over K, endowed with an involution, denoted c̃, that induces the Galois automorphism c on K; i.e., c̃ is an involution of the second kind. We will generally make the hypothesis

(D) At every finite place v of K, D_{v} is either split or a division algebra.

We define algebraic groups U(D) = U(D, c̃) and GU(D) = GU(D, c̃) over ℚ such that, for any ℚ-algebra R,

U(D)(R) = {g ∈ D^{opp} ⊗_{ℚ} R | g · c̃(g) = 1};
GU(D)(R) = {g ∈ D^{opp} ⊗_{ℚ} R | g · c̃(g) = ν(g) for some ν(g) ∈ R^{×}}.

Thus GU(D) admits a homomorphism ν : GU(D) → 𝔾_{m} with kernel U(D). There is an algebraic group U_{E}(D) over E such that U(D) ⟶^{∼} R_{E/ℚ}U_{E}(D), where R_{E/ℚ} denotes Weil’s restriction of scalars functor. This isomorphism identifies automorphic representations of U(D) and U_{E}(D).

The groups U(D) (resp. GU(D)) are all inner forms of the same quasi-split unitary group (resp. unitary similitude group), denoted U_{0} (resp. GU_{0}). The group U_{0} is of the form U(D_{0}, χ̃(∗)_{0}) where D_{0} is the matrix algebra and χ̃(∗)_{0} is an appropriate involution. Then U_{0,∞} ≅ U(\frac{n}{2}, \frac{n}{2})^{[E:ℚ]} if n is even, U_{0,∞} ≅ U(\frac{n+1}{2}, \frac{n−1}{2})^{[E:ℚ]} if n is odd.

Let G be a reductive algebraic group over the number field F. If v is a place of F we let G_{v} = G(ℚ_{v}); if v is archimedean we let 𝔤_{v} = Lie(G_{v})_{ℂ}. We let G_{∞} denote ∏_{v|∞} G_{v}, the product taken over all archimedean places of F, and let 𝔤_{∞} = ∏_{v|∞} 𝔤_{v}. Let π be an irreducible automorphic representation of G; i.e., an irreducible (𝔤_{∞}, K_{∞}) × G(A^{f})-module that embeds as a submodule of the space of automorphic forms relative to the maximal compact subgroup K_{∞}. We write π = π_{∞} ⊗ π_{f} as usual, and say π is cohomological if π is cuspidal and if the relative Lie algebra cohomology H^{•}(𝔤_{∞}, K_{∞}; π_{∞} ⊗ V) ≠ 0 for some finite dimensional representation V of 𝔤_{∞}. We let Coh(G) denote the set of cohomological cuspidal automorphic representations of G, Coh(G, V) ⊂ Coh(G) the subset of π for which H^{•}(𝔤_{∞}, K_{∞}; π_{∞} ⊗ V) ≠ 0, with V fixed. If K ⊂ G(A^{f}) is a compact open subgroup, σ a finite-dimensional irreducible representation of K, we let Coh(G, K, σ) ⊂ Coh(G) denote the subset of π such that Hom_{K}(σ, π) ≠ 0, Coh(G, K, σ, V) = Coh(G, K, σ) ∩ Coh(G, V). If σ = 1 we just write Coh(G, K) and Coh(G, K, V).

Let A_{0}(G) denote the space of cusp forms on G. Let Rep(G) denote the set of equivalence classes of irreducible (𝔤_{∞}, K_{∞}) × G(A^{f})-modules. If π ∈ Rep(G), we let m(π) = dim Hom(π, A_{0}(G)), where Hom denotes the space of homomorphisms of (𝔤_{∞}, K_{∞}) × G(A^{f})-modules. More generally, let S be a finite set of places of F, containing the archimedean places (for simplicity) and let Rep(G)^{S} denote the set of equivalence classes of irreducible G(A^{f,S})-modules, where G(A^{f,S}) ⊂ G(A^{f}) is the subgroup with trivial entry at every place in S. We say π^{S} ∈ Rep(G)^{S} is automorphic if

Hom_{G(A^{f,S})}(π^{S}, A_{0}(G)) ≠ 0;

i.e., if π^{S} can be extended to a cuspidal automorphic representation of G. We say π^{S} is cohomological if it can be extended to a cohomological cuspidal automorphic representation of G.

We now choose a central simple algebra D, over K satisfying (D). Let c̃ be an involution of the second kind of D and define J = GU(D, c̃), J′ = U(D, c̃), and view J′ alternatively as a group over ℚ or E. We assume that

(I.1.1)  J′_{∞} ≅ U(n)^{[E:ℚ]} (compact inner form);

Moreover, if v is a finite prime, we assume that

(I.1.3)  J′_{v} ≅ U_{0,v} if v does not split in K/F.

We will be working with a variant, denoted G̃, of the Langlands L-group of J over E. Let

(I.1.4) G̃^{0} = {g = (g_{1}, g_{2}) ∈ GL(n)×GL(n) | ∃a ∈ GL(1) such that g_{2} = a·^{t}g_{1}^{−1}}.

We let

(I.1.5)  G̃ = G̃^{0} ⋉ Gal(\overline{E}/E),

where the action of Gal(\overline{E}/E) on GL(n) factors through Gal(K/E), and the nontrivial element c acts by

(I.1.6)  c((g_{1}, g_{2})) = (g_{2}, g_{1}).

We regard G̃ as a group scheme over Spec(ℤ), in order to work with its points over finite fields. The map that to g associates the element a in the definition (I.1.4) defines a homomorphism of group schemes G̃^{0}→GL(1), and one verifies that it extends to a homomorphism ν : G̃→GL(1) by setting ν(c) = (−1)^{n−1}.

In the applications to Galois representations, we need to work with G̃, rather than the L-group of J′ or G′, because Galois representations of geometric origin are generally polarized but rarely self-dual. Moreover, G̃ does not incorporate the usual conjugation by a symmetric or anti-symmetric form. This simplifies our arguments. In Remark V.2.5, we reformulate our results in terms of the usual Langlands Lgroup.

I.2. Automorphic forms on a totally definite unitary group.

Let K ⊂ J(A^{f}) be a compact open subgroup,

KS(J) = J(ℚ)\J(A)/J_{∞} × K.

More generally, if U ⊂ J(A^{f}) is any closed subgroup, we let

US(J) = \underleftarrow{lim}_{K⊃U} _{K}S(J),

the limit over open compact subgroups K containing U. We define certain spaces of functions on J(A). In what follows, K as above is always assumed to be of the form ∏_{v} K_{v}, the product taken over finite places of ℚ, with K_{v} open compact in J_{v}. We fix a rational prime number ℓ and let K^{ℓ} = K ∩ J^{A^{f,ℓ}} (finite adeles of J trivial at ℓ). All functions are taken to be continuous.

Let (ρ, W) denote a finite-dimensional algebraic representation of J, rational over the number field L = L(ρ). For any place v of ℚ we use the same letter ρ to designate the representation of J_{v} on W (L_{v}), where L_{v} = L ⊗_{ℚ} ℚ_{v}. For each place v we choose a finite-dimensional representation σ_{v} of K_{v} with coefficients in L, factoring through a finite quotient of K_{v}, so that σ_{v} is trivial for all v dividing ℓ and for almost all v. We let σ = ⊗_{v}σ_{v}, and let W_{σ} = W ⊗ σ, on which J_{∞} × K acts via ρ ⊗ σ.

Let (I.2.1) A_{0}(J, ρ, σ, K) = {f ∈ C^{∞}(J(ℚ)\J(A), W_{σ}(ℂ))|f(g·g_{∞}k) = (ρ ⊗ σ)(g_{∞}×k)^{−1}f(g)},

where g ∈ J(A), g_{∞} ∈ J_{∞}, k ∈ K.

For any L-algebra L′, we let (I.2.2)

A_{f}(J, ρ, σ, K)(L′) = {f : J(A^{f}) → W (L′)|f(γ · gk) = (ρ ⊗ σ)(γ × k^{−1})f(g)},

where now g ∈ J(A), γ ∈ J(ℚ), k ∈ K. If L′ is an L_{ℓ} algebra, we define (I.2.3)

A_{ℓ}(J, ρ, σ, K)(L′) = {f : _{K^{ℓ}}S(J) → W (L′_{ℓ})|f(gk) = (ρ ⊗ σ)(k_{ℓ} × k)^{−1}f(g)},

for g ∈ J(A), k ∈ K, k_{ℓ} its J_{ℓ}-component. Here the standing continuity hypothesis needs to be specified: we assume f ∈ A_{ℓ}(J, ρ, K)(L′) to be locally constant on J(A^{f,ℓ}). Continuity of f in G_{ℓ} in the ℓ-adic topology is guaranteed by the functional equation.

For any L-algebra L′, there is a canonical isomorphism

(I.2.4)  A_{f}(J, ρ, K)(L′) = A_{f}(J, ρ, K)(L) ⊗_{L} L′.

Restriction to J(A^{f}) defines a natural isomorphism

res : A_{0}(J, ρ, σ, K) ⟶^{∼} A_{f}(J, ρ, σ, K)(ℂ)

and hence, via (I.2.4), a canonical L-structure on A_{0}(J, ρ, σ, K). Similarly, for any L_{ℓ}-algebra L′ we have an isomorphism

(I.2.5)  res_{ℓ} : A_{ℓ}(J, ρ, σ, K)(L′) ⟶^{∼} A_{f}(J, ρ, σ, K)(L′),

given by

f ↦ {g ↦ ρ(g_{ℓ})^{−1} · f(g), g ∈ J(A^{f})}.

Letting K vary, it is clear that these isomorphisms commute with the action of the prime-to-ℓ finite adeles of J.

When ρ or σ is the trivial one-dimensional representation, we drop it from the notation.

In what follows, we assume K^{ℓ} is sufficiently small so that

(I.2.6)  K_{ℓ} acts without fixed points on J(ℚ)\J(A)/K_{∞} · K^{ℓ}.

Let O denote the ring of integers of L_{ℓ}, and let Λ_{W_{σ}} be a K_{ℓ}-stable O-lattice in W_{σ}(L). We can define

(I.2.7) A_{ℓ}(J, Λ_{W_{σ}}, K)

= {f : _{K^{ℓ}}S(J) → Λ_{W_{σ}}|f(gk) = (ρ ⊗ σ)(k_{ℓ} × k)^{−1}f(g), g ∈ J(A), k ∈ K},

with the continuity hypothesis as above. Then there is a natural isomorphism

A_{ℓ}(J, Λ_{W_{σ}}, K) ⊗_{O} L_{ℓ} ≅ A_{ℓ}(J, ρ, σ, K)(L_{ℓ}).

When ρ and σ are trivial, we take the lattice O ⊂ L_{ℓ} and write A_{ℓ}(J, K)(O). I.2.8. Now we will make specific choices of (K_{v}, σ_{v}). We choose disjoint finite sets SC, Q, and {𝔯} of finite places of E dividing disjoint sets of rational primes that split in K_{0}; {𝔯} is a single prime. The set SC is assumed to include all primes divisible by primes of K at which D is not split. For any rational prime q that splits in K_{0}, we choose a maximal compact subgroup of J_{q} in the form

(I.2.8.1)  ℤ_{q}^{×} × ∏_{v | q} J_{v},

the product being taken over divisors v of q in E. Here if w is a divisor of q in K_{0} lying above v and if D_{w} is isomorphic to GL(a, B_{w}) for some factorization n = ab and some division algebra B_{w} of degree b^{2} over K_{w}, then J_{v} can be taken in the form GL(a, O_{B_{w}}), where O_{B_{w}} is the maximal order of B_{w}.

To define the (K_{v}, σ_{v}) at ramified places, we assume D satisfies hypothesis (D) of §I.1. For w ∈ SC, we fix an irreducible admissible representation π_{w} of GL(n, E_{w}). We assume there is a (not necessarily proper) parabolic subgroup P of G, with Levi quotient L_{P}, such that π_{w} can be realized as the full induced representation from a supercuspidal representation of L_{P}, inflated to P. Equivalently, π_{w} is associated to a fully decomposable representation of the Weil-Deligne group of E_{w} – i.e., a representation of the Weil group W_{E_{w}} of E_{w} – under the local Langlands correspondence π ↦ L(π) [HT1, He]. Such representations will be called non-monodromic. If the algebra D is split at w we let (K_{w}, σ_{w}) be a semisimple type of π_{w} in the sense of Bushnell and Kutzko [BK]. Thus any irreducible admissible representation π′ of GL(n, E_{w}) whose restriction to K_{w} contains σ_{w} is inertially equivalent to π_{w} [BK, Theorem 8.2]. In other words, if L(π_{w}) is the representation 𝔯_{w} = ⊕_{i=1}^{r}𝔯_{i,w} of W_{E_{w}}, with each 𝔯_{i,w} irreducible, and if π′ contains the type (K_{w}, σ_{w}), then L(π′) is of the form ⊕_{i=1}^{r}𝔯_{i,w} ⊗ χ_{i}, where each χ_{i} is an unramified character of W_{E_{w}}.

If D_{w} is a division algebra we assume π_{w} is supercuspidal, and let JL(π_{w}) be the irreducible representation of D_{w}^{×} associated to π_{w} by the Jacquet-Langlands correspondence [DKV,R]. We let K_{w} be the maximal compact subgroup of D_{w}^{×} and let σ_{w} be any irreducible component of the restriction of JL(π_{w}) to K_{w} (since JL(π_{w}) is irreducible, all choices of σ_{w} are conjugate under D_{w}^{×}. If q is a rational prime divisible by some w ∈ SC, let

K_{q} = ℤ_{q}^{×} × ∏_{w∉SC} GL(n, O_{w}) × ∏_{w∈SC} K_{w},

the products running only over divisors of q. We also take

σ_{q} = ξ_{q} ⊗ ⨂_{w∈SC} σ_{w},

where ξ_{q} is a character of ℤ_{q}^{×} and ∏_{w∉SC} GL(n, O_{w}) acts trivially.

Primes 𝔮 ∈ Q are assumed to have the property that, if q is the rational prime divisible by 𝔮, then q splits completely in E. We let Q(ℚ) denote the set of rational primes divisible by primes in Q, and assume each q ∈ Q(ℚ) is divisible by a unique 𝔮 ∈ Q. For 𝔮 ∈ Q we define

Γ_{0,𝔮} = \{k ∈ GL(n, ℤ_{q}) | k ≡ ( \begin{matrix}∗_{1} & ∗ \\ 0 & ∗_{n−1} \end{matrix} ) (mod q)\}

Γ_{1,𝔮} = \{k ∈ GL(n, ℤ_{q}) | k ≡ ( \begin{matrix}1 & ∗ \\ 0 & ∗_{n−1} \end{matrix} ) (mod q)\}

as subgroups of the 𝔮-factor of I.2.8.1. Then we let

K_{0,q} = ℤ_{q}^{×} × ∏_{v≠𝔮,v|q} GL(n, O_{v}) × Γ_{0,𝔮};

K_{1,q} = ℤ_{q}^{×} × ∏_{v≠𝔮,v|q} GL(n, O_{v}) × Γ_{1,𝔮},

viewed as subgroups of J_{q}.

Let q(𝔯) denote the residue characteristic of 𝔯. Let N_{0} denote the upper triangular unipotent subgroup of GL(n) and let

I_{1}(𝔯) = {k · n ∈ GL(n, O_{𝔯}) | k ≡ 1 (mod 𝔯), n ∈ N_{0}(O_{𝔯})};

I(𝔯) = ℤ_{q(𝔯)}^{×} × ∏_{v≠𝔯,v|q(𝔯)} GL(n, O_{w}) × I_{1}(𝔯).

We let

(I.2.8.2)  K_{q(𝔯)} = I_{𝔯}.

Finally, for the remaining primes q, we take maximal compact subgroups K_{q} that are very special in the sense of [L]. In particular, K_{q} is hyperspecial whenever J_{q} contains a hyperspecial maximal compact subgroup; i.e., if q is unramified in K_{0}. If q is ramified in K_{0} and n is odd, there are two conjugacy classes of special maximal compact subgroups, only one of which is very special. Here we are using the hypothesis (I.1.3). We let

K_{0,Q} = ∏_{q∉Q(ℚ)} K_{q} × ∏_{q∈Q(ℚ)} K_{0,q};

K_{1,Q} = ∏_{q∉Q(ℚ)} K_{q} × ∏_{q∈Q(ℚ)} K_{1,q}.

Lemma I.2.9. For q(𝔯) sufficiently large, the subgroups K_{0,Q} and K_{1,Q} satisfy (I.2.6). Moreover, for any s ∈ J(A^{f}) the groups s^{−1}J(ℚ)s ∩ K_{0,Q} and s^{−1}J(ℚ)s ∩ K_{1,Q} are trivial.

Proof. The first assertion follows from the second. Let K = K_{0,Q} or K_{1,Q}. Let x ∈ s^{−1}J(ℚ)s ∩ K for some s ∈ J(A^{f}). The subgroup of J(A) generated by x is both discrete and compact, hence finite. The group I_{𝔯} is pro-q(𝔯) and it follows that x is a root of unity of order a power of q(𝔯), lying in some extension field K′ of K that admits an embedding in D. The degree of K′ over ℚ is bounded by n[K : ℚ], hence for q(𝔯) sufficiently large we must have x = 1. Condition (I.2.6) is now immediate.

Let π be an automorphic representation of J. The restriction of π to the unitary group J′, which can be viewed as an algebraic group over E, decomposes as a direct sum of irreducible automorphic representations. Any two summands have the same local components at any finite place w dividing a rational prime q that splits in K_{0}, since at such places the similitude map splits as a product J(ℚ_{q}) ≃ J′(ℚ_{q}) × ℚ_{q}^{×}. For such a place w, we say π is non-monodromic at w if one component (hence every component) of the restriction of π to the unitary group J′ is non-monodromic at w (if J_{w} is split) or corresponds to a supercuspidal representation of GL(n, E_{w}) by the Jacquet-Langlands correspondence (if J_{w} is the multiplicative group of a division algebra).

Proposition I.2.10. Let π be an automorphic representation such that Hom_{K_{1,Q}}(σ, π) ≠ 0. Then π_{w} is non-monodromic for all w ∈ SC and, for every w ∈ Q, π_{w} is either (a) unramified; (b) principal series attached to an n-tuple (α, β_{1}, …, β_{n−1}) of characters of E_{w}^{×}, with α tamely ramified and each β_{i} unramified; or (c) the Langlands sum of a special representation of GL(2) and an unramified representation of GL(n − 2). In cases (a) and (c), but not in case (b), π_{w} has a Γ_{0,w}-fixed vector.

Moreover, if SC is non-empty, then π_{q} is generic for every q that splits in K_{0}.

Proof. The assertion regarding w ∈ SC is a consequence of the properties of Bushnell-Kutzko types. The first assertion regarding w ∈ Q follows easily from the Bernstein-Zelevinsky classification of admissible irreducible representations of GL(n) and from the theory of the conductor [JPSS]. The second assertion follows from the existence of base change of π to J_{K_{0}} ≃ GL(n)_{K} × GL(1)_{K_{0}} [CL], cf. [HT1,VI.2]. Let Π denote the base change. Then Π is cuspidal, hence globally generic by Shalika’s theorem. The second assertion of (I.2.10) then follows from the properties of base change, especially [L, Proposition IV.4.1].

Let 𝕂 = ∏_{v} 𝕂_{v} be a subgroup of finite index in K_{1,Q}, contained in the kernel of σ and normal in K_{0,Q}. Let Λ_{W} = Λ_{W_{σ}} with σ replaced by the trivial representation. We can write

(I.2.11)  J(A^{f}) = ∐_{i=1}^{r} J(ℚ)g_{i}𝕂,

for some finite set Ξ = {g_{1}, …, g_{r}} of elements of J(A^{f}). Then the map f ↦ (f(g_{i})) defines an injection

(I.2.12)  A_{ℓ}(J, Λ_{W} , 𝕂) ↪ ⊕_{i=1}^{r}Λ_{W} .

As in [DT,p. 442], this is an isomorphism provided g_{i}^{−1}J(ℚ)g_{i} ∩ 𝕂 is trivial for all i. In particular, this is true when q(𝔯) is sufficiently large, by Lemma I.2.9, even when SC is empty. We assume q(𝔯) to be sufficiently large from now on.

Now K_{1,Q} satisfies the same properties as 𝕂 above, hence A_{ℓ}(J, Λ_{W} , K_{1,Q}) has the decomposition (I.2.12). If Ξ′ = {g_{1}, …, g_{r′}} is a set of double coset representatives for J(ℚ)\J(A^{f})/K_{1,Q} then we can take Ξ = Ξ′ · [K_{1,Q}/𝕂] in (I.2.12). It follows that

(I.2.13)  A_{ℓ}(J, Λ_{W} , 𝕂) is a free O[K_{1,Q}/𝕂] − module.

