Ihara's lemma and level rising in higher dimension

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Pascal Boyer

To cite this version:

Pascal Boyer. Ihara’s lemma and level rising in higher dimension. 2015. �hal-01222967v3�

HIGHER DIMENSION by

Boyer Pascal

Abstract. — A key ingredient in the Taylor-Wiles proof of Fermat last theorem is the classical Ihara’s lemma which is used to rise the modu- larity property between some congruent galoisian representations. In their work on Sato-Tate, Clozel-Harris-Taylor proposed a generalization of the Ihara’s lemma in higher dimension for some similitude groups.

The main aim of this paper is then to prove some new instances of this generalized Ihara’s lemma by considering some particular non pseudo Eisenstein maximal ideals of unramified Hecke algebras. As a conse- quence, we prove a level rising statement.

Contents

Introduction. . . 2

1. Shimura variety of Kottwitz-Harris-Taylor type . . . 5

1.1. Geometry. . . 6

1.2. Jacquet-Langlands correspondence and %-type . . . . 8

1.3. Harris-Taylor local systems. . . 10

1.4. Free perverse sheaf. . . 12

1.5. Vanishing cycles perverse sheaf. . . 14

2. Cohomology of KHT Shimura varieties . . . 16

2.1. Localization at a non pseudo-Eisenstein ideal . . . 16

2.2. Freeness of the cohomology. . . 19

2010 Mathematics Subject Classification. — 11F70, 11F80, 11F85, 11G18, 20C08.

Key words and phrases. — Shimura varieties, torsion in the cohomology, maximal ideal of the Hecke algebra, localized cohomology, galois representation.

2.3. From Ihara’s lemma to the cohomology. . . 23

3. Non degeneracy property for global cohomology . . . 25

3.1. Global lattices are tensorial product. . . 25

3.2. Proof of the main result. . . 26

3.3. Level rising . . . 28

References. . . 30

Introduction

LetF “F^`E be a CM field withE{Qquadratic imaginary. For B{F a central division algebra with dimension d² equipped with a involution of second specie ˚ and β P B^˚“´1, consider the similitude group G{Q defined for any Q-algebra R by

GpRq:“ pλ, gq P R^ˆˆ pB^opb_QRq^ˆ such that gg^7β “λ(

withB^op “BbF,cF wherec“ ˚|F is the complex conjugation and7β the involution xÞÑx^7β “βx^˚β^´1. For p“uu^c decomposed in E, we have

GpQpq »Q^ˆp ˆź

w|u

pB^op_v q^ˆ

where w describes the places of F above u. We suppose:

– the associated unitary group G0pRq being compact,

– for any placexofQinert or ramified inE, thenGpQ^xqis quasi-split.

– There exists a place v₀ of F above u such that B_v0 » D_v0,d is the central division algebra over the completion F_v0 of F at v₀, with invariant ¹_d.

Fix a prime number l ‰ p and consider a finite set S of places of F containing the ramification places Bad of B. Let denote TS{Zl the un- ramified Hecke algebra of G outside S. For a cohomological minimal prime ideal mr of TS, we can associate both a near equivalence class of Ql-automorphic representation Π_mr and a Galois representation

ρ_mr :G_F :“GalpF¯{Fq ÝÑGL_dpQlq

such that the eigenvalues of the Frobenius morphism at an unramified placeware given by the Satake’s parameter of the local component Π_m,wr of Π_mr.The semi-simple class of the reduction modulolofρ_mr depends only of the maximal ideal m of Tcontaining m.r

Conjecture. — (Generalized Ihara’s lemma by Clozel-Harris- Taylor) Consider

– an open compact subgroup U of GpAq such that outside S, its local component is the maximal compact subgroup;

– a place w₀ RS decomposed in E;

– a maximal m of TS such that ρ_m is absolutely irreducible.

Let π¯ be an irreducible sub-representation of C⁸pGpQqzGpAq{U^w0,Flqm, where U “U_w0U^w0, then its local component π¯_w0 at w₀ is generic.

Remark. In its classical version for GL₂, Ihara’s lemma is used to rise the modularity property between some congruent galoisian representa- tions and so was the role of this higher dimensional version in Clozel- Harris-Taylor paper on the Sato-Tate conjecture. Shortly after Taylor found an argument to avoid Ihara’s lemma. However this conjecture re- mains highly interesting, see for example works of Clozel-Thorne [15], or Emerton-Helm [18].

The main result of this paper is the following instances of the previous conjecture.

Theorem. — The previous generalized Ihara’s conjecture is true if the maximal ideal m verifies the following extra properties.

(H1) ρ_m is absolutely irreducible andl ěd`1 except^p1q maybe for a finite number of exceptions depending of the extension F{Q.

(H2) The image of ρ_m,w0 in its Grothendieck group is multiplicity free and^p2q does not contain any full Zelevinsky line.^p3q

(H3) The image of ρ_m,v0 which by Jacquet-Langlands correspondence is associated to some superSpeh, Speh_sp%_v0q, for some supercuspidal Fl-representation %_v0 of GL_gpF_v0 with d “ sg, cf. theorem 3.1.4 of [17]. We then suppose that the set, cf. the notation in §1.2

%_v0, %_v0t1u,¨ ¨ ¨ , %_v0ts´1u( is of cardinal s.

Remark. Concerning (H1), you can replace it by any hypothesis which insures the freeness of the cohomology of the Kottwitz-Harris-Taylor

p1qcf. the main result of [10]

p2qUsing the main result of [9] we could take off the condition about not containing a full Zelevinsky line, cf. the last remark of§3.2.

p3qIn particularqw1 can’t be congruent to 1 modulol.

Shimura variety of §1.1. In [8] we gave two such conditions, the one given by (H1) is proved in [10]. We give more details before theorem 2.2.1.

