Modularity lifting theorems for Galois representations of unitary type
aa r X i v : . [ m a t h . N T ] J u l MODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OFUNITARY TYPE
LUCIO GUERBEROFF
Abstract.
We prove modularity lifting theorems for ℓ -adic Galois representations of any dimension satisfy-ing a unitary type condition and a Fontaine-Laffaille condition at ℓ . This extends the results of Clozel, Harrisand Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved byDiamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesseon stable base change and descent from unitary groups to GL n . Introduction
The goal of this paper is to prove modularity lifting theorems for Galois representations of any dimensionsatisfying certain conditions. We largely follow the articles [CHT08] and [Tay08], where an extra local con-dition appears. In this work we remove that condition, which can be done thanks to the latest developmentsof the trace formula. More precisely, let F be a totally imaginary quadratic extension of a totally real field F (cid:0) . Let Π be a cuspidal automorphic representation of GL n p A F q satisfying the following conditions. (cid:13) There exists a continuous character χ : A (cid:2) F (cid:0){p F (cid:0)q(cid:2) Ñ C (cid:2) such that χ v p(cid:1) q is independent of v |8 and Π _ (cid:21) Π c b p χ (cid:5) N F { F (cid:0) (cid:5) det q . (cid:13) Π is cohomological.Here, c is the non-trivial Galois automorphism of F { F (cid:0) , and cohomological means that Π has the sameinfinitesimal character as an algebraic, finite dimensional, irreducible representation of p Res F { Q GL n qp C q .Let ℓ be a prime number, and ι : Q ℓ (cid:18)ÝÑ C an isomorphism. Then there is a continuous semisimple Galoisrepresentation r ℓ,ι p Π q : Gal p F { F q Ñ GL n p Q ℓ q which satisfies certain expected conditions. In particular, for places v of F not dividing ℓ , the restriction r ℓ,ι p Π q| Gal p F v { F v q to a decomposition group at v should be isomorphic, as a Weil-Deligne representation, tothe representation corresponding to Π v under a suitably normalized local Langlands correspondence. Theconstruction of the Galois representation r ℓ,ι p Π q under these hypotheses is due to Clozel, Harris and Labesse([CHLa, CHLb]), Chenevier and Harris ([CH]), and Shin ([Shi]), although they only match the Weil partsand not the whole Weil-Deligne representation. In the case that Π satisfies the additional hypothesis thatΠ v is a square integrable representation for some finite place v , Taylor and Yoshida have shown in [TY07]that the corresponding Weil-Deligne representations are indeed the same, as expected. Without the squareintegrable hypothesis, this is proved by Shin in [Shi] in the case where n is odd, or when n is even and thearchimedean weight of Π is ’slightly regular’, a mild condition we will not recall here. We will not need thisstronger result for the purposes of our paper.We use the instances of stable base change and descent from GL n to unitary groups, proved by Labesse([Lab]) to attach Galois representations to automorphic representations of totally definite unitary groups. Keywords: modularity, Galois representations, unitary groups.The author was partially supported by a CONICET fellowship and a fellowship from the Ministerio de Educaci´on deArgentina & Ambassade de France en Argentine.
In this setting, we prove an R red (cid:16) T theorem, following the development of the Taylor-Wiles method usedin [Tay08]. Finally, using the results of Labesse again, we prove our modularity lifting theorem for GL n . Wedescribe with more detail the contents of this paper.Section 1 contains some basic preliminaries. We include some generalities about smooth representationsof GL n of a p -adic field, over Q ℓ or F ℓ , which will be used later in the proof of the main theorem. We notethat many of the results of this section are also proved in [CHT08], although in a slightly different way. Westress the use of the Bernstein formalism in our proofs; some of them are based on an earlier draft [HT] of[CHT08].In Section 2, we develop the theory of ( ℓ -adic) automorphic forms on totally definite unitary groups,and apply the results of Labesse and the construction mentioned above to attach Galois representations toautomorphic representations of unitary groups.In Section 3, we study the Hecke algebras of unitary groups and put everything together to prove the mainresult of the paper. More precisely, if T denotes the (localized) Hecke algebra and R is a certain universaldeformation ring of a mod ℓ Galois representation attached to T , we prove that R red (cid:16) T . In Section 4, we goback to GL n and use this result to prove the desired modularity lifting theorems. The most general theoremwe prove for imaginary CM fields is the following. For the terminology used in the different hypotheses, werefer the reader to the main text. Theorem.
Let F (cid:0) be a totally real field, and F a totally imaginary quadratic extension of F (cid:0) . Let n ¥ be an integer and ℓ ¡ n be a prime number, unramified in F . Let r : Gal p F { F q ÝÑ GL n p Q ℓ q be a continuous irreducible representation with the following properties. Let r denote the semisimplificationof the reduction of r .(i) r c (cid:21) r _p (cid:1) n q .(ii) r is unramified at all but finitely many primes.(iii) For every place v | ℓ of F , r | Γ v is crystalline.(iv) There is an element a P p Z n, (cid:0)q Hom p F, Q ℓ q such that (cid:13) for all τ P Hom p F (cid:0) , Q ℓ q , we have either ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ or ℓ (cid:1) (cid:1) n ¥ a τc, ¥ (cid:4) (cid:4) (cid:4) ¥ a τc,n ¥ (cid:13) for all τ P Hom p F, Q ℓ q and every i (cid:16) , . . . , n , a τc,i (cid:16) (cid:1) a τ,n (cid:0) (cid:1) i . (cid:13) for all τ P Hom p F, Q ℓ q giving rise to a prime w | ℓ , HT τ p r | Γ w q (cid:16) t j (cid:1) n (cid:1) a τ,j u nj (cid:16) . In particular, r is Hodge-Tate regular.(v) F ker p ad r q does not contain F p ζ ℓ q .(vi) The group r p Gal p F { F p ζ ℓ qqq is big.(vii) The representation r is irreducible and there is a conjugate self-dual, cohomological, cuspidal auto-morphic representation Π of GL n p A F q , of weight a and unramified above ℓ , and an isomorphism ι : Q ℓ (cid:18)ÝÑ C , such that r (cid:21) r ℓ,ι p Π q .Then r is automorphic of weight a and level prime to ℓ . We make some remarks about the conditions in the theorem. Condition (i) says that r is conjugate self-dual, and this is essential for the numerology behind the Taylor-Wiles method. Conditions (ii) and (iii) saythat the Galois representation is geometric in the sense of Fontaine-Mazur, although it says a little more. ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 3
It is expected that one can relax condition (iii) to the requirement that r is de Rham at places dividing ℓ .The stronger crystalline form, the hypothesis on the Hodge-Tate weights made in (iv) and the requirementthat ℓ ¡ n is unramified in F are needed to apply the theory of Fontaine and Laffaille to calculate the localdeformation rings. The condition that ℓ ¡ n is also used to treat non-minimal deformations. Condition(v) allows us to choose auxiliary primes to augment the level and ensure that certain level structures aresufficiently small. The bigness condition in (vi) is to make the Tchebotarev argument in the Taylor-Wilesmethod work. Hypothesis (vii) is, as usual, essential to the method. An analogous theorem can be provedover totally real fields. Acknowledgements.
I take this opportunity to express my gratitude to my thesis adviser Michael Harrisfor his many explanations and suggestions, and specially for his advices. I would also like to thank BrianConrad, Jean-Francois Dat, Alberto M´ınguez and Nicol´as Ojeda B¨ar for useful conversations. Finally, Iwould like to thank Roberto Miatello for his constant support and encouragement. I also thank the refereefor his corrections and helpful comments.0.
Some notation and definitions
As a general principle, whenever F is a field and F is a chosen separable closure, we write Γ F (cid:16) Gal p F { F q .We also write Γ F when the choice of F is implicit. If F is a number field and v is a place of F , we usuallywrite Γ v € Γ F for a decomposition group at v . If v is finite, we denote by q v the order of the residue fieldof v .0.1. Irreducible algebraic representations of GL n . Let Z n, (cid:0) denote the set of n -tuples of integers a (cid:16) p a , . . . , a n q such that a ¥ (cid:4) (cid:4) (cid:4) ¥ a n . Given a P Z n, (cid:0) , there is a unique irreducible, finite dimensional, algebraic representation ξ a : GL n Ñ GL p W a q over Q with highest weight given by diag p t , . . . , t n q ÞÑ n ¹ i (cid:16) t a i i . Let E be any field of characteristic zero. Tensoring with E , we obtain an irreducible algebraic representation W a,E of GL n over E , and every such representation arises in this way. Suppose that E { Q is a finite extension.Then the irreducible, finite dimensional, algebraic representations of p Res E { Q GL n { E qp C q are parametrizedby elements a P p Z n, (cid:0)q Hom p E, C q . We denote them by p ξ a , W a q .0.2. Local Langlands correspondence.
Let p be a rational prime and let F be a finite extension of Q p .Fix an algebraic closure F of F . Fix also a positive integer n , a prime number ℓ (cid:24) p and an algebraic closure Q ℓ of Q ℓ . Let Art F : F (cid:2) Ñ Γ ab F be the local reciprocity map, normalized to take uniformizers to geometricFrobenius elements. If π is an irreducible smooth representation of GL n p F q over Q ℓ , we will write r ℓ p π q forthe ℓ -adic Galois representation associated to the Weil-Deligne representation L p π b | |p (cid:1) n q{ q , where L denotes the local Langlands correspondence, normalized to coincide with the correspondence in-duced by Art F in the case n (cid:16)
1. Note that r ℓ p π q does not always exist. The eigenvalues of L p π b| |p n (cid:1) q{ qp φ F q must be ℓ -adic units for some lift φ F of the geometric Frobenius (see [Tat79]). Wheneverwe make a statement about r ℓ p π q , we will suppose that this is the case. Note that our conventions differfrom those of [CHT08] and [Tay08], where r ℓ p π q is defined to be the Galois representation associated to L p π _ b | |p (cid:1) n q{ q . LUCIO GUERBEROFF
Hodge-Tate weights.
Fix a finite extension L { Q ℓ and an algebraic closure L of L . Fix an algebraicclosure Q ℓ of Q ℓ and an algebraic extension K of Q ℓ contained in Q ℓ such that K contains every Q ℓ -embedding L ãÑ Q ℓ . Suppose that V is a finite dimensional K -vector space equipped with a continuous linear action ofΓ L . Let B dR be the ring of p -adic periods, as in [Ast94]. Then p B dR b Q ℓ V q Γ L is an L b Q ℓ K -module. Wesay that V is de Rham if this module is free of rank equal to dim K V . Since L b Q ℓ K (cid:20) p K q Hom Q ℓ p L,K q , if V is a K -representation of Γ L , we have that p B dR b Q ℓ V q Γ L (cid:20) ± τ P Hom Q ℓ p L,K qp B dR b Q ℓ V q Γ L b L b Q ℓ K,τ b K (cid:20) ± τ P Hom Q ℓ p L,K qp B dR b L,τ V q Γ L . It follows that V is de Rham if and only ifdim K p B dR b L,τ V q Γ L (cid:16) dim K V for every τ P Hom Q ℓ p L, K q . We use the convention of Hodge-Tate weights in which the cyclotomic characterhas 1 as its unique Hodge-Tate weight. Thus, for V de Rham, we let HT τ p V q be the multiset consisting ofthe elements q P Z such that gr (cid:1) q p B dR b L,τ V q Γ L (cid:24)
0, with multiplicity equal todim K gr (cid:1) q p B dR b L,τ V q Γ L . Thus, HT τ p V q is a multiset of dim K V elements. We say that V is Hodge-Tate regular if for every τ P Hom Q ℓ p L, K q , the multiplicity of each Hodge-Tate weight with respect to τ is 1. We make analogousdefinitions for crystalline representations over K .0.4. Galois representations of unitary type.
Let F be any number field. If ℓ is a prime number, ι : Q ℓ (cid:18)ÝÑ C is an isomorphism and ψ : A (cid:2) F { F (cid:2) Ñ C (cid:2) is an algebraic character, we denote by r ℓ,ι p ψ q theGalois character associated to it by Lemma 4.1.3 of [CHT08].Let F (cid:0) be a totally real number field, and F { F (cid:0) a totally imaginary quadratic extension. Denote by c P Gal p F { F (cid:0)q the non-trivial automorphism. Let Π be an irreducible admissible representation of GL n p A F q .We say that Π is essentially conjugate self dual if there exists a continuous character χ : A (cid:2) F (cid:0){p F (cid:0)q(cid:2) Ñ C (cid:2) with χ v p(cid:1) q independent of v |8 such thatΠ _ (cid:21) Π c b p χ (cid:5) N F { F (cid:0) (cid:5) det q . If we can take χ (cid:16)
1, that is, if Π _ (cid:21) Π c , we say that Π is conjugate self dual .Let Π be an automorphic representation of GL n p A F q . We say that Π is cohomological if there existsan irreducible, algebraic, finite-dimensional representation W of Res F { Q GL n , such that the infinitesimalcharacter of Π is the same as that of W . Let a P p Z n, (cid:0)q Hom p F, C q , and let p ξ a , W a q the irreducible, finitedimensional, algebraic representation of p Res F { Q GL n qp C q with highest weight a . We say that Π has weight a if it has the same infinitesimal character as p ξ _ a , W _ a q .The next theorem (in the conjugate self dual case) is due to Clozel, Harris and Labesse ([CHLa, CHLb]),with some improvements by Chenevier and Harris ([CH]), except that they only provide compatibility ofthe local and global Langlands correspondences for the unramified places. Shin ([Shi]), using a very slightlydifferent method, obtained the identification at the remaining places. The slightly more general versionstated here for an essentially conjugate self dual representation is proved in Theorem 1.2 of [BLGHT09]. Let F be an algebraic closure of F and let Γ F (cid:16) Gal p F { F q . For m P Z and r : Γ F Ñ GL n p Q ℓ q a continuousrepresentation, we denote by r p m q the m -th Tate twist of r , and by r ss the semisimplification of r . Fix aprime number ℓ , an algebraic closure Q ℓ of Q ℓ , and an isomorpshim ι : Q ℓ (cid:18)ÝÑ C . Theorem 0.1.