We can apply the same argument to K_{0,Q} instead of K_{1,Q}. Since

K_{0,Q}/𝕂 ≅ K_{1,Q}/𝕂 × ∏_{q∈Q}(ℤ/qℤ)^{×},

we obtain:

Lemma I.2.14. Assume q(𝔯) to be sufficiently large. Then A_{ℓ}(J, Λ_{W} , 𝕂) is a free O[K_{1,Q}/𝕂 × ∏_{q∈Q}(ℤ/qℤ)^{×}]-module.

In what follows, we assume Λ_{W_{σ}} to be of the form Λ_{W} ⊗ Λ_{σ}, where Λ_{σ} is a K_{1,Q}/𝕂-invariant lattice in σ.

Corollary I.2.15. Assume q(𝔯) to be sufficiently large. Suppose Λ_{σ} can be realized as a direct summand in O[K_{1,Q}/𝕂] as O[K_{1,Q}/𝕂]-module. (For example, suppose ℓ does not divide the order of ∏_{w∈SC} K_{w}/𝕂_{w}.) Then A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}) is a free O[∏_{q∈Q}(ℤ/qℤ)^{×}]-module.

Let D_{Q} = ∏_{q∈Q}(ℤ/qℤ)^{×}. For any ℤ_{ℓ}[D_{Q}]-module N let N_{D_{Q}} denote the module of coinvariants.

Corollary I.2.16. Let ℓ^{N} be the exact power of ℓ dividing the order of D_{Q}. Under the hypotheses of Corollary I.2.15, the natural inclusion

A_{ℓ}(J, Λ_{W_{σ}}, K_{0,Q}) → A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}),

followed by the canonical map

A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}) → A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q})_{D_{Q}},

induces an isomorphism between A_{ℓ}(J, Λ_{W_{σ}}, K_{0,Q}) and ℓ^{N} · A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q})_{D_{Q}}.

Proof. Indeed, Corollary I.2.15 reduces the assertion of I.2.16 to the corresponding statement for the group ring O[D_{Q}] itself; but this is clear.

Let K_{1,Q}^{[ℓ]} ⊂ K_{0,Q} be the largest subgroup containing K_{1,Q} such that the quotient

Δ_{Q} = K_{0,Q}/K_{1,Q}^{[ℓ]} is an ℓ-group, necessarily abelian. (In other words, K_{1,Q}^{[ℓ]} is generated by K_{1,Q} and by the diamond operators of order prime to ℓ; cf. III.1.3.8, below). We will be using Corollaries I.2.15 and I.2.16 with K_{1,Q} replaced by K_{1,Q}^{[ℓ]}:

Corollary I.2.17. Assume the hypotheses of Corollary I.2.15. Then (i) A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]}) is a free O[Δ_{Q}]-module. (ii) The natural inclusion

A_{ℓ}(J, Λ_{W_{σ}}, K_{0,Q}) → A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]}),

followed by the canonical map

A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]}) → A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]})_{Δ_{Q}},

induces an isomorphism between A_{ℓ}(J, Λ_{W_{σ}}, K_{0,Q}) and ℓ^{N} · A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]})_{Δ_{Q}}.

Proof. Since the kernel of D_{Q}→Δ_{Q} is of order prime to ℓ, this follows immediately from the previous corollaries.

I.3. Functoriality and multiplicities

We collect here some applications of the stable trace formula, established in [C1,CL,L,HL]. We let J̇′ denote the group R_{K/ℚ}GL(n)_{K}, J̇ = J̇′×R_{K_{0}/ℚ}𝔾_{m,K_{0}}, both viewed as algebraic groups over ℚ; J̇′ will also be viewed as an algebraic group over K or over E. Let J and J′ be the groups defined at the end of I.1. Then we have

(I.3.1)  J̇ is an inner form of R_{K_{0}/ℚ}G_{K_{0}} and R_{K_{0}/ℚ}J_{K_{0}};

(I.3.2)  J̇′ is an inner form of R_{K/E}G′_{K} and R_{K/E}J_{K}.

Let π ∈ Coh(J), resp. π ∈ Coh(J′), and let S be the set of places of ℚ, resp. of E, containing all archimedean places and all finite places where either π or the group J is ramified. Then we can formally define BC^{S}(π) ∈ Rep(J̇)^{S} (resp. Rep(J̇′)^{S}) to be the representation such that BC^{S}(π)_{w} = BC(π_{w}) for all w ∉ S, with BC(π_{w}) the local unramified base change map with respect to the isomorphism I.3.1.

Theorem I.3.3 [C1,L,HL]. Let π ∈ Coh(J) (resp. Coh(J′)). Suppose either (i) J is the unitary similitude group of a division algebra; or (ii) There is a rational prime p split in K_{0} and a prime v_{0} of E dividing p such

that π is supercuspidal at v_{0}, in the sense of I.2.10. Then BC^{S}(π) is the prime-to-S part of a cohomological automorphic representation of J̇ (resp. J̇′).

Proof. The proof in case (i) is basically contained in [C1,C2], and is completed in [CL] using the results of [L]. Case (ii) is worked out in [HL].

Under the hypotheses of Theorem I.3.3, strong multiplicity one implies that BC^{S}(π) extends to a unique cuspidal automorphic representation of J̇ (resp. J̇′), which we denote BC(π); it is necessarily cohomological. In the case of J̇, we write BC(π) = BC(π)_{1}×BC(π)_{2}, where BC(π)_{1} is an automorphic representation of J̇′ and BC(π)_{2} is a Hecke character of K_{0}.

Proposition I.3.4 [L]. We retain the hypotheses of Theorem I.3.3. Let q be a rational prime that ramifies in K_{0}, and suppose π_{q} is spherical with respect to a very special maximal compact subgroup of G_{q} (resp. G′_{q}). Then BC(π) is unramified at all primes dividing q.

Moreover, in the case of G′, let q be a rational prime that splits in K_{0}, q = v_{1}·v_{2}. Then we have

BC(π)_{q} = BC(π)_{v_{1}} ⊗ BC(π)_{v_{2}} ≃ π_{q} ⊗ π̌_{q}

via the isomorphism G′(K_{0,q}) ⟶^{∼} G′(ℚ_{q}) × G′(ℚ_{q}).

Proof. The first assertion follows from [L, Proposition IV.6.4], the second from [L,Lemma IV.4.1]. The application of these results to unitary groups is carried out in detail in [HL].

Let E′/E be a totally real cyclic extension, K′ = K_{0} · E′, and let

BC_{K′}(π) = BC(π)_{1,K′}×BC(π_{2}),

where BC(π)_{1,K′} is the base change, as in [AC], of BC(π)_{1}, for the cyclic extension K′/K. We let J^{E′}, (resp. J^{′,E′}) be unitary similitude groups (resp. unitary groups) defined as in §I.1 relative to K′/E′ and satisfying the analogues of (I.1.1)-(I.1.2).

Theorem I.3.5 [C1]. Suppose BC(π)_{1,K′} is supercuspidal at a non-empty set of places, including every finite place w at which J^{′,E′} is anisotropic. Then there is an automorphic representation BC_{E′}(π) ∈ Coh(J^{E′}) whose base change to K′, in the sense of Theorem I.3.3, is isomorphic to BC_{K′}(π).

Moreover, BC_{E′}(π)_{∞} belongs to the discrete series.

Proof. The first assertion for the unitary group J^{′,E′} is a special case of [C1, Proposition 4.11]. Indeed, it suffices to show that BC_{K′}(π) is dual to its Gal(K′/E′)- conjugate, and this is immediate from the corresponding fact for BC(π). The assertion for the similitude groups follows easily from this case.

The final assertion is [C1, Corollaire 5.6].

(I.3.6) Clozel’s theorem actually asserts that BC(π)_{1,K′} descends to an L-packet of cohomological automorphic representations of J^{E′}, which is stable at archimedean places in the sense that the one can switch freely within the archimedean discrete series L-packet. Under additional hypotheses we can say more. Let π ∈ Coh(J). We now suppose

(I.3.6.1)  The extension K/E is unramified at all finite primes.

(I.3.6.2) π_{q} is spherical for every rational prime q that does not split in K_{0}.

The proof of the following theorem will appear in [HL]:

Theorem I.3.7 [HL]. Under hypotheses (I.3.6.1) and (I.3.6.2), the multiplicity m(π) = 1.

**[THIS IS NO LONGER REALLY NECESSARY]

II. Hecke operators and modular representation theory

II.1. Calculations with Jacquet modules.

In the present section we fix a rational prime q and a q-adic field F, with maximal order O_{F}, maximal ideal m_{F}, and residue field k(F), and uniformizing parameter ϖ. Let H = GL(n, F) and let B be its standard upper triangular Borel subgroup. Let P ⊂ GL(n) be the standard parabolic subgroup with Levi decomposition P = LN, L = GL(1) × GL(n − 1) with its standard embedding in H. Let Γ = GL(n, O_{F}) and let r : Γ→GL(n, k(F)) denote the reduction map. We define Γ_{1} ⊂ Γ_{0} ⊂ Γ as in §I.2: the parahoric subgroup Γ_{0} is the inverse image under r of P (k(F)), and

Γ_{1} = \{g ∈ Γ_{0} : r(g) = ( \begin{matrix}1 & ∗ \\ 0 & ∗_{n−1} \end{matrix} )\}.

We fix a commutative ring R in which q is invertible and let H, H_{0}, and H_{1} denote the Hecke algebras of H with respect to Γ, Γ_{0}, and Γ_{1}, respectively, in each case with R-coefficients. We also let H_{L} denote the Hecke algebra of L(F) with respect to its maximal compact subgroup Γ_{L} = Γ ∩ L(F).

Let z : F^{×}→H denote the embedding of F^{×} as diagonal matrices. The map

a →< a >= z(a)K,

where K = Γ, Γ_{0}, or Γ_{1}, maps F^{×} to a subgroup of the multiplicative group of the corresponding Hecke algebra. The map <> identifies its image with F^{×}/O_{F}^{×} in the case of H and H_{0} and with F^{×}/(1 + O_{F}) in the case of H_{1}.

Let χ = (χ_{1}, χ_{2}, …, χ_{n}) be an ordered n-tuple of R-valued characters of F^{×}. We may regard χ as a character of T = GL(1, F)^{n}, viewed as the Levi factor of B. Then, by composition with the natural projection, χ defines a character of B. We let i_{H,T} χ denote the non-normalized induced representation

i_{H,T} χ = {f : H→R | f(bh) = χ(b)f(h), ∀b ∈ B, h ∈ H}.

Let | |_{F} : F^{×}→q^{ℤ} be the absolute value character:

|x|_{F} = |k(F)|^{−v_{F} (x)},

where v_{F} is the valuation on F. Composing with the natural map q^{ℤ}→R^{×} we obtain a character ν : F^{×}→R^{×}. Define the modulus character δ : B→R^{×} as the composition ν ∘ det_{ad} where det_{ad} : B→F^{×} is the determinant of the adjoint representation of B on its Lie algebra. We assume q to be a square in R^{×} and choose a square root \sqrt{q}. Then we can define δ^{\frac{1}{2}} : B→R^{×}, which we write as an n-tuple of characters of F^{×}:

δ^{\frac{1}{2}} = (δ_{1}, δ_{2}, …, δ_{n}).

The Weyl group W of GL(n), which we identify with the symmetric group 𝔖_{n}, acts on the set X_{R}(T) of R^{×}-valued characters of T by permutation:

w(χ) = (χ_{w}(1), χ_{w}(2), …, χ_{w}(n)).

We define the twisted action

w ∗ χ = w(χ) · δ^{\frac{1}{2}} w(δ^{− \frac{1}{2}} ).

The product δ^{\frac{1}{2}} w(δ^{− \frac{1}{2}} ) does not depend on the choice of \sqrt{q}.

We will need the following theorem, due to Bernstein and Zelevinski when R is a field of characteristic zero and to Vignéras when R is a field of positive characteristic ℓ ≠ q [BZ;V1, III. 1.15]:

Theorem II.1.1. Suppose R = \overline{ℚ}_{ℓ} or R = \overline{𝔽}_{ℓ}. Suppose χ ∈ X_{R}(T) has the property that, for all 1 ≤ i, j ≤ n, we have

(II.1.1.2)  χ_{i}δ_{i}^{−1} ≠ χ_{j}δ_{j}^{−1}ν.

Then i_{H,T} χ is irreducible.

We note that the irreducibility criterion of Theorem II.1.1 does not depend on the choice of \sqrt{q}.

Let Q be a standard parabolic subgroup, with Levi decomposition Q = L_{Q}N_{Q}. For any smooth R[G]-module π we define the (non-normalized) Jacquet module r_{H,L_{Q}}π to be π_{N_{Q}}. Then r_{H,L_{Q}}π is a smooth L_{Q} module and is admissible if π is.

The parahoric subgroup Γ_{0} admits an Iwahori decomposition

(II.1.2)  Γ_{0} = (Γ_{0} ∩ \overline{N})(Γ_{L})(Γ_{0} ∩ N).

Note that Γ_{L} ≃ GL(1, O_{F}) × GL(n − 1, O_{F}). Let α_{0} be an R^{×}-valued character of GL(1, O_{F})/(1 + m_{F}). Then α_{0} can be extended to an R^{×}-valued character σ of Γ_{0} that is trivial on (Γ_{0} ∩ \overline{N})×GL(n − 1, O_{F})×(Γ_{0} ∩ N). Let χ ∈ X_{R}(T) have the property that χ_{i} is unramified for i > 1, and such that χ_{1} is trivial on 1 + m_{F}. Write I(χ) = i_{H,T} (χ). We are interested in Hom_{Γ_{0}}(σ, I(χ)).

Lemma II.1.3. Suppose R = \overline{ℚ}_{ℓ} or R = \overline{𝔽}_{ℓ}. Let χ ∈ X_{R}(T) be as above, and suppose χ_{1}|_{GL(1,O_{F})} = α_{0}. If χ_{1} is unramified then dim_{R} Hom_{Γ_{0}}(σ, I(χ)) = n; if χ_{1} is ramified then dim_{R} Hom_{Γ_{0}}(σ, I(χ)) = 1.

Proof. The calculation is standard but we include it for completeness. By restriction to Γ we obtain an isomorphism

I(χ) ⟶^{∼} {f : Γ→R | f(bk) = χ(b)f(k), ∀b ∈ B ∩ Γ, k ∈ Γ}.

Let

M(σ) = {f : Γ→R | f(bkβ) = χ(b)σ(β)f(k), ∀b ∈ B ∩ Γ, k ∈ Γ, β ∈ Γ_{0}}.

One checks easily that dim_{R} Hom_{Γ_{0}}(σ, I(χ)) = dim_{R} M(σ). In the unramified case σ is trivial and we find that dim_{R} M(σ) = #(B ∩ Γ\Γ/Γ_{0}) = n by the Bruhat decomposition for GL(n, k(F)). Indeed, let W_{n−1} denote the Weyl group of GL(n− 1), viewed as a subgroup of W. Let B(m_{F}) = r(B ∩ Γ) ⊂ GL(n, k(F)), P (m_{F}) = r(Γ_{0}) ⊂ GL(n, k(F)). Then the Bruhat decomposition yields

B ∩ Γ\Γ/Γ_{0} ≃ B(m_{F})\GL(n, k(F))/P (m_{F}) ≃ W/W_{n−1}.

Let W^{P} ⊂ W denote a set of coset representatives of W/W_{n−1}, so that

Γ = ∪_{w∈W^{P}} C(w); C(w) = (B ∩ Γ) · w · Γ_{0}.

Thus f ∈ M(σ) is determined by its restrictions f_{w} to each C(w). Let a be the diagonal matrix diag(a_{1}, …, a_{n}) ∈ B ∩ Γ, with a_{i} ∈ O_{F}^{×}. In the ramified case we have

χ_{1}(a_{1})f(w) = f(aw) = f(wa^{w}) = σ(a^{w})f(w) = f(w)

unless w ∈ W_{n−1}. Thus f_{w} ≡ 0 except on the identity coset and dim M(σ) = 1.

Let N_{0} ⊂ Lie(N(F)) be a lattice in the abelian Lie algebra of N(F). Let H_{L}^{−} ⊂ H_{L} denote the subspace of Hecke operators supported on the union of cosets Γ_{L}T^{−}Γ_{L} where

T^{−} = {t ∈ T | ad(t^{−1})(N_{0}) ⊂ N_{0}}.

The elements of T^{−} are said to be expanding.

More generally, we let

H_{L}(σ) = End_{L}ind_{L,Γ_{L}}σ;

here ind denotes compact induction. Here and in what follows, all endomorphisms are assumed to commute with R. We can identify H_{L}(σ) with the R-module of compactly-supported functions b : L(F)→R such that b(khk′) = σ(k)b(h)σ(k′) for h ∈ L(F), k, k′ ∈ Γ_{L}. Then the natural algebra structure on H_{L}(σ) inherited from its definition as endomorphism ring induces the convolution algebra structure on the space of functions. Similarly, we let

H_{0}(σ) = End_{H}ind_{H,Γ_{0}}σ;

this is isomorphic to the R-module of compactly-supported functions

b : H(F)→R; b(khk′) = σ(k)b(h)σ(k′), h ∈ H(F), k, k′ ∈ Γ_{0}.

Then we let H_{L}(σ)^{−} ⊂ H_{L}(σ) be the R-submodule of functions supported on Γ_{L}T^{−}Γ_{L}; this algebra is stable under product. (For all this, cf. [V1,I.8.6]).

For any smooth R[H] module π, the space Hom_{H}(ind_{H,Γ_{0}}σ, π) is naturally a right module over H_{0}(σ). By the universal property of compact induction,

Hom_{H}(ind_{H,Γ_{0}}σ, π) = Hom_{Γ_{0}}(σ, π),

and therefore the right-hand side is also a right H_{0}(σ)-module. In the same way, Hom_{Γ_{L}}(σ, π) is a right H_{L}(σ)-module when π is a smooth R[L]-module.

The proof of the first part of the following proposition was provided by M.-F. Vignéras, as were the references for the second part.

Proposition II.1.4. The notation is as in Lemma II.1.3.

(i) The natural projection I(χ)→r_{H,L}I(χ) induces an isomorphism of R-modules

(II.1.4.1)  p : Hom_{Γ_{0}}(σ, I(χ))→Hom_{Γ_{L}}(α_{0} ⊗ 1, r_{H,L}I(χ)).

(ii) The subset H_{L}^{−} ⊂ H_{L} is a subalgebra and there exists a homomorphism T_{H,L}^{−} : H_{L}(σ)^{−}→H_{0}(σ) such that

p(fT_{H,L}^{−}(b)) = p(f)b.

Here the right actions of the Hecke algebras are defined as above. The homomorphism is given on functions b ∈ H_{L}^{−} by

T_{H,L}^{−}(b)(khk′) = \{ \begin{matrix}0 if khk′ ∉ Γ_{0}T^{−}Γ_{0} \\ σ(k)b(h)σ(k′) if h ∈ T^{−}, k, k′ ∈ Γ_{0}. \end{matrix}

Proof. (i) We denote by p the natural map (II.1.4.1). Maps like p are known to be surjective in great generality, and in particular in the present case [V1,II.3.3]. Thus we have to prove injectivity. We first prove injectivity over \overline{ℚ}_{ℓ}. Note that the dimension of the left-hand side is n for χ_{1} unramified, 1 otherwise. On the other hand, write

(II.1.4.2)  L(F) = GL(1, F) × H_{n−1}, H_{n−1} = GL(n − 1, F),

and let T_{n−1} = T ∩H_{n−1}, the intersection taking place in L(F). With this notation, the semisimplification of r_{H,L}I(χ) as admissible H-module is the direct sum

(II.1.4.3)  r_{H,L}I(χ)_{ss} ⟶^{∼} ⊕_{i=1}^{n} χ_{i} ⊗ i_{H_{n−1},T_{n−1}}(χ^{i})_{ss},

where χ^{i} = (χ_{1}, …, χ̂_{i}, …, χ_{n}), the tensor product corresponding to the decomposition (II.1.4.2). In the unramified case each summand has a one-dimensional Γ_{L}-fixed subspace. Over \overline{ℚ}_{ℓ} the action of Γ_{L} is semisimple, hence the right hand side of (II.1.4.1) has dimension n, and p is an isomorphism for reasons of dimension. In the ramified case we have

dim Hom_{Γ_{L}}(α_{0} ⊗ 1, χ_{1} ⊗ i_{H_{n−1},T_{n−1}}(χ^{1})_{ss}) = 1;
dim Hom_{Γ_{L}}(α_{0} ⊗ 1, χ_{i} ⊗ i_{H_{n−1},T_{n−1}}(χ^{i})_{ss}) = 0, i ≠ 1.