Remark. About (H2) note that the first condition also appears in section 4.5 of [14] in the statements of level raising. Concerning the second condition of (H2), using the main result of [9], we can remove it, cf. the remark after the proof of lemma 3.2.3.

To prove this result, we first translate such a property to the coho- mology group of middle degree of the Kottwitz-Harris-Taylor Shimura variety XI associated to the similitude group G{Qsuch that

– GpA⁸q “GpA^8,pq ˆGL_dpF_v0q ˆś

w|u w‰v0

pB^op_wq^ˆ,

– the signatures ofGpRq are p1, d´1q ˆ p0, dq ˆ ¨ ¨ ¨ ˆ p0, dq.

In particular to each prime ideal mr of T^S, is associated a Ql-irreducible automorphic representation Π_mr of GpAQq whose Satake parameters at finite places outside S are prescribed by m. We then compute the coho-r mology groups of the geometric generic fiber ofX_I through the spectral sequence of vanishing cycles at the placev₀. Thanks to (H1), we know, cf.

[10], theHⁱpX_U,Zlqm to be free and so, HⁱpX_U,Flqm “ p0qfor i‰d´1.

Remark. Moreover (H2) (resp. (H3)) insures that the graduates of the filtration of H^d´1pX_U,Flqm, given by the entire version of the weight- monodromy filtration, at the place w₀ (resp. v₀) are also free.

The contribution of the supersingular points of the special fiber atv₀, using (H3), allows us to associate to an irreducible sub-representation π of C⁸pGpQqzGpAq{U^w0,Flqm, an irreducible sub-representation π of H^d´1pX_U,Flqm, such that π^8,v0 »¯π^8,v0. We then try to prove the gener- icness of π_w0 by proving, using (H2), such genericness property of irre- ducible submodules ofH^d´1pX_U,Flqm. On ingredient in§3.1, comes from [21] §5, where the hypothesis that ρ_m is absolutely irreducible, insures that the lattices of isotypic components of H^d´1pX_U,Qlqm given by the Zl-cohomology, can be written as a tensorial product of stable lattices for the GpA⁸qand the Galois actions.

Finally (H2) is needed to control the combinatorics.

Remark. As pointed out to us by M. Harris, the case where the cardinal qw0 of the residue field at w0, is congruent to 1 modulo l should be of crucial importance for the application. Meanwhile our strategy relies on the construction of a filtration of H^d´1pX_U,F^lqm which each graduates

verify the genericness property of irreducible submodule and where these graduates are parabolically induced. Whenq_w0 ”1 mod l, parabolically inducedFl-representations are often semi-simple and so they can’t verify such genericness property of irreducible submodule. It seems that our approach is not well adapted to treat this fundamental case.

To state our application to level rising, denote S_w0pmq the supercus- pidal support of the modulo l reduction of Π_{m,wr 0} for any prime ideal mr Ă m: it depends only on m. By (H2) this support is multiplicity free and we first break itSw0pmq “ š

%S%pmqaccording to the inertial equiva- lence classes% of irreducibleF^l-representations of someGLgp%qpFw0qwith 1ď gp%q ďd. For any such % we then denote l₁p%q ě ¨ ¨ ¨ ěl_rp%qp%q ě 1, such thatS_%pmqcan be written as the union of rp%qZelevinsky unlinked segments of length l_ip%q

r%ν_i^k,ρν¯ ^{k`lip%q</sup>s “ %ν<sup>k</sup>, %ν<sup>k`1},¨ ¨ ¨ , %ν^k`lip%q( .

Then for any minimal prime ideal mr Ăm, and ΠPΠ_mr, we write its local component Πw0 » Ś

%Πw0p%q and Πw0p%q » Śrp%q

i“1 Πw0p%, iq where for each 1 ďi ďrp%q, the modulo l reduction of the supercuspidal support of Π_w0p%, iq is, with the notations of the previous section, those of the Zelevinsky segment r%ν^δi, %ν^δi`lip%qs.

Proposition. — Take a maximal ideal m verifying the hypothesis (H1) and (H2). Let %₀ such that S_%0pmq is non empty and consider 1 ď i ď rp%₀q. Then there exists a minimal prime ideal mr Ăm and an automor- phic representationΠPΠ_mr such that with the previous notationΠ_w0p%₀, iq is non degenerate, i.e. isomorphic toSt_lip%qpπw0qfor some irreducible cus- pidal Ql-representation π_w0.

In particular when there is only one segment, which is always the case for GL₂, then the result is optimal.

Remark. In the previous proposition, we could also prove that for any such mr and any Π P Π_mr, then Π_w0p%₀, iq is non degenerate, which looks similar to theorem 2.1 of [1] where ρ_m is supposed to be absolutely ir- reducible and decomposed generic which also imposes the cohomology groups to be free.

1. Shimura variety of Kottwitz-Harris-Taylor type

1.1. Geometry. — Recall from the introduction that a prime number lis fixed, distinct from all others prime numbers which will be considered in the following. Let F “ F^{`</sup>E be a CM field with E{Q imaginary quadratic such that l is unramified, and F<sup>`} totally real with a fixed embeddingτ :F^` ãÑR. For a placev ofF, we denoteF_v the completion ofF atv, with ring of integers Ov, uniformizer$v and residual fieldκpvq with cardinal qv.