Let Π be an essentially conjugate self dual, cohomological, cuspidal automorphic representa-tion of GL n p A F q . More precisely, suppose that Π _ (cid:21) Π c b p χ (cid:5) N F { F (cid:0) (cid:5) det q for some continuous character ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 5 χ : A (cid:2) F (cid:0){p F (cid:0)q(cid:2) Ñ C (cid:2) with χ v p(cid:1) q independent of v |8 . Then there exists a continuous semisimple repre-sentation r ℓ p Π q (cid:16) r ℓ,ι p Π q : Γ F Ñ GL n p Q ℓ q with the following properties.(i) For every finite place w ∤ ℓ , p r ℓ p Π q| Γ w q ss (cid:20) (cid:0) r ℓ p ι (cid:1) Π w q(cid:8) ss . (ii) r ℓ p Π q c (cid:21) r ℓ p Π q_p (cid:1) n q b r ℓ p χ (cid:1) q| Γ F .(iii) If w ∤ ℓ is a finite place such that Π w is unramified, then r ℓ p Π q is unramified at w .(iv) For every w | ℓ , r ℓ p Π q is de Rham at w . Moreover, if Π w is unramified, then r ℓ p Π q is crystalline at w .(v) Suppose that Π has weight a . Then for each w | ℓ and each embedding τ : F ãÑ Q ℓ giving rise to w , theHodge-Tate weights of r ℓ p Π q| Γ w with respect to τ are given by HT τ p r ℓ p Π q| Γ w q (cid:16) t j (cid:1) n (cid:1) a ιτ,j u j (cid:16) ,...,n , and in particular, r ℓ p Π q| Γ w is Hodge-Tate regular. The representation r ℓ,ι p Π q can be taken to be valued in the ring of integers of a finite extension of Q ℓ .Thus, we can reduce it modulo its maximal ideal and semisimplify to obtain a well defined continuoussemisimple representation r ℓ,ι p Π q : Γ F ÝÑ GL n p F ℓ q . Let a be an element of p Z n, (cid:0)q Hom p F, Q ℓ q . Let r : Γ F ÝÑ GL n p Q ℓ q be a continuous semisimple representation. We say that r is automorphic of weight a if there is an isomor-phism ι : Q ℓ (cid:18)ÝÑ C and an essentially conjugate self dual, cohomological, cuspidal automorphic representationΠ of GL n p A F q of weight ι (cid:6) a such that r (cid:21) r ℓ,ι p Π q . We say that r is automorphic of weight a and level primeto ℓ if moreover there exists such a pair p ι, Π q with Π ℓ unramified. Here ι (cid:6) a P p Z n, (cid:0)q Hom p F, C q is defined as p ι (cid:6) a q τ (cid:16) a ι (cid:1) τ .There is an analogous construction for a totally real field F (cid:0) . The definition of cohomological is the same,namely, that the infinitesimal character is the same as that of some irreducible algebraic finite dimensionalrepresentation of p Res F (cid:0){ Q GL n qp C q . Theorem 0.2.
Let Π be a cuspidal automorphic representation of GL n p A F (cid:0)q , cohomological of weight a ,and suppose that Π _ (cid:21) Π b p χ (cid:5) det q , where χ : A (cid:2) F (cid:0){p F (cid:0)q(cid:2) Ñ C (cid:2) is a continuous character such that χ v p(cid:1) q is independent of v |8 . Let ι : Q ℓ (cid:18)ÝÑ C . Then there is a continuous semisimple representation r ℓ p Π q (cid:16) r ℓ,ι p Π q : Γ F (cid:0) Ñ GL n p Q ℓ q with the following properties.(i) For every finite place v ∤ ℓ , p r ℓ p Π q| Γ v q ss (cid:20) (cid:0) r ℓ p ι (cid:1) Π v q(cid:8) ss . (ii) r ℓ p Π q (cid:21) r ℓ p Π q_p (cid:1) n q b r ℓ p χ (cid:1) q .(iii) If v ∤ ℓ is a finite place such that Π v is unramified, then r ℓ p Π q is unramified at v .(iv) For every v | ℓ , r ℓ p Π q is de Rham at v . Moreover, if Π v is unramified, then r ℓ p Π q is crystalline at v .(v) For each v | ℓ and each embedding τ : F (cid:0) ãÑ Q ℓ giving rise to v , the Hodge-Tate weights of r ℓ p Π q| Γ v with respect to τ are given by HT τ p r ℓ p Π q| Γ v q (cid:16) t j (cid:1) n (cid:1) a ιτ,j u j (cid:16) ,...,n , and in particular, r ℓ p Π q| Γ v is Hodge-Tate regular. LUCIO GUERBEROFF
Moreover, if ψ : A (cid:2) F (cid:0){p F (cid:0)q(cid:2) Ñ C (cid:2) is an algebraic character, then r ℓ p Π b p ψ (cid:5) det qq (cid:16) r ℓ p Π q b r ℓ p ψ q . Proof.
This can be deduced from the last theorem in exactly the same way as Proposition 4.3.1 of [CHT08]is deduced from Proposition 4.2.1 of loc. cit. (cid:3)
We analogously define what it means for a Galois representation of a totally real field to be automorphicof some weight a .1. Admissible representations of GL n of a p -adic field over Q ℓ and F ℓ Let p be a rational prime and let F be a finite extension of Q p , with ring of integers O F , maximal ideal λ F and residue field k F (cid:16) O F { λ F . Let q (cid:16) k F . Let ω be a generator of λ F . We will fix an algebraic closure F of F , and write Γ F (cid:16) Gal p F { F q . Corresponding to it, we have an algebraic closure k F of k F , and we willlet Frob F be the geometric Frobenius in Gal p k F { k F q and I F be the inertia subgroup of Γ F . Usually we willalso write Frob F for a lift to Γ F . Fix also a positive integer n , a prime number ℓ (cid:24) p , an algebraic closure Q ℓ of Q ℓ and an algebraic closure F ℓ of F ℓ . We will let R be either Q ℓ or F ℓ . Denote by | | : F (cid:2) Ñ q Z € Z r q s the absolute value normalized such that | ω | (cid:16) q (cid:1) . We denote by the same symbol the composition of | | and the natural map Z r q s Ñ R , which exists because q is invertible in R . For the general theory of smoothrepresentations over R , we refer the reader to [Vig96]. Throughout this section, representation will alwaysmean smooth representation.For a locally compact, totally disconnected group G , a compact open subgroup K € G and an element g P G , we denote by r KgK s the operator in the Hecke algebra of G relative to K corresponding to the( R -valued) characteristic function of the double coset KgK .Given a tuple t (cid:16) p t p q , . . . , t p n qq of elements in any ring A , we denote by P q, t P A r X s the polynomial P q, t (cid:16) X n (cid:0) n ¸ j (cid:16) p(cid:1) q j q j p j (cid:1) q{ t p j q X n (cid:1) j . We use freely the terms Borel, parabolic, Levi, and so on, to refer to the F -valued points of the correspond-ing algebraic subgroups of GL n . Write B for the Borel subgroup of GL n p F q consisting of upper triangularmatrices, and B (cid:16) B X GL n p O F q . Let T (cid:20) p F (cid:2)q n be the standard maximal torus of GL n p F q . Let N be thegroup of upper triangular matrices whose diagonal elements are all 1. Then B (cid:16) T N (semi-direct product).Let r : GL n p O F q Ñ GL n p k F q denote the reduction map. We introduce the following subgroups of GL n p O F q : (cid:13) U (cid:16) t g P GL n p O F q : r p g q (cid:16) (cid:2) (cid:6) n (cid:1) ,n (cid:1) (cid:6) n (cid:1) , ,n (cid:1) (cid:6) (cid:10)u ; (cid:13) U (cid:16) t g P GL n p O F q : r p g q (cid:16) (cid:2) (cid:6) n (cid:1) ,n (cid:1) (cid:6) n (cid:1) , ,n (cid:1) (cid:10)u ; (cid:13) Iw (cid:16) t g P GL n p O F q : r p g q is upper triangular u ; (cid:13) Iw (cid:16) t g P Iw : r p g q ii (cid:16) i (cid:16) , . . . , n u .Thus, U is a normal subgroup of U and we have a natural identification U { U (cid:20) k (cid:2) F , and similarly, Iw is a normal subgroup of Iw and we have a natural identificationIw { Iw (cid:20) p k (cid:2) F q n . We denote by H the R -valued Hecke algebra of GL n p F q with respect to GL n p O F q . We do not include R in the notation. For every smooth representation π of GL n p F q , π GL n p O F q is naturally a left module over H . For j (cid:16) , . . . , n , we will let T p j q F P H denote the Hecke operator (cid:18) GL n p O F q (cid:2) ω j
00 1 n (cid:1) j (cid:10) GL n p O F q(cid:26) . ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 7
Let π be a representation of GL n p F q over Q ℓ . We say that π is essentially square-integrable if, underan isomorphism Q ℓ (cid:21) C , the corresponding complex representation is essentially square integrable in theusual sense. It is a non trivial fact that the notion of essentially square integrable complex representation isinvariant under an automorphism of C , which makes our definition independent of the chosen isomorphism Q ℓ (cid:21) C . This can be shown using the Bernstein-Zelevinsky classification of essentially square integrablerepresentations in terms of quotients of parabolic inductions from supercuspidals (see below).Let n (cid:16) n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n r be a partition of n and P (cid:129) B the corresponding parabolic subgroup of GL n p F q .The modular character δ P : P Ñ Q (cid:2) takes values in q Z € R (cid:2) . Choosing once and for all a square root of q in R , we can consider the square root character δ { P : P Ñ R (cid:2) . For each i (cid:16) , . . . r , let π i be a representationof GL n i p F q . We denote by π (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) π r the normalized induction from P to GL n p F q of the representation π b (cid:4) (cid:4) (cid:4) b π r . Whenever we write | | we will mean | | (cid:5) det. For any R -valued character β of F (cid:2) and anypositive integer m , we denote by β r m s the one dimensional representation β (cid:5) det of GL m p F q .Suppose that R (cid:16) Q ℓ . Let n (cid:16) rk and σ be an irreducible supercuspidal representation of GL r p F q . By atheorem of Bernstein ([Zel80, 9.3]), (cid:1) σ b | | (cid:1) k (cid:9) (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) (cid:1) σ b | | k (cid:1) (cid:9) has a unique irreducible quotient denoted St k p σ q , which is essentially square integrable. Moreover, everyirreducible, essentially square integrable representation of GL n p F q is of the form St k p σ q for a unique pair p k, σ q . Under the local Langlands correspondence L , St k p σ q corresponds to Sp k b L p σ b | | (cid:1) k q (see page252 of [HT01] or Section 4.4 of [Rod82]), where Sp k is as in [Tat79, 4.1.4]. Suppose now that n (cid:16) n (cid:0)(cid:4) (cid:4) (cid:4)(cid:0) n r and that π i is an irreducible essentially square integrable representation of GL n i p F q . Then π (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) π r has a distinguished constituent appearing with multiplicity one, called the Langlands subquotient, which wedenote by π ` (cid:4) (cid:4) (cid:4) ` π r . Every irreducible representation of GL n p F q over Q ℓ is of this form for some partition of n , and the π i arewell determined modulo permutation ([Zel80, 6.1]). The π i can be ordered in such a way that the Langlandssubquotient is actually a quotient of the parabolic induction.If χ , . . . , χ n are unramified characters then χ ` (cid:4) (cid:4) (cid:4) ` χ n is the unique unramified constituent of χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n , and every irreducible unramified representation ofGL n p F q over Q ℓ is of this form. Let π be such a representation, corresponding to a Q ℓ -algebra morphism λ π : H Ñ Q ℓ . For j (cid:16) , . . . , n , let s j denote the j -th elementary symmetric polynomial in n variables. Ifwe define unramified characters χ i : F (cid:2) Ñ Q (cid:2) ℓ in such a way that λ π p T p j q F q (cid:16) q j p n (cid:1) j q{ s j p χ p ω q , . . . , χ n p ω qq , then π (cid:20) χ ` (cid:4) (cid:4) (cid:4) ` χ n . Moreover, by the Iwasawa decomposition GL n p F q (cid:16) B GL n p O F q , we have that dim Q ℓ π GL n p O F q (cid:16)
1. Wedenote t π (cid:16) p λ π p T p q F q , . . . , λ π p T p n q F qq . Lemma 1.1.
Let π be an irreducible unramified representation of GL n p F q over Q ℓ . Then the characteristicpolynomial of r ℓ p π qp Frob F q is P q, t π .Proof. Suppose that π (cid:16) χ ` (cid:4) (cid:4) (cid:4) ` χ n . Then r ℓ p π q (cid:16) n à i (cid:16) p χ i b | |p (cid:1) n q{ q (cid:5) Art (cid:1) F . LUCIO GUERBEROFF
Thus, the characteristic polynomial of r ℓ p π qp Frob F q is n ¹ i (cid:16) p X (cid:1) χ i p ω q q p n (cid:1) q{ q (cid:16) n ¸ j (cid:16) p(cid:1) q j s j p χ p ω q q p n (cid:1) q{ , . . . , χ n p ω q q p n (cid:1) q{ q X n (cid:1) j (cid:16) P q, t π . (cid:3) Let n (cid:16) n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n r be a partition of n and let β , . . . β r be distinct unramified F ℓ -valued characters of F (cid:2) . Suppose that q (cid:17) p mod ℓ q . Then the representation β r n s (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) β r r n r s is irreducible and unramified,and every irreducible unramified F ℓ -representation of GL n p F q is obtained in this way. This is proved byVigneras in [Vig98, VI.3]. Moreover, if π (cid:16) β r n s(cid:2)(cid:4) (cid:4) (cid:4)(cid:2) β r r n r s , then π is an unramified subrepresentation ofthe principal series β (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) β (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) β r (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) β r , where β i appears n i times. The Iwasawa decompositionimplies that the dimension of the GL n p O F q -invariants of this unramified principal series is one, and thus thesame is true for π .A character χ of F (cid:2) is called tamely ramified if it is trivial on 1 (cid:0) λ F , that is, if its conductor is ¤