This is because χ^{i} is ramified for i ≠ 1. Thus again we have equality of dimensions in (II.1.4.1). This completes the proof of (i) over \overline{ℚ}_{ℓ}.

We complete the proof for \overline{𝔽}_{ℓ} in the unramified case, the ramified case being similar. Note first that the result for \overline{ℚ}_{ℓ} implies the corresponding result over R = \overline{ℤ}_{ℓ}, the integral closure of ℤ_{ℓ} in \overline{ℚ}_{ℓ}. Indeed, let χ take values in \overline{ℤ}_{ℓ}^{×} and write I(χ)_{\overline{ℤ}_{ℓ}} for the corresponding induced representation, viewed as a free \overline{ℤ}_{ℓ}-module. Let I(χ)_{\overline{ℚ}_{ℓ}} denote the induced representation with \overline{ℚ}_{ℓ}-coefficients. It is known that I(χ)_{\overline{ℤ}_{ℓ}}^{Γ_{0}} is a lattice in I(χ)_{\overline{ℚ}_{ℓ}}^{Γ_{0}}, hence is free of rank n. It is also known that r_{H,L}I(χ)_{\overline{ℤ}_{ℓ}}^{Γ_{0}} is a lattice in r_{H,L}I(χ)_{\overline{ℚ}_{ℓ}}^{Γ_{0}} [V1, II.4.14(d)], hence is also free of rank n. Moreover,

the map p with coefficients in \overline{ℤ}_{ℓ} is still surjective (cf. [V1, II 3.3]), hence is an isomorphism by considerations of rank.

Finally, we suppose R = \overline{𝔽}_{ℓ}. Reduction modulo ℓ defines an injection

(II.1.4.4)  r_{H,L}I(χ)_{\overline{ℤ}_{ℓ}}^{Γ_{0}} ⊗ \overline{𝔽}_{ℓ} → r_{H,L}I(χ)_{\overline{𝔽}_{ℓ}}^{Γ_{0}}.

But we have seen that r_{H,L}I(χ)_{\overline{ℤ}_{ℓ}}^{Γ_{0}} is free of rank n. On the other hand, the righthand side of (II.1.4.4) is of dimension ≤ n. Indeed, the analogue of (II.1.4.3) remains true over \overline{𝔽}_{ℓ}, and the Γ_{L}-fixed subspace of the right-hand side of (II.1.4.3) is always of dimension n. Thus dim r_{H,L}I(χ)_{\overline{𝔽}_{ℓ}}^{Γ_{0}} = n. But the dimension over \overline{𝔽}_{ℓ} of the left-hand side of (II.1.4.1) still equals n, and we have already seen that p is surjective. This completes the proof of (i).

Assertion (ii) is a special case of [V2,Theorem II.4] and [V2,Lemma II.10.1].

For i = 1, …, n, we write t(ϖ)_{i} for the diagonal matrix with entry ϖ^{−1} in the ith place and 1 elsewhere, and let U_{L} ∈ H_{L} denote the function equal to 1 on Γ_{L}t(ϖ)_{1}Γ_{L} and 0 elsewhere. Then t(ϖ)_{1} belong to T^{−}, hence U_{L} ∈ H_{L}^{−}. We let U = T_{H,L}^{−}(U_{L}) ∈ H_{?}(σ), for ? = 0, 1.

For the remainder of this section we assume q ≡ 1 (mod ℓ). Then the characters ν and δ_{i}, i = 1, …, n are trivial mod ℓ. We restrict attention to the case of unramified χ for the time being, so σ is the trivial representation. We also work over R = \overline{𝔽}_{ℓ}. Let χ and I(χ) be as in the statement of Proposition II.1.4. The operator U_{L} has support in the center of L(F), and acts on each constituent of the righthand side of II.1.4.3 as the scalar χ_{i}(ϖ)^{−1}. It follows from (ii) of Proposition II.1.4 that U, viewed as a linear operator on the n-dimensional space Hom_{Γ_{0}}(σ, I(χ)) has the n generalized eigenvalues χ_{i}(ϖ)^{−1} (counted with multiplicity). Let β_{1}, …, β_{r} denote the r distinct χ_{i}, m_{r} the multiplicity of β_{r} in χ, and assume the χ_{i} are ordered so that

(II.1.5) χ_{1} = ⋯ = χ_{m_{1}} = β_{1}; …, χ_{m_{1}+···+m_{j−1}} = ⋯ = χ_{m_{1}+···+m_{j}} = β_{j}; …

For any R-valued character β of F^{×}, and any positive integer m, let β[m] be the 1-dimensional representation β ∘ det of GL(m, F). Following the notation of [V2], we write

χ_{1}×χ_{2}× ⋯ ×χ_{n} = i_{H,T} χ.

More generally, we let n = m_{1} + ⋯ + m_{r} be a partition, P = P (m_{1}, …, m_{r}) the corresponding standard parabolic subgroup of H, and let

β_{1}[m_{1}]× … ×β_{r}[m_{r}]

denote the full induced representation (when q ≡ 1 (mod ℓ) normalized and nonnormalized induction coincide) from the representation β_{1}[m_{1}] ⊗ ⋯ ⊗ β_{r}[m_{r}] of the standard Levi factor GL(m_{1}, F)× … GL(m_{r}, F) of P.

A representation of H is spherical if it contains a non-zero fixed vector under Γ. Obviously β_{1}[m_{1}]× … ×β_{r}[m_{r}] is a spherical representation. Conversely, Proposition II.1.6 [V2]. Assume R = \overline{𝔽}_{ℓ} and q ≡ 1 (mod ℓ). Assume β_{i} ≠ β_{j} for i ≠ j. Then

(i) β_{1}[m_{1}]× … ×β_{r}[m_{r}] is irreducible;

(ii) Every irreducible spherical R-representation of H is obtained in this way. (iii) dim_{R} Hom_{Γ_{L}}(1, r_{H,L}β_{1}[m_{1}]× … ×β_{r}[m_{r}]) = r, and the operator U_{L} has the

r distinct eigenvalues β_{i}(ϖ)^{−1}, i = 1, …, r.

(iv) The natural projection β_{1}[m_{1}]× … ×β_{r}[m_{r}]→r_{H,L} β_{1}[m_{1}]× … ×β_{r}[m_{r}] induces an isomorphism of R-modules

Hom_{Γ_{0}}(1, β_{1}[m_{1}]× … ×β_{r}[m_{r}]) ⟶^{∼} Hom_{Γ_{L}}(1, r_{H,L}β_{1}[m_{1}]× … ×β_{r}[m_{r}])

and the operator U on the left-hand side has the r-distinct eigenvalues β_{i}(ϖ)^{−1}, i = 1, …, r.

Proof. Assertions (i) and (ii) are proved in VI.3 of [V2]. We prove (iii) and (iv) together. First, we can lift β_{i} to a character β̃_{i} with values in \overline{ℤ}_{ℓ}^{×} and define the representations β̃_{i}[m_{i}] and β̃_{1}[m_{1}]× … ×β̃_{r}[m_{r}] over \overline{ℚ}_{ℓ}, by the same procedure as above. It is well known that, for appropriate characteristic zero lifts χ̃_{j} of the characters χ_{j} of (II.1.5), there is an embedding

f : β̃_{1}[m_{1}]× … ×β̃_{r}[m_{r}] ↪ χ̃_{1}× … ×χ̃_{n}.

Write π̃ = β̃_{1}[m_{1}]× … ×β̃_{r}[m_{r}], π̃′ = χ̃_{1}× … ×χ̃_{n}, and let π̃′′ denote π̃′/f(π̃). There is a commutative diagram (II.1.6.1)

0 ⟶ π̃^{Γ_{0}}  ⟶^{f} (π̃′)^{Γ_{0}} ⟶ (π̃′′)^{Γ_{0}} ⟶ 0
g↓  ^{g′}↓  ^{g′′}↓
0 ⟶ r_{H,L}(π̃)^{Γ_{L}} ⟶ r_{H,L}(π̃′)^{Γ_{L}} ⟶ r_{H,L}(π̃′′)^{Γ_{L}} ⟶ 0

Exactness of the bottom row follows from exactness of the Jacquet functors. Now we have already seen in Proposition II.1.4 that g′ is an isomorphism. On the other hand, all the vertical arrows are surjective [V1,II.3.3]; hence they are isomorphisms. This proves the analogue of the first part of (iv) over \overline{ℚ}_{ℓ}. The proof over \overline{𝔽}_{ℓ} then proceeds as in the proof of Proposition II.1.4.

In what follows, we let π = β_{1}[m_{1}]× … ×β_{r}[m_{r}]. It follows from what we have shown thus far that the dimension in (iii) is equal to

dim_{R}Hom_{Γ_{0}}(1, π).

Using the fact that H = Γ·P, it follows by standard arguments that this dimension equals the cardinality of Γ ∩ P\Γ/Γ_{0}. The Bruhat decomposition then shows that this is equal to

(II.1.6.2)  |𝔖_{m_{1}}× ⋯ 𝔖_{m_{r}}\𝔖_{n}/𝔖_{1}×𝔖_{n−1}|,

where 𝔖_{m} is the symmetric group on m letters. It is elementary that (II.1.6.2) equals r.

It remains to prove the assertions about the eigenvalues of U_{L} and U. Indeed, Proposition II.1.4 shows that the assertions for U and for U_{L} are equivalent. Following a suggestion of Vignéras, we first replace Γ_{0} by the standard Iwahori subgroup I_{0}; this is the subgroup of Γ of matrices whose reductions (mod m_{F}) are upper triangular. There is a commutative diagram analogous to (II.1.6.1) when Γ_{0} is replaced by I_{0} and L is replaced by the diagonal torus T; thus Γ_{T} is the maximal compact subgroup of T. In this case, the above arguments (cf. [V2, Lemma VI.2 (b)]) show that

(II.1.6.3)  π^{I_{0}} = |𝔖_{m_{1}}× ⋯ 𝔖_{m_{r}}\𝔖_{n}| = \frac{n!}{m_{1}! ⋯ m_{r}!}.

Let H_{R}(n, 1) denote the Iwahori-Hecke algebra for GL(n) over R = \overline{𝔽}_{ℓ}, with the parameter q = 1. Then π^{I_{0}} is an H_{R}(n, 1)-module. Let {X_{i}|i = 1, …, n}, {S_{j}|j = 1, …, n − 1} be the generators of H_{R}(n, 1) defined in [V1,I.2.14]. Here X_{i} is the product T_{i} · (T_{i−1})^{−1}, where T_{j} is the I_{0}-double coset of the diagonal matrix ∏_{i≤j} t(ϖ)_{i}, with t(ϖ)_{i} as above, multiplied by a power of q which under our hypotheses equals 1. Similarly S_{j} is the I_{0}-double coset of the standard transposition s_{j} in the Weyl group of GL(n) which exchanges j and j + 1. The S_{j} generate the Hecke algebra H_{R}^{0}(n, 1) of GL(n, k(w)) relative to its Borel subgroup. Since q = 1 in R, this is just the R-group algebra R[𝔖_{n}]. The (X_{i})^{±} generate a commutative subalgebra A ⊂ H_{R}(n, 1), normalized by H_{R}^{0}(n, 1), and we let π_{ss}^{I_{0}} denote the semisimplification of π^{I_{0}} as A-module. Then π_{ss}^{I_{0}} is a sum of characters of A, the weights of π, which we identify with unramified characters of the diagonal torus T.

By construction, the character χ occurs non-trivially in π_{ss}^{I_{0}} as the character on the standard function in the induced representation β_{1}[m_{1}]× … ×β_{r}[m_{r}] which equals 1 on Γ. Now the Weyl group 𝔖_{n} acts on A by permuting the X_{i}, and it follows from Corollary 2.12 of [V3] that, if s ∈ 𝔖_{n}, then s(χ) is also a weight of π. More precisely, the reference cited only applies explicitly when s is a simple transposition, but we may induct on the number of simple transpositions occuring in a minimal factorization of an arbitrary s. The set of distinct s(χ), as s varies through 𝔖_{n}, has cardinality precisely equal to \frac{n!}{m_{1}!····m_{r}!}, which is just the dimension of π^{I_{0}}. In particular, each character of A in the set {s(χ)} occurs with multiplicity one, and the representation π^{I_{0}} of A is already semi-simple.

Note that the set of {s(χ)} is just the set of n-tuples of characters in which β_{i} occurs m_{i} times, with arbitrary order. On the other hand, let H_{T} denote the Hecke algebra of T relative to its maximal compact subgroup. Then there is a homomorphism t_{H,T}^{−} : H_{T} →H_{R}(n, 1) satisfying the analogue of Proposition II.1.4 (ii) (cf. [V2, Theorem II.6]); in particular, we have

(II.1.6.4)  t_{H,T}^{−} (t(ϖ)_{1}) = X_{1}.

It follows that, for each i = 1, …, n, there is at least one s ∈ 𝔖_{n} for which s(χ)(X_{1}) = β_{i}(ϖ).

Now π^{Γ_{0}} ⊂ π^{I_{0}} is the subspace of invariants under the subgroup generated by the transpositions s_{j}, j = 2, …, n − 1, hence of the double cosets S_{j}, j = 2, …, n − 1 (since q = 1 in R). On the other hand, the operator U is the Γ_{0} double coset containing X_{1}; it is of the form ∑ S · X_{1} · S′, where S and S′ run through finite sets of finite products of elements of {S_{j}|j > 1}. But all such S commute with X_{1}. The set of eigenvalues of U on π^{Γ_{0}} thus equals the set of eigenvalues of X_{1} on π^{Γ_{0}}, hence is contained in the set of eigenvalues of X_{1} on π^{I_{0}}. It follows from the above description in terms of the {s(χ)} that the latter set is precisely the set indicated in (iv).

Note that if we assume ℓ > n − 1 we can conclude immediately. Indeed, by projecting π^{I_{0}} on its 𝔖_{n−1}-invariant subspace π^{Γ_{0}}, where 𝔖_{n−1} is generated by the s_{j} with j > 1, we see that X_{1} already has r distinct eigenvalues on the latter space. But the above argument shows that every eigenvalue of X_{1} on π^{Γ_{0}} is also an eigenvalue of U.

In general, we will show in the course of the proof of the following lemma – the reader can check that this involves no circularity – that

(II.1.6.5)  U has at least r eigenvalues on π^{Γ_{0}}.

This suffices to complete the proof.

The above proof actually provides more information about the spherical vector.

Lemma II.1.7. We retain the hypotheses of the previous proposition.

(i) Let π = β_{1}[m_{1}]× … ×β_{r}[m_{r}] and let v_{0} ∈ π^{Γ} be a non-zero spherical vector. Then v_{0} generates the Iwahori-fixed subspace π^{I_{0}} as a module over the commutative algebra A.

(ii) The spherical vector v_{0} generates the Γ_{0}-fixed subspace π^{Γ_{0}} over the algebra R[U] of polynomials in the operator U.

Proof. It follows from the irreducibility of π that v_{0} generates π^{I_{0}} as module over H_{R}(n, 1). We may characterize R·v_{0} ⊂ π^{I_{0}} as the fixed subspace under H_{R}^{0}(n, 1) = R[𝔖_{n}]. But we have seen in the proof of Proposition II.1.6 that 𝔖_{n} acts transitively on the weights of π. It then follows formally that every weight of π occurs in the A-module generated by v_{0}. Since the weights have multiplicity one, this implies (i).

Now let V_{1} ⊂ π^{I_{0}} denote the R[X_{1}]-module generated by v_{0}. Since the remaining X_{i}’s commute with X_{1}, the above argument shows that the r distinct eigenvalues β_{i}(X_{1}), i = 1, …, r, all occur in V_{1}. Now there is a commutative diagram

(II.1.7.1)  \begin{matrix}π^{Γ_{0}} & ⟶^{i} & π^{I_{0}} \\ ^{p_{G,L}}↓ & ^{p_{G,T}} ↓ \\ r_{H,L}π^{Γ_{L}} & ⟶^{p_{L,T}} & r_{H,T} π. \end{matrix}

Here i is the inclusion, p_{G,L} is the isomorphism of Proposition II.1.4, and the other two maps are constructed analogously via the corresponding Jacquet functors. Applying the analogue of Proposition II.1.4 (ii) for the Levi subgroup T of L (cf. [V2, loc. cit.]), and recalling (II.1.6.4), we find that

(II.1.7.2)  \begin{aligned}p_{L,T} ∘ p_{G,L}(v_{0}U) & = p_{L,T} ∘ p_{G,L}(v_{0}t(ϖ)_{1}) \\ & = p_{G,T} (i(v_{0})X_{1}). \end{aligned}

We now see that U has r distinct eigenvalues on the subspace R[U]·v_{0} ⊂ π^{I_{0}}, which implies II.1.6.5 and completes the proof of Proposition II.1.6. Since U fixes π^{Γ_{0}}, this also completes the proof of the lemma.

We now replace \overline{𝔽}_{ℓ} by the finite field k of characteristic ℓ, and let R be a finite local ℤ_{ℓ}-algebra with maximal ideal m_{R} and residue field k; let x ↦ \overline{x} : R→k denote reduction mod m_{R}. Let K_{R} = R ⊗ ℚ_{ℓ}; we assume K_{R} to be a finite direct sum ⊕_{i}K^{i} of ℓ-adic fields with the same residue field, although there is probably no reason to exclude nilpotents. Let \overline{m}_{j} denote the multiplicity of the character \overline{β}_{j} := \overline{•} ∘ β_{j}. Suppose \overline{m}_{1} = 1, so that χ_{1} = β_{1} is distinct from the other characters mod m_{R}, and let λ = χ_{1}(ϖ) ∈ R^{×}. Let π be an admissible R[H]-module generated by a spherical vector, say v, corresponding to the same character of H as I(χ). We assume π ⊗ ℚ_{ℓ} is a direct sum ⊕π^{i} where π^{i} is an irreducible spherical K^{i}[H]- module corresponding to the K^{i,×}-valued character χ^{i}, say. Thus each χ^{i} takes values in the integers of K^{i} and reduces modulo the maximal ideal to \overline{χ}.

It follows from Proposition II.1.6 that there is a surjective morphism

(II.1.8)  ϕ : \overline{π}→\overline{β}_{1}[m_{1}]× … ×\overline{β}_{r}[m_{r}]

of k[H]-modules, the right-hand side being the irreducible quotient of \overline{π}, generated by the image of v. Let \overline{v_{0}} generate the 1-dimensional U-eigenspace of [\overline{β}_{1}[m_{1}]× … ×\overline{β}_{r}[m_{r}]^{Γ_{0}} with eigenvalue \overline{λ}. It follows from Lemma II.1.7 that \overline{v_{0}} ∈ ϕ(k[U]\overline{v}). Then, letting

(II.1.9)  X = ∏_{i>1} \frac{U − β_{i}(ϖ)^{−1}}{λ − β_{i}(ϖ)^{−1}} ,

it is clear that

(II.1.10)  X(v) ≡ \overline{v_{0}} (mod m_{R} + ker(ϕ)).

In particular, X(v) defines a lifting of the \overline{λ}-eigenspace of U to \overline{π}.

Let V_{0} ⊂ π denote the R-submodule on which U acts with eigenvalue λ; i.e., V_{0} is the intersection of π with the corresponding U-eigenspaces in π^{i}.

Lemma II.1.11. . Define π, χ, and β_{i} as above, and assume χ_{1} has multiplicity one mod m_{R}.

(i) The R-submodules π^{Γ} and V_{0} of π^{Γ_{0}} are free of rank 1.

(ii) The limit X_{∞} = lim_{m→∞}X^{ℓ^{m}} exists as a continuous operator on π^{G_{0}} and defines a projection onto v_{0}. The restriction of X_{∞} to the spherical subspace π^{G} defines an isomorphism

X_{∞} : π^{G}→v_{0}

of R-modules.

Proof. Part (i) follows from Nakayama’s lemma and the corresponding facts for induced representations of H over k. Part (ii) follows from (II.1.10) by successive approximation modulo increasing powers of m_{R}.

II.2. Notation for Hecke algebras.

Let CS(ℚ) denote the set of all primes of ℚ, and let CS^{+}(ℚ) be the subset of finite primes that split in K_{0} and are unramified in K. For v ∈ CS^{+}(ℚ) of residue characteristic p we choose a place v_{1} of K_{0} above v, and let Σ = Σ(v_{1}) be the set of primes of K dividing v_{1}. Then we have

(II.2.1)  G_{v} ≅ ∏_{w∈Σ} GL(n, K_{w})×ℚ_{v}^{×}.