Let B be a central division algebra over F of dimension d² such that at any place x of F, either Bx is split either it’s a division algebra. We moreover suppose the existence of an involution of second specie ˚on B such that ˚|F is the complex conjugation c. For β P B^˚“´1, we denote 7β :xÞÑβx^˚β^´1 and let G{Q such that for any Q-algebra R,

GpRq “ pλ, gq PR^ˆˆ pB^opb_QRq^ˆ such that gg^7β “λ( , with B^op “BbcF. If x“yy^c is split in E then

GpQ^xq » pB_y^opq^ˆˆQ^ˆ_x »Q^ˆx ˆ ź

zi

pB^op_ziq^ˆ

where, identifying the places of F^` abovex with those ofF abovey, we write x“ś

iz_i. Moreover we can impose that – if x is inert in E then GpQxqis quasi-split,

– the signature of GpRq are p1, d´1q ˆ p0, dq ˆ ¨ ¨ ¨ ˆ p0, dq.

With the notations of the introduction we have GpA⁸q “ GpA^8,pq ˆ

´

Q^ˆpv0GL_dpF_v0q ˆ ź

w|u w‰v0

pB^op_wq^ˆ

¯ .

1.1.1. Definition. — We denote Bad the set of places w of F such that B_w is non split. Let Spl the set w of finite places of F not in Bad such that w_|Q is split in E. For such a place w with p “w_|Q, we write abusively

GpA^wq “GpA^pq ˆQ^ˆp ˆ ź

u|p u‰w

pB_u^opq^ˆ,

and GpF_wq “ GL_dpF_wq.

Remark. With the notations of the introduction, the role of w in the previous definition will be taken by v₀, v₁ orw₀.

1.1.1. Notation. — For all open compact subgroupU^p of GpA^8,pqand m“ pm₁,¨ ¨ ¨ , m_rq PZ^rě0, we consider

U^ppmq “U^pˆZ^ˆp ˆ

r

ź

i“1

KerpO_B^ˆ

vi ÝÑ pO_Bvi{P_v^m_iⁱq^ˆq.

For w₀ one of the v_i and n P N, we also introduce U^w0pnq :“ U^pp0,¨ ¨ ¨ ,0, n,0,¨ ¨ ¨,0q.

We then denote I for the set of these U^ppmq such that there exists a place x for which the projection from U^p to GpQxq doesn’t contain any element with finite order except the identity, cf. [19] bellow of page 90.

Attached to each I P I is a Shimura variety X_I Ñ SpecO_v of type Kottwitz-Harris-Taylor and we denote XI “ pX_IqIPI the projective sys- tem: recall that the transition morphisms r_J,I : X_J ÑX_I are finite flat and even etale when m₁pJq “ m₁pIq. This projective system is then equipped with a Hecke action of GpA⁸q ˆZ, where the action ofz in the Weil group W_v of F_v is given par ´degpzq PZ where deg “val˝Art^´1_v , where Art^´1_v :W_v^ab »F_v^ˆis the Artin isomorphism which sends geometric Frobenius to uniformizers.

1.1.2. Notations. — (cf. [3] §1.3) Let I PI,

– the special fiber of X_I will be denoted X_I,s and its geometric special fiber X_I,¯s :“X_I,sˆSpecFp.

– For 1 ď h ď d, let X_I,¯^ěh_s (resp. X_I,¯^“h_s) be the closed (resp. open) Newton stratum of height h, defined as the subscheme where the connected component of the universal Barsotti-Tate group is of rank greater or equal to h (resp. equal to h).

Remark: X_I,¯^ěh_s is of pure dimension d´h. For 1 ď h ă d, the Newton stratum X_I,¯^“h_s is geometrically induced under the action of the parabolic subgroup P_h,d´hpF_vq, defined as the stabilizer of the first h vectors of the canonical basis of F_v^d. Concretly this means there exists a closed subscheme X_I,¯^“h_s,1

h stabilized by the Hecke action of P_h,d´hpF_vq and such that

X_I,¯^“h_s »X_I,¯^“h_s,1

hˆPh,d´hpFvqGL_dpF_vq.

1.1.3. Notations. — Let denote

i^h :X_I,¯^ěh_s ãÑX_I,¯^ě1_s, j^ěh :X_I,¯^“h_s ãÑX_I,¯^ěh_s and j^“h “i^hj^ěh.

1.2. Jacquet-Langlands correspondence and %-type. — For a representation π_v of GL_dpF_vq and n P ¹₂Z, set π_vtnu :“ π_v bq^´n_v ^val˝det. Recall that the normalized induction of two representations π_v,1 and π_v,2 of respectively GL_n1pF_vqand GL_n2pF_vqis

π1ˆπ2 :“ind^GL_P ^n1`n2pFvq

n1,n1`n2pFvqπv,1tn2

2 u bπv,2t´n1

2 u.

A representation π_v of GL_dpF_vq is called cuspidal (resp. supercuspidal) if it’s not a subspace (resp. subquotient) of a proper parabolic induced representation. When the field of coefficients is of characteristic zero then these two notions coincides, but this is no more true for Fl.

Remark. The modulo l reduction of an irreducible Ql-representation is still irreducible and cuspidal but not necessary supercuspidal. In this last case, its supercuspidal support is a Zelevinsky segment associated to some unique inertial equivalent classe%, where%is an irreducible supercuspidal Fl-representation. Thanks to (H2) we will not be concerned by this subtility.

1.2.1. Definition. — We say that π_v is of type %when the supercusp- idal support of its modulo l reduction is contained in the Zelevinsky line of %.

1.2.1. Definition. — (see [24] §9 and [4] §1.4) Let g be a divisor of d “ sg and π_v an irreducible cuspidal Ql-representation of GL_gpF_vq.

Then the normalized induced representation π_vt1´s

2 u ˆπ_vt3´s

2 u ˆ ¨ ¨ ¨ ˆπ_vts´1 2 u

holds a unique irreducible quotient (resp. subspace) denotedStspπvq(resp.

Speh_spπvq); it’s a generalized Steinberg (resp. Speh) representation.

Remark. If χ_v is a character of F_v^ˆ then Speh_spχ_vq “χ_v ˝det.