1. Inthis case, χ | O (cid:2) F has a natural extension to U , which we denote by χ . Lemma 1.2.
Let χ , . . . , χ n be R -valued characters of F (cid:2) such that χ , . . . , χ n (cid:1) are unramified and χ n istamely ramified. Then dim R Hom U p χ n , χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q (cid:16) " n if χ n is unramified otherwise.Furthermore, if χ n is ramified then p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U (cid:16) .Proof. Let M p χ n q (cid:16) t f : GL n p O F q Ñ R : f p bku q (cid:16) χ p b q χ n p u q f p k q b P B , k P GL n p O F q , u P U u , where we write χ for the character of p F (cid:2)q n given by χ , . . . , χ n . Then, Hom U p χ n , χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q (cid:16) p χ (cid:2)(cid:4) (cid:4) (cid:4)(cid:2) χ n q U (cid:16) χ n , which by the Iwasawa decomposition is isomorphic to M p χ n q . By the Bruhat decomposition, B z GL n p O F q{ U (cid:20) r p B qz GL n p k F q{ r p U q (cid:20) W n { W n (cid:1) , where W j is the Weyl group of GL j with respect to its standard maximal split torus. Here we see W n (cid:1) inside W n in the natural way. Let X denote a set of coset representatives of W n { W n (cid:1) , so thatGL n p O F q (cid:16) º w P X B wU . Thus, if f P M p χ n q , f is determined by its restriction to the cosets B wU . We have that M p χ n q (cid:20) ¹ w P X M w , where M w is the space of functions on B wU satisfying the transformation rule of M p χ n q . It is clear thatdim R M w ¤ w . Moreover, if χ n is unramified, then M w is non-zero, a non-zero function beinggiven by f p w q (cid:16)
1. Thus, in this case, dim R M p χ n q (cid:16) n .In the ramified case, let a (cid:16) diag p a , . . . , a n q P B , with a i P O (cid:2) F and a n such that χ n p a n q (cid:24)
1. Then χ n p a n q f p w q (cid:16) f p aw q (cid:16) f p wa w q (cid:16) χ n p a w q f p w q (cid:16) f p w q unless w P W n (cid:1) . Thus, only the identity coset survives, and dim R M p χ n q (cid:16) f P p χ (cid:2) (cid:4) (cid:4) (cid:4)(cid:2) χ n q be U -invariant. To see that it is zero, it is enough to see that f p w q (cid:16) w P X . Choosing a P GL n p O F q to be a scalar matrix corresponding to an element a P O (cid:2) F for which χ n p a q (cid:24)
1, we see that a is in B (and hence in U ), thus f p aw q (cid:16) χ n p a q f p w q (cid:16) f p wa q (cid:16) f p w q , so f p w q (cid:16) w P X . (cid:3) ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 9
Let P M denote the parabolic subgroup of GL n p F q containing B corresponding to the partition n (cid:16)p n (cid:1) q (cid:0)
1, and let U M denote its unipotent radical. Take the Levi decomposition P M (cid:16) M U M , where M (cid:20) GL n (cid:1) p F q (cid:2) GL p F q . Consider the opposite parabolic subgroup P M with Levi decomposition P M (cid:16) M U M .Let U ,M (cid:16) U X M (cid:20) GL n (cid:1) p O F q (cid:2) GL p O F q . Let χ n be a tamely ramified character of F (cid:2) , and let χ n be its extension to U . Let H M p χ n q (cid:16) End M p ind MU ,M χ n q , where ind denotes compact induction and χ n is viewed as a character of U ,M via projection to the lastelement of the diagonal. Thus, H M p χ n q can be identified with the R -vector space of compactly supportedfunctions f : M Ñ R such that f p kmk
1q (cid:16) χ n p k q f p m q χ n p k for m P M and k, k U ,M . Similarly, let H p χ n q (cid:16) End GL n p F qp ind GL n p F q U χ n q . This is identified with the R -vector space of compactly supported functions f : GL n p F q Ñ R such that f p kgk
1q (cid:16) χ n p k q f p g q χ n p k for every g P GL n p F q , k, k U . There is a natural injective homomorphism of R -modules T : H M p χ n q Ñ H p χ n q , which can be described as follows (see [Vig98, II.3]). Let m P M . Then T p U ,M mU ,M q (cid:16) U mU , where1 U ,M mU ,M is the function supported in U ,M mU ,M whose value at umu is χ n p u q χ n p u , and similarly for1 U mU . Define U (cid:0) (cid:16) U X U M and U (cid:1) (cid:16) U X U M . Then U (cid:16) U (cid:1) U ,M U (cid:0) (cid:16) U (cid:0) U ,M U (cid:1) , and χ n is trivial on U (cid:1) and U (cid:0) . Let M (cid:1) (cid:16) t m P M { m (cid:1) U (cid:0) m € U (cid:0) and mU (cid:1) m (cid:1) € U (cid:1) u . We denote by H (cid:1) M p χ n q the subspace of H M p χ n q consisting of functions supported on the union of cosets ofthe form U ,M mU ,M with m P M (cid:1) . Proposition 1.3.
The subspace H (cid:1) M p χ n q € H M p χ n q is a subalgebra, and the restriction T (cid:1) : H (cid:1) M p χ n q Ñ H p χ n q is an algebra homomorphism.Proof. This is proved in [Vig98, II.5]. (cid:3)
Let π be a representation of GL n p F q over R . Then Hom GL n p F qp ind GL n p F q U χ n , π q is naturally a rightmodule over H p χ n q . By the adjointness between compact induction and restriction,Hom GL n p F qp ind GL n p F q U χ n , π q (cid:16) Hom U p χ n , π q , and therefore the right hand side is also a right H p χ n q -module. There is an R -algebra isomorphism H p χ n q (cid:20) H p χ (cid:1) n q opp given by f ÞÑ f (cid:6) , where f (cid:6)p g q (cid:16) f p g (cid:1) q . We then see Hom U p χ n , π q as a left H p χ (cid:1) n q -module in this way. Similarly, Hom U ,M p χ n , π q is a left H M p χ (cid:1) n q -module when π is a represen-tation of M over R . For a representation π of GL n p F q , let π U M be the representation of M obtained by(non-normalized) parabolic restriction. Then the natural projection π Ñ π U M is M -linear. Remark . Let B n (cid:1) denote the subgroup of lower triangular matrices of GL n (cid:1) p F q , so that B n (cid:1) (cid:2) GL p F q is a parabolic subgroup of M , with the standard maximal torus T € M of GL n p F q as a Levi factor. Let χ , . . . , χ n be characters of F (cid:2) . Then(1.0.1) (cid:0)p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M (cid:8) ss (cid:20) n à i (cid:16) (cid:1) i MB n (cid:1) (cid:2) GL p F qp χ w i q(cid:9) ss b δ { P M , where ss denotes semisimplification and i MB n (cid:1) (cid:2) GL p F q is the normalized parabolic induction. Here, w i is thepermutation of n letters such that w i p n q (cid:16) n (cid:0) (cid:1) i and w i p q ¡ w i p q ¡ (cid:4) (cid:4) (cid:4) ¡ w i p n (cid:1) q . This follows fromTheorem 6.3.5 of [Cas74] when R (cid:16) Q ℓ . As Vign´eras points out in [Vig98, II.2.18], the same proof is validfor the R (cid:16) F ℓ case. Proposition 1.5.
Let χ , . . . , χ n be R -valued characters of F (cid:2) , such that χ , . . . , χ n (cid:1) are unramified and χ n is tamely ramified.(i) The natural projection χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n Ñ p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M induces an isomorphism of R -modules (1.0.2) p : Hom U p χ n , p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n qq Ñ Hom U ,M p χ n , p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M q . (ii) For every φ P Hom U p χ n , p χ (cid:4) (cid:4) (cid:4) (cid:2) . . . χ n qq and every m P M (cid:1) , p p U mU .φ q (cid:16) δ P M p m q U ,M mU ,M .p p φ q . Proof.
The last assertion is proved in [Vig98, II.9]. The fact that p is surjective follows by [Vig96, II.3.5].We prove injectivity now. By Lemma 1.2, the dimension of the left hand side is n if χ n is unramified and 1otherwise. Suppose first that R (cid:16) Q ℓ . If χ n is unramified, each summand of the right hand side of (1.0.1)has a one dimensional U ,M -fixed subspace, while if χ n is ramified, only the summand corresponding to theidentity permutation has a one dimensional U ,M -fixed subspace, all the rest being zero. This implies thatdim Q ℓ (cid:0)p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M (cid:8) U ,M (cid:16) " n if χ n is unramified1 otherwise,Therefore p is an isomorphism for reasons of dimension. This completes the proof of the injectivity of p over Q ℓ .We give the proof over F ℓ only in the unramified case, the ramified case being similar. First of all, notethat the result for Q ℓ implies the corresponding result over Z ℓ , the ring of integers of Q ℓ . Indeed, supposeeach χ i takes values in Z (cid:2) ℓ , and let p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Z ℓ (respectively, p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Q ℓ ) denote the parabolicinduction over Z ℓ (respectively, Q ℓ ). Then p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Z ℓ is a lattice in p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Q ℓ , that is, a free Z ℓ -submodule which generates p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Q ℓ and is GL n p F q -stable ([Vig96, II.4.14(c)]). It then followsthat pp χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Z ℓ q U is a lattice in p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q Q ℓ q U ([Vig96, I.9.1]), and so is free of rank n over Z ℓ . Simiarly, pp χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M , Z ℓ q U ,M is a lattice in pp χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M , Q ℓ q U ,M ([Vig96, II.4.14(d)]), andthus it is free of rank n over Z ℓ . Moreover, the map p with coefficients in Z ℓ is still surjective ([Vig96, II3.3]), hence it is an isomorphism by reasons of rank.Finally, consider the F ℓ case. Choose liftings r χ i of χ i to Z ℓ -valued characters. Then there is a naturalinjection pr χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) r χ n q U M b Z ℓ F ℓ ãÑ p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M inducing an injection(1.0.3) ppr χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) r χ n q U M q U ,M b Z ℓ F ℓ ãÑ pp χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M q U ,M . Now, we have seen that the left hand side of (1.0.3) has dimension n over F ℓ . We claim that the right handside of (1.0.3) has dimension ¤ n . Indeed, by looking at the right hand side of (1.0.1), this follows from thefact that the U ,M -invariants of the semisimplification have dimension n . Thus, (1.0.3) is an isomorphismand dim F ℓ p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U M q U ,M (cid:16) n . Since the left hand side of (1.0.2) has dimension n and p is surjective,it must be an isomorphism. (cid:3) Let H (respectively, H ) be the R -valued Hecke algebra of GL n p F q with respect to U (respectively, U ).Thus, H (cid:16) H p q . If π is a representation of GL n p F q over R , then π U is naturally a left H -module. Forany α P F (cid:2) with | α | ¤
1, let m α P M be the element m α (cid:16) (cid:2) n (cid:1) α (cid:10) . ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 11
For i (cid:16) V α,i P H i be the Hecke operators r U i m α U i s . If π is a representation of GL n p F q , then π U € π U and the action of the operators defined above are compatible with this inclusion.Let H M (cid:16) H M p q , and let V ω,M (cid:16) r U ,M m ω U ,M s P H M . Since m ω P M (cid:1) , V ω,M P H (cid:1) M , and T (cid:1)p V ω,M q (cid:16) V ω, P H . As above, if π is a representation of M over R , we consider the natural leftaction H M on π U ,M . Corollary 1.6.
Let χ , . . . , χ n be Q ℓ -valued unramified characters of F (cid:2) . Then the set of eigenvalues of V ω, acting on the n -dimensional space p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U is equal (counting multiplicities) to t q p n (cid:1) q{ χ i p ω qu ni (cid:16) .Proof. Note that V ω,M acts on the U ,M -invariants of each summand of the right hand side of (1.0.1) bythe scalar χ i p ω q q p (cid:1) n q{ . Thus, the eigenvalues of V ω,M in p χ (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) χ n q U ,M U M are the q p (cid:1) n q{ χ i p ω q . Thecorollary follows then by Proposition 1.5. (cid:3) Proposition 1.7.
Let π be an irreducible unramified representation of GL n p F q over R . Then π U (cid:16) π U and the following properties hold.(i) If R (cid:16) Q ℓ and π (cid:16) χ ` (cid:4) (cid:4) (cid:4) ` χ n , with χ i unramified characters of F (cid:2) , then dim R π U ¤ n and theeigenvalues of V ω, acting on π U are contained in t q p n (cid:1) q{ χ i p ω qu ni (cid:16) (counting multiplicities).(ii) If R (cid:16) F ℓ , q (cid:17) p mod ℓ q and π (cid:16) β r n s (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) β r r n r s with β i distinct unramified characters of F (cid:2) ,then dim R π U (cid:16) r and V ω, acting on π U has the r distinct eigenvalues t β j p ω qu rj (cid:16) .Proof. The fact that π U (cid:16) π U follows immediately because the central character of π is unramified.Since taking U -invariants is exact in characteristic zero, part (i) is clear from the last corollary. Letus prove (ii). Let P be the parabolic subgroup of GL n p F q containing B corresponding to the partition n (cid:16) n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n r . As usual, since GL n p F q (cid:16) P GL n p O F q , the F ℓ -dimension of π U is equal to the cardinalityof p GL n p O F q X P q z GL n p O F q{ U . By the Bruhat decomposition, this equals the cardinality of S n (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) S n r z S n { S n (cid:1) (cid:2) S , where S i is the symmetric group on i letters. This cardinality is easily seen to be r .It remains to prove the assertion about the eigenvalues of V ω, on π U . Let us first replace U by Iw (thiswas first suggested by Vign´eras). By the Iwasawa decomposition and the Bruhat decomposition,GL n p F q (cid:16) º s P S P s Iw , where S € GL n p F q is a set of representatives for p S n (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) S n r qz S n . Then π Iw has as a basis the set t ϕ s u s P S , where ϕ s is supported on P s
Iw and ϕ s p s q (cid:16) H F ℓ p n, q denote the Iwahori-Hecke algebra for GL n p F q over F ℓ , that is, the Hecke algebra for GL n p F q with respect to the compact open subgroup Iw. Thus, π Iw is naturally a left module over H F ℓ p n, q . For i (cid:16) , . . . , n (cid:1)
1, let s i denote the n by n permutation matrix corresponding to the transposition p i i (cid:0) q , andlet S i (cid:16) r Iw s i Iw s P H F ℓ p n, q . For j (cid:16) , . . . , n , let t j denote the diagonal matrix whose first j coordinatesare equal to ω , and whose last n (cid:1) j coordinates are equal to 1. Let T j (cid:16) r Iw t j Iw s P H F ℓ p n, q , and for j (cid:16) , . . . , n , let X j (cid:16) T j p T (cid:1) j (cid:1) q . Then H F ℓ p n, q is generated as an F ℓ -algebra by t S i u n (cid:1) i (cid:16) Y t X , X (cid:1) u ([Vig96,I.3.14]). We denote by H F ℓ p n, q the subalgebra generated by t S i u n (cid:1) i (cid:16) , which is canonically isomorphic tothe group algebra F ℓ r S n s of the symmetric group ([Vig96, I.3.12]). It can also be identified with the Heckealgebra of GL n p O F q with respect to Iw ([Vig96, I.3.14]). The subalgebra A (cid:16) F ℓ rt X (cid:8) i u ni (cid:16) s is commutative,and characters of T can be seen as characters on A . Let χ , . . . , χ n : F (cid:2) Ñ F (cid:2) ℓ be the characters defined by χ (cid:16) (cid:4) (cid:4) (cid:4) (cid:16) χ n (cid:16) β ; (cid:4) (cid:4) (cid:4) ; χ n (cid:0)(cid:4)(cid:4)(cid:4)(cid:0) n j (cid:1) (cid:0) (cid:16) (cid:4) (cid:4) (cid:4) (cid:16) χ n (cid:0)(cid:4)(cid:4)(cid:4)(cid:0) n j (cid:16) β j ; (cid:4) (cid:4) (cid:4) . Then the action of A on ϕ s is given by the character s p χ q . Note that the set t s p χ qu s P S is just the set of n -tuples of characters in which β i occurs n i times, with arbitrary order. It is clear that for each j (cid:16) , . . . , r ,there is at least one s P S for which s p n q P t n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n j (cid:1) (cid:0) , . . . , n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n j u , so that X n ϕ s (cid:16) β j p ω q ϕ s .Let ϕ (cid:16) ¸ s P S ϕ s . Then ϕ generates π GL n p O F q . For j (cid:16) , . . . , r , let ψ j (cid:16) ¸ s P S,χ s p n q(cid:16) β j ϕ s . We have seen above that ψ j (cid:24)
0. Moreover, X n ψ j (cid:16) β j p ω q ψ j . Let P j P F ℓ r X s be a polynomial such that P j p β j p ω qq (cid:16) P j p β i p ω qq (cid:16) i (cid:24) j . Then ψ j (cid:16) P j p X n q ϕ , and it follows that the r distincteigenvalues t β j p ω qu rj (cid:16) of X n on π Iw already occur on the subspace F ℓ r X n s ϕ .Consider now the map p T : π Iw Ñ p π N q T , where N is the unipotent radical of the parabolic subgroup ofGL n p F q containing T , opposite to B , and T (cid:16) T X GL n p O F q . By [Vig96, II.3.5], p T is an isomorphism. Onthe other hand, there is a commutative diagram π U i / / p M (cid:15) (cid:15) π Iw p T (cid:15) (cid:15) p π U M q U ,M p M,T / / p π N q T , where i is the inclusion and p M and p M,T are the natural projection to the coinvariants. The analogues ofpart (ii) of Proposition 1.5 for p M , p T and p M,T are still valid ([Vig98, II.9]). Thus, for f P π U , p T p i p V ω, f qq (cid:16) p M,T p p M p V ω, f qq (cid:16) p M,T pr U ,M m ω U ,M s p M p f qq (cid:16)(cid:16) r T m ω T s p M,T p p M p f qq (cid:16) r T m ω T s p T p i p f qq (cid:16) p T p X n i p f qq . It follows that V ω, (cid:16) X n on π U . In particular, F ℓ r X n s ϕ (cid:16) F ℓ r V ω, s ϕ € π U . By what we have seen above,we conclude that the eigenvalues of V ω, on the r dimensional space π U are t β j p ω qu rj (cid:16) , as claimed. (cid:3) Corollary 1.8.