The Hecke algebra T_{v} of G_{v} relative to any maximal compact subgroup (conjugate to ∏_{w∈Σ} GL(n, O_{K_{w}})×ℤ_{v}^{×}) is isomorphic to a polynomial algebra over ℤ[\frac{1}{p}] in the variables

{T_{i,w}, i = 1, …, n, T_{n,w}^{−1}, w ∈ Σ; T_{0,v}, T_{0,v}^{−1}}.

Here ℤ[T_{0,v}, T_{0,v}^{−1}] is the Hecke algebra of the factor ℚ_{v}^{×} in II.2.1; by abuse of language we refer to the T_{i,w} as the Hecke operators at w, or at the prime of E below w. The Hecke operators at w are normalized so that

(II.2.1)  P_{w}(q^{−s}) = 1 + ∑_{i=1}^{n}(−1)^{i}T_{i,w}q^{−is}

is the local Euler factor at w of the motivically normalized standard L-function of GL(n). Here q is the order of the residue field k(w) and the inverse roots of P_{w}(X) are the Satake parameters, multiplied by q^{(n−1)/2}. Up to canonical isomorphism the algebra T_{v} does not depend on the choice of v_{1} above v.

The global Hecke algebra T is the tensor product over v ∈ CS^{+}(ℚ) of the T_{v}. If S is a finite subset of CS^{+}(ℚ), we let T^{S} ⊂ T be the subalgebra generated by the T_{v} for v ∉ S.

Suppose q ∈ CS^{+}(ℚ), v_{1} a divisor of q in K_{0} and let Σ(v_{1}) be as above. For 𝔮 ∈ Σ(v_{1}), we consider the parabolic subgroup P = LN of type (1, n − 1) in H = GL(n, K_{w}), as in §II.1, and let Γ_{0} be the corresponding parahoric subgroup. Let T′_{i,𝔮}, 1 ≤ i ≤ n−1, denote the Hecke operator as above for the factor GL(n−1, K_{𝔮}) of L, and define

(II.2.2)  V_{i,𝔮} = T_{H,L}^{−}(T_{n−1,𝔮}^{′,−1} · T′_{n−i,𝔮}) ∈ H_{?}, 1 ≤ i ≤ n − 1, ? = 0, 1.

Here we have realized T_{n−1,𝔮}^{′,−1} · T′_{n−i,𝔮} as an element of H_{L}^{−} in the obvious way. II.2.3. Let ϖ ∈ K_{w} be a uniformizer. Suppose R = \overline{ℚ}_{ℓ} and χ ∈ X_{R}(T) are as in §II.1, and let I(χ) be the induced representation with coefficients in R. Let a ∈ r_{H,L}I(χ) be a non-zero eigenvector in the χ_{1}(ϖ^{−1})-eigenspace for the operator U. Assume a is a spherical vector for Γ_{L}; then a is unique up to scalar multiples. Moreover, a is an eigenvector for the Hecke operators T′_{i,𝔮} defined above. We denote by b^{i}(χ; χ_{1}) the eigenvalue of T_{n−1,𝔮}^{′,−1} · T′_{n−i,𝔮}, acting on a.

II.2.4: The U-operator. Choose 𝔮 ∈ Q, and define Γ_{1,𝔮} ⊂ GL(n, ℤ_{q}) as in §I.2. We consider the double coset U_{𝔮} = Γ_{1,𝔮} · t(ϖ)_{1}Γ_{1,𝔮} ⊂ GL(n, ℚ_{q}). Here and in what follows we are identifying GL(n, ℚ_{q}) = U(D^{#})_{𝔮} with D_{𝔮^{(2)}}^{#,×}. The identification with D_{𝔮^{(1)}}^{#,×} identifies U_{𝔮} with the double coset Γ′_{1,𝔮} · (t(ϖ)_{1})^{−1}Γ′_{1,𝔮}, where Γ′_{1,𝔮} consists

of invertible matrices over ℤ_{q} congruent to ( \begin{matrix}1 & 0 \\ ∗ & ∗_{n−1} \end{matrix} ) (mod q).

One verifies immediately that

(II.2.4.1)  U_{𝔮} = ∐ Γ_{1,𝔮} · t(ϖ)_{1} · ( \begin{matrix}1 & b \\ 0 & I_{n−1} \end{matrix} ) ,

where b = (b_{1}, …, b_{n−1}) runs through (ℤ_{q}/qℤ_{q})^{n−1}. Write W = ℚ_{q}^{n}. Let Λ = ℤ_{q}^{n} ⊂ W be the standard lattice, and let λ denote the row vector (q^{−1}, 1, …, 1), so that

Γ_{1,𝔮} = {g ∈ GL(W) | g(Λ) = Λ, g(λ) = λ (mod ()Λ)}.

Let Λ′ be the lattice generated by Λ and λ. Then the description (III.2.2) shows that, in the standard right action on lattices, we have

(II.2.4.2)  (Λ, λ) · U_{𝔮} = {(Λ′, λ′) |qλ′ = λ (mod Λ))}.

III. Unitary automorphic forms and λ-adic representations

III.1. Review of λ-adic representations.

We return to the situation of §I. Let SC, Q, 𝔯, K_{1,Q}, V, and σ be as in I.2. We assume throughout that ℓ is unramified in K and that K_{1,Q} contains a hyperspecial maximal compact subgroup of J(ℚ_{ℓ}). The primes in Q are denoted w rather than 𝔮, and the primes above them in K are denoted w_{1} and w_{2}. The identification Γ_{w} ≃ GL(n, ℚ_{q}) for w ∈ Q is made via w_{2}, so that K_{0,Q} is upper triangular parahoric. Let π ∈ Coh(J, V), in the notation of I.3, and assume π satisfies the hypothesis of Proposition I.2.10:

(III.1.1)  Hom_{K_{1,Q}}(σ, π) ≠ 0.

At primes that ramify in K/ℚ we assume π_{v} to be spherical with respect to a very special compact open subgroup K_{v}, cf. I.3.4.

We assume π satisfies either condition (i) or (ii) of Theorem I.3.3. Then π admits an automorphic base change. Let Π be the automorphic representation of GL(n)_{K} denoted BC(π)_{1} in I.3. It follows as in [CL], *** that Π satisfies the following three hypotheses:

(i) Π_{∞} is a cohomological representation (ii) Π ∘ c ⟶^{∼} Π^{∨} (Π is θ stable, where θ is the involution of J′_{E} corresponding

to the group J′); (iii) For some finite place v of K, Π_{v} is supercuspidal Let E(π) be a number field which is simultaneously a field of definition for π_{f}

and V, and let λ be a prime of E(π) dividing ℓ. Let O = O_{λ} be the ring of integers of E_{λ}, 𝔪_{λ} its maximal ideal, and k = k(λ) its residue field. Under these hypotheses, it is shown in [HT1], following Clozel and Kottwitz, how to associate a continuous representation^{1}

r_{ρ}(π) : G_{K}→GL(n, O).

where henceforth we write G_{K} for Gal(\overline{K}/K). In our applications the residual representation over k will be irreducible, so the lattice will be unique up to scalar multiplication.

Next, we let G_{E} denote Gal(\overline{E}/E), and define

ρ(π) = Ind_{K/E}r_{ρ}(π)

to be the induced 2n-dimensional representation of G_{E}, acting on the O-module Ind_{K/E}A_{λ}[π_{f}]. Finally, let

\overline{ρ} = \overline{ρ}(π) : Gal(\overline{E}/E) → Aut(Ind_{K/E}A_{λ}[π_{f}] ⊗_{O} k)

denote the residual representation, and let r\overline{ρ} = r\overline{ρ}(π) denote the restriction of \overline{ρ} to G_{K}.

1More precisely, [HT1] defines a Galois representation over some finite extension of E(π). Since the Galois group is compact, it necessarily stabilizes an integer lattice. For the purposes of this paper the descent to E(π) is not strictly necessary, however COMPLETE???

Proposition III.1.3. Suppose SC is non-empty and contains a place w such that π_{w} is either supercuspidal (if J′_{w} ≅ GL(n, E_{w})) or corresponds under the Jacquet-Langlands correspondence to a supercuspidal representation of GL(n, E_{w}) (if J′_{w} is isomorphic to the multiplicative group of a division algebra). Then r_{ρ}(π) is absolutely irreducible. Moreover, suppose ℓ ∤ |GL(n, k(w))| (i.e., ℓ is banal for GL(n, E_{w}), in the sense of Vignéras [V1]). Then r\overline{ρ}(π) is absolutely irreducible.

The proof will be given at the end of this section.

For any prime v of ℚ that splits in K_{0} and is unramified in K, and such that π_{v} is unramified we let ϕ_{v,π} : T_{v} → E(π) denote the character by which the local unramified Hecke algebra acts on π_{v}^{K_{v}}. Let S(π) be the set of all such unramified primes, let T^{S} be the corresponding global Hecke algebra, as in §II.2, and let ϕ_{π} : T^{S} → E(π) be the corresponding character; ϕ_{π} gives the natural action of T^{S(π)} on π^{K}. If w is a prime of E dividing some prime in S(π), we let

(III.1.4)  ϕ_{π}(P_{w})(X) = 1 + ∑_{i=1}^{n}(−1)^{i}ϕ(π)(T_{i,w})X^{i}

in the notation of (II.2.1).

Theorem III.1.5. Suppose π contains a fixed vector for a hyperspecial maximal compact subgroup of J_{q(𝔯)} (i.e., not only a fixed vector for K_{q(𝔯)}, as implied by (III.1.1). Let S^{bad} be the set of primes of K dividing primes in Q ∪ SC or of residue characteristic ℓ. Then the representation r_{ρ}(π) is unramified outside S^{bad}. Moreover, for all but finitely many primes w of E dividing rational primes in S(π), there is a prime w_{1} of K dividing w and such that the arithmetic Frobenius F rob_{w_{1}} satisfies

ϕ_{π}(P_{w})(F rob_{w_{1}}) = 0.

Remark. The choice of w_{1} in the above theorem is determined as in II.2 by the choice of identification of J′_{w} with a general linear group. The base change of π to K is conjugate self-dual, and one verifies that, if w_{2} is the other prime dividing w, then ϕ_{π}(P_{w})(q^{n−1}F rob_{w_{2}}^{−1}) = 0. Bear in mind also that the natural action on the cohomology of the Shimura variety is that of the Galois group of \overline{ℚ} over the reflex field, and that there is an implicit identification of the reflex field with K.

An alternative way of phrasing this theorem is in terms of partial L-functions: there is a finite set S of finite primes such that we have the equality of Euler products

(III.1.5.1)  L^{S}(s, r_{ρ}(π)) = L^{S}(s − \frac{n − 1}{2}, BC_{K}(π)).

Here the right-hand side is the standard L-function with the unitary (Langlands) normalization; the superscript ^{S} indicates that factors at S have been removed. As mentioned above, we have normalized r_{ρ} to make (III.1.5.1) true; cf. the discussion on p. 100 ff. of [H1].

Proof. The proof of this theorem is mainly due to Kottwitz, and is contained in [K2]. Specifically, Kottwitz proves there that r_{ρ}(π) is unramified outside S^{bad}, except possibly at primes of K ramified over ℚ, and shows that the arithmetic Frobenius satisfies the Hecke polynomial for all but finitely many unramified places. The proof is completed in Theorem VII.1.9 of [HT1]; by (2) of that theorem, it suffices to verify that Π_{v} is unramified for all v ∉ Q∪SC, and this follows from Proposition I.3.4.

For each archimedean place τ of E, let

\underline{a}(τ) = (a_{1}(τ) ≥ ⋯ ≥ a_{n}(τ))

be the highest weight of the τ-component V_{τ} of the representation V of J′(ℝ), with respect to some maximal torus; let \underline{a} = (\underline{a}(τ))_{τ∈Σ_{E}}. For each τ, let

(III.1.6)  n(τ, V) = inf{n_{1} + n_{2} | V_{τ} ⊂ St^{⊗n_{1}} ⊗ St^{⊗n_{2},∗}}

where St denotes the standard representation of GL(n) and inclusion is as a direct summand. Let n(V) = n − 1 + ∑_{τ∈Σ_{E}} n(τ, V).

Proposition III.1.7. Under the hypotheses of Theorem III.1.5, the representation ρ(π) is crystalline at all primes of K dividing ℓ. Suppose moreover that ℓ > n(V)+1. Then the representation on A_{λ}[π_{f}] is crystalline in the sense of belonging to the category Rep_{O,cris,[0,ℓ−1[} defined in IV.4, below.

Proof. Let w be a prime of K dividing ℓ, O_{w} the w-adic completion of O_{K}. By ℓ-adic Hodge theory [FM,F] it suffices to show that A_{λ}[π_{f}] occurs as a Grothendieck submotive of the cohomology of a smooth scheme A_{V} of dimension n(V) over Spec O_{w}, where the projectors are given by correspondences with ℓ-integral coefficients. If V is the trivial representation then this is clear. The general case requires a standard argument to relate M_{λ}[π_{f}] to the cohomology of a Kuga fiber variety. We omit this argument, since only the case of trivial V will be considered in the remainder of this paper.

Theorem III.1.8. Let w ∈ Q of residue characteristic q. Let w_{1} be the prime of K above w and let Z_{w} ⊂ G_{E} denote a decomposition group. Suppose π_{w} is in case (b) of Proposition I.2.10. Let (α, β_{1}, …, β_{n−1}) be the corresponding n-tuple of characters. Then r_{ρ}(π)|_{Z_{w}} breaks up as a direct sum

r_{ρ}(π)|_{Z_{w}} ⟶^{∼} A ⊕ B.

Here B is an unramified representation and the inertia subgroup I_{w} of Z_{w} acts on A via the restriction to I_{w} of the character associated to α via local class field theory.

This is a special case of Theorem VII.1.7 of [HT1]; an earlier proof has been published in [HT2]. Proof of Proposition III.1.3 Now let w ∈ SC be the place mentioned in the statement of the proposition. Let σ(π_{w}) denote the ℓ-adic representation of the decomposition group Z_{w} associated to π_{w} by the local Langlands correspondence [HT1,He]. Proposition III.1.3 is a consequence of the following two theorems and the multiplicity one theorem (Theorem I.3.7).

Theorem III.1.9 [H1,§3; HT1]. Under the hypotheses of this section, let w ∈ SC. Then r_{ρ}(π)|_{Z_{w}} ≅ σ(π_{w}).

Theorem III.1.10. Let F be a p-adic field, p ≠ ℓ, and let τ be a supercuspidal representation of GL(n, F) with coefficients in the ℓ-adic field L, with continuous central character ω_{τ}.

(a)([BHK], Theorem 3.3) Let σ(τ) be the ℓ-adic representation of W_{F} associated to τ by the correspondence of [H1]. Then σ(τ) is irreducible.

(b) ([V3], 1.20) Moreover, suppose ω_{τ} is ℓ-adically integral, and suppose ℓ is banal for GL(n, F). Then the image of σ(τ) fixes a lattice A over the integer ring O_{L} ⊂ L, and the residual representation \overline{σ}(τ) on A ⊗ k(L) is irreducible; here k(L) is the residue field of O_{L}.

Proof. The article [BHK], where part (a) is stated as Theorem 3.3, only treats the case of characteristic zero coefficients. The argument in 1.20 of [V3] shows that the semisimplified reduction (mod ℓ) decomposes as the sum of m irreducible representations of dimension e = \frac{n}{m}. Here if m > 1 then m is a divisor of n of the form ℓ^{y} · ϵ(|k(E)|^{d}), where y is a non-negative integer, necessarily equal to 0 since ℓ > n (ℓ is banal). Next, d is a divisor of e, E is a tamely ramified extension of F, with residue field k(E), such that [E : F] = \frac{e}{d}, and ϵ(•) is the order of the integer • in 𝔽_{ℓ}^{×}. Thus |k(F)|^{e} = |k(E)|^{d}, hence m = ϵ(|k(E)|^{d}) = ϵ(|k(F)|^{\frac{n}{m}}). The hypothesis that ℓ is banal implies that ϵ(|k(F)|) > n, thus that ϵ(|k(F)|^{\frac{n}{m}} > m, which is a contradiction unless m = 1. In other words, \overline{σ}(τ) is irreducible.

III.2. Representations over the Hecke algebra

We now introduce the ℓ-adic Hecke algebras of operators on spaces of automorphic forms; these will be indicated by 𝕋’s (as opposed to T’s). Fix a π ∈ Coh(J, V) as above, and let S be a finite set of rational primes including ℓ and all primes bad for π. We will allow S to grow as necessary; in particular, when we choose a finite set Q of auxiliary primes we will assume that S contains Q. Let T^{S} be the tensor product over all T_{v} as in §II.2, where v ∉ S runs through rational primes that split in K_{0}. We define σ and Λ_{W_{σ}} for π as in §I.2.8. Then T^{S} acts on the modules A_{ℓ}(J, Λ_{W_{σ}}, K) of ℓ-adic automorphic forms, introduced in §I.2, when K = K_{0,Q} or K_{1,Q}. We let 𝕋_{K,σ} denote the complete ℤ_{ℓ}-subalgebra of End(A_{ℓ}(J, Λ_{W_{σ}}, K)) generated by T^{S} and by the operators U_{𝔮} and V_{i,𝔮}, defined as in §§III.2 and II.2, respectively, for 𝔮 ∈ Q, i = 1, …, n − 1.

Define K_{1,Q}^{[ℓ]} ⊂ K_{0,Q} as in §I.2. When K = K_{0,Q}, (resp. K_{1,Q}^{[ℓ]}) we write 𝕋_{0}(Q) (resp. 𝕋_{1}(Q)) for 𝕋_{K,σ}. It follows from the characteristic zero theory that the algebras 𝕋_{i}(Q) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} are semisimple, i = 0, 1. The choice of a component 𝔪 of 𝕋_{i}(Q) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} determines the local Galois representation r_{ρ}(π) locally at all but finitely many primes that split in K/E, for any π that corresponding to 𝔪. The set of such primes has Dirichlet density one, hence determines r_{ρ}(π), and therefore the partial L-function L^{S}(s − \frac{n−1}{2}, BC_{K}(π)) via formula III.1.5.1, for some finite set S of bad primes.

We now fix i = 0 or 1 and let 𝕋 denote the complete subalgebra of 𝕋_{i}(Q) for i = 0, 1 generated only by the unramified Hecke operators in T^{S}. Let 𝔪 denote a maximal ideal of 𝕋_{i}(Q), and denote by L_{𝔪} the localization of A_{ℓ}(J, Λ_{W_{σ}}, K) at 𝔪; in §V this will be denoted L_{i,Q}. Let M(𝔪) denote the set of distinct irreducible components of L_{𝔪} ⊗ ℚ_{ℓ} and let C(𝔪) denote the corresponding set of automorphic representations of J; we assume our chosen π belongs to C(𝔪). For π′ ∈ C(𝔪) we let E(π′) denote the corresponding direct factor of 𝕋 ⊗ ℚ_{ℓ}. Then the λ-adic representation ρ(π′) can be realized over E(π′).

Say there are r distinct elements in M(𝔪), and let {π_{1}, …, π_{r}} be a corresponding set of elements of C(𝔪). The sum of the ℓ-adic representations r_{ρ}(π_{i}) for i = 1, …, r can be viewed ad hoc as a E(π_{1}) × ⋯ × E(π_{r}) = 𝕋 ⊗ ℚ_{ℓ}-module. Denote this Hecke module V (𝔪), with action r_{ρ}(𝔪) of Gal(\overline{ℚ}/K).

Proposition III.2.1. Assume the residual representation r\overline{ρ}(π) is absolutely irreducible. Then the action of Gal(\overline{ℚ}/K) on V (𝔪) can be realized over the localization 𝕋_{𝔪} of 𝕋 at 𝔪.

Proof. This is a simple application of Carayol’s theorem [Ca, Théorème 1.2]. By construction, the traces of r_{ρ}(𝔪)(F rob_{w}) lie in 𝕋_{𝔪} for almost all w that split in K/E. This is a dense subset of Gal(\overline{ℚ}/K). It is now clear that r_{ρ}(𝔪) satisfies the hypotheses of Carayol’s theorem, the most important being absolute irreducibility of the residual representation.

III.3. Action of complex conjugation.

Let k be a field of characteristic ≠ 2, V an n-dimensional vector space over k, and let r_{ϕ} : G_{K} → GL(V) be an irreducible representation. Let c ∈ G_{E} be any complex conjugation, and let ϕ^{c} denote the representation g ↦ ϕ(cgc^{−1}). Assume there exists a character χ : G_{E}→k^{×} such that

(III.3.1)  r_{ϕ}^{c} ≅ ř_{ϕ} ⊗ χ

as representations of G_{E}. Choosing dual bases, we identify V and V̌ with k^{n}. Then for g ∈ G_{K},

(III.3.2)  ϕ(g) = (r_{ϕ}(g), ř_{ϕ} · χ(g)) ∈ GL(n, k) × GL(n, k)

lies in the group G̃(k) defined in (I.1.5).