The local Jacquet-Langlands correspondance is a bijection between ir- reducible essentially square integrable representations of GL_dpF_vq, i.e.

representations of the type St_spπ_vqforπ_v cuspidal, and irreducible repre- sentations of D_v,d^ˆ where D_v,d is the central division algebra over F_v with invariant ¹_d.

1.2.2. Notation. — We will denote πvrssD the irreductible representa- tion of D_v,d^ˆ associated to Stspπ^__vq by the local Jacquet-Langlands corre- spondance.

We denote R

Flpdq the set of equivalence classes of irreducible Fl- representations of D_v,d^ˆ . For ¯τ P R

Flpdq, let C_¯τ be the sub-category of smooth Z^nrl -representations of D^ˆ_v,d with objects whose irreducible sub-quotients are isomorphic to a sub-quotient of ¯τ_|Dˆ

v,d. Note that C_τ¯ is a direct factor inside Rep⁸

Z^nrl pD^ˆ_v,dq so that every Z^nrl -representation V_Z^nr

l

of D^ˆ_v,d can be decomposed as a direct sum V_Z^nr

l » à

¯ τPRFlpdq

V_Z^nr

l,¯τ

where V_Z^nr

l,¯τ is an object of C_τ¯.

Letπ_v be an irreducible cuspidal representation of GL_gpF_vq and fix an integer s ě 1. Then the modulo l reduction of Speh_spπq is irreducible, cf. [17] §2.2.3.

1.2.3. Notation. — When the modulo l reduction of π, denoted %, is supercuspidal, then we will denote Speh_sp%q the modulo l reduction of Speh_spπq: we call it a F^l- superSpeh representation.

By [17] 3.1.4, we have a bijection

!F¯l´superspeh irreducible representations of GL_dpF_vq )

»

!F¯l´representations irreducible ofD^ˆ_v,d )

(1.2.1) compatible with the modulol reduction in the sense that ifπ_v is a lifting of%, then the modulolreduction ofπ^_rss_D matches through the previous bijection, with the superSpeh Speh_sp%q.

1.2.2. Definition. — AF^l-representation ofD^ˆ_v,d (resp. an irreducible cuspidal representation of GL_dpF_vq) is said of type % if all its irreducible

sub-quotient are, through the previous bijection, associated to some su- perSpeh Speh_sp%q(resp. its supercuspidal support belongs to the Zelevin- sky line of %).

Recall that if p%q is the cardinal of the Zelevinsky line associated to

% and, cf. [23] p.51, then let mp%q “

"

p%q, si p%q ą1;

l, sinon.

1.2.4. Notation. — Let rp%q the biggest integer i such that lⁱ divides

d

mp%qg and define

g_´1p%q “ g and @0ďiďrp%q, g_ip%q “mp%qlⁱg.

We also denote s_ip%q:“t_g^d

ip%qu.

Then ifπ_v is an irreducible cuspidalQl-representation ofGL_kpF_vqwith type %, then there exists i such that k “ g_i. We say that π_v is of %- type i and we denote Scusp_ip%q the set of inertial equivalence classes of irreducible cuspidal representations of %-typei.

1.2.5. Notation. — For % an irreducible supercuspidalF^l- representa- tion of GLgpFvq, we denote R% “ š

sě1R%psgq where R%psgq is the set of equivalence classes of irreducible F^l-representations of D_v,sg^ˆ of type %.

1.3. Harris-Taylor local systems. — Letπ_v be an irreducible cuspi- dal Ql-representation of GL_gpF_vqand fix tě1 such thattgďd. Thanks to Igusa varieties, Harris and Taylor constructed a local system onX_I,¯^“tg_s,1

h

LQlpπ_vrtsDq_1h “

e_πv

à

i“1

LQlpρ_v,iq_1h

where pπ_vrts_Dq_|D^ˆ

v,h “ Àe_πv

i“1ρ_v,i with ρ_v,i irreductible. The Hecke action ofP_tg,d´tgpF_vqis then given through its quotientGL_d´tgˆZ. These local systems have stable Zl-lattices and we will write simply Lpπ_vrts_Dq_1h for any Z^l-stable lattice that we don’t want to specify.

1.3.1. Notations. — For Π_t any representation of GL_tg and Ξ :

1

2ZÝÑZ^ˆl defined by Ξp¹₂q “q^1{2, we introduce

HTĄ1pπv,Πtq:“LpπvrtsDq_1hbΠtbΞ^tg´d2 and its induced version

HTĄpπ_v,Π_tq:“

´

Lpπ_vrtsDq_1hbΠ_tbΞ^tg´d2

¯

ˆP_tg,d´tgpFvqGL_dpF_vq, where the unipotent radical of P_tg,d´tgpF_vq acts trivially and the action of

pg^8,v,

ˆ g_v^c ˚ 0 g^et_v

˙

, σ_vq P GpA^8,vq ˆP_tg,d´tgpF_vq ˆW_v

is given

– by the action of g^c_v on Π_t and degpσ_vq PZ on Ξ^tg´d2 , and

– the action ofpg^8,v, g_v^et,valpdetg_v^cq´degσ_vq PGpA^8,vqˆGL_d´tgpF_vqˆ Z on L

Qlpπ_vrtsDq_1hbΞ^tg´d2 . We also introduce

HTpπ_v,Π_tq_1h :“HTĄpπ_v,Π_tq_1hrd´tgs, and the perverse sheaf

Ppt, π_vq_1h :“j₁^“tg

h,!˚HTpπ_v,St_tpπ_vqq_1hbLpπ_vq, and their induced version, HTpπv,Πtq and Ppt, πvq, where

j “h“i^h˝j^ěh :X_I,¯^“h_s ãÑX_I,¯^ěh_s ãÑXI,¯s

and L^_ is the local Langlands correspondence.