Suppose that q (cid:17) p mod ℓ q and let π be an irreducible unramified representation of GL n p F q over F ℓ . Let ϕ P π GL n p O F q be a non-zero spherical vector. Then ϕ generates π U as a module over the algebra F ℓ r V ω, s .Proof. This is actually a corollary of the proof of the above proposition. Indeed, V ω, has r distinct eigenvalueson F ℓ r V ω, s ϕ € π U , and dim F ℓ π U (cid:16) r . (cid:3) Lemma 1.9.
Let π be an irreducible representation of GL n p F q over Q ℓ with a non-zero U -fixed vector butno non-zero GL n p O F q -fixed vectors. Then dim Q ℓ π U (cid:16) and there is a character V π : F (cid:2) Ñ Q (cid:2) ℓ with open kernel such that for every α P F (cid:2) with non-negative valuation, V π p α q is the eigenvalue of V α, on π U . Moreover, there is an exact sequence ÝÑ s ÝÑ r ℓ p π q ÝÑ V π (cid:5) Art (cid:1) F ÝÑ , where s is unramified. If π U (cid:24) then q (cid:1) V π p ω q is a root of the characteristic polynomial of s p Frob F q . If,on the other hand, if π U (cid:16) , then r ℓ p π qp Gal p F { F qq is abelian. ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 13
Proof.
This is Lemma 3.1.5 of [CHT08]. The proof basically consists in noting that if π U (cid:24)
0, then either π (cid:20) χ ` (cid:4) (cid:4) (cid:4) ` χ n with χ , . . . , χ n (cid:1) unramified and χ n tamely ramified, or π (cid:20) χ ` (cid:4) (cid:4) (cid:4) ` χ n (cid:1) ` St p χ n (cid:1) q with χ , . . . , χ n (cid:1) unramified. Then one just analyzes the cases separately, and calculates explicitly theaction of the operators U p j q F, (see [CHT08] for their definition) and V α, . (cid:3) Lemma 1.10.
Suppose that q (cid:17) p mod ℓ q , and let π be an irreducible unramified representation of GL n p F q over F ℓ . Let λ π p T p j q F q be the eigenvalue of T p j q F on π GL n p O F q , and t π (cid:16) p λ π p T p q F q , . . . , λ π p T p n q F qq . Supposethat P q, t π (cid:16) p X (cid:1) a q m F p X q in F ℓ r X s , with m ¡ and F p a q (cid:24) . Then F p V ω, q , as an operator acting on π U , is non-zero on the subspace π GL n p O F q .Proof. Suppose on the contrary that F p V ω, qp π GL n p O F qq (cid:16)
0. Let ϕ P π GL n p O F q be a non-zero element.Suppose that π (cid:16) β r n s (cid:2) (cid:4) (cid:4) (cid:4) (cid:2) β r r n r s , with β i distinct unramified F (cid:2) ℓ -valued characters of F (cid:2) . Then, since q (cid:16) F ℓ , P q, t π (cid:16) r ¹ i (cid:16) p X (cid:1) β i p ω qq n i . Suppose that a (cid:16) β j p ω q , so that F p X q (cid:16) ± i (cid:24) j p X (cid:1) β i p ω qq n i . By Proposition 1.7 (ii), π U has dimen-sion r and V ω, is diagonalizable on this space, with distinct eigenvalues β i p ω q . Let ϕ j P π U denote aneigenfunction of V ω, of eigenvalue β j p ω q . By Corollary 1.8, there exists a polynomial P j P F ℓ r X s such that ϕ j (cid:16) P j p V ω, qp ϕ q . Since polynomials in V ω, commute with each other, we must have F p V ω, qp ϕ j q (cid:16)
0, butthis also equals F p β j p ω qq ϕ j (cid:24)
0, which is a contradiction. (cid:3) Automorphic forms on unitary groups
Totally definite groups.
Let F (cid:0) be a totally real field and F a totally imaginary quadratic extensionof F (cid:0) . Denote by c P Gal p F { F (cid:0)q the non-trivial Galois automorphism. Let n ¥ V an n -dimensional vector space over F , equipped with a non-degenerate c -hermitian form h : V (cid:2) V Ñ F . Tothe pair p V, h q there is attached a reductive algebraic group U p V, h q over F (cid:0) , whose points in an F (cid:0) -algebra R are U p V, h qp R q (cid:16) t g P Aut p F b F (cid:0) R q(cid:1) lin p V b F (cid:0) R q : h p gx, gy q (cid:16) h p x, y q x, y P V b F (cid:0) R u . By an unitary group attached to F { F (cid:0) in n variables, we shall mean an algebraic group of the form U p V, h q for some pair p V, h q as above. Let G be such a group. Then G F (cid:16) G b F (cid:0) F is isomorphic to GL V , and infact it is an outer form of GL V . Let G p F (cid:0)8 q (cid:16) ± v |8 G p F (cid:0) v q , and if v is any place of F (cid:0) , let G v (cid:16) G b F (cid:0) F (cid:0) v .We say that G is totally definite if G p F (cid:0)8 q is compact (and thus isomorphic to a product of copies of thecompact unitary group U p n q ).Suppose that v is a place of F (cid:0) which splits in F , and let w be a place of F above v , correspondingto an F (cid:0) -embedding σ w : F ãÑ F (cid:0) v . Then F (cid:0) v (cid:16) σ w p F q F (cid:0) v is an F -algebra by means of σ w , and thus G v is isomorphic to GL V b F (cid:0) v , the tensor product being over F . Note that if we choose another place w c of F above v , then σ w and σ w c give F (cid:0) v two different F -algebra structures. If we choose a basis of V , weobtain two isomorphisms i w , i w c : G v Ñ GL n { F (cid:0) v . If X P GL n p F q is the matrix of h in the chosen basis,then for any F (cid:0) v -algebra R and any g P G v p R q , i w c p g q (cid:16) X (cid:1) p t i w p g q(cid:1) q X , where we see X P GL n p R q via σ w : F Ñ F (cid:0) v Ñ R .The choice of a lattice L in V such that h p L (cid:2) L q € O F gives an affine group scheme over O F (cid:0) , stilldenoted by G , which is isomorphic to G after extending scalars to F (cid:0) . We will fix from now on a basis for L over O F , giving also an F -basis for V ; with respect to these, for each split place v of F (cid:0) and each place w of F above v , i w gives an isomorphism between G p F (cid:0) v q and GL n p F w q taking G p O F (cid:0) v q to GL n p O F w q . Automorphic forms.
Let G be a totally definite unitary group in n variables attached to F { F (cid:0) . Welet A denote the space of automorphic forms on G p A F (cid:0) q . Since the group is totally definite, A decomposes,as a representation of G p A F (cid:0)q , as A (cid:21) à π m p π q π, where π runs through the isomorphism classes of irreducible admissible representations of G p A F (cid:0)q , and m p π q is the multiplicity of π in A , which is always finite. This is a well known fact for any reductive groupcompact at infinity, but we recall the proof as a warm up for the following sections and to set some notation.The isomorphism classes of continuous, complex, irreducible (and hence finite dimensional) representationsof G p F (cid:0)8 q are parametrized by elements b (cid:16) p b τ q P p Z n, (cid:0)q Hom p F (cid:0) , R q . We denote them by W b . Since G p F (cid:0)8 q is compact and every element of A is G p F (cid:0)8 q -finite, A decomposes as a direct sum of irreducible G p A F (cid:0) q -submodules. Moreover, we can write A (cid:21) à b W b b Hom G p F (cid:0)8qp W b , A q as G p A F (cid:0) q -modules. Denote by A F (cid:0) the ring of finite ad`eles. For any b , let S b be the space of smooth(that is, locally constant) functions f : G p A F (cid:0)q Ñ W _ b such that f p γg q (cid:16) γ f p g q for all g P G p A F (cid:0)q and γ P G p F (cid:0)q . Then the map f ÞÑ (cid:0) w ÞÑ (cid:0) g ÞÑ p g (cid:1) f p g w q(cid:8)(cid:8) induces a G p A F (cid:0) q -isomorphism between Hom G p F (cid:0)8qp W b , A q and S b , where the action on this last space isby right translation. For every compact open subgroup U € G p A F (cid:0) q , the space G p F qz G p A F (cid:0) q{ U is finite,and hence the space of U -invariants of S b is finite-dimensional. In particular, every irreducible summand of W b b Hom G p F (cid:0)8qp W b , A q is admissible and appears with finite multiplicity. Thus, every irreducible summandof A is admissible, and appears with finite multiplicity because its isotypic component is contained in W b b Hom G p F (cid:0)8qp W b , A q for some b .2.3. ℓ -adic models of automorphic forms. Let ℓ be an odd prime number. We will assume, from nowon to the end of this section, that every place of F (cid:0) above ℓ splits in F . Let K be a finite extension of Q ℓ .Fix an algebraic closure K of K , and suppose that K is big enough to contain all embeddings of F into K .Let O be its ring of integers and λ its maximal ideal. Let S ℓ denote the set of places of F (cid:0) above ℓ , and I ℓ the set of embeddings F (cid:0) ãÑ K . Thus, there is a natural surjection h : I ℓ ։ S ℓ . Let r S ℓ denote a setof places of F such that r S ℓ ² r S cℓ consists of all the places above S ℓ ; thus, there is a bijection S ℓ (cid:20) r S ℓ . For v P S ℓ , we denote by r v the corresponding place in r S ℓ . Also, let r I ℓ denote the set of embeddings F ãÑ K giving rise to a place in r S ℓ . Thus, there is a bijection between I ℓ and r I ℓ , which we denote by τ ÞÑ r τ . Also,denote by τ ÞÑ w τ the natural surjection r I ℓ Ñ r S ℓ . Finally, Let F (cid:0) ℓ (cid:16) ± v | ℓ F (cid:0) v .Let a P p Z n, (cid:0)q Hom p F,K q . Consider the following representation of G p F (cid:0) ℓ q (cid:20) ±r v P r S ℓ GL n p F r v q . For each r τ P r I ℓ , we have an embedding GL n p F w r τ q ãÑ GL n p K q . Taking the product over r τ and composing with theprojection on the w r τ -coordinates, we have an irreducible representation ξ a : G p F (cid:0) ℓ q ÝÑ GL p W a q , where W a (cid:16) br τ Pr I ℓ W a r τ ,K . This representation has an integral model ξ a : G p O F (cid:0) ℓ q Ñ GL p M a q . In order tobase change to automorphic representations of GL n , we need to impose the additional assumption that a τc,i (cid:16) (cid:1) a τ,n (cid:0) (cid:1) i for every τ P Hom p F, K q and every i (cid:16) , . . . , n .Besides the weight, we will have to introduce another collection of data, away from ℓ , for defining ourautomorphic forms. This will take care of the level-raising arguments needed later on. Let S r be a finite set ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 15 of places of F (cid:0) , split in F and disjoint from S ℓ . For v P S r , let U ,v € G p F (cid:0) v q be a compact open subgroup,and let χ v : U ,v Ñ O (cid:2) be a morphism with open kernel. We will use the notation U r (cid:16) ± v P S r U ,v and χ (cid:16) ± v P S r χ v .Fix the data t a , U r , χ u . Let M a ,χ (cid:16) M a b O p v P S r O p χ v qq . Let U € G p A F (cid:0)q be a compact open subgroupsuch that its projection to the v -th coordinate is contained in U ,v for each v P S r . Let A be an O -algebra.Suppose either that the projection of U to G p F (cid:0) ℓ q is contained in G p O F (cid:0) ℓ q , or that A is a K -algebra. Thendefine S a ,χ p U, A q to be the space of functions f : G p F (cid:0)qz G p A F (cid:0) q Ñ M a ,χ b O A such that f p gu q (cid:16) u (cid:1) ℓ,S r f p g q g P G p A F (cid:0) q , u P U, where u ℓ,S r denotes the product of the projections to the coordinates of S ℓ and S r . Here, u S r acts alreadyon M a ,χ by χ , and the action of u ℓ is via ξ a .Let V be any compact subgroup of G p A F (cid:0)q such that its projection to G p F (cid:0) v q is contained in U ,v foreach v P S r , and let A be an O -algebra. If either A is a K -algebra or the projection of V to G p F (cid:0) ℓ q iscontained in G p O F (cid:0) ℓ q , denote by S a ,χ p V, A q the union of the S a ,χ p U, A q , where U runs over compact opensubgroups containing V for which their projection to G p F (cid:0) v q is contained in U ,v for each v P S r , and forwhich their projection to G p F (cid:0) ℓ q is contained in G p O F (cid:0) ℓ q if A is not a K -algebra. Note that if V € V then S a ,χ p V , A q € S a ,χ p V, A q .If U is open and we choose a decomposition G p A F (cid:0)q (cid:16) º j P J G p F (cid:0)q g j U, then the map f ÞÑ p f p g j qq j P J defines an injection of A -modules(2.3.1) S a ,χ p U, A q ãÑ ¹ j P J M a ,χ b O A. Since G p F (cid:0)qz G p A F (cid:0)q{ U is finite and M a ,χ is a free O -module of finite rank, we have that S a ,χ p U, A q is afinitely generated A -module.We say that a compact open subgroup U € G p A F (cid:0) q is sufficiently small if for some finite place v of F (cid:0) ,the projection of U to G p F (cid:0) v q contains only one element of finite order. Note that the map (2.3.1) is notalways surjective, but it is if, for example, U is sufficiently small. Thus, in this case, S a ,χ p U, A q is a free A -module of rank p dim K W a q . (cid:0) G p F (cid:0)qz G p A F (cid:0)q{ U (cid:8) . Moreover, if either U is sufficiently small or A is O -flat, we have that S a ,χ p U, A q (cid:16) S a ,χ p U, O q b O A. Let U and V be compact subgroups of G p A F (cid:0)q such that their projections to G p F (cid:0) v q are contained in U ,v for each v P S r . Suppose either A is a K -algebra or that the projections of U and V to G p F (cid:0) ℓ q arecontained in G p O F (cid:0) ℓ q . Also, let g P G p A S r , F (cid:0) q (cid:2) U r ; if A is not a K -algebra, we suppose that g ℓ P G p O F (cid:0) ℓ q .If V € gU g (cid:1) , then there is a natural map g : S a ,χ p U, A q ÝÑ S a ,χ p V, A q defined by p gf qp h q (cid:16) g ℓ,S r f p hg q . In particular, if V is a normal subgroup of U , then U acts on S a ,χ p V, A q , and we have that S a ,χ p U, A q (cid:16) S a ,χ p V, A q U . Let U and U be compact subgroups of G p A F (cid:0) q such that their projections to G p F (cid:0) v q are contained in U ,v for all v P S r . Let g P G p A S r , F (cid:0) q (cid:2) U r . If A is not a K -algebra, we suppose that the projections of U and U to G p F (cid:0) ℓ q are contained in G p O F (cid:0) ℓ q , and that g ℓ P G p O F (cid:0) ℓ q . Suppose also that U gU { U (thiswill be automatic if U and U are open). Then we can define an A -linear map r U gU s : S a ,χ p U , A q ÝÑ S a ,χ p U , A q by pr U gU s f qp h q (cid:16) ¸ i p g i q ℓ,S r f p hg i q , if U gU (cid:16) ² i g i U . Lemma 2.1.