There is an isomorphism B : V̌ →V intertwining the representation ř_{ϕ} ⊗ χ of G_{K} on V̌ with the representation r_{ϕ}^{c} on V:

(III.3.3)  Bř_{ϕ} · χ(g)B^{−1} = r_{ϕ}^{c}(g).

By Schur’s Lemma B is unique up to scalar multiples.

Lemma III.3.4. The intertwining map B is either symmetric or skew-symmetric, and this is independent of the choice of B and of the complex conjugation c.

Proof.. When χ is the trivial character, this follows from Lemma 15.1.1 of [R2], bearing in mind that c^{2} = 1. The same proof works for general χ.

Let ϵ(r_{ϕ}) = 1 if B is symmetric, = −1 if B is skewsymmetric. Evidently ϵ(r_{ϕ}) = 1 if n is odd. In terms of the chosen bases, we identify B with an invertible n × n matrix. Then it follows from Lemma III.3.4 that the element (III.3.5) ϕ(c) = (B, B^{−1}) ⋉ c = (B,^{t}B^{−1} · ϵ(r_{ϕ})) ⋉ c ∈ [GL(n, k)×GL(n, k)] ⋉ Gal(K/E)

actually belongs to G̃(k). One verifies immediately that

ϕ(c)^{2} = 1

and (III.3.3) translates into the relation

ϕ(c)ϕ(g)ϕ(c)^{−1} = ϕ(cgc^{−1}).

Thus

Lemma III.3.6. The map

ϕ : G_{E} ≅ G_{K} ⋉ {1, c} → G̃(k)

is a homomorphism of groups.

We return to the notation of §III.1. It follows from (III.1.5.1) and Chebotarev density that r_{ϕ} = r\overline{ρ}(π) satisfies (III.3.1) with χ = ω^{1−n}, where ω is the cyclotomic character. Let r_{ρ}(π)→GL(n, O) be as in §III.1. Then the argument above applies to yield

Corollary III.3.7. Suppose SC is non-empty, so r_{ρ}(π) is absolutely irreducible. Then ρ(π) factors through a homomorphism to the L-group G̃(O). More precisely, there is a sequence of maps

Gal(\overline{E}/E) → G̃(O) → Aut(Ind_{K/E}A_{λ}[π_{f}])

whose composite is equivalent to ρ(π). Moreover, the restriction to G_{K} of the composite ν ∘ ρ(π) : Gal(\overline{E}/E)→O^{×} equals the (1 − n)th power of the cyclotomic character; here ν is the similitude character defined at the end of §I.1.

Let

(III.3.8)  ω_{π}(1 − n) = ν ∘ ρ(π)

be the character of G_{E} defined by Corollary III.3.7. It is determined by its restriction to G_{K} and by the formula

(III.3.9)  ω_{π}(1 − n)(c) = ϵ(r_{ρ}(π)),

which is immediate from (III.3.5).

Let ad r\overline{ρ}(π) (resp. ad r_{ρ}) denote the adjoint representation G_{E} on the Lie algebra M(n, k) of G̃(k) (resp. on M(n, O)). It is immediate that

ad r\overline{ρ}(π)(c)(X) = −B^{t}XB^{−1}, X ∈ M(n, k);

the same formula holds for ad r_{ρ}.

Lemma III.3.10. Let c denote any complex conjugation in G_{E}. Then the dimension of the +1-eigenspace of ad r\overline{ρ}(π)(c) equals \frac{n(n−1)}{2} if ϵ(r_{ϕ}) = 1, \frac{n^{2}}{2} if ϵ(r_{ϕ}) = −1. The same formula holds for ad r_{ρ}.

In particular, the dimension is always ≥ \frac{n(n−1)}{2}.

Proof. This is an elementary calculation.

The results of the present section apply to the representations over the Hecke algebras constructed in §III.2. In particular, the representation of G_{K} constructed in Proposition III.2.1 extends to a homomorphism from G_{E} to G̃(𝕋_{𝔪}).

IV. Deformation of Galois representations

IV.1. Definition of the deformation problem. As in the previous sections, we let G_{E} = Gal(\overline{ℚ}/E), G_{K} = Gal(\overline{ℚ}/K). A decomposition group at the prime v is denoted Z_{v}, the inertia subgroup by I_{v}. The L-group G̃, viewed as a ℤ-group scheme, is as in (I.1.5). We fix a prime ℓ, unramified in K, and a representation \overline{ρ} : G_{E} → G̃(\overline{𝔽}_{l}), and let r\overline{ρ} denote the restriction of \overline{ρ} to G_{K}. The representation \overline{ρ} is assumed to satisfy the following conditions:

IV.1.1.0. There is a finite subfield k ⊂ \overline{𝔽}_{l} such that \overline{ρ} takes values in G̃(k).

IV.1.1.1. The composite G_{E} → G̃(\overline{𝔽}_{l}) → \{1, c\} cuts out K/E.

IV.1.1.2. r\overline{ρ} is unramified except at primes above ℓ and above a non-empty finite set of primes SC of E. At primes above ℓ, r\overline{ρ} is crystalline. If 𝔭 ∈ SC then 𝔭 = vv^{c} splits in K and r\overline{ρ}|_{Z_{v}} breaks up as a direct sum of irreducible representations 𝔯_{i,v}. Moreover, there is at least one 𝔭 ∈ SC such that r\overline{ρ}|_{Z_{v}} is irreducible, with v as above.

IV.1.1.3. Denote by c any lifting of c to a complex conjugation in G_{E}. In the adjoint representation ad \overline{ρ} of G_{E} on Lie(G̃), the +1-eigenspace of c has dimension ≥ \frac{n(n−1)}{2}.

IV.1.1.4. The composite ω\overline{ρ} = ν∘\overline{ρ} : G_{E}→k^{×}, restricted to G_{K}, equals the (1−n)th power of the cyclotomic character, where ν : G̃ → GL(1) is the similitude character defined in §I.1.

Here and in what follows the term ”crystalline,” applied to ℓ-torsion modules, is used to refer to Galois representations obtained by the Fontaine-Laffaille construction (see IV.4.3, below). The details of this theory are recalled in §IV.4.

Lemma IV.1.1.5. In the situation of IV.1.1.2, suppose ℓ is banal for GL(n, E_{𝔭}) (cf. Proposition III.1.3). Then ℓ ∤ #\overline{ρ}(I_{𝔭}) and, for any lifting ρ of \overline{ρ} to characteristic zero, ℓ ∤ #ρ(I_{𝔭}).

Proof. Let ρ be a lifting as in the statement of the lemma. It is irreducible, hence ρ(I_{𝔭}) is a finite group. Let q = Nv. Let I = I_{v}, the inertia subgroup of Z_{v}, and let P ⊂ I be the wild ramification subgroup (i.e., the p-Sylow subgroup, where p is the residue characteristic of 𝔭). By the banality assumption ℓ > n ≥ 2, so it suffices to show that ℓ is prime to the image of r\overline{ρ}(I). Let g ∈ ℤ_{ℓ}(1) ⊂ I/P be a topological generator and assume some lift g̃ ∈ I has non-trivial image under ρ. Let σ ∈ Z_{v} be a Frobenius element. It acts on I by conjugation, and g̃^{σ} = g̃^{q} · b for some b ∈ P. Then the set of eigenvalues of ρ(g̃) and of ρ(g̃^{q} · b) are the same. The closed subgroup < g̃^{q} · b >⊂ I topologically generated by g̃^{q} · b maps bijectively to ℤ_{ℓ}(1) under projection to I/P; let g′ ∈< g̃^{q} · b > be the lifting of g, so that (g′)^{q} = g̃^{q} · b. By conjugacy of Sylow ℓ-subgroups, g′ is conjugate to g in I, hence has the same eigenvalues. It follows that, if s ∈ μ_{ℓ^{∞}} is an eigenvalue of ρ(g), then so is s^{q}. The set of eigenvalues has at most n elements, hence there is m ≤ n such that s^{q^{m}} = s. Thus ℓ divides q^{m} − 1, which contradicts the banality hypothesis.

We note the following consequence of (IV.1.1.2):

IV.1.1.7. The intersection K ∩ ℚ(ζ_{ℓ}) = ℚ.

Let O denote the ring of integers in a totally ramified finite extension 𝕂 of the fraction field of the Witt ring W (k). Let C_{O} denote the category of complete noetherian local O-algebras with residue field k; morphisms in C_{O} are assumed to be local (take maximal ideals to maximal ideals). If R is an object of C_{O} we let m_{R} denote its maximal ideal. Since ℓ > 2 by the banality hypothesis, the character ω\overline{ρ} defined by IV.1.1.4 has a unique lift ω_{\overline{ρ},R} : G_{E} → R^{×} for any object R of C_{O}.

IV.1.2. Let R be an object of C_{O}. A deformation of \overline{ρ} to R is a homomorphism ρ : G_{E} → G̃(R) such that

(IV.1.2.1)  \overline{ρ} ≡ ρ (mod m_{R}).

(IV.1.2.2)  ν ∘ ρ(g) = ω_{\overline{ρ},R}.

Here ν : G̃(R)→R^{×} is the similitude character.

We assume

IV.1.3. \overline{ρ} has a deformation ρ_{0} to O such that for each prime λ of K dividing ℓ r_{ρ_{0}}|_{G_{λ}} is crystalline and the filtered module has n graded pieces, each free of rank one over O, and of weights 0, 1, …, n − 1.

IV.1.4. We will be considering deformations of \overline{ρ} with conditions at certain auxiliary sets of primes. Let Q denote a finite set of height one primes 𝔮 of E disjoint from SC ∪ {ℓ} [divisors of ℓ] which satisfy

IV.1.4.1. 𝔮 splits in K and the division algebras D and D^{#} are split above 𝔮;

IV.1.4.2. The residue characteristic q of 𝔮 satisfies q ≡ 1 (mod ℓ);

IV.1.4.3. \overline{ρ}(F rob_{𝔮}) has a distinguished eigenvalue α_{𝔮} of multiplicity one.

As representations of Z_{𝔮}, we write

(IV.1.4.4)  \overline{ρ} = \overline{ρ}_{α} ⊕ \overline{ρ}_{β},

where \overline{ρ}_{α} is the α_{𝔮}-eigenspace of \overline{ρ}(F rob_{𝔮}) and \overline{ρ}_{β} is the direct sum of the remaining eigenspaces. Let Δ_{𝔮} denote the maximal ℓ-power quotient of (ℤ/qℤ)^{×} and Δ_{Q} = ∏_{𝔮∈Q} Δ_{𝔮}.

By a deformation of \overline{ρ} of type Q we shall mean a pair (R, ρ) as in Definition IV.1.2 such that:

IV.1.5.1. For each prime λ of K dividing ℓ, r_{ρ}|_{Z_{λ}} is crystalline and the filtered module has n graded pieces, each free of rank one over R, and of weights 0, 1, …, n − 1.

IV.1.5.2. If 𝔮 ∈ Q then r_{ρ}|_{Z_{𝔮}} = χ ⊕ r′ where r′ = r′_{𝔮} is unramified and χ = χ_{𝔮} : Z_{𝔮}→R^{×} is a character whose reduction modulo m_{R} is unramified and takes F rob_{𝔮} to α_{𝔮}.

IV.1.5.3. If v ∉ Q ∪ \{ℓ\} then ρ(I_{v}) ⟶^{∼} \overline{ρ}(I_{v}).

Proposition IV.1.6. There exists a universal deformation (R_{Q}, ρ_{Q}) of \overline{ρ} of type Q.

Proof. When \overline{ρ} is replaced by the absolutely irreducible representation r\overline{ρ}, the existence of a universal deformation follows from Theorem 1.1 of [R], as in [DDT, Lemma 2.37]; cf. the proof of Proposition IV.2.3 below. The extension to the disconnected group G̃ can be seen easily from the approach (using “well-placed liftings”) described in [loc. cit.]. The crucial point is that the lifting of \overline{ρ}(c) to G̃(R) is unique, up to an element of the center of GL(n, R). Indeed, if c_{1} and c_{2} are any two liftings, then (c_{2})^{−1}c_{1} intertwines r_{ρ} with itself, hence is a scalar matrix congruent to 1 (mod m_{R}).

IV.1.7 For 𝔮 ∈ Q we let χ_{𝔮} : Z_{𝔮}→R_{Q}^{×} be the character defined in (IV.1.5.2). Then χ_{𝔮} necessarily factors through a natural map Δ_{𝔮}→R_{Q}^{×}. Thus R_{Q} is tautologically an O[Δ_{Q}]-module.

IV.2. Bounding the Selmer group.

Henceforward, we assume ℓ > n. We fix a finite set Q of primes of E as in IV.1.4. Let ad r\overline{ρ} denote the composition of \overline{ρ} with the adjoint representation ad : G̃→Aut(𝔤𝔩(n)), where 𝔤𝔩(n) ⊂ Lie(G̃) is viewed as the kernel of the similitude map. For each place v of E we fix a k-subspace L_{Q,v} ⊂ H^{1}(Z_{v}, ad r\overline{ρ}). The L_{Q,v} are chosen as follows:

IV.2.1.1. For v dividing ℓ, L_{Q,v} is the Bloch-Kato group H_{f}^{1}(Z_{v}, ad r\overline{ρ}).

In [BlK], Bloch and Kato work with characteristic zero coefficients. The ℓ-torsion group H_{f}^{1}(Z_{v}, ad r\overline{ρ}) will be defined in IV.4, below.

IV.2.1.2. For v = 𝔮 ∈ Q, write

ad r\overline{ρ} = ad \overline{ρ}_{α} ⊕ ad \overline{ρ}′_{α},

where

ad \overline{ρ}′_{α} = ad \overline{ρ}_{β} ⊕ Hom(\overline{ρ}_{α}, \overline{ρ}_{β}) ⊕ Hom(\overline{ρ}_{β}, \overline{ρ}_{α}),

(notation IV.1.4.4). We set

L_{Q,𝔮} = H^{1}(Z_{𝔮}, ad \overline{ρ}_{α}) ⊕ H^{1}(Z_{𝔮}/I_{𝔮}, ad \overline{ρ}′_{α}).

IV.2.1.3. At all other finite primes v L_{Q,v} = H^{1}(Z_{v}/I_{v}, ad rI_{v} ; \overline{ρ}).

IV.2.1.4. At archimedean primes we take L_{Q,v} = 0.

There is a natural isomorphism (Poincaré duality)

ad r\overline{ρ} ⟶^{∼} ad r∗ ; \overline{ρ},

hence natural non-degenerate pairings for each place v

(IV.2.1.5)  H^{i}(Z_{v}, ad r\overline{ρ}) × H^{2−i}(Z_{v}, ad r\overline{ρ}(1)) → ℚ/ℤ

(Tate’s local duality), where (1) denotes Tate twist. For each v we let L_{Q,v}^{⊥} ⊂ H^{1}(Z_{v}, ad r\overline{ρ}(1)) be the annihilator of L_{Q,v} with respect to (IV.2.1.5), and define the Selmer group of ad r\overline{ρ}(1), relative to the data L_{Q,v}^{⊥}:

(IV.2.1.6)  H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1)) = \{h ∈ H^{1}(E, ad r\overline{ρ}(1)) | ∀ v r_{v}(h) ∈ L_{Q,v}^{⊥}\}

We write 𝔐_{Q} for m_{R_{Q}}. The objective of this section is to prove the following theorem.

Theorem IV.2.2. The Selmer group H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1)) is finite and we have the inequality

dim_{k} 𝔐_{Q}/(𝔐_{Q}^{2}, ℓ) ≤ #Q + dim_{k} H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1)).

In particular, if dim H_{Q^{∗}}^{1}(E, adr\overline{ρ}) = 0 then the O-algebra R_{Q} can be topologically generated by #Q elements.

This theorem generalizes Lemma 5 of [TW]. Henceforward we write dim instead of dim_{k}. We begin by translating the theorem into a statement purely in terms of Galois cohomology.

Proposition IV.2.3. Define the Selmer group of ad r\overline{ρ}, relative to the data L_{Q,v}:

H_{Q}^{1}(E, ad r\overline{ρ}) = \{h ∈ H^{1}(E, ad r\overline{ρ}) | ∀ v r_{v}(h) ∈ L_{Q,v}\}.

Then

dim 𝔐_{Q}/(𝔐_{Q}^{2}, ℓ) = dim H_{Q}^{1}(E, ad r\overline{ρ}).

Proof. This is proved as in [DDT,Theorem 2.41]. Let D denote the category of k[G_{E}]-modules M finite over k with dimension divisible by n, satisfying the analogues of properties IV.1.5.1-3:

IV.2.3.1. As a module over Z_{v}, v above ℓ, M is a Fontaine-Laffaille representation (cf. §IV.4, below).

IV.2.3.2. As a module over Z_{𝔮}, 𝔮 in Q, M is a the sum of an unramified module B and a module A whose semisimplification is isotypic for the unramified character α_{𝔮}.

IV.2.3.3. If v ∉ Q ∪ {ℓ} then the action of I_{v} on M is a direct sum of copies of irreducible direct summands of \overline{ρ}(I_{v}).

The category D is closed under products and taking subobjects and quotient objects. Obviously it contains \overline{ρ}. Thus Lemma 2.39 of [DDT] applies and yields

dim 𝔐_{Q}/(𝔐_{Q}^{2}, ℓ) = dim H_{D}^{1}(E, ad r\overline{ρ}),

where H_{D}^{1}(E, ad r\overline{ρ}) ⊂ H^{1}(E, ad r\overline{ρ}) ≃ Ext_{G_{E}}^{1}(\overline{ρ}, \overline{ρ}) is the subspace of classes whose corresponding extensions lie in D.

Now we have to verify that conditions IV.2.3.1-3 for extensions translate into the cohomological conditions IV.2.1.1-3. Specifically, the equivalence of IV.2.3.1 and IV.2.1.1 is proved below in IV.4.7. The equivalence of IV.2.3.2 with IV.2.1.2 is easy to verify. At finite places v ∉ Q∪ℓ∪SC, and such that v is unramified in K, IV.2.3.3 says the action of I_{v} is trivial, which is obviously equivalent to IV.2.1.3. Now suppose v in SC. The compatibility of IV.2.3.3 and IV.2.1.3 is equivalent to the condition

H^{1}(Z_{v}/I_{v}, Hom_{I_{v}}(\overline{ρ}, \overline{ρ})) ≃ Ker[H^{1}(Z_{v}, Hom(\overline{ρ}, \overline{ρ})) → H^{1}(I_{v}, Hom(\overline{ρ}, \overline{ρ}))],

and this is just the inflation-restriction sequence. For v ramified in K, the argument is similar.

We thus need to prove the inequality

(IV.2.4)  dim H_{Q}^{1}(E, ad r\overline{ρ}) − dim H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1)) ≤ #Q.

Following Wiles [W,Prop. 1.6], the left hand side of (IV.2.4) can be expressed as a sum of local terms. We write the formula as in [DDT, Theorem 2.19], where it is stated for a general number field:

Proposition IV.2.5. Let h^{0} = dim H^{0}(E, ad r\overline{ρ}), h^{0,∗} = dim H^{0}(E, ad r\overline{ρ}(1)). For any place v of E let h_{v}^{0} = dim H^{0}(Z_{v}, ad r\overline{ρ}). Then we have the formula

dim H_{Q}^{1}(E, ad r\overline{ρ}) − dim H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1)) = h^{0} − h^{0,∗} + ∑_{v}(dim L_{Q,v} − h_{v}^{0}).

Lemma IV.2.6. Under the hypotheses of Proposition IV.2.5, the local terms are computed as follows:

(a) For v real, h_{v}^{0} ≥ \frac{n(n−1)}{2}, dim L_{Q,v} = 0.

(b) For v ∈ Q, dim L_{Q,v} − h_{v}^{0} = 1.

(c) For v above ℓ, dim L_{Q,v} − h_{v}^{0} = [k(v) : 𝔽_{ℓ}] · \frac{n(n−1)}{2}.

(d) For all other places v, dim L_{Q,v} − h_{v}^{0} = 0.

Finally, the global terms are given by h^{0} = h^{0,∗} = 0.

Admit this lemma for the moment. Comparing Proposition IV.2.5 with Lemma IV.2.6, we find

(IV.2.7) dim H_{Q}^{1}(E, ad r\overline{ρ}) − dim H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1))

≤ #Q − ∑_{v real} \frac{n(n − 1)}{2} + ∑_{v|ℓ}[k(v) : 𝔽_{ℓ}] · \frac{n(n − 1)}{2}

≤ #Q − [E : ℚ]\frac{n(n − 1)}{2} + [E : ℚ]\frac{n(n − 1)}{2} ≤ #Q

Theorem IV.2.2 now follows by comparing (IV.2.7) with Proposition IV.2.3.