Remark. Recall that π_v¹ is said inertially equivalent to π_v if there exists a characterζ :ZÝÑQ^ˆl such that π_v¹ »π_vb pζ˝val˝detq. Note, cf. [3]

2.1.4, that Ppt, π_vq depends only on the inertial class ofπ_v and Ppt, π_vq “ e_πvPpt, π_vq

where Ppt, πvq is an irreducible perverse sheaf. When we want to speak of the Ql-versions we will add it on the notations.

1.3.1. Definition. — We will say thatHTpπ_v,Π_tqorPpt, π_vqis of type

% if π_v is.

1.3.2. Lemma. — If ρbσ is a GL_dpF_vq ˆW_v equivariant irreducible sub-quotient of HⁱpX_I,¯sv,Ppπ_v, tq b_ZlF^lq then

– σ is an irreducible sub-quotient of the modulo l reduction of Lpπ_vb χ_vq, where χ_v is an uramified character of F_v, and

– ρis an irreducible sub-quotient of the modulolreduction of a induced representation of the following shape St_tpπ_v bχ_vq ˆψ_v where ψ_v is an entire irreducible representation of GL_d´tgpF_vq.

Proof. — The result follows directly from the description of the actions given previously.

As usually for σ a representation of W_v and n P ¹₂Z, we will denote σpnq the twisted representation g ÞÑ σpgq|Art^´1_v pgq|ⁿ where | ´ | is the absolute value of F_v.

1.4. Free perverse sheaf. — LetS “SpecF^qandX{S of finite type, then the usual t-structure onDpX,Z^lq:“D^b_cpX,Z^lqis

AP ^pD^ď0pX,Zlq ô @xP X, H^ki^˚_xA“0, @k ą ´dimtxu AP ^pD^ě0pX,Zlq ô @xP X, H^ki^!_xA“0, @k ă ´dimtxu

wherei_x : SpecκpxqãÑX and H^kpKq is thek-th sheaf of cohomology of K.

1.4.1. Notation. — Let denote ^pCpX,Zlq the heart of this t-structure with associated cohomology functors ^pHⁱ. For a functor T we denote

pT :“^pH⁰˝T.

The category ^pCpX,Zlq is abelian equipped with a torsion theory pT,Fq where T (resp. F) is the full subcategory of objects T (resp.

F) such that l^N1_T is trivial for some large enough N(resp. l.1_F is a monomorphism). Applying Grothendieck-Verdier duality, we obtain

p`D^ď0pX,Zlq:“ tAP ^pD^ď1pX,Zlq: ^pH¹pAq PTu

p`D<sup>ě0</sup>pX,Zlq:“ tAP <sup>p</sup>D<sup>ě0</sup>pX,Zlq: <sup>p</sup>H<sup>0</sup>pAq PFu with heart<sup>p`</sup>CpX,Zlq equipped with its torsion theory pF,Tr´1sq. 1.4.2. Definition. — (cf. [6] §1.3) Let

FpX,Z^lq:“ ^pCpX,Z^lq X^{p`</sup>CpX,Z<sup>l</sup>q “ <sup>p</sup>D<sup>ď0</sup>pX,Z<sup>l</sup>q X <sup>p`}D^ě0pX,Z^lq

the quasi-abelian category of free perverse sheaves over X.

Remark: for an object L of FpX,Zlq, we will consider filtrations L₁ ĂL₂ Ă ¨ ¨ ¨ ĂL_e “L

such that for every 1 ďiďe´1, L_i ãÑL_{i`1</sub> is a strict monomorphism, i.e. L<sub>i`1}{L_i is an object of FpX,Zlq.

For a free LPFpX,Λq, we consider the following diagram L

can˚,L

**p`j_!j^˚L

can!,L

99

` //// ^pj_!˚j^˚L ^

`// // <sup>p`</sup>j_!˚j^˚L ^{ ` // p}j_˚j^˚L

where below is, cf. the remark following 1.3.12 of [6], the canonical factorisation of ^p`j_!j^˚LÝÑ ^pj_˚j^˚Land where the maps can_!,Land can_˚,L are given by the adjonction property. Consider now X equipped with a stratification

X “X^ě1 ĄX^ě2 Ą ¨ ¨ ¨ ĄX^ěd,

and let L P FpX,Zlq. For 1ď hă d, let denote X^1ďh :“ X^ě1 ´X^ěh`1 and j^1ďh :X^1ďh ãÑX^ě1. We then define

Fil^r_!pLq:“Im_F

´p`j_!^1ďrj^1ďr,˚LÝÑL

¯ , which gives a filtration

0“Fil⁰_!pLq Ă Fil¹_!pLq ĂFil¹_!pLq ¨ ¨ ¨ Ă Fil^d´1_! pLq ĂFil^d_!pLq “L.

Dually CoFil˚,rpLq “CoimF

´

LÝÑ ^pj_˚^1ďrj^1ďr,˚L

¯

,define a cofiltration L“CoFil_S,˚,dpLqCoFil_S,˚,d´1pLq¨ ¨ ¨

¨ ¨ ¨CoFilS,˚,1pLqCoFilS,˚,0pLq “0, and a filtration

0“Fil^´d_˚ pLq ĂFil^1´d_˚ pLq Ă ¨ ¨ ¨ ĂFil⁰_˚pLq “ L where Fil^´r_˚ pLq:“Ker_F`

LCoFil_˚,rpLq˘ .

Remark.These two constructions are exchanged by Grothendieck-Verdier duality, D

´

CoFil_S,!,´rpLq

¯

» Fil^´r_S,˚pDpLqq and D

´

CoFil_S,˚,rpLq

¯

» Fil^r_S,!pDpLqq.

