CONSTRUCTION OF TORSION COHOMOLOGY CLASSES FOR KHT SHIMURA VARIETIES

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CLASSES FOR KHT SHIMURA VARIETIES

Pascal Boyer

To cite this version:

Pascal Boyer. CONSTRUCTION OF TORSION COHOMOLOGY CLASSES FOR KHT SHIMURA

VARIETIES. 2019. �hal-02080467�

COHOMOLOGY CLASSES FOR KHT SHIMURA VARIETIES

by Boyer Pascal

Abstract. — Let Sh

pG, µq be a Shimura variety of KHT type, as introduced in [11], associated to some similitude group G{ Q and a open compact subgroup K of Gp A q. For any irreducible algebraic Q

- representation ξ of G, let V

be the Z

-local system on Sh

pG, µq. From [7], we know that if we allow the local component K

of K to be small enough, then there must exists some non trivial cohomology classes with coefficient in V

. The aim of this paper is then to construct explicitly such torsion classes with the control of K

. As an application we ob- tain the construction of some new automorphic congruences between tempered and non tempered automorphic representations of the same weight and same level at l.

Contents

1. Introduction . . . . 2 2. Cohomology of Harris-Taylor perverse sheaves . . . . 4 2.1. Shimura varieties of Kottwitz-Harris-Taylor type . 4 2.2. Harris-Taylor perverse sheaves. . . . 6 2.3. Cohomology groups over Q

. . . . 8 3. Torsion in the cohomology of KHT-Shimura varieties . 10 3.1. Torsion for Harris-Taylor perverse sheaves . . . 11 3.2. Perverse sheaf of vanishing cycles and lattices . . . . 14 3.3. Main result . . . 17

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, galoisian representation.

References. . . 18

1. Introduction

The class number formula for number fields (resp. the Birch- Swinnerton-Dyer conjecture) asserts that the order of vanishing of the Dedekind zeta function at s “ 0 of a number field K (resp. the order of vanishing at s “ 1 of the L-function of some elliptic curve E over a number field K) with the rank of its group of units (resp. with the rank of the Mordell-Weil group EpKq). Both of these statements can be restated in terms of rank of Selmer groups and is generalized for p-adic motivic Galois representations in the Bloch-Kato conjecture.

Since the work of Ribet, one strategy to realize a part of this con- jecture is to consider some automorphic tempered representation Π of a reductive group G{ Q and take a prime divisor l of some special values of its L-function. We try then to construct an automorphic non tempered representation Π

of G congruent to Π modulo l in some sense so that such an automorphic congruence produces a non trivial element in some Selmer group.

In [9] we show how to produce automorphic congruences from torsion classes in the cohomology of Kottwitz-Harris-Taylor Shimura varieties associated to G, with coefficients in a local system L

indexed by irre- ducible algebraic representations ξ, called weight, of Gp Q q. For example, see corollary 2.9 of [9], to each non trivial torsion cohomology class of level I, we can associate an infinite collection of non isomorphic weakly con- gruent irreducible automorphic representations of the same weight and level but each of them being tempered. In section 3 of [9], we obtained automorphic congruences between tempered and non tempered automor- phic representations but with distinct weights. In [7], using completed cohomology, we constructed automorphic congruences between tempered and non tempered automorphic representations of the same weight but without any control of their respective level at l which might be an issue to construct then non trivial element in some Selmer groups, cf. loc. cit.

Another way to interpret the computations of [7], is to say that, what- ever is the weight ξ, if you take the level at l small enough, then the cohomology groups of your KHT Shimura variety with coefficients in L

can’t be all free, there must exist some non trivial cohomology classes.

The main aim of this paper is then to find explicit conditions for the ex- istence of non trivial cohomology classes with coefficients in L

, without playing with the level at l.