IV.2.8. We begin by calculating the global terms in Lemma IV.2.6. The hypothesis that SC is non-empty implies that \overline{ρ} is already irreducible when restricted to a decomposition group of G_{K} above a prime in SC. Thus H^{0}(K, ad r\overline{ρ}) is onedimensional and given by the trace of r\overline{ρ}. But complex conjugation c acts as −1 on the center of the GL(n)-component of the L-group, so H^{0}(E, ad r\overline{ρ}) is trivial. In the same way, and using IV.1.1.5, we see that h^{0,∗} = 0.

The local terms will be computed in the next two sections.

IV.3. Local calculations, char v ≠ ℓ.

In this section we carry out the calculations summarized in Lemma IV.2.6. For any place v and any finite 𝔽_{ℓ}[Z_{v}]-module M we set

h^{i}(M) = dim H^{i}(G_{v}, M); h^{i,unr}(M) = dim H^{i}(Z_{v}/I_{v}, M^{I_{v}}),

i = 0, 1, 2. IV.3.1. If M is an unramified Z_{v}-module then of course h^{0,unr}(M) = h^{0}(M). On the other hand, M is always assumed to be F rob_{v}-semi-simple when ℓ is not equal to the residue characteristic of v. Then M is the sum of characters of Z_{v}/I_{v} and

(IV.3.1)  h^{1,unr}(M) = h^{0}(M) = dim M^{Z_{v}}.

It follows that, for v unramified, v ∉ Q, we have

dim L_{Q,v} − h_{v}^{0} = 0.

This verifies IV.2.6 (d) at unramified places. IV.3.2. Now take v ∈ Q. We have

dim L_{Q,v} − h_{v}^{0} = h^{1}(ad \overline{ρ}_{α}) − h^{0}(ad \overline{ρ}_{α}) + h^{1,unr}(ad \overline{ρ}_{α})′ − h^{0}(ad \overline{ρ}_{α})′.

Since ad(\overline{ρ}_{α})′ is unramified the last two terms cancel, by (IV.3.1). On the other hand, the first two terms give

h^{0}(ad \overline{ρ}_{α}(1))

by the local Euler characteristic formula and local duality (cf. [W,p. 473]). But \overline{ρ}_{α} is one-dimensional, so ad(\overline{ρ}_{α}) is the trivial Z_{v} module. Since q ≡ 1 (mod ℓ) the Tate twist is also trivial, and we find

dim L_{Q,v} − h_{v}^{0} = 1,

which verifies IV.2.6 (b). IV.3.3. For v real, we have dim L_{Q,v} = 0, by hypothesis. On the other hand,

h_{v}^{0} = dim[ad r\overline{ρ}]^{c=1},

independently of v. Then (a) follows immediately from hypothesis IV.1.1.3. IV.3.4. Now suppose v is ramified, but of residue characteristic ≠ ℓ. By hypothesis, either v ∈ SC, or v ramifies in K/E and r\overline{ρ} is unramified at the prime above v. We need to calculate

dim L_{Q,v} − h_{v}^{0} = h^{1,unr}(ad r\overline{ρ}) − h^{0}(ad r\overline{ρ}).

First, suppose v ∈ SC, and r\overline{ρ} = ⊕_{i=1}^{r}(𝔯_{i})^{a_{i}}, where the 𝔯_{i} are irreducible and distinct. Returning to IV.1.1.2, we find that

dim[ad r\overline{ρ}]^{Z_{v}} = ∑_{i}(a_{i})^{2}.

Let L_{ij} = H^{1}(Z_{v}/I_{v}, 𝔯_{i} ⊗ 𝔯_{j}^{∗}), where ∗ denotes dual. It suffices to show that dim L_{ij} = δ_{ij}. Suppose 𝔯_{i}|_{I_{v}} breaks up as the sum of d irreducible representations τ_{ik}. Then

(IV.3.5)  (𝔯_{i} ⊗ 𝔯_{i}^{∗})^{I_{v}} = ⊕_{k=1}^{d}[ad τ_{k}]^{I_{v}}

has dimension d. As a representation of the cyclic group Z_{v}/I_{v}, the right-hand side of IV.3.5 is isomorphic to the sum ⊕χ of the distinct characters of Z_{v}/H, where H ⊃ I_{v} is the stabilizer in Z_{v} of τ_{1}, say. Thus

dim L_{ii} = ∑_{χ} dim H^{1}(Z_{v}/H, χ) = 1,

since only the trivial character has non-trivial cohomology. The verification for L_{ij} with i ≠ j breaks up into two cases. If 𝔯_{j} is not an unramified twist of r_{i}, then (𝔯_{i} ⊗ 𝔯_{j}^{∗})^{I_{v}} = 0. If 𝔯_{i} = 𝔯_{j} ⊗ ξ, with ξ an unramified character, then we find

(𝔯_{i} ⊗ 𝔯_{j}^{∗})^{I_{v}} = ⊕_{k=1}^{d}χ · ξ

where χ runs through the characters of Z_{v}/H, as above. We conclude that dim L_{ij} = 0 by observing that the non-isomorphy of 𝔯_{i} and 𝔯_{j} implies that ξ does not factor through Z_{v}/H.

Now suppose v ramifies in K/E. Let w denote the prime above v. In this case Z_{v} acts via the abelian group Gal(K/E) × Z_{w}/I_{w}. Let M denote the subspace of ad \overline{ρ} fixed by Gal(K/E). Then dim L_{Q,v} − h_{v}^{0} = h^{1,unr}(M) − h^{0}(M) = 0 as in IV.3.1. This completes the verification of (d).

To complete the proof of Lemma IV.2.6, it remains to estimate the local terms at primes dividing ℓ. This is the subject of the next section.

IV.4 Local calculations, crystalline case.

In this section we complete the proof of Lemma IV.2.6 by all the facts we need about comparison theorems between crystalline and étale cohomology. As above, we work in residue characteristic ℓ and restrict our attention to the situation of good reduction. We begin by defining crystalline representations of Gal(\overline{E_{v}}/E_{v}), when v is of residue characteristic ℓ, with coefficients in \overline{𝔽}_{l}, in terms of the Fontaine-Laffaille construction [FL]. Recall that ℓ is assumed unramified in E. Next we define crystalline deformations of mod ℓ crystalline representations and show that the crystalline condition is captured, in terms of local Galois cohomology, by an ℓ-torsion analogue of the Bloch-Kato group H_{f}^{1}. The inequality of Lemma IV.2.6 (c) is then proved by a calculation in the Fontaine-Laffaille category. We conclude by showing that under a certain regularity hypothesis (satisfied in our applications) the image of tame inertia contains an element whose eigenvalues have multiplicity one.

Our base ring will be the Witt ring W (k(v)), where k(v) is the residue field of v; thus W (k(v)) = O_{v} in our previous notation. We let O_{ℂ_{ℓ}} denote the ring of integers in the completion ℂ_{ℓ} of the algebraic closure of the fraction field of W (k(v)). Let A_{cris} = A_{cris}(O_{ℂ_{ℓ}}) be the ring defined in [Fo, Asterisque, 2.3], an ℓ-integral form of B_{cris}. The W (k(v))-algebra A_{cris} is endowed with a decreasing filtration F il^{i}A_{cris}, i ≥ 0 and with a σ-linear operator ϕ such that

(IV.4.1)  ϕ(F il^{i}A_{cris}) ⊂ ℓ^{i}A_{cris}, 0 ≤ i ≤ ℓ − 1.

Here σ is absolute Frobenius.

Recall the ℓ-adic coefficient ring O from IV.1. Let Rep_{W (k(v))⊗O} denote the category of O[Gal(\overline{E_{v}}/E_{v})]-modules of finite type. By analogy with [FPR, p. 638], we define Rep_{W (k(v))⊗O,cris,[0,ℓ−1[} to be the full subcategory of Rep_{W (k(v))⊗O} whose objects are isomorphic to subquotients of crystalline ℚ_{ℓ} ⊗_{ℤ_{ℓ}} O[Gal(\overline{E_{v}}/E_{v})]- modules. For any object Λ of Rep_{W (k(v))⊗O,cris,[0,ℓ−1[} we can define the filtered module (Fontaine-Laffaille module) M_{cris}(Λ). It is an object of the category MF_{W (k(v))⊗O,[0,ℓ−1[} consisting of

(i) a W (k(v)) ⊗ O-module M of finite type, a decreasing filtration F il^{i}(M) by W (k(v)) ⊗ O-submodules which are direct factors, with F il^{0}M = M, F il^{ℓ}M = 0; and

(ii) a family ϕ^{i} : F il^{i}(M) → M of σ-linear maps such that ϕ^{i}(x) = ℓϕ^{i+1}(x) for x ∈ F il^{i+1}M, and such that M is the sum of the images of the ϕ^{i}, as i ranges over ℤ.

It is further assumed that M contains no non-trivial subobject M′ with F il^{ℓ−1}M′ = M′.

The definition is (IV.4.2)

M_{cris}(Λ) = ⋃ M | M ⊂ A_{cris} ⊗_{W (k)} Λ)^{Gal(\overline{E_{v}}/E_{v})}; M ∈ MF_{W (k(v)),[0,ℓ−1[}

(cf. [Niz, p. 750]; [Wa,Remarque 2.4.4]). The filtration is inherited from the filtration on A_{cris}, the O-action on Λ is left undisturbed, and ϕ^{i} is inherited from ℓ^{−i}ϕ on F il^{i}A_{cris}, which makes sense by (IV.4.1). Then M_{cris} : Rep_{W (k),cris,[0,ℓ−1[} → MF_{W (k),[0,ℓ−1[} is an equivalence of categories. An inverse equivalence [FL] is given by the formula

(IV.4.3)  Λ(M) = F il^{0}(A_{cris} ⊗_{W (k)} M)^{ϕ=1}.

For our purposes, crystalline Galois representations are those of the form Λ(M), for some M ∈ MF_{W (k(v)),[0,ℓ−1[}. More generally, let A be a W (k(v)) algebra of finite type with a σ-linear automorphism σ_{A}. Then we can define the categories MF_{A,[0,ℓ−1[} and Rep_{A,cris,[0,ℓ−1[} by analogy with (i), (ii) above, taking A as coefficient ring. The functors M_{cris} and Λ can be defined as inverse equivalences between these two categories. In the applications, A will be a W (k(v)) ⊗ O-algebra, where σ_{W (k(v))⊗O} is defined to be σ ⊗ 1 : W (k(v)) ⊗ O→W (k(v)) ⊗ O.

Let MF_{tor,[0,ℓ−1[} denote the subcategory of MF_{W (k(v)),[0,ℓ−1[} of objects of finite length (as W (k(v))-modules). By [FL,Prop. 1.8], MF_{tor,[0,ℓ−1[} is an abelian category. Let

Rep_{tor,cris,[0,ℓ−1[} ⊂ Rep_{W (k(v)),cris,[0,ℓ−1[}

denote the essential image of the functor Λ, restricted to MF_{tor,[0,ℓ−1[}. Then Rep_{tor,cris,[0,ℓ−1[} is a full subcategory of Rep_{W (k(v)),cris,[0,ℓ−1[}, itself a full subcategory of Rep_{W (k(v))}. The functors M_{cris} and Λ define inverse equivalences between the abelian categories Rep_{tor,cris,[0,ℓ−1[} and MF_{tor,[0,ℓ−1[}. Thus, given two objects M, N ∈ MF_{tor,[0,ℓ−1[}, there is a natural isomorphism

(IV.4.4)  Ext_{MF}^{1} (M, N) ⟶^{∼} Ext_{cris}^{1}(Λ(M), Λ(N)),

where Ext_{MF}^{1} (resp. Ext_{cris}^{1}) is shorthand for extensions in MF_{tor,[0,ℓ−1[} (resp. Rep_{tor,cris,[0,ℓ−1[}). Composing the isomorphism (IV.4.4) with the forgetful functor, we obtain a homomorphism

(IV.4.5)  Ext_{MF}^{1} (M, N) → Ext_{Rep_{W (k(v))}}^{1}(Λ(M), Λ(N)).

The isomorphism (IV.4.4) and the homomorphism (IV.4.5) respect O-structures; i.e., take extensions with compatible O-structure to extensions with O-structures.

Write k(v) ⊗ k = k(v) ⊗_{ℤ_{ℓ}} k, and suppose now that M and N are free k(v) ⊗ k-modules. In particular, ℓ · M = 0, ℓ · N = 0. Let Ext_{MF,k(v)⊗k}^{1}(M, N) ⊂ Ext_{MF,k(v)⊗k}^{1}(M, N) denote the subgroup of extensions in the category of Fontaine-Laffaille modules which are free k(v) ⊗ k-modules. Likewise, let Rep_{k(v)⊗k} denote the category of k(v) ⊗ k[Gal(\overline{E_{v}}/E_{v})]-modules free over k(v) ⊗ k. Then

Ext_{Rep_{k(v)⊗k}}^{1}(Λ(M), Λ(N)) ⟶^{∼} Ext_{Rep_{k(v)⊗k}}^{1}(k(v) ⊗ k, Hom_{k(v)⊗k}(Λ(M), Λ(N)))

⟶^{∼} H^{1}(Gal(\overline{E_{v}}/E_{v}), Hom_{k(v)⊗k}(Λ(M), Λ(N))).

With respect to this isomorphism, we let

H_{f}^{1}(Gal(\overline{E_{v}}/E_{v}), Hom_{k(v)⊗k}(Λ(M), Λ(N))

⊂ H^{1}(Gal(\overline{E_{v}}/E_{v}), Hom_{k(v)⊗k}(Λ(M), Λ(N)))

denote the image of Ext_{MF,k(v)⊗k}^{1}(M, N).

Say M ∈ MF_{W (k(v))⊗O,[0,ℓ−1[} is regular if gr^{i}(M) = F il^{i}(M)/F il^{i+1}(M) is a free k(v) ⊗ k-module of rank ≤ 1 for all i.

Lemma IV.4.6. Let Λ be a crystalline k(v) ⊗ k[Gal(\overline{E_{v}}/E_{v})]-module, and let Ad(Λ) denote the k(v) ⊗ k[Gal(\overline{E_{v}}/E_{v})]-module Hom_{k(v)⊗k}(Λ, Λ). Suppose Λ = Λ(M), with M a regular Fontaine-Laffaille module of rank n over k(v) ⊗ k. Then

rank_{k(v)⊗k}H_{f}^{1}(Gal(\overline{E_{v}}/E_{v}), Ad(Λ)) − dim H^{0}(Gal(\overline{E_{v}}/E_{v}), Ad(Λ)) = \frac{1}{2}n(n − 1).

Proof. By definition, the left-hand side equals

rank_{k(v)⊗k}Ext_{MF,rank_{k(v)⊗k}}^{1}(M, M) − rank_{k(v)⊗k}Hom_{MF,rank_{k(v)⊗k}}(M, M).

This is unchanged when we extend scalars from k to a finite extension k′ and replace M by M_{k′}. We thus may assume k ⊃ k(v), and then by projecting on irreducible components we may replace k(v) ⊗ k by k.

Since M is a k-module, its structure as Fontaine-Laffaille module reduces to a triple consisting of a k-vector space V, a decreasing filtration F il^{•}V, and an

isomorphism ϕ_{V} : gr^{•}V ⟶^{∼} V. Let gr^{•}V = ⊕_{i=1}^{n}gr^{a_{i}}V, where 0 ≤ a_{1} < a_{2} ⋯ <

a_{n} ≤ ℓ − 1 is an n-tuple of positive integers. Fix a basis e_{1}, …, e_{n} of V such that F il^{i}V is the span of e_{i+1}, …, e_{n}. Suppose E is an extension of M by itself in Ext_{MF,k}^{1}(M, M) and let E be the underlying k-module. There is a short exact sequence

0 → V →^{i} E →^{π} V → 0

compatible with filtrations and the morphisms ϕ_{E} and ϕ_{V}. Let s : V → E denote any splitting of π as filtered module; s is determined uniquely up to an element α ∈ F^{0}End(V), where F^{0}End(V) ⊂ End_{k}(V) denotes the subspace of filtrationpreserving endomorphisms. Then gr^{•}E = gr(i)(gr^{•}V) ⊕ gr(π)(gr^{•}V). In terms of

this basis, ϕ_{E} can be written ( \begin{matrix}ϕ_{V} & μ \\ 0 & ϕ_{V} \end{matrix} ), for some μ ∈ Hom(gr^{•}V, V). We have

μ = ϕ_{E} ∘ gr(s) − s ∘ ϕ_{V} .

Moreover, replacing s by s + α changes μ to μ + ϕ_{V} ∘ gr(α) − α ∘ ϕ_{V}. Thus the map M ↦ μ (mod equivalence) defines an isomorphism

Ext_{MF,k}^{1}(M, M) → {ϕ ∈ Hom(V, V)}/{ϕ ∘ gr(α) − α ∘ ϕ|α ∈ F^{0}End(V))}.

Moreover,

Hom_{MF,k}(M, M) → {α ∈ F^{0}End(V)|ϕ ∘ gr(α) = α ∘ ϕ}.

This yields an exact sequence

0 → Hom_{MF,k}(M, M) → F^{0}End(V) →^{j} Hom_{k}(gr^{•}V, V) → Ext_{MF,k}^{1}(M, M) → 0,

where the map j takes α to ϕ ∘ gr(α) − α ∘ ϕ.

It follows that

dim Ext_{MF,k}^{1}(M, M) − dim Hom_{MF,k}(M, M) = dim Hom_{k}(gr^{•}V, V) − dim F^{0}End(V)

= n^{2} − \frac{1}{2}n(n + 1) = \frac{1}{2}n(n − 1).

IV.4.7. Let A = k[ε]/(ε^{2}), let Λ ∈ Rep_{k,cris,[0,ℓ−1[}, and let Λ̃ be a deformation of Λ to A. Multiplication by ε defines an isomorphism

Λ ≅ Λ̃/εΛ̃ ⟶^{∼} εΛ̃

of k[Gal(\overline{E_{v}}/E_{v})]-modules. Thus Λ̃ defines an extension

(IV.4.7.1)  0 → Λ → Λ̃ → Λ → 0,

and this correspondence defines a bijection between the equivalence classes of deformations of Λ to A and Ext_{Rep_{k}}^{1}(Λ, Λ) (cf. [DDT], p. 67). By definition, the deformation Λ̃ is crystalline if and only if (IV.4.7.1) is an extension in the category of crystalline representations; in other words, crystalline deformations are in bijection with Ext_{cris}^{1}(Λ, Λ) ≅ H_{f}^{1}(Gal(\overline{E_{v}}/E_{v}), Ad(Λ). We have thus verified the equivalence of IV.2.3.1 and IV.2.1.1 in the definition of H_{D}^{1}(E, adr\overline{ρ}). IV.4.8. It remains to verify Lemma IV.2.6 (c). Thus let v be a prime of E dividing ℓ. Set h_{f}^{1} = dim_{𝔽_{ℓ}} H_{f}^{1}(Z_{v}, ad(r\overline{ρ})). By hypothesis IV.1.3 r\overline{ρ} is the crystalline representation over k(v) associated to a regular Fontaine-Laffaille module. Thus Lemma IV.2.6 (c) follows directly from Lemma IV.4.6. This completes the proof of Lemma IV.2.6.

For M ∈ MF_{W (\overline{𝔽}_{l}),[0,ℓ−1[} the action of tame inertia on Λ(M) is calculated explicitly in [FL]. As a consequence of that calculation, we can prove

Lemma IV.4.9. Let M ∈ MF_{W (k),[0,ℓ−1[} and suppose M is regular. Then the eigenspaces of tame inertia on Λ(M) have dimension 1. More precisely, let \overline{Λ}(M)_{ss} denote the semi-simplification of the k[Gal(\overline{E_{v}}/E_{v})]-module Λ/ℓΛ(M). Then the action of the inertia subgroup on \overline{Λ}(M)_{ss} factors through the tame quotient, and the latter has dim M/ℓM-distinct eigenvalues.

Proof. The proof is a simple combinatorial exercise, using the results of [FL,§§4-5]. Without loss of generality we may assume M to be a semi-simple ℓ-torsion module, so that Λ(M) = \overline{Λ}(M)_{ss}. Let n = dim_{\overline{𝔽}_{l}}M and let b_{i}, i = 1, …, n be the integers such that dim gr^{b_{i}}(M) = 1; let B = {b_{1}, …, b_{n}} ⊂ ℤ. Let q = |k| = ℓ^{r}, for some r. Then M is determined, up to isomorphism, by the following set of data:

(a) A partition n = h_{1} + ⋯ + h_{s}, with h_{j} ≥ 1 for all j;

(b) For each j a map ι_{j} : ℤ/h_{j}ℤ → B, with image B_{j}, so that B = ∪_{j}B_{j};

(c) The period of the map ι_{j} is exactly h_{j}; i.e. for any h < h_{j} the map ι_{j}(a) ↦ ι_{j}(a + h) is a non-trivial permutation of B_{j}.