We can also refine the previous filtrations, cf. [6] proposition 2.3.3, to obtain exhaustive filtrations

0“Fill^´2_! ^d´1pLq ĂFill^´2_! ^d´1`1pLq Ă ¨ ¨ ¨

¨ ¨ ¨ Ă Fill⁰_!pLq Ă ¨ ¨ ¨ ĂFill²_!^d´1´1pLq “L, (1.4.0) such that the graduate grr^kpLq are simple over Ql, i.e. verify

pj_!˚^“hj^“h,˚grr^kpLq ã` grr^kpLq for some h. Dually we can construct a cofiltration

L“CoFill_˚,2d´1pLqCoFill_˚,2d´1´1pLq¨ ¨ ¨CoFill_˚,´2d´1pLq “ 0 and a filtration Fill^´r_˚ pLq:“Ker_F`

LCoFill_˚,rpLq˘ . 1.5. Vanishing cycles perverse sheaf. —

1.5.1. Notation. — For I PI, let

Ψ_I,Λ:“RΨ_ηv,IpΛrd´1sqpd´1 2 q

be the vanishing cycle autodual perverse sheaf on X_I,¯sv. When Λ “ Zl, we will simply write ΨI.

Recall the following result of [19] relating ΨI with Harris-Taylor local systems.

1.5.2. Proposition. — (cf. [19] proposition IV.2.2 and §2.4 of [3]) There is an isomorphism GpA^8,vq ˆPh,d´hpFvq ˆWv-equivariant

ind^D

ˆ v,h

pD^ˆ_v,hq⁰$^Z_v

`H^h´d´iΨ_I,

Zl

˘

|X_I,¯^“h

s,1h

» à

τPR¯

Flphq

LZl,1hpU_{τ ,¯}^h´1´i

N q,

where – R

Flphqis the set of equivalence classes of irreducibleFl-representations of D_v,h^ˆ ;

– for τ¯P R

Flphq and V a Zl-representation of D^ˆ_v,h, then V_τ¯ denotes, cf. [16] §B.2, the direct factor of V whose irreducible subquotients are isomorphic to a subquotient of ¯τ_|D^ˆ

v,h where D_v,h is the maximal order of D_v,h.

– With the previous notation, U_{¯τ ,N}ⁱ :“` U_Fⁱ

v,Zl,d

˘

¯ τ.

– The matching between the system indexed by I and those by N is given by the application m₁ :I ÝÑ N.

Remark. For ¯τ P R

Flphq, and a lifting τ which by Jacquet-Langlands correspondence can be written τ »πrtsD for π irreductible cuspidal, let

% P Scusp

Flpgq be in the supercuspidal support. Then the inertial class of % depends only on ¯τ and we will use the following notation.

1.5.3. Notation. — With the previous notation, we denote V_% for V_τ¯. Remark. Ψ_I,

Zl is an object of FpXI,¯s,Zlq. Indeed, by [2] proposition 4.4.2, Ψ_I,Z

l is an object of ^pD^ď0pXI,¯s,Z^lq. By [20] variant 4.4 of theorem 4.2, we have DΨ_I,Z

l »Ψ_I,Z

l, so that Ψ_I,

Zl P ^pD^ď0pX_I,¯s,Z^lq X^p`D^ě0pX_I,¯s,Z^lq “ FpX_I,¯s,Z^lq. 1.5.4. Proposition. — (cf. [9] §3.2) We have a decomposition

ΨI »

d

à

g“1

à

%PScusp

Flpgq

Ψ_%

where all the Harris-Taylor perverse sheaves of Ψ_%b_ZlQl are of type %.

Remark. In [3], we decomposed Ψb_ZlQl as a direct sum À

πvΨ_πv where πv described the set of equivalent inertial classes of irreducible cuspidal representations. The Ψ%b_ZlQl »À

πvPCuspp%qΨπv where Cuspp%q is the set of equivalent inertial classes of irreducible cuspidal representation of type %in the sense of definition 1.2.2.

In [11], we give the precise description of the gr^r_S,!pΨI,%q which is de- fined other Zl. By construction they are supported on X_I,¯^ěr_s

v and trivial if g does not divide r. Otherwise for r“tg, we have

ind^D

ˆ v,tg

pD^ˆ_v,tgq⁰$^Z_v

´

j^“tg,˚gr^tg_S,!pΨI,%q b_ZlQl

¯

»

rp%q

à

i“´1 tigip%q“tg

à

πvPScusp_ip%q

HTpπ_v,St_tipπ_vqq bLpπ_vqp´t_i´1 2 q.

We can then consider the following naive %-filtration Fil^˚_%,rp%q,Q

lpΨ, tgq Ă ¨ ¨ ¨ ĂFil^˚_%,´1,Q

lpΨ, tgq “ j^“tg,˚gr^tg_S,!pΨ_I,%q b_ZlQl

where ind^D

ˆ v,tg

pD^ˆ_v,tgq⁰$_v^Z

´ Fil^˚_%,k,

QlpΨ, tgq

¯

is isomorphic to

rp%q

à

i“k tigip%q“tg

à

πvPScusp_ip%q

HTpπv,Sttipπvqq bLpπvqp´t_i´1 2 q,

and the associated entire filtration ofj^“tg,˚gr^tg_S,!pΨI,%qdefined by pullback Fil^˚_%,kpΨ, tgq ^{  ////}

 _

Fil^˚_%,k,

QlpΨ, tgq

 _

j^“tg,˚gr^tg_S,!pΨI,%q ^{ //} j^“tg,˚gr^tg_S,!pΨI,%q b_ZlQl.

For k “ ´1,¨ ¨ ¨ , rp%q, the graduates gr_%,kpΨ, tgq are then of %-type k.