Let U € G p A ,S r F (cid:0) q (cid:2) ± v P S r U ,v be a sufficiently small compact open subgroup and let V € U be a normal open subgroup. Let A be an O -algebra. Suppose that either A is a K -algebra or the projectionof U to G p F (cid:0) ℓ q is contained in G p O F (cid:0) ℓ q . Then S a ,χ p V, A q is a finite free A r U { V s -module. Moreover, let I U { V € A r U { V s be the augmentation ideal and let S a ,χ p V, A q U { V (cid:16) S a ,χ p V, A q{ I U { V S a ,χ p V, A q be the moduleof coinvariants. Define Tr U { V : S a ,χ p V, A q U { V Ñ S a ,χ p U, A q (cid:16) S a ,χ p V, A q U as Tr U { V p f q (cid:16) ° u P U { V uf . Then Tr U { V is an isomorphism.Proof. This is the analog of Lemma 3.3.1 of [CHT08], and can be proved in the same way. (cid:3)
Choose an isomorphism ι : K (cid:20)ÝÑ C . The choice of r I ℓ gives a bijection(2.3.2) ι (cid:0)(cid:6) : p Z n, (cid:0)q Hom p F,K q c (cid:18)ÝÑ p Z n, (cid:0)q Hom p F (cid:0) , R q , where p Z n, (cid:0)q Hom p F,K q c denotes the set of elements a P p Z n, (cid:0)q Hom p F,K q such that a τc,i (cid:16) (cid:1) a τ,n (cid:0) (cid:1) i for every τ P Hom p F, K q and every i (cid:16) , . . . , n . The map is given by p ι (cid:0)(cid:6) a q τ (cid:16) a ƒ ι (cid:1) τ . We have anisomorphism θ : W a b K,ι C Ñ W ι (cid:0)(cid:6) a , . Then the map S a , Hpt u , C q ÝÑ S p ι (cid:0)(cid:6) a q_ given by f ÞÑ p g ÞÑ θ p g ℓ f p g qqq is an isomorphism of C r G p A F (cid:0)qs -modules, where, p ι (cid:0)(cid:6) a q_ τ,i (cid:16) (cid:1)p ι (cid:0)(cid:6) a q τ,n (cid:0) (cid:1) i . Its inverse is given by φ ÞÑ p g ÞÑ g (cid:1) ℓ θ (cid:1) p φ p g qqq . It follows that S a , Hpt u , C q is a semi-simple admissible module. Hence, S a , Hpt u , K q is also semi-simpleadmissible, and this easily implies that S a ,χ p U r , K q is a semi-simple admissible G p A ,S r F (cid:0) q -module. If π € S a , Hpt u , K q is an irreducible G p A ,S r F (cid:0) q (cid:2) U r -constituent such that the subspace on which U r acts by χ (cid:1) isnon-zero, then this subspace is an irreducible constituent of S a ,χ p U r , K q , and every irreducible constituentof it is obtained in this way. ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 17
Base change and descent.
Keep the notation as above. We will assume from now on the followinghypotheses. (cid:13) F { F (cid:0) is unramified at all finite places. (cid:13) G v is quasi-split for every finite place v .It is not a very serious restriction for the applications we have in mind, because we will always be able tobase change to this situation. First, note that given F { F (cid:0) , if n is odd there always exists a totally definiteunitary group G in n variables with G v quasi-split for every finite v . If n is even, such a G exists if and onlyif r F (cid:0) : Q s n { G (cid:6) n (cid:16) Res F { F (cid:0)p GL n q . Let v be a finite place of F (cid:0) , so that G v is an unramified group. In particular,it contains hyperspecial maximal compact subgroups. Let σ v be any irreducible admissible representationof G p F (cid:0) v q . If v is split in F , or if v is inert and σ v is spherical, there exists an irreducible admissiblerepresentation BC v p σ v q of G (cid:6) n p F (cid:0) v q , called the local base change of σ v , with the following properties. Supposethat v is inert and σ v is a spherical representation of G p F (cid:0) v q ; then BC v p σ v q is an unramified representationof G (cid:6) n p F (cid:0) v q , whose Satake parameters are explicitly determined in terms of those of σ v ; the formula is givenin [Min], where we take the standard base change defined there. If v splits in F as ww c , the local base changein this case is BC v p σ v q (cid:16) σ v (cid:5) i (cid:1) w b p σ v (cid:5) i (cid:1) w c q_ as a representation of G (cid:6) n p F (cid:0) v q (cid:16) GL n p F w q (cid:2) GL n p F w c q .In this way, if we see BC v p σ v q as a representation of G p F (cid:0) v q (cid:2) G p F (cid:0) v q via the isomorphism i w (cid:2) i w c : G p F (cid:0) v q (cid:2) G p F (cid:0) v q (cid:18)Ñ GL n p F w q (cid:2) GL n p F w c q , then BC v p σ v q (cid:16) σ v b σ _ v . The base change for ramified finiteplaces is being treated in the work of Mœglin, but for our applications it is enough to assume that F { F (cid:0) isunramified at finite places.In the global case, if σ is an automorphic representation of G p A F (cid:0) q , we say that an automorphic repre-sentation Π of G (cid:6) n p A F (cid:0)q (cid:16) GL n p A F q is a (strong) base change of σ if Π v is the local base change of σ v forevery finite v , except those inert v where σ v is not spherical, and if the infinitesimal character of Π is thebase change of that of σ . In particular, since G p F (cid:0)8 q is compact, Π is cohomological.The following theorem is one of the main results of [Lab], and a key ingredient in this paper. We use thenotation ` for the isobaric sum of discrete automorphic representations, as in [Clo90]. Theorem 2.2 (Labesse) . Let σ be an automorphic representation of G p A F (cid:0) q . Then there exists a partition n (cid:16) n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n r and discrete, conjugate self dual automorphic representations Π i of GL n i p A F q , for i (cid:16) , . . . , r , such that Π ` (cid:4) (cid:4) (cid:4) ` Π r is a base change of σ .Conversely, let Π be a conjugate self dual, cuspidal, cohomological automorphic representation of GL n p A F q .Then Π is the base change of an automorphic representation σ of G p A F (cid:0)q . Moreover, if such a σ satisfies that σ v is spherical for every inert place v of F (cid:0) , then σ appears with multiplicity one in the cuspidal spectrumof G .Proof. The first part is Corollaire 5.3 of [Lab] and the second is Th´eor`eme 5.4. (cid:3)
Remarks. (1) In [Lab] there are two hypothesis to Corollaire 5.3, namely, the property called (*) by Labesseand that σ is a discrete series, which are automatically satisfied in our case because the group is totallydefinite.(2) Since Π ` (cid:4) (cid:4) (cid:4) ` Π r is a base change of σ , it is a cohomological representation of GL n p A F q . However,this doesn’t imply that each Π i is cohomological, although it will be if n (cid:1) n i is even.(3) The partition n (cid:16) n (cid:0) (cid:4) (cid:4) (cid:4) n r and the representations Π i are uniquely determined by multiplicity onefor GL n , because the Π i are discrete. Galois representations of unitary type via unitary groups.
Keep the notation and assumptionsas in the last sections.
Theorem 2.3.
Let π be as above. Let π (cid:16) b v R S r π v be an irreducible constituent of the space S a ,χ p U r , K q .Then there exists a unique continuous semisimple representation r ℓ p π q : Gal p F { F q Ñ GL n p K q satisfying the following properties.(i) If v R S ℓ Y S r is a place of F (cid:0) which splits as v (cid:16) ww c in F , then r ℓ p π q| ssΓ w (cid:20) (cid:0) r ℓ p π v (cid:5) i (cid:1) w q(cid:8) ss . (ii) r ℓ p π q c (cid:21) r ℓ p π q_p (cid:1) n q .(iii) If v is an inert place such that π v is spherical then r ℓ p π q is unramified at v .(iv) If w | ℓ then r ℓ p π q is de Rham at w , and if moreover π w | F (cid:0) is unramified, then r ℓ p π q is crystalline at w .(v) For every τ P Hom p F, K q giving rise to an place w | ℓ of F , the Hodge-Tate weights of r | Γ w with respectto τ are given by HT τ p r | Γ w q (cid:16) t j (cid:1) n (cid:1) a τ,j u j (cid:16) ,...,n . In particular, r is Hodge-Tate regular.Proof. For the uniqueness, note that the set of places w of F which are split over a place v of F (cid:0) which is not in S ℓ Y S r has Dirichlet density 1, and hence, if two continuous semisimple representations Gal p F { F q Ñ GL n p Q ℓ q satisfy property (i), they are isomorphic.Take an isomorphism ι : K (cid:18)ÝÑ C . By the above argument, the representation we will construct willnot depend on it. By means of ι and the choice of r I ℓ , we obtain a (necessarily cuspidal) automorphicrepresentation σ (cid:16) b v σ v of G p A F (cid:0)q , such that σ v (cid:16) ιπ v for v R S r finite and σ is the representation of G p F (cid:0)8 q given by the weight p ι (cid:0)(cid:6) a q_ P p Z n, (cid:0)q Hom p F (cid:0) , R q . By Theorem 2.2, there is a partition n (cid:16) n (cid:0) (cid:4) (cid:4) (cid:4) (cid:0) n r and discrete automorphic representations Π i of GL n i p A F q such thatΠ (cid:16) Π ` (cid:4) (cid:4) (cid:4) ` Π r is a strong base change of σ . Moreover, Π is cohomological of weight ι (cid:6) a , where p ι (cid:6) a q τ (cid:16) a ι (cid:1) τ for τ P Hom p F, C q . For each i (cid:16) , . . . , r , let S i (cid:129) S ℓ be any finite set of finite primes of F (cid:0) , unramified in F . Foreach i (cid:16) , . . . , r , let ψ i : A (cid:2) F { F (cid:2) Ñ C (cid:2) be a character such that (cid:13) ψ (cid:1) i (cid:16) ψ ci ; (cid:13) ψ i is unramified above S i , and (cid:13) for every τ P Hom p F, C q giving rise to an infinite place w , we have ψ i,w p z q (cid:16) p τ z {| τ z |q δ i,τ , where | z | (cid:16) zz and δ i,τ (cid:16) n (cid:1) n i is even, and δ i,τ (cid:16) (cid:8) n (cid:1) n i is even, we may just choose ψ i (cid:16)
1. The proof of the existence of such a character follows froma similar argument used in the proof of [HT01, Lemma VII.2.8]. With these choices, it follows that Π i ψ i is cohomological. Also, by the classification of Mœglin and Waldspurger ([MW89]), there is a factorization n i (cid:16) a i b i , and a cuspidal automorphic representation ρ i of GL a i p A F q such thatΠ i ψ i (cid:16) ρ i ` ρ i | | ` (cid:4) (cid:4) (cid:4) ` ρ i | | b i (cid:1) . Moreover, ρ i | | bi (cid:1) is cuspidal and conjugate self dual. Let χ i : A (cid:2) F { F (cid:2) Ñ C (cid:2) be a character such that (cid:13) χ (cid:1) i (cid:16) χ ci ; (cid:13) χ i is unramified above S i , and ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 19 (cid:13) for every τ P Hom p F, C q giving rise to an infinite place w , we have χ i,w p z q (cid:16) p τ z {| τ z |q µ i,τ , where µ i,τ (cid:16) a i is odd or b i is odd, and µ i,τ (cid:16) (cid:8) ρ i | | bi (cid:1) χ i is cuspidal, cohomological and conjugate self dual. Note that χ (cid:1) i | |p a i (cid:1) qp b i (cid:1) q{ and ψ (cid:1) i | | ni (cid:1) n are algebraic characters. Let r ℓ p π q (cid:16) À ri (cid:16) (cid:1) r ℓ (cid:1) ρ i χ i | | bi (cid:1) (cid:9) b ǫ a i (cid:1) n i b r ℓ (cid:0) χ (cid:1) i | |p a i (cid:1) qp b i (cid:1) q{ (cid:8)b (cid:0) ` ǫ ` (cid:4) (cid:4) (cid:4) ` ǫ b i (cid:1) (cid:8) b r ℓ (cid:1) ψ (cid:1) i | | ni (cid:1) n (cid:9)(cid:9) , where r ℓ (cid:16) r ℓ,ι and ǫ is the ℓ -adic cyclotomic character. This is a continuous semisimple representation whichsatisfies all the required properties. We use the freedom to vary the sets S i to achieve property (iii). (cid:3) Remark . In the proof of the above theorem, if r (cid:16) r ℓ p π q (cid:21) r ℓ,ι p Π q . Asa consequence, suppose that ι : Q ℓ (cid:18)ÝÑ C is an isomorphism and Π is a conjugate self dual, cohomological,cuspidal automorphic representation of GL n p A F q of weight ι (cid:6) a . Then, by Theorem 2.2, we can find anirreducible constituent π € S a , Hpt u , K q such that r ℓ,ι p Π q (cid:21) r ℓ p π q . Remark . If r ℓ p π q is irreducible, then the base change of π is already cuspidal. Indeed, from the con-struction made in the proof and Remark 2.4, (2), we see that r ℓ p π q is a direct sum of r representations r i of dimension n i . If r ℓ p π q is irreducible, we must have r (cid:16)
1. Similarly, the discrete base change Π must becuspidal, because otherwise there would be a factorization n (cid:16) ab with a, b ¡ r ℓ p π q would be a directsum of b representations of dimension a . This proves our claim.3. An R red (cid:16) T theorem for Hecke algebras of unitary groups Hecke algebras.