Indeed, we can write M = ⊕_{j}M(j) as a sum of simple objects, and the object M(j) is determined up to isomorphism by a pair (h_{j}, ι_{j}) as above. Let Λ(j) = Λ(M(j)). For any positive integer h, let α ∈ \overline{ℚ}_{ℓ} satisfy

α^{q^{h}−1} = ℓ

and set χ_{h}(g) = g(α)/α (mod ℓ) ∈ \overline{𝔽}_{l}^{×}, for g in the tame inertia group I^{t}. Let

(IV.4.9.1)  C(j) = { ∑_{a=0}^{h_{j}−1} ι_{j}(a + t)q^{a} | t = 0, 1, …, h_{j} − 1}.

It follows from (c) above, and from the fact that B ⊂ {0, …, p − 1}, that C(j) has h_{j} distinct elements. The calculation in [FL] shows that the action of I^{t} on Λ(j) factors through the character χ_{h_{j}}, and g ∈ I^{t} has eigenvalues

χ_{h_{j}}(g)^{c}, c ∈ C(j).

The exponents in C(j) being distinct, the action of I^{t} on Λ(j) is multiplicity-free. On the other hand, it is easy to see, that I^{t} has no common eigenvalues on Λ(j_{1}) and Λ(j_{2}), first if h_{j_{1}} ≠ h_{j_{2}}, then in the general case.

IV.5. Capturing ramification by tame classes.

In order to make Theorem IV.2.2 effective, we need to find sets Q for which dim H_{Q^{∗}}^{1}(E, ad \overline{ρ}) = 0. We follow the strategy of [TW]. For this additional hypotheses are needed. Unfortunately, we have not found an optimal set of hypotheses. In the coordinates of (I.1.4) the map

(IV.5.1)  G̃^{0} → GL(n) × GL(1); g ↦ (g_{1}, a = ν(g))

is an isomorphism. Let ri ; \overline{ρ}, i = 1, 2, denote the composition of r\overline{ρ} with projection on the i-th factor in (IV.5.1.1). Thus Ker(r1 ; \overline{ρ}) determines an extension K^{1} of K with Galois group naturally a subgroup of GL(n, k); Ker(r^{2}(\overline{ρ})) determines the extension K(ζ_{ℓ}^{n−1}) of K, of degree [ℚ(ζ_{ℓ}^{n−1}) : ℚ] (cf. (IV.1.1.4) and (IV.1.1.7)). We consider the following conditions.

Hypotheses IV.5.2.

(a) K^{1} ∩ K(ζ_{ℓ}) = K.

(b) The group Im(\overline{ρ}) has no quotient of order ℓ.

(c) Let V ⊂ ad \overline{ρ} be an irreducible subrepresentation. Then there is s ∈ G_{K} such that r\overline{ρ}(s) has n distinct eigenvalues and such that ad (\overline{ρ})(s) has eigenvalue 1 on V.

Theorem IV.5.3. Assume Hypotheses IV.5.2. Then there is an integer r such that, for any m ≥ 1 there is a set Q_{m} satisfying the hypotheses of IV.1.4, and such that moreoever

(a) #Q_{m} = r;

(b) For all 𝔮 ∈ Q_{m} we have q = N𝔮 ≡ 1 (mod ℓ^{m});

(c) H_{Q_{m}^{∗}}^{1}(E, ad \overline{ρ}(1)) = 0.

(d) r\overline{ρ}(F rob_{𝔮}) has n distinct eigenvalues, and in particular a distinguished eigenvalue α_{𝔮} of multiplicity one.

Proof. We begin by recalling that, for any Q as in IV.1.4, and any 𝔮 ∈ Q, the subspace L_{Q,𝔮}^{⊥} ⊂ H^{1}(Z_{𝔮}, ad \overline{ρ}(1)) is defined by

H^{1}(Z_{𝔮}/I_{𝔮}, ad \overline{ρ}′_{α}(1))

in the notation of IV.2.1.2. In other words, L_{Q,𝔮}^{⊥} consists of unramified classes with trivial ad (\overline{ρ}_{α})(1)-component. Thus (IV.5.3.1)

H_{Q^{∗}}^{1}(E, ad r\overline{ρ}(1)) = Ker[H_{∅}^{1}(E, ad \overline{ρ}(1))→ ⊕_{𝔮∈Q_{m}} H^{1}(Z_{𝔮}/I_{𝔮}, ad \overline{ρ}_{α}(1))].

For r we take the dimension of H_{∅}^{1}(E, ad^{0}\overline{ρ}(1)). As in [TW,p. 567] we need to find sets Q_{m} satisfying conditions (a), (b), (d), and the hypotheses of IV.1.4, and such that the natural map

(IV.5.3.2)  H_{∅}^{1}(E, ad\overline{ρ}(1))→ ⊕_{𝔮∈Q_{m}} H^{1}(Z_{𝔮}/I_{𝔮}, ad(\overline{ρ}_{α})(1))

is injective, hence an isomorphism for dimension reasons. Condition (b) asserts that 𝔮 splits completely in K(ζ_{ℓ^{m}}).

Let [ψ] ∈ H_{∅}^{1}(E, ad\overline{ρ}(1)) be a non-zero class. The objective is to find 𝔮 as above satisfying condition (b), (d), and IV.1.4 and such that

(IV.5.3.3)  res_{𝔮}[ψ] ∈ H^{1}(Z_{𝔮}/I_{𝔮}, ad \overline{ρ}_{α}(1)) is nontrivial.

By Chebotarev density it thus suffices to find σ ∈ G_{E} such that

IV.5.3.4. (i) σ fixes E(ζ_{ℓ^{m}}); (ii) \overline{ρ}(σ) has n distinct eigenvalues; (iii) There is a distinguished eigenvalue α of \overline{ρ}(σ) such that ψ(σ) ∉ ad \overline{ρ}′_{α}(1)

where ad \overline{ρ}′_{α} ⊂ ad \overline{ρ} is the codimension one subspace defined with respect to α by analogy with IV.2.1.2.

Let E_{m} = E(ζ_{ℓ^{m}}), and let F_{m} denote the extension of E_{m} fixed by the kernel of ad \overline{ρ}. We claim ψ restricts to non-trivially to H_{∅^{∗}}^{1}(F_{m}, ad \overline{ρ}(1)). The kernel of the restriction map is H^{1}(Gal(F_{m}/E), ad \overline{ρ}(1)). It suffices to show

(IV.5.3.5)  H^{1}(Gal(F_{m}/E), ad \overline{ρ}(1)) = 0.

We argue as in [DDT], p. 84. The inflation-restriction sequence for F_{m} ⊃ F_{1} ⊃ E is an exact sequence

H^{1}(Gal(F_{1}/E), ad \overline{ρ}(1)^{G_{F_{1}}} ) ↪ H^{1}(Gal(F_{m}/E), ad \overline{ρ}(1))

→[H^{1}(Gal(F_{m}/F_{1}), ad \overline{ρ}(1))]^{G_{E}}.

Now G_{F_{1}} acts trivially on ad \overline{ρ}(1). Hence

[H^{1}(Gal(F_{m}/F_{1}), ad \overline{ρ}(1))]^{G_{E}} ≅ Hom(Gal(F_{m}/F_{1}), [ad \overline{ρ}(1))]^{G_{E}}).

Moreover, it follows from Condition IV.5.2 (a) that Gal(F_{1}/E) breaks up as the direct product Gal(F_{1}/F_{0}) × Gal(F_{0}/E). Thus

(IV.5.3.6)  [ad \overline{ρ}(1))]^{G_{E}} ⊂ [ad \overline{ρ}(1))]^{Gal(F_{1}/F_{0})} = \{0\}.

Indeed, Gal(F_{1}/F_{0}) acts on ad \overline{ρ}(1)) as a direct sum of copies of the natural action on the ℓth roots of unity. But Gal(F_{1}/F_{0}) can be identified with the subgroup of Aut(μ_{ℓ}) that acts trivially on μ_{ℓ}^{⊗(n−1)}. The hypothesis ℓ > n implies that this subgroup is non-trivial.

Thus the above exact sequence simplifies to yield

(IV.5.3.7)  H^{1}(Gal(F_{1}/E), ad \overline{ρ}(1)) ⟶^{∼} H^{1}(Gal(F_{m}/E), ad \overline{ρ}(1)).

On the other hand, applying the inflation restriction sequence for F_{1} ⊃ F_{0} ⊃ E to the left-hand side of (IV.5.3.7), we find

H^{1}(Gal(F_{0}/E), ad \overline{ρ}(1)^{Gal(F_{1}/F_{0})}) ↪ H^{1}(Gal(F_{1}/E), ad \overline{ρ}(1))

→[H^{1}(Gal(F_{1}/F_{0}), ad \overline{ρ}(1))]^{Gal(F_{0}/E)}.

Here the right-hand side vanishes because [F_{1} : F_{0}] is prime to ℓ, while the left-hand side vanishes as in (IV.5.3.6). This completes the verification of (IV.5.3.5).

Now it follows from IV.5.2 (a) and (b) that \overline{ρ} remains absolutely irreducible upon restriction to G_{E_{m}} for all m. Thus, to verify (IV.5.3.2), it suffices to find sets of height one primes of E_{m} satisfying conditions (b), (d), IV.1.4, and (IV.5.3.3), with E replaced by E_{m}. Conditions IV.1.4.1-2 are already satisfied, and IV.1.4.3 concerns only a finite set of primes, which we can avoid. We have

H_{∅}^{1}(F_{m}, ad r\overline{ρ}(1)) ⊂ Hom(G_{F_{m}}, ad r\overline{ρ}(1))

is the subset satisfying various ramification conditions. Thus let ψ ∈ H_{∅}^{1}(E_{m}, ad r\overline{ρ}(1)). Its restriction to F_{m} is a homomorphism from G_{F_{m}} to ad r\overline{ρ} whose image is a Gal(F_{m}/E_{m})-submodule, say V_{ψ}. Moreover, Gal(F_{m}/E_{m}) = Gal(F_{0}/E) by IV.5.2 (a). Let s ∈ Gal(F_{m}/E_{m}) satisfy the conditions of IV.5.2 (c), and let σ_{0} be a lifting of s to G_{E_{m}}. It already satisfies conditions (i) and (ii) of IV.5.3.4, and so does σ = τσ_{0} for any τ ∈ G_{F_{m}}. It remains to show that we can choose α and τ so that σ satisfies condition (iii). Now the eigenvalues of ad r\overline{ρ}(s) are of the form α_{i} · α_{j}^{−1}, where α_{i}, i = 1, …, n are the n distinct eigenvalues of r\overline{ρ}(s). Let v_{ij} be the corresponding eigenvectors. By hypothesis IV.5.2 (c) the fixed subspace V_{ψ}^{s} is non-trivial and is spanned by r non-trivial linear combinations v_{k} = ∑_{i} a_{ik}v_{ii}, 1 ≤ k ≤ r. Now ψ(σ) = ψ(τ) + ψ(σ_{0}). Write ψ(σ_{0}) = ∑ b_{ij}v_{ij}, ψ(τ) = ∑ c_{k}(τ)v_{k} + v′, where v′ is a linear combination of the v_{ij} with i ≠ j. Thus the coefficient of v_{ii} in ψ(σ) is

b_{i}(τ) = ∑ c_{k}(τ)a_{ik} + b_{ii}.

But we may vary the c_{k}(τ) freely, and it is clear that by doing so we can arrange that at least one b_{i}(τ) is non-zero. Taking α = α_{i}, we then see that σ satisfies condition (iii). This completes the proof.

IV.6. Eliminating tame deformations

Let q be a rational prime, q ≠ ℓ, and let v be a prime of E dividing q. The maximal ℓ-power quotient I_{v,ℓ} of the inertia group I_{v} is isomorphic to ℤ_{ℓ}(1) as a module over Z_{v}/I_{v}, where the (1) denotes Tate twist. Let P^{ℓ} ⊂ I_{v} be the kernel of the canonical map to I_{v,ℓ}; it is a profinite group with pro-order prime to ℓ. Thus, for any Z_{v}-module M, the canonical inflation map H^{1}(Z_{v}/P^{ℓ}, M)→H^{1}(Z_{v}, M) is an isomorphism.

Now let (\overline{ρ}, V) be an n-dimensional semi-simple unramified representation of Z_{v} with coefficients in a finite field k of characteristic ℓ, and let M = ad \overline{ρ}.

Lemma IV.6.1. Suppose \overline{ρ} is trivial and Nv ≠ 1 (mod ℓ). Then the inflation map

(IV.6.2)  H^{1}(Z_{v}/I_{v}, M)→H^{1}(Z_{v}/P^{ℓ}, M)

is an isomorphism.

Proof. We use the inflation-restriction sequence for the inclusion of I_{v,ℓ} in Z_{v}/P^{ℓ}:

(IV.6.3)  \begin{aligned}0 → H^{1}(Z_{v}/I_{v}, M) → H^{1}(Z_{v}/P^{ℓ}, M) & → Hom(I_{v,ℓ}, M)^{Z_{v}/I_{v}} \\ & = Hom_{Z_{v}/I_{v}}(𝔽_{ℓ}(1), M) \end{aligned}

By our hypothesis, Z_{v}/I_{v} acts non-trivially on 𝔽_{ℓ}(1) but trivially on M. Thus the right-hand term in (IV.6.3) vanishes.

V. The main theorem

V.1. Representations on global Hecke algebras.

We now return to the language of automorphic forms on the group J. Fix ℓ, σ, 𝔯, Q and Λ_{W_{σ}} as in §I; ℓ is assumed unramified in K. Let K = ∏_{p} K_{p} be a level subgroup satisfying the hypotheses of I.2.8. We assume Q satisfies the hypotheses of IV.1.4 and Theorem IV.5.3. Denote by D_{K} the set of primes of ℚ divisible by primes of E that ramify in K. Let S_{bad} = SC ∪ Q ∪ D_{K} ∪ {q(𝔯), ℓ}. We define T^{S_{bad}} and the completed Hecke algebras 𝕋_{1}(Q) and 𝕋_{0}(Q) as in §III.2. Recall that 𝕋_{1}(Q) is attached to the level group K_{1,Q}^{[ℓ]} rather than K_{1,Q}.

For 𝔮 ∈ Q, we choose a lifting α̃_{𝔮} of the Frobenius eigenvalue α_{𝔮} to the ring O. Similarly, let β_{i,𝔮}, i = 1, …, n − 1, be the eigenvalues of Frobenius in \overline{ρ}_{β} at 𝔮, and let β̃_{i,𝔮} be liftings to characteristic zero. We define unramified characters χ_{i}, i = i, …, n of K_{𝔮}^{×} by setting

χ_{1}(ϖ) = α_{𝔮}; χ_{i}(ϖ) = β_{i−1,𝔮}, i = 1, …, n − 1.

Let \overline{ρ} be a residual representation as in §IV.1.1. Let 𝔪 = 𝔪(\overline{ρ}) be the maximal ideal of 𝕋_{1}(Q) generated by ℓ, by U_{𝔮} − α_{𝔮} and by

V_{i,𝔮} − b^{i}(α) i = 1, …, n − 1

for each 𝔮 ∈ Q, and by

trace ∧^{i} r\overline{ρ}(F rob_{𝔭}) − T_{i,𝔭} i = 1, …, n

for all divisors 𝔭 of p, as p runs through the set of primes not in S_{bad} that split in K_{0}. Here F rob_{𝔭} denotes geometric Frobenius, T_{i,𝔭} is the standard Hecke operator defined in §II.2, V_{i,𝔮} is defined by (II.2.2), and b^{i}(χ; χ_{1}) is the eigenvalue defined in II.2.3.

Note that 𝕋_{1}(Q) stabilizes the subspace A_{ℓ}(J, Λ_{W_{σ}}, K_{0,Q}) ⊂ A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]}). As in §III.2, we let L_{0,Q} (resp. L_{1,Q}) denote the localization of A_{ℓ}(J, Λ_{W_{σ}}, K_{0,Q}) (resp. A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}^{[ℓ]})) at 𝔪. Let 𝕋_{0,Q} (resp. 𝕋_{1,Q}) denote the Hecke algebra acting on L_{0,Q} (resp. L_{1,Q}) generated only by the unramified Hecke operators at primes p ∉ S_{bad} that split in K_{0}.

Hypothesis V.1.1. The space L_{0,∅} is non-trivial.

Hypothesis V.1.1 can be regarded as a strong form of Serre’s conjecture for the group J. We are assuming that the residual representation r\overline{ρ} is modular of minimal level. In particular we are completely setting aside the problem of lowering the level solved by Ribet in the case of GL(2). We let π denote an irreducible automorphic representation of J – with characteristic 0 coefficients – contained in the space A_{𝔪} of automorphic forms on J(A) generated by L_{0,Q}. By the Hecke eigenvalues of π we will mean the eigenvalues of the standard Hecke operators T_{𝔭}^{(i)}, for 𝔭 dividing p as above, acting on the space of K_{p}-fixed vectors. By definition of L_{0,Q}, if π′ is another irreducible automorphic representation contained in A_{𝔪} then the Hecke eigenvalues of π and those of π′ are congruent modulo ℓ.

Hypothesis V.1.2.

(i) The automorphic representation π is unramified at Q ∪ q(𝔯).

(ii) We assume 𝔯 has been chosen so that q(𝔯) splits completely in K and is not congruent to 1 (mod ℓ), and so that the Frobenius at 𝔯 acts trivially on the extension K^{1} of K (notation as in §IV.5).

(iii) The local component of π at 𝔮 is isomorphic to the induced representation I(χ).

The assumption that π is unramified at q(𝔯) implies that r\overline{ρ} is unramified at primes dividing q(𝔯), so the last condition of (ii) makes sense. Moreover, the characters χ_{i} are distinct (mod ℓ), by condition (d) of Theorem IV.5.3, so (iii) also makes sense.

Proposition V.1.3. For an appropriate choice of 𝔯:

1. There is a representation ρ_{0,Q} : G_{E} → G̃(𝕋_{0,Q}) which is a deformation of \overline{ρ} of type ∅ and satisfies trace ∧^{i} r_{ρ_{0,Q}}(F rob_{𝔭}) = T_{𝔭}^{(i)} for all divisors 𝔭 of rational primes not in S_{bad} that split in K_{0}, and for i = 1, …, n.

2. There is a representation ρ_{1,Q} : G_{E}→G̃(𝕋_{1,Q}) which is a deformation of \overline{ρ} of type Q and satisfies

trace ∧^{i} r_{ρ_{1,Q}}(F rob_{𝔭}) = T_{𝔭}^{(i)}

for all divisors 𝔭 of rational primes not in S_{bad} that split in K_{0} and for i = 1, …, n.

Moreover, the residual representations \overline{ρ}_{0,Q} and \overline{ρ}_{1,Q} mod 𝔪 are isomorphic and satisfy conditions IV.1.1.0-4.

Proof. The existence of the representations ρ_{?,Q}, with ? = 0 or 1, is a consequence of Proposition III.2.1. That the residual representations are isomorphic follows from the relation between Frobenius elements and Hecke operators at unramified split places (Theorem III.1.5) and Chebotarev density. Conditions IV.1.1.0 and IV.1.1.1 are obvious, IV.1.1.2 follows from Propositions III.1.7 and III.1.3 and Theorem III.1.5, fe.1.1.3 is Lemma III.3.10, and IV.1.1.4 is Corollary III.3.7.

It remains to prove that ρ_{0,Q} is a deformation of type ∅ and that ρ_{1,Q} is a deformation of type Q. First, ρ_{?,Q} is crystalline because each of its components is crystalline and because ℓ is assumed greater than n and unramified in K. The second condition of IV.1.5.1 follows from the corresponding fact for the residual representation, and from Nakayama’s lemma. Condition IV.1.5.3 for primes in SC is an immediate consequence of the last assertion of Proposition III.1.3. The only other possible ramification is at primes of E, and at primes above 𝔯. For the former set of primes, IV.1.5.3 follows easily from Theorem III.1.5 (since ℓ > n > 2 and the ramification group at such primes is of order 2). The non-ramification at 𝔯 follows from Lemma IV.6.1, in view of Hypothesis V.1.2 (ii).

This leaves IV.1.5.2. Let π be an automorphic representation contributing to 𝕋_{?,Q}. For 𝔮 ∈ Q, π_{𝔮} is in (a) or (c) of Proposition I.2.10 for ? = 0; for 𝕋_{1,Q} we also have case (b). We claim

(V.1.3.1)  Under hypothesis IV.5.3 (d), case (c) cannot occur.