We can then refine these filtrations by separating the π_v P Scusp_kp%q to obtain

p0q “Fil^˚,0_% pΨ, tgq ĂFil^˚,1_% pΨ, tgq Ă ¨ ¨ ¨ ĂFil^˚,r_% pΨ, tgq “ j^“tg,˚gr^tg_S,!pΨI,%q. By taking the iterated images of j_!^“tgFil^k_%pΨ, tgq ÝÑgr^tg_S,!pΨI,%q, we then construct a filtration

p0q “Fil⁰_%pΨ, tgq ĂFil¹_%pΨ, tgq Ă ¨ ¨ ¨ ĂFil^r_%pΨ, tgq “ gr^tg_S,!pΨI,%q.

Finally we can filtrate each of these graduate using an exhaustive fil- tration of stratification to obtain a filtration of ΨI,% whose graduates are the Ppπ_v, tqp^1´sip%q`2k₂ q for π_v P Scusp_ip%q with i ě ´1, and k “ 0,¨ ¨ ¨, s_ip%q ´1.

2. Cohomology of KHT Shimura varieties 2.1. Localization at a non pseudo-Eisenstein ideal. —

2.1.1. Definition. — Define Spl the set of places v of F such that p_v :“ v_|Q ‰ l is split in E and B_v^ˆ » GL_dpF_vq. For each I P I, write SplpIq the subset of Spl of places which doesn’t divide the level I.

Let UnrpIq be the union of

– places q‰l of Q inert inE not below a place of Bad and where I_q is maximal,

– and places wPSplpIq.

2.1.2. Notation. — For I PI a finite level, write T^I :“

ź

xPUnrpIq

T^x

where for x a place of Q (resp. x P SplpIq), Tx is the unramified Hecke algebra of GpQxq (resp. of GL_dpF_xq) over Zl.

Example: for wPSplpIq, we have Tw “Zl

“T_w,i : i“1,¨ ¨ ¨ , d‰ , where T_w,i is the characteristic function of

GLdpOwqdiagp

i

hkkkkkkikkkkkkj

$w,¨ ¨ ¨, $w,

d´i

hkkkikkkj

1,¨ ¨ ¨ ,1qGLdpOwq ĂGLdpFwq.

More generally, the Satake isomorphism identifiesTxwithZlrX^unpT_xqs^Wx where

– T_x is a split torus,

– W_x is the spherical Weyl group

– and X^unpT_xq is the set of Z^l-unramified characters of T_x.

Consider a fixed maximal ideal m of TI and for every x P UnrpIq let denote S_mpxq be the multi-set^p4q of modulo l Satake parameters at x associated to m.

Example: for every w P SplpIq, the multi-set of Satake parameters at w corresponds to the roots of the Hecke polynomial

P_m,wpXq:“

d

ÿ

i“0

p´1qⁱq

ipi´1q

w 2 T_w,iX^d´i PFlrXs

i.e. S_mpwq :“ λ P TI{m » Fl such that P_m,wpλq “ 0(

. For a maximal ideal mr of TI^lb_ZlQl, we also have the multi-set of Satake parameters

S_mrpwq:“ λP TIb_Zl Ql{mr »Ql such thatP_m,wĄpλq “0( .

p4qA multi-set is a set with multiplicities.

2.1.3. Notation. — Let Π be an irreducible automorphic representa- tion of GpAq of level I which means here, that Π has non trivial invari- ants under I and for every x P UnrpIq, then Π_x is unramified. Then Π defines

– a maximal ideal mpΠqr of TI^lb_ZlQl, or

– a minimal prime ideal mpΠqr of TI^l contained in a maximal ideal mpΠqofTI^l which corresponds to its the modulolSatake parameters.

A minimal prime ideal mr of TI^l is said cohomological if there exists a cohomological automorphic Ql-representation Π of GpAq of levelI with mr “mpΠq. Such a Π is not unique butr mr defines a unique near equiva- lence class in the sense of [22], we denote it Π_mr. Let then

ρm,Qr l : GalpF¯{Fq ÝÑGL_dpQlq

be the galoisian representation associated to such a Π thanks to [19]

and [22], which by Cebotarev theorem, can be defined over some finite extension K_mr, i.e. ρ

m,rQl »ρ_mrbK

mr Ql.

It’s well known that ρ_mr has stable lattices and the semi-simplication of its modulo l reduction is independent of the chosen stable lattice.

Moreover it depends only of the maximal ideal m, we denote ρ_m :G_F ÝÑGL_dpFlq,

its extension to Fl. For every w P SplpIq, recall that the multiset of eigenvalues ofρ_mpFrob_wqisS_mpwqobtained fromS_mrpwqby taking modulo l reduction.

Assume moreover that ρ_m is absolutely irreducible. Then the Ql- cohomology groupH^d´1pX_U,¯η,Qlqm gives a continuousd-dimensional Ga- lois representation

ρ_m : Gal_F,S ÝÑGL_dpTS,mr1{lsq,

where Gal_F,S is the Galois group of the maximal extension of F which is unramified outsideS. As all characteristic polynomials of Frobenius takes values in T^S,m and as ρ_m is absolutely irreducible, using classical theory of pseudo-representations, we know that ρ_m takes values in GL_dpT^S,mq.

2.2. Freeness of the cohomology. — From now on we fix such a maximal ideal mof T^I verifying one of the following three conditions:

(1) There exists w₁ P SplpIq such that S_mpw₁q does not contain any sub-multi-set of the shapetα, q_w1αuwhereq_w1 is the cardinal of the residue field. Note also that this hypothesis is closed but weaker than the notion of decomposed genericness introduced in [12].

(2) Withl ěd`2 we askmto verify one of the two following hypothesis

‚ either ρ_m is induced form a character of GK for a cyclic ga- loisian extensionK{F;

‚ eitherSL_npkq Ă ρ_mpG_Fq ĂF^ˆl GL_npkqfor a sub-field k ĂFl. (3) ρ_m is absolutely irreducible and l ěd`1 except maybe for a finite

number of them depending only of the extensionF{Q, cf. the main theorem of [10].