Keep the notation and assumptions as in the last section. For each place w of F ,split above a place v of F (cid:0) , let Iw p w q € G p O F (cid:0) v q be the inverse image under i w of the group of matrices inGL n p O F w q which reduce modulo w to an upper triangular matrix. Let Iw p w q be the kernel of the naturalsurjection Iw p w q Ñ p k (cid:2) w q n , where k w is the residue field of F w . Similarly, let U p w q (resp. U p w q ) bethe inverse image under i w of the group of matrices in GL n p O F w q whose reduction modulo w has last row p , . . . , , (cid:6)q (resp. p , . . . , , q ). Then U p w q is a normal subgroup of U p w q , and the quotient U p w q{ U p w q is naturally isomorphic to k (cid:2) w .Let Q be a finite (possibly empty) set of places of F (cid:0) split in F , disjoint from S ℓ and S r , and let T (cid:129) S r Y S ℓ Y Q be a finite set of places of F (cid:0) split in F . Let r T denote a set of primes of F above T suchthat r T ² r T c is the set of all primes of F above T . For v P T , we denote by r v the corresponding element of r T , and for S € T , we denote by r S the set of places of F consisting of the r v for v P T . Let U (cid:16) ¹ v U v € G p A F (cid:0)q be a sufficiently small compact open subgroup such that: (cid:13) if v R T splits in F then U v (cid:16) G p O F (cid:0) v q ; (cid:13) if v P S r then U v (cid:16) Iw pr v q ; (cid:13) if v P Q then U v (cid:16) U pr v q ; (cid:13) if v P S ℓ then U v € G p O F (cid:0) v q .We write U r (cid:16) ± v P S r U v . For v P S r , let χ v be an O -valued character of Iw pr v q , trivial on Iw pr v q . SinceIw pr v q{ Iw pr v q (cid:20) p k (cid:2)r v q n , χ v is of the form g ÞÑ n ¹ i (cid:16) χ v,i p g ii q , where χ v,i : k (cid:2)r v Ñ O (cid:2) . Let w be a place of F , split over a place v of F (cid:0) which is not in T . We translate the Hecke operators T p j q F w for j (cid:16) , . . . , n on GL n p O F w q to G via the isomorphism i w . More precisely, let g p j q w denote the elementof G p A F (cid:0)q whose v -coordinate is i (cid:1) w (cid:2) ω w j
00 1 n (cid:1) j (cid:10) , and with all other coordinates equal to 1. Then we define T p j q w to be the operator r U g p j q w U s of S a ,χ p U, A q . Wewill denote by T T a ,χ p U q the O -subalgebra of End O p S a ,χ p U, O qq generated by the operators T p j q w for j (cid:16) , . . . , n and p T p n q w q(cid:1) , where w runs over places of F which are split over a place of F (cid:0) not in T . The algebra T T a ,χ p U q is reduced, and finite free as an O -module (see [CHT08]). Since O is a domain, this also implies that T T a ,χ p U q is a semi-local ring. If v P Q , we can also translate the Hecke operators V α, of Section 1, for α P F (cid:2)r v withnon-negative valuation, in exactly the same manner to operators in S a ,χ p U, A q , and similarly for V α, if U v (cid:16) U pr v q .Write(3.1.1) S a ,χ p U, K q (cid:16) ` π π U , where π runs over the irreducible constituents of S a ,χ p U r , K q for which π U (cid:24)
0. The Hecke algebra T T a ,χ p U q acts on each π U by a scalar, say, by λ π : T T a ,χ p U q ÝÑ K. Then, ker p λ π q is a minimal prime ideal of T T a ,χ p U q , and every minimal prime is of this form. If m € T T a ,χ p U q is a maximal ideal, then S a ,χ p U, K q m (cid:24) , and localizing at m kills all the representations π such that ker p λ π q ‚ m . Note also that T T a ,χ p U q{ m is a finiteextension of k . For w a place of F , split over a place v R T , we will denote by T w the n -tuple p T p q w , . . . , T p n q w q of elements of T T a ,χ p U q . We denote by T w its reduction modulo m . We use the notation of section 2.4.1 of[CHT08] regarding torsion crystalline representations and Fontaine-Laffaille modules. Proposition 3.1.
Suppose that m is a maximal ideal of T T a ,χ p U q with residue field k . Then there is a uniquecontinuous semisimple representation r m : Gal p F { F q Ñ GL n p k q with the following properties. The first two already characterize r m uniquely.(i) r m is unramified at all but finitely many places.(ii) If a place v R T splits as ww c in F then r m is unramified at w and r m p Frob w q has characteristicpolynomial P q w , T w p X q .(iii) r c m (cid:21) r _ m p (cid:1) n q .(iv) If a place v of F (cid:0) is inert in F and if U v is a hyperspecial maximal compact subgroup of G p F (cid:0) v q , then r m is unramified above v .(v) If w P r S ℓ is unramified over ℓ , U w | F (cid:0) (cid:16) G p O F (cid:0) w q and for every τ P r I ℓ above w we have that ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ , then r m | Γ w (cid:16) G w p M m ,w q for some object M m ,w of M F k,w . Moreover, for every τ P r I ℓ over w , we have dim k p gr (cid:1) i M m ,w q b O Fw b Z ℓ O ,τ b O (cid:16) if i (cid:16) j (cid:1) n (cid:1) a τ,j for some j (cid:16) , . . . , n , and otherwise. ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 21
Proof.
Choose a minimal prime ideal p € m and an irreducible constituent π of S a ,χ p U r , K q such that π U (cid:24) T T a ,χ p U q acts on π U via T T a ,χ p U q{ p . Choose an invariant lattice for r ℓ p π q and definethen r m to be the semi-simplification of the reduction of r ℓ p π q . This satisfies all of the statements of theproposition, except for the fact that a priori it takes values on the algebraic closure of k . Since all thecharacteristic polynomials of the elements on the image of r m have coefficients in k , we may assume (because k is finite) that, after conjugation, r m actually takes values in k . (cid:3) We say that a maximal ideal m € T Ta,χ p U q is Eisenstein if r m is absolutely reducible. We define (seeChapter 2 of [CHT08]) G n as the group scheme over Z given by the semi-direct product of GL n (cid:2) GL bythe group t , u acting on GL n (cid:2) GL by p g, µ q (cid:1) (cid:16) p µ t g (cid:1) , µ q . There is a homomorphism ν : G n Ñ GL which sends p g, µ q to µ and to (cid:1) Proposition 3.2.
Let m be a non-Eisenstein maximal ideal of T Ta,χ p U q , with residue field equal to k . Then r m has an extension to a continuous morphism r m : Gal p F { F (cid:0)q Ñ G n p k q . Pick such an extension. Then there is a unique continuous lifting r m : Gal p F { F (cid:0)q Ñ G n p T Ta,χ p U q m q of r m with the following properties. The first two of these already characterize the lifting r m uniquely.(i) r m is unramified at almost all places.(ii) If a place v R T of F (cid:0) splits as ww c in F , then r m is unramified at w and r m p Frob w q has characteristicpolynomial P q w , T w p X q .(iii) ν (cid:5) r m (cid:16) ǫ (cid:1) n δ µ m F { F (cid:0) , where δ F { F (cid:0) is the non-trivial character of Gal p F { F (cid:0)q and µ m P Z { Z .(iv) If v is an inert place of F (cid:0) such that U v is a hyperspecial maximal compact subgroup of G p F (cid:0) v q then r m is unramified at v .(v) Suppose that w P r S ℓ is unramified over ℓ , that U w | F (cid:0) (cid:16) G p O F (cid:0) w q , and that for every τ P r I ℓ above w wehave that ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ . Then for each open ideal I € T T a ,χ p U q m , (cid:1) r m b T T a ,χ p U q m T T a ,χ p U q m { I (cid:9) | Γ w (cid:16) G w p M m ,I,w q for some object M m ,I,w of M F O ,w .(vi) If v P S r and σ P I F r v then r m p σ q has characteristic polynomial n ¹ j (cid:16) p X (cid:1) χ (cid:1) v,j p Art (cid:1) F r v σ qq . (vii) Suppose that v P Q . Let φ r v be a lift of Frob r v to Gal p F r v { F r v q . Suppose that α P k is a simple root of thecharacteristic polynomial of r m p φ r v q . Then there exists a unique root r α P T T a ,χ p U q m of the characteristicpolynomial of r m p φ r v q which lifts α .Let ω r v be the uniformizer of F r v corresponding to φ r v via Art F r v . Suppose that Y € S a ,χ p U, K q m is a T T a ,χ p U qr V ̟ r v , s -invariant subspace such that V ̟ r v , (cid:1) r α is topologically nilpotent on Y , and let T T p Y q denote the image of T T a ,χ p U q in End K p Y q . Then for each β P F (cid:2)r v with non-negative valuation, V β, (in End K p Y q ) lies in T T p Y q , and β ÞÑ V p β q extends to a continuous character V : F (cid:2)r v Ñ T T p Y q(cid:2) .Further, p X (cid:1) V ̟ r v , q divides the characteristic polynomial of r m p φ r v q over T T p Y q . Finally, if q v (cid:17) ℓ then r m | Γ r v (cid:21) s ` p V (cid:5) Art (cid:1) F r v q , where s is unramified.Proof. This is the analogue of Proposition 3.4.4 of [CHT08], and can be proved exactly in the same way. (cid:3)
Corollary 3.3.
Let Q denote a finite set of places of F (cid:0) , split in F and disjoint from T . Let m be anon-Eisenstein maximal ideal of T T a ,χ p U q with residue field k , and let U p Q
1q (cid:16) ± v R Q U v (cid:2) ± v P Q U pr v q .Denote by ϕ : T T Y Q a ,χ p U
1q Ñ T T a ,χ p U q the natural map, and let m ϕ (cid:1) p m q , so that m is also non-Eisensteinwith residue field k . Then the localized map ϕ : T T Y Q a ,χ p U p Q m T T a ,χ p U q m is surjective.Proof. It suffices to see that T p j q w { ϕ for j (cid:16) , . . . , n and w a place of F over Q , whichfollows easily because r m (cid:16) ϕ (cid:5) r m , and so T p j q w (cid:16) ϕ (cid:3) q j p (cid:1) j q{ w Tr (cid:3) j © r m φ w q(cid:11) , where φ w is any lift of Frobenius at w . (cid:3) The main theorem.
In this section we will use the Taylor-Wiles method in the version improved byDiamond, Fujiwara, Kisin and Taylor. We will recapitulate the running assumptions made until now, andadd a few more. Thus, let F (cid:0) be a totally real field and F { F (cid:0) a totally imaginary quadratic extension. Fixa positive integer n and an odd prime ℓ ¡ n . Let K { Q ℓ be a finite extension, let K be an algebraic closure of K , and suppose that K is big enough to contain the image of every embedding F ãÑ K . Let O be the ringof integers of K , and k its residue field. Let S ℓ denote the set of places of F (cid:0) above ℓ . Let r S ℓ denote a setof places of F above ℓ such that r S ℓ ² r S cℓ are all the places above ℓ . We let r I ℓ denote the set of embeddings F ãÑ K which give rise to a place in r S ℓ . We will suppose that the following conditions are satisfied. (cid:13) F { F (cid:0) is unramified at all finite places; (cid:13) ℓ is unramified in F (cid:0) ; (cid:13) every place of S ℓ is split in F ;Let G be a totally definite unitary group in n variables, attached to the extension F { F (cid:0) such that G v isquasi-split for every finite place v (cf. Section 2.4 for conditions on n and r F (cid:0) : Q s to ensure that such agroup exists). Choose a lattice in F (cid:0) giving a model for G over O F (cid:0) , and fix a basis of the lattice, so thatfor each split v (cid:16) ww c , there are two isomorphisms i w : G v ÝÑ GL n { F w and i w c : G v ÝÑ GL n { F wc taking G p O F (cid:0) v q to GL n p O F w q and GL n p O F wc q respectively.Let S a denote a finite, non-empty set of primes of F (cid:0) , disjoint from S ℓ , such that if v P S a then (cid:13) v splits in F , and (cid:13) if v lies above a rational prime p then v is unramified over p and r F p ζ p q : F s ¡ n .Let S r denote a finite set of places of F (cid:0) , disjoint from S a Y S ℓ , such that if v P S r then (cid:13) v splits in F , and (cid:13) q v (cid:17) ℓ . ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 23
We will write T (cid:16) S ℓ Y S a Y S r , and r T (cid:129) r S ℓ for a set of places of F above those of T such that r T ² r T c isthe set of all places of F above T . For S € T , we will write r S to denote the set of r v for v P S . We will fix acompact open subgroup U (cid:16) ¹ v U v of G p A F (cid:0)q , such that (cid:13) if v is not split in F then U v is a hyperspecial maximal compact subgroup of G p F (cid:0) v q ; (cid:13) if v R S a Y S r splits in F then U v (cid:16) G p O F (cid:0) v q ; (cid:13) if v P S r then U v (cid:16) Iw pr v q , and (cid:13) if v P S a then U v (cid:16) i (cid:1) r v ker p GL n p O F r v q Ñ GL n p k r v qq .Then, U is sufficiently small ( U v has only one element of finite order if v P S a ) and its projection to G p F (cid:0) ℓ q is contained in G p O F (cid:0) ℓ q . Write U r (cid:16) ¹ v P S r U v . For any finite set Q of places of F (cid:0) , split in F and disjoint from T , we will write T p Q q (cid:16) T Y Q . Also, wewill fix a set of places r T p Q q (cid:129) r T of F over T p Q q as above, for each Q . We will also write U p Q q (cid:16) ¹ v R Q U v (cid:2) ¹ v P Q U pr v q and U p Q q (cid:16) ¹ v R Q U v (cid:2) ¹ v P Q U pr v q . Thus, U p Q q and U p Q q are also sufficiently small compact open subgroups of G p A F (cid:0)q .Fix an element a P p Z n, (cid:0)q Hom p F,K q such that for every τ P r I ℓ we have (cid:13) a τc,i (cid:16) (cid:1) a n (cid:0) (cid:1) i and (cid:13) ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ m € T T a , p U q be a non-Eisenstein maximal ideal with residue field equal to k . Write T (cid:16) T T a , p U q m .Consider the representation r m : Gal p F { F (cid:0)q Ñ G n p k q and its lifting r m : Gal p F { F (cid:0)q Ñ G n p T q given by Proposition 3.2. For v P T , denote by r m ,v the restriction of r m to a decomposition group Γ r v at r v .We will assume that r m has the following properties. (cid:13) r m p Gal p F { F (cid:0)p ζ ℓ qqq is big (see Definition 2.5.1 of [CHT08], where the same notion is also defined forsubgroups of GL n p k q ); (cid:13) if v P S r then r m ,v is the trivial representation of Γ r v , and (cid:13) if v P S a then r m is unramified at v and H p Γ r v , p ad r m qp qq (cid:16) . We will use the Galois deformation theory developed in Section 2 of [CHT08], to where we refer the readerfor the definitions and results. Consider the global deformation problem S (cid:16) p F { F (cid:0) , T, r T , O , r m , ǫ (cid:1) n δ µ m F { F (cid:0) , t D v u v P T q , where the local deformation problems D v are as follows. For v P T , we denote by r univ v : Γ r v Ñ GL n p R loc v q the universal lifting ring of r m ,v , and by I v € R loc v the ideal corresponding to D v . (cid:13) For v P S a , D v consists of all lifts of r m ,v , and thus I v (cid:16) (cid:13) For v P S ℓ , D v consist of all lifts whose artinian quotients all arise from torsion Fontaine-Laffaillemodules, as in Section 2.4.1 of [CHT08]. (cid:13) For v P S r , D v corresponds to the ideal I p , ,..., q v of R loc v , as in Section 3 of [Tay08]. Thus, D v consists of all the liftings r : Γ r v Ñ GL n p A q such that for every σ in the inertia subgroup I r v , thecharacteristic polynomial of r p σ q is n ¹ i (cid:16) p X (cid:1) q . Let r univ S : Gal p F { F (cid:0)q Ñ G n p R univ S q denote the universal deformation of r m of type S . By Proposition 3.2, r m gives a lifting of r m which is oftype S ; this gives rise to a surjection R univ S ÝÑ T . Let H (cid:16) S a , p U, O q m . This is a T -module, and under the above map, a R univ S -module. Our main result isthe following. Theorem 3.4.