As in previous work, we compute the cohomology groups of the Shimura variety ShpG, µq through the vanishing cycles spectral se- quence, i.e. as the cohomology of the special fiber of ShpG, µq at some place v not dividing l, with coefficients in L

b Ψ

where Ψ

designates the perverse sheaf of vanishing cycles at v. In [4] we explained how to, using the Newton stratification of the special fiber of ShpG, µq, construct a Z

-filtration of Ψ

which graduates are some intermediate extensions of the Z

-Harris-Taylor local systems constructed in [11]. These local systems are indexed by irreducible Q

-cuspidal entire representations π

of the linear group of rank g ď d, where d is the dimension of Sh p G, µ q : among the data is some lattice of the Steinberg representation St

p π

q with tg ď d. We then have a spectral sequence E

computing the H

p Sh

p G, µ q , L

q : up to translation we may suppose that E

“ 0 for all p ă 0.

– The first idea to construct torsion could be to find some non trivial torsion classes in the E

-page, i.e. in the cohomology of the Harris- Taylor perverse sheaves. For example in [5] proposition 4.5.1, we prove that if the modulo l reduction of such π

is cuspidal but not supercuspidal, then, for a well chosen level K, the cohomology groups of the associated Harris-Taylor perverse sheaves, can’t be all free, so there is torsion on the E

page. Unfortunately it seems not so easy to prove that such torsion cohomology classes remains on the E

-page.

– We can then try to produce torsion in the E

page by finding a map d

with

E

b

Q

d^p,q₁ b

ZlQl

//

E

b

Q

Q

^// Q ^?

OO

such that the Z

-lattices of Q and Q

respectively induced by E

and E

, are not isomorphic.

The idea to realize this last point, is to use the main result of [8] where

we describe some of the lattices of the St

p π

q which appears as some

data of the Harris-Taylor perverse sheaves as graduates of the filtration of stratification of Ψ

. These lattices verify some non degeneracy persitence property in the following sense: the socle of their modulo l reduction is irreducible and non degenerate. In particular when the local system is concentrated in the supersingular locus, which means with the previous notations that St

pπ

q is a representation of GL

pF

q, then this persitence of non degeneracy is also true for E

while all the E

for p ą 0 containing St

pπ

q are non trivially parabolically induced. If we manage so that this non degenerate socle is cuspidal, necessary these induced lattices don’t satisfy the persitence of non degeneracy property, so that the E

-page has non trivial cohomology classes and it’s then quite easy to prove that it remains at the E

-page.

In terms of L-functions, our assumption to realize this program, corre- sponds to the following. Find an irreducible automorphic representation Π of level K such that its local L-factor at p modulo l, has a pole at s “ 1.

2. Cohomology of Harris-Taylor perverse sheaves

2.1. Shimura varieties of Kottwitz-Harris-Taylor type. — Let F “ F

E be a CM field where E { Q is quadratic imaginary and F

{ Q totally real with a fixed real embedding τ : F

ã Ñ R . For a place v of F , we will denote

– F

the completion of F at v, – O

the ring of integers of F

, – $

a uniformizer,

– q

the cardinal of the residual field κ p v q “ O

{p $

q .

Let B be a division algebra with center F , of dimension d

such that at every place x of F , either B

is split or a local division algebra and suppose B provided with an involution of second kind ˚ such that ˚

is the complexe conjugation. For any β P B

, denote 7

the involution x ÞÑ x

“ βx

β

and G {Q the group of similitudes, denoted G

in [11], defined for every Q -algebra R by

GpRq » tpλ, gq P R

ˆ pB

b

Rq

such that gg

“ λu with B

“ B b

F . If x is a place of Q split x “ yy

in E then

Gp Q

q » pB

q

ˆ Q

» Q

ˆ ź

zi

pB

i

q

, (2.1.1)

where, identifying places of F

over x with places of F over y, x “ ś

i

z

in F

.

Convention: for x “ yy

a place of Q split in E and z a place of F over y as before, we shall make throughout the text, the following abuse of notation by denoting GpF

q in place of the factor pB

q

in the formula (2.1.1).