2.2.1. Theorem. — (cf. [8]) For m as above, the localized cohomology groups HⁱpX_I,¯η,Zlqm are free.

As X_I ÝÑ SpecO_v is proper, we have a GpA⁸q ˆ W_v-equivariant isomorphism H^{d´1`i</sup>pXI,¯ηv,Zlq » H<sup>i</sup>pX<sub>I,¯sv</sub>,ΨIq. Using the previous fil- tration of ΨI, we can compute H<sup>p`q}pX_I,¯sv,ΨI,%qm through a spectral sequence whose entries E₁^p,q are the H^jpXI,¯sv,Ppπ_v, tqp^{1´sip%q`2k</sup><sub>2</sub> qqm for π<sub>v</sub> P Scusp<sub>i</sub>p%q with i ě ´1, and k “ 0,¨ ¨ ¨ , s<sub>i</sub>p%q ´1. Over Ql, it fol- lows from [4], thanks to the hypothesis (1) above on m, that all these cohomology groups are concentrated in degree 0 so that this Ql-spectral sequence degenerates in E<sub>1</sub>. In this section, under (H2), we want to prove the same result on Fl which is equivalent to the freeness of the H<sup>j</sup>pXI,¯sv,Ppπ<sub>v</sub>, tqp<sup>1´sip%q`2k}₂ qqm.

We need first some notations from [4] §1.2. For all tě0, we denote Γ^t :“

!

pa₁,¨ ¨ ¨ , a_r, ₁,¨ ¨ ¨, _rq PN^rˆ t˘u^r : r ě1,

r

ÿ

i“1

a_i “t )

. A element of Γ^t will be denoted pÐaÝ₁,¨ ¨ ¨ ,ÝÑa_rq where for the arrow above each integera_i isÐaÝ_i (resp. ÝÑa_i) if_i “ ´(resp. _i “ `). We then consider on Γ^t the equivalence relation induced by

p¨ ¨ ¨ ,ÐÝa ,ÐÝ

b ,¨ ¨ ¨ q “ p¨ ¨ ¨ ,ÐÝÝÝ

a`b,¨ ¨ ¨ q, p¨ ¨ ¨ ,ÝÑa ,ÝÑ

b ,¨ ¨ ¨ q “ p¨ ¨ ¨ ,ÝÝÝÑ a`b,¨ ¨ ¨ q,

and p¨ ¨ ¨ ,ÐÝ

0,¨ ¨ ¨ q “ p¨ ¨ ¨ ,ÝÑ

0,¨ ¨ ¨ q. We denoteÑÝ

Γ^t the set of these equiv- alence classes whose elements are denoted rÐaÝ₁,¨ ¨ ¨ ,ÝÑa_rs.

Remark. In each class rÐaÝ1,¨ ¨ ¨,ÝÑaks P ÝÑ

Γ^t, there exists a unique reduced element pb1,¨ ¨ ¨ , br, 1,¨ ¨ ¨, rq PΓ^t such that for all 1ďiďr bi ą0 and for 1ďiăr, ii`1 “ ´.

2.2.2. Definition. — Let pb₁,¨ ¨ ¨, b_r, ₁,¨ ¨ ¨ , _rq be the reduced ele- ment in rÐaÝ₁,¨ ¨ ¨ ,ÑÝa_ks PÝÑ

Γ^t. We then define S´

rÐaÝ₁,¨ ¨ ¨ ,ÑÝa_ks

¯

as the subset of permutationsσoft0,¨ ¨ ¨ , t´1usuch that for all 1ďiďr with_i “ `(resp. <sub>i</sub> “ ´) and for allb<sub>1</sub>`¨ ¨ ¨`b<sub>i´1</sub> ďk ăk<sup>1</sup> ďb<sub>1</sub>`¨ ¨ ¨`b_i then σ^´1pkq ă σ^´1pk¹q (resp. σ^´1pkq ąσ^´1pk¹q).

We also introduceS^op´

rÐaÝ₁,¨ ¨ ¨ ,ÑÝa_ks

¯

, by imposing under the same con- ditions, σ^´1pkq ąσ^´1pk¹q(resp. σ^´1pkq ă σ^´1pk¹q).

2.2.3. Proposition. — (cf. [24] §2) Let g be a divisor of d “ sg and π an irreducible cuspidal representation of GL_gpF_vq. There exists a bi- jection

rÐaÝ₁,¨ ¨ ¨ ,ÝÑa_rs PÝÑ

Γ^s´1 ÞÑ rÐaÝ₁,¨ ¨ ¨,ÝÑa_rsπ

into the set of irreducible sub-quotients of the induced representation πt1´s

2 u ˆπt3´s

2 u ˆ ¨ ¨ ¨ ˆπts´1 2 u characterized by the following property

J_{Pg,2g,¨¨¨,sg}prÐaÝ₁,¨ ¨ ¨ ,ÝÑa_rsπq “ ÿ

σPS

´

rÐaÝ1,¨¨¨,ÑÝars

¯

πt1´s

2 `σp0qub¨ ¨ ¨bπt1´s

2 `σps´1qu,

or equivalently by J_P^op

g,2g,¨¨¨,sgprÐaÝ₁,¨ ¨ ¨ ,ÝÑa_rsπq “

ÿ

σPS^op

´

rÐaÝ₁,¨¨¨,ÑÝa_rs

¯

πt1´s

2 `σp0qub¨ ¨ ¨bπt1´s

2 `σps´1qu.

Remark. With these notations St_spπq (resp. Speh_spπq) is rÐsÝÝ´Ý1sπ (resp.

rÝsÝÝ´Ñ1sπ).