Keep the notation and assumptions of the start of this section. Then p R univ S q red (cid:20) T . Moreover, µ m (cid:17) n mod 2 .Proof. The proof is essentially the same as Taylor’s ([Tay08]), except that here there are no primes S p B q and S p B q , in his notation. One has just to note that his argument is still valid in our simpler case. Theidea is to use Kisin’s version ([Kis09]) of the Taylor-Wiles method in the following way, in order to avoiddealing with non-minimal deformations separately. There are essentially two moduli problems to considerat places in S r . One of them consists in considering all the characters χ v to be trivial. This is the case inwhich we are ultimately interested, but the local deformation rings are not so well behaved (for example,they are not irreducible). We call this the degenerate case . On the other hand, we can also consider thecharacters χ v in such a way that χ v,i (cid:24) χ v,j for all v P S r and all i (cid:24) j . This is the non-degenerate case , andwe can always consider such a set of characters by our assumption that ℓ ¡ n . Note that both problems areequal modulo ℓ . The Taylor-Wiles-Kisin method doesn’t work with the first moduli problem, but it worksfine in the non-degenerate case. Taylor’s idea is to apply all the steps of the method simultaneosly for thedegenerate and non-degenerate cases, and obtain the final conclusion of the theorem by means of comparingboth processes modulo λ , and using the fact that in the degenerate case, even if the local deformation ring isnot irreducible, every prime ideal which is minimal over λ contains a unique minimal prime, and this sufficesto proof what we want. We will reproduce most of the argument in the following pages. What we will provein the end is that H is a nearly faithful R univ S -module, which by definition means that the ideal Ann R univ S p H q is nilpotent. Since T is reduced, this proves the main statement of the theorem.We will be working with several deformation problems at a time. Consider a set Q of finite set of placesof F (cid:0) , disjoint from T , such that if v P Q , then (cid:13) v splits as ww c in F , (cid:13) q v (cid:17) ℓ , and (cid:13) r m ,v (cid:16) ψ v ` s v , with dim ψ v (cid:16) s v does not contain ψ v as a sub-quotient.Let T p Q q and r T p Q q be as in the start of the section. Also, let t χ v : Iw pr v q{ Iw pr v q Ñ O (cid:2)u v P S r be aset of characters of order dividing ℓ . To facilitate the notation, we will write χ v (cid:16) p χ v, , . . . , χ v,n q and χ (cid:16) t χ v u v P S r . Consider the deformation problem given by S χ,Q (cid:16) p F { F (cid:0) , T p Q q , r T p Q q , O , r m , ǫ (cid:1) n δ µ m F { F (cid:0) , t D v u v P T p Q qq , where: (cid:13) for v P S a Y S ℓ , D v (cid:16) D v ; ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 25 (cid:13) for v P S r , D v consists of all the liftings r : Γ r v Ñ GL n p A q such that the characteristic polynomial of r p σ q for σ P I r v is n ¹ i (cid:16) p X (cid:1) χ (cid:1) v,i p Art (cid:1) F r v σ qq (see Section 3 of [Tay08]). (cid:13) for v P Q , D v consists of all Taylor-Wiles liftings of r m ,v , as in Section 2.4.6 of [CHT08]. Moreprecisely, D v consists of all the liftings r : Γ r v Ñ GL n p A q which are conjugate to one of the form ψ v ` s v with ψ v a lift of ψ v and s v an unramified lift of s v .Denote by I χ v v the corresponding ideal of R loc v for every v P T p Q q . Let r univ S χ,Q : Gal p F { F (cid:0)q Ñ G n p R univ S χ,Q q denote the universal deformation of r of type S χ,Q , and let r (cid:5) T S χ,Q : Gal p F { F (cid:0)q Ñ G n p R (cid:5) T S χ,Q q denote the universal T -framed deformation of r of type S χ,Q (see [CHT08, 2.2.7] for the definition of T -framed deformations; note that it depends on r T ). Thus, by definition of the deformation problems, we havethat R univ S , H (cid:16) R univ S . As we claimed above, both problems are equal modulo ℓ . We have isomorphisms(3.2.1) R univ S χ,Q { λ (cid:21) R univ S ,Q { λ and(3.2.2) R (cid:5) T S χ,Q { λ (cid:21) R (cid:5) T S ,Q { λ, compatible with the natural commutative diagrams R univ S χ,Q / / / / (cid:15) (cid:15) R univ S χ, H (cid:15) (cid:15) R (cid:5) T S χ,Q / / / / R (cid:5) T S χ, H and R univ S ,Q / / / / (cid:15) (cid:15) R univ S , H (cid:15) (cid:15) R (cid:5) T S ,Q / / / / R (cid:5) T S , H Also, let R loc χ,T (cid:16) xâ v P T R loc v { I χ v v . Then(3.2.3) R loc χ,T { λ (cid:21) R loc1 ,T { λ. To any T -framed deformation of type S χ,Q and any v P T we can associate a lifting of r m ,v of type D v , andhence there are natural maps R loc χ,T ÝÑ R (cid:5) T S χ,Q which modulo λ are compatible with the identifications (3.2.3) and (3.2.2).Let T (cid:16) O rr X v,i,j ss v P T ; i,j (cid:16) ,...,n . Then a choice of a lifting r univ S χ,Q of r m over R univ S χ,Q representing theuniversal deformation of type S χ,Q gives rise to an isomorphism of R univ S ,Q -algebras(3.2.4) R (cid:5) T S χ,Q (cid:20) R univ S χ,Q ˆ b O T , so that p r univ S χ,Q ; t n (cid:0) p X v,i,j qu v P T q represents the universal T -framed deformation of type S χ,Q (see Proposition 2.2.9 of [CHT08]). Moreover,we can choose the liftings r univ S χ,Q so that r univ S χ,Q b O k (cid:16) r univ S ,Q b O k under the natural identifications (3.2.1). Then the isomorphisms (3.2.4) for χ and 1 are compatible with theidentifications (3.2.2) and (3.2.1).For v P Q , let ψ v denote the lifting of ψ r v to p R univ S χ,Q q(cid:2) given by the lifting r univ S χ,Q . Also, write ∆ Q forthe maximal ℓ -power order quotient of ± v P Q k (cid:2)r v , and let a Q denote the ideal of T r ∆ Q s generated by theaugmentation ideal of O r ∆ Q s and by the X v,i,j for v P T and i, j (cid:16) , . . . , n . Since the primes of Q aredifferent from ℓ and ψ r v is unramified, ψ v is tamely ramified, and then ¹ v P Q p ψ v (cid:5) Art F r v q : ∆ Q ÝÑ p R univ S χ,Q q(cid:2) makes R univ S χ,Q an O r ∆ Q s -algebra. This algebra structure is compatible with the identifications (3.2.1), becausewe chose the liftings r univ S χ,Q and r univ S ,Q compatibly. Via the isomorphisms (3.2.4), R (cid:5) T S χ,Q are T r ∆ Q s -algebras,which are compatible modulo λ for the different choices of χ . Finally, we have an isomorphism(3.2.5) R (cid:5) T S χ,Q { a Q (cid:20) R univ S χ, H , compatible with the identifications (3.2.2) and (3.2.1), the last one with Q (cid:16) H .Note that since S a , p U, k q (cid:16) S a ,χ p U, k q we can find a maximal ideal m χ, H € T T a ,χ p U q with residue field k such that for a prime w of F split over aprime v R T of F (cid:0) , the Hecke operators T p j q w have the same image in T T a ,χ p U q{ m χ, H (cid:16) k as in T T a , p U q{ m (cid:16) k .It follows that r m χ, H (cid:21) r m , and in particular m χ, H is non-Eisenstein. We define m χ,Q € T T p Q q a ,χ p U p Q qq asthe preimage of m χ, H under the natural map T T p Q q a ,χ p U p Q qq ։ T T p Q q a ,χ p U p Q qq ։ T T p Q q a ,χ p U q ãÑ T T a ,χ p U q . Then T T p Q q a ,χ p U p Q qq{ m χ,Q (cid:16) k , and if a prime w of F splits over a prime v R T p Q q of F (cid:0) , then the Heckeoperators T p j q w have the same image in T T p Q q a ,χ p U p Q qq{ m χ,Q (cid:16) k as in T T a , p U q{ m (cid:16) k . Hence, r m χ,Q (cid:21) r m and m χ,Q is non-Eisenstein. Let r m χ,Q : Gal p F { F (cid:0)q Ñ G n p T T p Q q a ,χ p U p Q qq m χ,Q q be the continuous representation attached to m χ,Q as in Proposition 3.2. Write T χ (cid:16) T T a ,χ p U q m χ, H and H χ (cid:16) S a ,χ p U, O q m χ, H . We have the following natural surjections(3.2.6) T T p Q q a ,χ p U p Q qq m χ,Q ։ T T p Q q a ,χ p U p Q qq m χ,Q ։ T T p Q q a ,χ p U q m χ,Q (cid:16) T χ . The last equality follows easily from Corollary 3.3.For each v P Q , choose φ r v P Γ r v a lift of Frob r v , and let ω r v P F (cid:2)r v be the uniformizer corresponding to φ r v via Art F r v . Let P r v P T T p Q q a ,χ p U p Q qq m χ,Q r X s denote the characteristic polynomial of r m χ,Q p φ r v q . Since ψ v p φ r v q is a simple root of the characteristic polyno-mial of r m p φ r v q , by Hensel’s lemma, there exists a unique root A r v P T T p Q q a ,χ p U p Q qq m χ,Q of P r v lifting ψ v p φ r v q .Thus, there is a factorisation P r v p X q (cid:16) p X (cid:1) A r v q Q r v p X q ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 27 over T T p Q q a ,χ p U p Q qq m χ,Q , where Q r v p A r v q P T T p Q q a ,χ p U p Q qq(cid:2) m χ,Q . By part (i) of Proposition 1.7 and Lemma 1.9, P r v p V ̟ r v , q (cid:16) S a ,χ p U p Q q , O q m χ,Q . For i (cid:16) ,
1, let H i,χ,Q (cid:16) (cid:3)¹ v P Q Q r v p V ̟ r v ,i q(cid:11) S a ,χ p U i p Q q , O q m χ,Q € S a ,χ p U i p Q q , O q m χ,Q , and let T i,χ,Q denote the image of T T p Q q a ,χ p U p Q qq m χ,Q in End O p H i,χ,Q q . We see that H ,χ,Q is a directsummand of S a ,χ p U p Q q , O q as a T T p Q q a ,χ p U p Q qq -module. Also, we have an isomorphism (cid:3)¹ v P Q Q r v p V ω r v , q(cid:11) : H χ (cid:21) H ,χ,Q . This can be proved using Proposition 1.7 and Lemmas 1.9 and 1.10, as in [CHT08, 3.2.2].For all v P Q , V ̟ r v , (cid:16) A r v on H ,χ,Q . By part (vii) of Proposition 3.2, for each v P Q there is a characterwith open kernel V v : F (cid:2)r v ÝÑ T (cid:2) ,χ,Q such that (cid:13) if α P O F r v is non-zero, then V α, (cid:16) V v p α q on H ,χ,Q and (cid:13) p r m χ,Q b T ,χ,Q q| Γ r v (cid:21) s v ` p V v (cid:5) Art (cid:1) F r v q , where s v is unramified.It is clear that V v (cid:5) Art (cid:1) F r v is a lifting of ψ v and s v is a lifting of s v . It follows by (v) and (vi) of the sameproposition that r m χ,Q b T ,χ,Q gives rise to a deformation of r m of type S χ,Q , and thus to a surjection R univ S χ,Q ։ T ,χ,Q , such that the composition ¹ v P Q O (cid:2) F r v ։ ∆ Q Ñ p R univ S χ,Q q(cid:2) Ñ T (cid:2) ,χ,Q coincides with ± v P Q V v . We then have that H ,χ,Q is an R univ S χ,Q -module, and we set H (cid:5) T ,χ,Q (cid:16) H ,χ,Q b R univ S χ,Q R (cid:5) T S χ,Q (cid:16) H ,χ,Q b O T . Since ker p± v P Q k (cid:2)r v Ñ ∆ Q q acts trivially on H ,χ,Q and H ,χ,Q is a T T p Q q a ,χ p U p Q qq -direct summand of S a ,χ p U p Q q , O q , Lemma 2.1 implies that H ,χ,Q is a finite free O r ∆ Q s -module, and that p H ,χ,Q q ∆ Q (cid:21) H ,χ,Q (cid:21) H χ . Since U is sufficiently small, we get isomorphisms S a ,χ p U, O q b O k (cid:21) S a ,χ p U, k q (cid:16) S a , p U, k q (cid:21) S a , p U, O q b O k and S a ,χ p U p Q q , O q b O k (cid:21) S a ,χ p U p Q q , k q (cid:16) S a , p U p Q q , k q (cid:21) S a , p U p Q q , O q b O k. Thus we get identifications H χ { λ (cid:21) H { λ,H ,χ,Q { λ (cid:21) H , ,Q { λ and H (cid:5) T ,χ,Q { λ (cid:21) H (cid:5) T , ,Q { λ, compatible with all the pertinent identifications modulo λ made before.Let ε (cid:1) p(cid:1) q µ m (cid:1) n q{ q (cid:16) r F (cid:0) : Q s n p n (cid:1) q{ (cid:0) r F (cid:0) : Q s nε . By Proposition 2.5.9 of [CHT08], there is an integer q ¥ q , such that for every natural number N , we canfind a set of primes Q N (and a set of corresponding ψ v and s v for r m ) such that (cid:13) Q N (cid:16) q ; (cid:13) for v P Q N , q v (cid:17) p mod ℓ N q and (cid:13) R (cid:5) T S ,QN can be topologically generated over R loc1 ,T by q q (cid:1) q elements.Define R (cid:5) T χ, R loc χ,T rr Y , . . . , Y q . Then there is a surjection R (cid:5) T , ։ R (cid:5) T S ,QN extending the natural map R loc1 ,T Ñ R (cid:5) T S ,QN . Reducing modulo λ and lifting the obtained surjection, via theidentifications R (cid:5) T χ, λ (cid:20) R (cid:5) T , λ, we obtain a surjection R (cid:5) T χ, ։ R (cid:5) T S χ,QN extending the natural map R loc χ,T Ñ R (cid:5) T S χ,QN .For v P S a , R loc v { I χ v v is a power series ring over O in n variables (see Lemma 2.4.9 of [CHT08]), and for v P S ℓ it is a power series ring over O in n (cid:0) r F r v : Q ℓ s n p n (cid:1) q{ loc. cit. ).Suppose that χ v,i (cid:24) χ v,j for every v P S r and every i, j (cid:16) , . . . , n with i (cid:24) j . Then, by Proposition 3.1 of[Tay08], for every v P S r , R loc v { I χ v v is irreducible of dimension n (cid:0) p R loc v { I χ v v q red is geometrically integral (in the sense that p R loc v { I χ v v q red b O O is anintegral domain for every finite extension K K , where O is the ring of integers of K ) and flat over O .Moreover, by part 3. of Lemma 3.3 of [BLGHT09], p R (cid:5) T χ, red (cid:20) (cid:1)(cid:1) xâ v P S r p R loc v { I χ v v q red (cid:9) xâ (cid:1) xâ v P S a Y S ℓ R loc v { I v (cid:9)(cid:9) rr Y , . . . , Y q , and the same part of that lemma implies that p R (cid:5) T χ, red is geometrically integral. We conclude that in thenon-degenerate case, R (cid:5) T χ, is irreducible, and, by part 2., its Krull dimension is1 (cid:0) q (cid:0) n T (cid:1) r F (cid:0) : Q s nε . Suppose now that we are in the degenerate case, that is, χ v (cid:16) v P S r . Then (see Proposition3.1 of [Tay08]) for every such v , R loc v { I χ v v is pure of dimension n (cid:0)