Admit (V.1.3.1) for the moment. Theorem III.1.5 then implies that r_{ρ_{0,Q}} is unramified outside SC ∪ ℓ, hence is a deformation of type ∅. Now let M_{1,Q} denote the free rank n T_{1,Q}-module on which r_{ρ_{1,Q}} acts. Fix 𝔮 ∈ Q and a Frobenius element F_{𝔮} in the decomposition group Z_{𝔮}, and let M′ ⊂ M_{1,Q} be the submodule of eigenvectors for F_{𝔮} with eigenvalue congruent to α_{𝔮} (mod 𝔪). Then it follows from Corollary IV.5.10.3 and Nakayama’s lemma, and our hypothesis that α_{𝔮} has multiplicity one in the residual representation, that M′ is a free rank one direct summand of M_{1,Q} and that the quotient M_{1,Q}/M′ is unramified. This is what we need to verify IV.1.5.1.

The claim (V.1.3.1) is a consequence of the following Lemma.

Lemma V.1.4. Fix 𝔮 ∈ Q and suppose π′ is a cohomological automorphic representation of J(A) such that π′_{𝔮} has a K_{0,𝔮}-fixed vector but no GL(n, O_{𝔮})-fixed vector. Then π′ does not contribute to L_{0,Q}.

Proof. Assume π′ contributes to L_{0,Q}, and let π′_{𝔮} be its local component at 𝔮. It is a subquotient of the representation I(χ′) induced from some unramified n-tuple of characters χ′. Now (π′_{𝔮})^{K_{0,𝔮}} has a non-trivial U_{𝔮}-eigenspace V′ with eigenvalue α′, where α′ ≡ α (mod 𝔪_{O}). Moreover, the Hecke operators V_{i,𝔮} act on V′ with eigenvalues congruent to b^{i}(χ; χ_{1}) (mod 𝔪_{O}). It follows that the reductions (mod 𝔪_{O}) of I(χ′) and I(χ) have a common subquotient. But the characters χ_{i} are distinct (mod 𝔪_{O}), hence the reduction (mod 𝔪_{O}) of I(χ) is irreducible by Proposition II.1.6 (i). It follows that I(χ′) is irreducible, hence π′_{𝔮} is spherical.

Corollary V.1.5. For any set Q as in IV.1.4, there are surjective homomorphisms

ϕ_{0,Q} : R_{∅}→𝕋_{0,Q};
ϕ_{1,Q} : R_{Q}→𝕋_{1,Q}.

Proof. The existence of the homomorphisms ϕ_{0,Q} and ϕ_{1,Q} is a consequence of Proposition V.1.2 and the universal property of R_{Q}. To prove surjectivity it suffices to prove surjectivity mod 𝔪, by Nakayama’s lemma. But mod 𝔪 the generator T_{𝔭}^{(i)} of 𝕋_{?,Q} is given by trace ∧^{i} r\overline{ρ}(γ) for some γ ∈ Gal(\overline{E}/E). To show that T_{𝔭}^{(i)} is in the image of ϕ_{?,Q} mod 𝔪, it therefore suffices to recall that the coefficients of the characteristic polynomial of ρ_{Q}(γ) belong to R_{Q}. Indeed, it follows as in [W, p. 510] from [M, §1.8] and IV.2.8 that R_{Q} is generated by traces, but we do not need this fact.

We also have to show that U_{𝔮} and the V_{i,𝔮} are in the image of ϕ_{?,Q} mod 𝔪. But these are given respectively by χ_{𝔮}(F rob_{𝔮}) and by the coefficients of the characteristic polynomial of r′_{𝔮}(F rob_{𝔮}), in the notation of IV.1.5.2.

Via ϕ_{0,Q} (resp. ϕ_{1,Q}) the module L_{0,Q} (resp. L_{1,Q}) becomes an R_{∅}-module (resp. an R_{Q}-module). Write 𝕋 = 𝕋_{0,Q}, R = R_{∅}. We can finally state our main theorem.

Theorem V.1.6. Let \overline{ρ} : G_{E}→G̃(\overline{𝔽}_{l}) be a representation satisfying Hypotheses V.1.1 and IV.5.2. We assume ℓ > n and, for all v ∈ SC, we assume ℓ is banal for GL(n, E_{v}). Finally, assume 𝔯 has been chosen to satisfy Hypothesis V.1.2 (ii). Then the homomorphism

ϕ_{0,∅} : R→𝕋

is an isomorphism. Moreover, R (and therefore 𝕋) is a complete intersection of dimension zero, and L_{0,∅} is a free R-module.

The proof of this theorem, which occupies the folllowing section, is based on a recent improvement of the method of [TW], found independently and nearly simultaneously by Diamond and Fujiwara [D,Fu]. The interest of Theorem V.1.6, of course, lies in the following corollary:

Theorem V.1.7. Let \overline{ρ} : G_{E}→G̃(\overline{𝔽}_{l}) be a representation satisfying conditions the hypotheses of Theorem V.1.6. Then any deformation of \overline{ρ} of type ∅, as in IV.1.5.1-3, is modular. Specifically, suppose ρ : G_{E}→G̃(O′) is a deformation of \overline{ρ} of type ∅, where O′ is the ring of integers in a finite extension 𝕂′of the fraction field 𝕂 of O. Then there is an automorphic representation π′ of G such that ρ′ = ρ(π′).

Other corollaries are derived in §V.4.

V.2. Application of the theorem of Diamond and Fujiwara.

For each integer m > 0 we now choose a set Q_{m} as in Theorem IV.5.3. In particular, there is an integer r such that each Q_{m} consists of r elements. For each such Q = Q_{m} we have an R_{Q} module L_{1,Q} which is free and finite over ℤ_{ℓ}. Define Δ_{Q} as in IV.1.4, and for any ℤ_{ℓ}[Δ_{Q}]-module N let N_{Δ_{Q}} denote the module of coinvariants. Then we have:

Theorem V.2.1. ( [D, Lemma 2.1;Fu, Theorem 1.2]) Suppose that

(i) L_{0,∅} ≠ {0}; (ii) For each Q = Q_{m},

(a) R_{Q} can be topologically generated by r elements,

(b) L_{1,Q} is a free O[Δ_{Q}]-module and

(c) there are isomorphisms

(L_{1,Q})_{Δ_{Q}} ⟶^{∼} L_{0,∅}

as R_{Q}-modules, via the natural map R_{Q}→R = R_{∅}.

Then R is a complete intersection and L_{0,∅} is a free R-module. In particular R is isomorphic to its image 𝕋 in the endomorphism algebra of L_{0,∅}.

The theorem as stated here is closer to Fujiwara’s formulation. In order to apply Fujiwara’s axioms, we need

Lemma V.2.2. For each 𝔮 ∈ Q let δ_{𝔮} be a generator of Δ_{𝔮}. Let m_{Q} ⊂ O[Δ_{Q}] be the ideal generated by {δ_{𝔮}, 𝔮 ∈ Q}. Then the projection R_{Q}→R defines a natural isomorphism

R_{Q}/m_{Q} ⟶^{∼} R.

Proof. The quotient R_{Q}/m_{Q} is tautologically the universal ring for ℓ-adic deformations of \overline{ρ} of type Q on which the inertia groups at primes in Q act trivially. This is just R.

We turn to the proof of Theorem V.1.6. Condition (i) of Theorem V.2.1 is Hypothesis V.1.1. Condition (ii) (a) is a consequence of Theorems IV.2.2 and IV.5.3, combined with Proposition V.1.3. Theorem V.1.6 is now a consequence of Theorem V.2.1 and the following two propositions.

Proposition V.2.3. For each Q = Q_{m}, L_{1,Q} is a free O[Δ_{Q}]-module.

Proposition V.2.4. For each Q = Q_{m},

(L_{1,Q})_{Δ_{Q}} ⟶^{∼} L_{∅}

as R_{Q}-modules, via the natural map R_{Q}→R.

Proof of Proposition V.2.3. We return to the notation of §I.2. The hypotheses of Theorem V.1.6 imply that ℓ does not divide the order of K_{1,Q}/𝕂, hence the hypotheses of Corollary I.2.15 are verified in our situation. It follows from Corollary I.2.17 that A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}) is a free O[Δ_{Q}]-module. But the localization L_{1,Q} at 𝔪 is a direct summand in A_{ℓ}(J, Λ_{W_{σ}}, K_{1,Q}) (cf. [DDT, p. 98]), hence L_{1,Q} is also free over O[Δ_{Q}].

Proof of Proposition V.2.4. Corollary I.2.17 and Proposition V.2.3 imply that (L_{1,Q})_{Δ_{Q}} ⟶^{∼} L_{0,Q} as R_{Q}-modules. So we need to construct an R_{Q}-isomorphism L_{∅}→L_{0,Q}.

For each 𝔮 ∈ Q, let X_{∞,𝔮} be the operator constructed in Lemma II.1.11, and let X_{∞,Q} = ∏ 𝔮 ∈ QX_{∞,𝔮}. This defines a homomorphism

(V.2.3.1)  X_{∞,Q} : L_{∅}→L_{0,Q}

that commutes with all unramified Hecke operators outside S_{bad}, hence with 𝕋_{0,Q} and, via ϕ_{0,Q}, with R_{∅}. Each X_{∞,𝔮} is injective (mod ℓ) by construction, since it just takes the spherical vector to the selected U_{𝔮}-eigenvector. So to prove X_{∞,Q} defines an isomorphism it suffices to check that it becomes an isomorphism after tensoring with \overline{ℚ}_{ℓ}.

Now, Lemma V.1.3 implies that the J(A^{f})-representation generated by L_{0,Q}⊗\overline{ℚ}_{ℓ} is generated by the vectors spherical at 𝔮 for all 𝔮 ∈ Q. So it suffices to prove that, if π′ is an irreducible cohomological automorphic representation of J(A) contained in A_{𝔪} (cf. the discussion following Hypothesis V.1.1) and spherical at all 𝔮 ∈ Q, then the the operator X_{∞,𝔮} maps the spherical subspace of π′_{𝔮} onto the space of U_{𝔮}-eigenvectors with eigenvalue congruent (mod ℓ) to α_{𝔮}. But for any π′ ⊂ A_{𝔪}, it follows from IV.1.4.3 that the space of U_{𝔮}-eigenvectors of π′_{𝔮} with eigenvalue congruent (mod ℓ) to α_{𝔮}, is one-dimensional. Thus it suffices to show that the operator X_{∞,𝔮} is non-trivial on the spherical subspace, and this follows from the injectivity of (V.2.3.1).

Remark V.2.5. Looking more closely at the results in [D] and [Fu], it follows from Proposition V.2.3 that, with our choice of Q = Q_{m}, equality holds in Theorem IV.2.2. Tracing through the proof of Theorem IV.2.2, it follows that the +1-eigenspace of c in ad r\overline{ρ} must have dimension precisely equal to \frac{n(n−1)}{2} in IV.1.1.3. It follows that in Lemma III.3.4 the matrix B must be symmetric. We can thus improve Corollary III.3.7, and conclude that the composite ν ∘ ρ(π) : Gal(\overline{E}/E)→G̃(O) equals ω^{1−n}.

This provides an unexpected proof of the fact that the representations ρ(π) considered here correspond to Langlands parameters of the unitary group. Indeed, let Φ_{n} be the n × n anti-diagonal matrix with entries

(Φ_{n})_{ij} = (−1)^{1+i}δ_{i,n+1−j}.

Define the L-group ^{L}G by analogy with the definition of G̃ in (I.1.4-6); we replace G̃^{0} by

Ĝ = {g = (g_{1}, g_{2}) ∈ GL(n) × GL(n) | ∃a ∈ GL(1) such that g_{2} = a · Φ_{n} ·^{t}g_{1}^{−1}Φ_{n}^{−1}}.

Let r̂_{ρ}(π) : G_{K}→Ĝ be defined by

r̂_{ρ}(π)(g) = (r_{ρ}(π)(g), Φ_{n} · (r_{ρ}(π)^{c}(g)Φ_{n}^{−1} · ω^{1−n}(g)).

It follows from Lemma 15.1.2 of [R1], as corrected on p. 419 of [R2], that the existence of an L-homomorphism ^{L}ρ(π) : G_{E}→^{L}G, extending r̂_{ρ}(π), is equivalent to the symmetry of the matrix B.

V.3. Examples for Hypotheses IV.5.2.

Our hypotheses IV.5.2 on the image of the residual representation are certainly stronger than necessary. We chose them for convenience, as the shortest list of conditions we found that guarantee existence of the sets Q_{m} and the prime 𝔯.

Hypothesis IV.5.2 (a) can in some instances be checked purely locally. For example, this is the case if the local Galois extension generated by r\overline{ρ} at some v ∈ SC is disjoint from E_{v}(ζ_{ℓ}). Hypothesis IV.5.2 (b) seems the most unnatural, and ought to be a consequence of a more general condition.

If the complement Ad(\overline{ρ})^{0} of the trace subspace of Ad \overline{ρ} is absolutely irreducible, then IV.5.2 (c) follows immediately from Lemma IV.4.9. We do not have a good criterion for irreducibility of Ad(\overline{ρ})^{0}. On the other hand, for applications, we are interested in cases when Ad(\overline{ρ})^{0} is highly reducible. The following Lemma will be applied to the problem of constructing automorphic tensor products.

Lemma V.3.1. Let G be a group, H ⊂ G a normal subgroup of index b, with G/H cyclic of order b, and let τ : G→GL(a, k), χ : H→GL(1, k) be representations of degree a and 1, respectively. Suppose r\overline{ρ} = τ ⊗ Ind_{H}^{G}χ is absolutely irreducible and that the map G→Im(τ)×Im(Ind_{H}^{G}χ) is surjective. Suppose Ad τ satisfies the analogue of Hypothesis IV.5.2(c), and suppose the elements s with distinct eigenvalues in τ have eigenvalues of order prime to b. Then Ad r\overline{ρ} satisfies Hypothesis IV.5.2 (c).

Proof. Let D be a set of coset representatives for H in G, and for δ ∈ D let χ^{δ} be the δ-conjugate of χ. Our hypotheses imply that the χ^{δ} are all distinct. Then we see easily that

(V.3.2)  Ad r\overline{ρ} ⟶^{∼} Ad τ ⊗ ⨁_{δ∈D} Ind_{H}^{G}χ^{1−δ},

Say Ad τ = ⊕_{i∈I}V_{i}. For any δ ∈ D, let G(δ) ⊃ H be the stabilizer of χ^{1−δ}, and let X(δ) denote the set of extensions of χ^{1−δ} to a character of G(δ), so that

Ind_{H}^{G}χ^{1−δ} = ⊕_{x∈X(δ)}Ind_{G(δ)}^{G}x.

Then the irreducible summands of Ad r\overline{ρ} are indexed by \{(i, δ, x)|i ∈ I, δ ∈ D, x ∈ X(δ)}. Let G_{1} = Im(τ), G_{2} = Im(Ind_{H}^{G}χ). For each triple (i, δ, x), it suffices to find pairs (s_{1}, s_{2}) ∈ G_{1} × G_{2} where s_{1} satisfies Hypothesis IV.5.2 (c) for V_{i} with eigenvalues of order prime to b, and s_{2} satisfies Hypothesis IV.5.2 (c) for Ind_{G(δ)}^{G}x. But the existence of s_{1} is given, and we can take always take s_{2} to be a generator of G/H.

The awkward hypothesis about the orders of the eigenvalues is easy to verify in examples.

V.4. Examples of automorphic tensor products, and other applications

We now introduce two positive integers, n and n′, and define unitary similitude groups G_{n} and G_{n′} attached to division algebras D_{n} and D_{n′} of dimension n^{2} and (n′)^{2}, respectively, over K, as in I.1. Let π and π′ be cohomological automorphic representations of G_{n} and G_{n′}, respectively. Let Π and Π′ denote the base changes to GL(n, K) and GL(n′, K), respectively; these are the representations denoted BC(π)_{1} and BC(π′)_{1} in I.3.

We suppose ℓ > n · n′. Define the modular representations r\overline{ρ}(π) and r\overline{ρ}(π′) over an appropriate finite field k of characteristic ℓ. Our hypothesis on ℓ ensures that both r\overline{ρ}(π) and r\overline{ρ}(π′) are Fontaine-Laffaille representations, with (Hodge-Tate) weights (a_{1} < ⋯ < a_{n}) and (b_{1} < ⋯ < b_{n′}), respectively. Suppose

(V.4.1)  {a_{i} + b_{j} | 1 ≤ i ≤ n; 1 ≤ j ≤ n′} = {0, 1, …, n · n′ − 1}.

For example, this is the case if π has cohomology with respect to the trivial representation, so \{a_{1}, …, a_{n}\} = \{0, …, n − 1\}, and if \overline{𝔟}_{j} = (n − 1)j + 1 for all j. Next, we suppose there is a cyclic extension K′ of K of degree n′, a complete local W (k)-algebra O with residue field k, and an algebraic Hecke character χ of K′, with values in O, such that

(V.4.2)  r\overline{ρ}(π′) ≅ Ind_{K′/K}\overline{χ}.

Here \overline{χ} is the reduction of (the λ-adic representation associated to) χ modulo the maximal ideal of O.

For any cyclic extension K′/K of number fields and any integer n, we let BC_{K′/K} (resp. AI_{K′/K}) denote the base change (resp. automorphic induction) map of Arthur-Clozel [AC] from automorphic representations of GL(n, K) to automorphic representations of GL(n, K′) (resp. from automorphic representations of GL(n, K′) to automorphic representations of GL(n[K′ : K], K′). The same notation is used for cyclic extensions of local fields. We make the following additional hypotheses.

Hypotheses V.4.2. (a) Let π(χ) denote the automorphic induction of χ from K′ to K, as an automorphic representation of GL(n′, K). Then π(χ) is cuspidal and satisfies π(χ)^{c} ⟶^{∼} π̌(χ)).

(b) For every finite place v of K, Π_{v} is either unramified or supercuspidal. If Π_{v} is supercuspidal, then v is split over E, and π(χ)_{v} is also supercuspidal. At such places, ℓ is prime to |GL(n · n′, k(v))|.

(c) The set of places v at which Π_{v} is supercuspidal is non-empty and contains at least two elements non-conjugate over E if [E : ℚ] is odd and n · n′ is congruent to 2 (mod 4).

(d) If Π_{v} is unramified but v is ramified over E, let v_{0} denote the restriction of v to E. Then the automorphic induction AI_{K′/K}(BC_{K′/K}π_{v}) ⊗ χ_{v} descends to a representation π_{0} of the quasi-split unitary group G_{nn′,v_{0}} of degree n · n′ over E_{v_{0}} such that π_{0} is spherical with respect to the conjugacy class of very special maximal compact subgroups.

Theorem V.4.3. Suppose π, π′, χ, and ℓ satisfy Hypotheses V.4.2. Suppose r\overline{ρ}(π) and r\overline{ρ}(π′) satisfy (V.4.1), r\overline{ρ}(π) satisfies Hypothesis IV.5.2, and the tensor product r\overline{ρ}(π)⊗r\overline{ρ}(π′) satisfies Hypotheses IV.5.2 (a) and (b) and the hypotheses of Lemma V.3.1. Then there exists a cohomological cuspidal automorphic representation Π ⊠ Π′ of GL(n · n′, K) such that, for almost all places v of K, the local Euler factor of Π ⊠ Π′ satisfies

L(s, (Π ⊠ Π′)_{v}) = L(s, Π_{v} × Π′_{v}),

where the right-hand side is the local Rankin-Selberg convolution.

Proof. Let ρ = r_{ρ}(π) ⊗ r_{ρ}(π′), and let r_{ρ} denote the residual representation on GL(n · n′, k). We have set things up so that Π ⊠ Π(χ) =^{def} AI_{K′/K}(BC_{K′/K}π) ⊗ χ is a cuspidal automorphic representation of GL(n · n′, K), cohomological with respect to the trivial representation. Our hypotheses V.4.2 (a) and (b) guarantee that (Π ⊠ Π(χ))_{v} is either unramified or supercuspidal at all places v, and V.4.2 (c) guarantees the existence of a unitary similitude group G_{n·n′} of degree n · n′, as in §I.1, such that Π ⊠ Π(χ) descends to an automorphic representation Π′′ of G_{n·n′}. Lemma V.3.1 then implies that r_{ρ}(Π′′) satisfies Hypothesis IV.5.2. Moreover,

r_{ρ} ⟶^{∼} r_{ρ}(Π′′).

The theorem is thus a consequence of Theorem V.1.7.

In a later draft, we hope to include numerical examples of Theorem V.4.3. We will also indicate applications to non-solvable base change and automorphic induction, generalizing the results of Clozel and Hida for GL(2).

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