1, its generic points have characteristiczero, and every prime of R loc v { I χ v v which is minimal over λ p R loc v { I χ v v q contains a unique minimal prime.After eventually replacing K by a finite extension K (which we are allowed to do since the main theoremfor one K implies the same theorem for every K ), R loc v { I χ v v satisfies that for every prime ideal p which isminimal (resp. every prime ideal q which is minimal over λ p R loc v { I χ v v q ), the quotient p R loc v { I χ v v q{ p (resp. p R loc v { I χ v v q{ q ) is geometrically integral. It follows then by parts 2., 5. and 7. of Lemma 3.3 of [BLGHT09]that every prime ideal of R (cid:5) T , which is minimal over λR (cid:5) T , contains a unique minimal prime, the genericpoints of R (cid:5) T , have characteristic zero and R (cid:5) T , is pure.Let ∆ Z qℓ , S T rr ∆ and a (cid:16) ker p S ։ O q , where the map sends ∆ to 1 and the variables X v,i,j to 0. Thus, S is isomorphic to a power series ring over O in q (cid:0) n T variables. For every N , choosea surjection ∆ ։ ∆ Q N . We have an induced map on completed group algebras O rr ∆ ։ O r ∆ Q N s . and thus a map(3.2.7) S ։ T r ∆ Q N s Ñ R (cid:5) T S χ,QN ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 29 which makes R (cid:5) T S χ,QN an algebra over S . The map S ։ T r ∆ Q N s sends the ideal a to a Q N . Let c N (cid:16) ker p S ։ T r ∆ Q N sq . Note that every open ideal of S contains c N for some N . The following propertieshold. (cid:13) H (cid:5) T ,χ,Q N is finite free over S c N . (cid:13) R (cid:5) T S χ,QN { a (cid:20) R univ S χ, H . (cid:13) H (cid:5) T ,χ,Q N { a (cid:20) H χ .In what follows, we will use that we can patch the R (cid:5) T S χ,QN to obtain in the limit a copy of R (cid:5) T χ, , andsimultaneously patch the H ,χ,Q N to form a module over R (cid:5) T χ, , finite free over S . The patching constructionis carried on in exactly the same way as in [Tay08]. The outcome of this process is a family of R (cid:5) T χ, O S -modules H (cid:5) T ,χ, with the following properties.(1) They are finite free over S , and the S -action factors through R (cid:5) T χ, , in such a way that the obtainedmaps S R (cid:5) T χ, ։ R (cid:5) T S χ,QN are the maps defined in (3.2.7) for every N ; in particular, there is asurjection R (cid:5) T χ, a ։ R univ S χ,QN { a (cid:16) R univ S χ, H . (2) There are isomorphism H (cid:5) T ,χ, λ (cid:20) H (cid:5) T , , λ of R (cid:5) T χ, λ (cid:20) R (cid:5) T , λ -modules.(3) There are isomorphisms H (cid:5) T ,χ, a (cid:20) H χ of R (cid:5) T χ, a -modules, where we see H χ as a module over R (cid:5) T χ, a by means of the map in (1). Moreover, these isomorphisms agree modulo λ via the identifications of (2).Let us place ourselves in the non-degenerate case. That is, let us choose the characters χ such that χ v,i (cid:24) χ v,j for every v P S r and every i (cid:24) j . This is possible because ℓ ¡ n and q v (cid:17) p mod ℓ q for v P S r .Since the action of S on H (cid:5) T ,χ, factors through R (cid:5) T χ, ,(3.2.8) depth R (cid:5) Tχ, H (cid:5) T ,χ,
8q ¥ depth S H (cid:5) T ,χ, . Also, since H (cid:5) T ,χ, is finite free over S , which is a Cohen-Macaulay ring, by the Auslander-Buchsbaumformula we have that(3.2.9) depth S H (cid:5) T ,χ,
8q (cid:16) dim S (cid:0) q (cid:0) n T. Since the depth of a module is at most its Krull dimension, by equations (3.2.8) and (3.2.9) we obtain that(3.2.10) dim (cid:1) R (cid:5) T χ, Ann R (cid:5) Tχ, H (cid:5) T ,χ, (cid:0) q (cid:0) n T. Recall that R (cid:5) T χ, is irreducible of dimension(3.2.11) 1 (cid:0) q (cid:0) n T (cid:1) r F (cid:0) : Q s nε . Then, (3.2.10), (3.2.11) and Lemma 2.3 of [Tay08] imply that ε (cid:16) µ m (cid:17) n p mod 2 q ) and that H (cid:5) T ,χ, is a nearly faithful R (cid:5) T χ, -module. This implies in turn that H (cid:5) T ,χ, λ (cid:20) H (cid:5) T , , λ is a nearly faithful R (cid:5) T χ, λ (cid:20) R (cid:5) T , λ -module (this follows from Nakayama’s Lemma, as in Lemma 2.2 of [Tay08]). Since thegeneric points of R (cid:5) T , have characteristic zero, R (cid:5) T , is pure and every prime of R (cid:5) T , which is minimal over λR (cid:5) T , contains a unique minimal prime of R (cid:5) T , , the same lemma implies that H (cid:5) T , , is a nearly faithful R (cid:5) T , -module. Finally, using the same Lemma again, this implies that H (cid:5) T , , a (cid:20) H is a nearly faithful R (cid:5) T , a -module, and since R (cid:5) T , a ։ R univ S , H is a nearly faithful R univ S -module. (cid:3) Modularity lifting theorems
In this section we apply the results of the previous sections to prove modularity lifting theorems for GL n .We deal first with the case of a totally imaginary field F . Theorem 4.1.
Let F (cid:0) be a totally real field, and F a totally imaginary quadratic extension of F (cid:0) . Let n ¥ be an integer and ℓ ¡ n be a prime number, unramified in F . Let r : Gal p F { F q ÝÑ GL n p Q ℓ q be a continuous irreducible representation with the following properties. Let r denote the semisimplificationof the reduction of r .(i) r c (cid:21) r _p (cid:1) n q .(ii) r is unramified at all but finitely many primes.(iii) For every place v | ℓ of F , r | Γ v is crystalline.(iv) There is an element a P p Z n, (cid:0)q Hom p F, Q ℓ q such that (cid:13) for all τ P Hom p F (cid:0) , Q ℓ q , we have either ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ or ℓ (cid:1) (cid:1) n ¥ a τc, ¥ (cid:4) (cid:4) (cid:4) ¥ a τc,n ¥ (cid:13) for all τ P Hom p F, Q ℓ q and every i (cid:16) , . . . , n , a τc,i (cid:16) (cid:1) a τ,n (cid:0) (cid:1) i . (cid:13) for all τ P Hom p F, Q ℓ q giving rise to a prime w | ℓ , HT τ p r | Γ w q (cid:16) t j (cid:1) n (cid:1) a τ,j u nj (cid:16) . In particular, r is Hodge-Tate regular.(v) F ker p ad r q does not contain F p ζ ℓ q .(vi) The group r p Gal p F { F p ζ ℓ qqq is big.(vii) The representation r is irreducible and there is a conjugate self-dual, cohomological, cuspidal auto-morphic representation Π of GL n p A F q , of weight a and unramified above ℓ , and an isomorphism ι : Q ℓ (cid:18)ÝÑ C , such that r (cid:21) r ℓ,ι p Π q .Then r is automorphic of weight a and level prime to ℓ .Proof. Arguing as in [Tay08, Theorem 5.2], we may assume that F contains an imaginary quadratic field E with an embedding τ E : E ãÑ Q ℓ such that ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ τ : F ãÑ Q ℓ extending τ E . This will allow us to choose the set r S ℓ (in the notation of Section 2.3) insuch a way that the weights a τ,i are all within the correct range for τ P r I ℓ . Let ι : Q ℓ (cid:20)ÝÑ C and let Π be aconjugate self dual, cuspidal, cohomological automorphic representation of GL n p A F q of weight ι (cid:6) a , with Π ℓ unramified, such that r (cid:21) r ℓ,ι p Π q . Let S r denote the places of F not dividing ℓ at which r or Π is ramified.Since F ker p ad r q does not contain F p ζ ℓ q , we can choose a prime v of F with the following properties. (cid:13) v R S r and v ∤ ℓ . (cid:13) v is unramified over a rational prime p , for which r F p ζ p q : F s ¡ n . (cid:13) v does not split completely in F p ζ ℓ q . (cid:13) ad r p Frob v q (cid:16) L (cid:0){ F (cid:0) with the following properties. (cid:13) |r L (cid:0) : Q s . (cid:13) L (cid:0){ F (cid:0) is Galois and soluble. (cid:13) L (cid:16) L (cid:0) E is unramified over L (cid:0) at every finite place. (cid:13) L is linearly disjoint from F ker p r qp ζ ℓ q over F . (cid:13) ℓ is unramified in L . (cid:13) All primes of L above S r Y t v u are split over L (cid:0) . (cid:13) v and cv split completely in L { F . (cid:13) Let Π L denote the base change of Π to L . If v is a place of L above S r , then – N v (cid:17) p mod ℓ q ; ODULARITY LIFTING THEOREMS FOR GALOIS REPRESENTATIONS OF UNITARY TYPE 31 – r p Gal p L v { L v qq (cid:16) – r | ss I v (cid:16)
1, and – Π Iw p v q L,v (cid:24) r L (cid:0) : Q s is even, there exists a unitary group G in n variables attached to L { L (cid:0) which is totallydefinite and such that G v is quasi-split for every finite place v of L (cid:0) . Let S ℓ p L (cid:0)q denote the set of primesof L (cid:0) above ℓ , S r p L (cid:0)q the set of primes of L (cid:0) lying above the restriction to F (cid:0) of an element of S r , and S a p L (cid:0)q the set of primes of L (cid:0) above v | F (cid:0) . Let T p L (cid:0)q (cid:16) S ℓ p L (cid:0)q Y S r p L (cid:0)q Y S a p L (cid:0)q . It follows fromRemarks 2.4 and 2.5 and Theorem 3.4 that r | Gal p F { L q is automorphic of weight a L and level prime to ℓ , where a L P p Z n, (cid:0)q Hom p L, Q ℓ q is defined as a L,τ (cid:16) a τ | F . By Lemma 1.4 of [BLGHT09] (note that the hypothesesthere must say “ r _ (cid:21) r c b χ ” rather than “ r _ (cid:21) r b χ ”), this implies that r itself is automorphic of weight a and level prime to ℓ . (cid:3) We can also prove a modularity lifting theorem for totally real fields F (cid:0) . The proof goes exactly like thatof Theorem 5.4 of [Tay08], using Lemma 1.5 of [BLGHT09] instead of Lemma 4.3.3 of [CHT08]. Theorem 4.2.
Let F (cid:0) be a totally real field. Let n ¥ be an integer and ℓ ¡ n be a prime number,unramified in F . Let r : Gal p F (cid:0){ F (cid:0)q ÝÑ GL n p Q ℓ q be a continuous irreducible representation with the following properties. Let r denote the semisimplificationof the reduction of r .(i) r _ (cid:21) r p n (cid:1) q b χ for some character χ : Gal p F (cid:0){ F (cid:0)q Ñ Q (cid:2) ℓ with χ p c v q independent of v |8 (here c v denotes a complex conjugation at v ).(ii) r is unramified at all but finitely many primes.(iii) For every place v | ℓ of F , r | Γ v is crystalline.(iv) There is an element a P p Z n, (cid:0)q Hom p F (cid:0) , Q ℓ q such that (cid:13) for all τ P Hom p F (cid:0) , Q ℓ q , we have either ℓ (cid:1) (cid:1) n ¥ a τ, ¥ (cid:4) (cid:4) (cid:4) ¥ a τ,n ¥ or ℓ (cid:1) (cid:1) n ¥ a τc, ¥ (cid:4) (cid:4) (cid:4) ¥ a τc,n ¥ (cid:13) for all τ P Hom p F (cid:0) , Q ℓ q and every i (cid:16) , . . . , n , a τc,i (cid:16) (cid:1) a τ,n (cid:0) (cid:1) i . (cid:13) for all τ P Hom p F (cid:0) , Q ℓ q giving rise to a prime v | ℓ , HT τ p r | Γ v q (cid:16) t j (cid:1) n (cid:1) a τ,j u nj (cid:16) . In particular, r is Hodge-Tate regular.(v) p F (cid:0)q ker p ad r q does not contain F (cid:0)p ζ ℓ q .(vi) The group r p Gal p F (cid:0){ F (cid:0)p ζ ℓ qqq is big.(vii) The representation r is irreducible and automorphic of weight a .Then r is automorphic of weight a and level prime to ℓ . References [Ast94]
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