In [11], the author justify the existence of some G like before such that moreover

– if x is a place of Q non split in E then G pQ

q is quasi split;

– the invariants of Gp R q are p1, d ´ 1q for the embedding τ and p0, dq for the others.

As in [11] bottom of page 90, a compact open subgroup U of G p A

q is said small enough if there exists a place x such that the projection from U

to G pQ

q does not contain any element of finite order except identity.

2.1.2. Notation. — Denote I the set of open compact subgroup small enough of Gp A

q. For I P I, write X

ÝÑ Spec F the associated Shimura variety of Kottwitz-Harris-Taylor type.

2.1.3. Definition. — Define Spl the set of places v of F such that p

:“ v

‰ l is split in E and B

» GL

pF

q. For each I P I , write SplpIq the subset of Spl of places which doesn’t divide the level I.

In the sequel, v will denote a place of F in Spl. For such a place v the scheme X

has a projective model X

over Spec O

with special fiber X

. For I going through I, the projective system pX

q

is naturally equipped with an action of Gp A

q ˆ Z such that w

in the Weil group W

of F

acts by ´ deg p w

q P Z , where deg “ val ˝ Art

and Art

: W

» F

is Artin’s isomorphism which sends geometric Frobenius to uniformizers.

2.1.4. Notations. — (see [2] §1.3) For I P I, the Newton stratification of the geometric special fiber X

is denoted

X

“: X

Ą X

Ą ¨ ¨ ¨ Ą X

where X

:“ X

v

´ X

v

is an affine scheme

, smooth of pure dimension d ´ h built up by the geometric points whose connected part of

p1q

see for example [12]

its Barsotti-Tate group is of rank h. For each 1 ď h ă d, write i

: X

ã Ñ X

, j

: X

ã Ñ X

, and j

“ i

˝ j

.

For 1 ď h ă d, the Newton stratum X

is geometrically induced under the action of the parabolic subgroup P

pO

q in the sense where there exists a closed subscheme X

h

stabilized by the Hecke action of P

pO

q and such that

X

» X

h

ˆ

GL

pO

q.

For a P GL

p F

q{ P

p F

q , we denote X

the image of X

h

by a and X

its closure in X

: they are stable under P

pF

q :“

aP

pF

qa

.

2.2. Harris-Taylor perverse sheaves. — From now on, we fix a prime number l unramified in E and suppose that for every place v of F considered after, its restriction v

is not equal to l. For a representation π

of GL

pF

q and n P

Z , set π

tnu :“ π

b q

. Recall that the normalized induction of two representations π

and π

of respectively GL

pF

q and GL

pF

q is

π

ˆ π

: “ ind

n1,n1`n2pFvq

π

t n

2 u b π

t´ n

2 u .

A representation π

of GL

pF

q is called cuspidal (resp. supercuspidal ) if it’s not a subspace (resp. subquotient) of a proper parabolic induced representation.

2.2.1. Definition. — (see [14] §9 and [3] §1.4) Let g be a divisor of d “ sg and π

an irreducible cuspidal Q

-representation of GL

pF

q.

Then the normalized induced representation π

t 1 ´ s

2 u ˆ π

t 3 ´ s

2 u ˆ ¨ ¨ ¨ ˆ π

t s ´ 1 2 u

holds a unique irreducible quotient (resp. subspace) denoted St

p π

q (resp.

Speh

p π

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

The local Jacquet-Langlands correspondance is a bijection between ir-

reducible essentially square integrable representations of GL

p F

q , i.e.

representations of the type St

p π

q for π

cuspidal, and irreducible repre- sentations of D

where D

is the central division algebra over F

with invariant

and with maximal order D

.

2.2.2. Notation. — We will denote π

rss

the irreductible representa- tion of D

associated to St

pπ

q by the local Jacquet-Langlands corre- spondance.

Let π

be an irreducible cuspidal Q

-representation of GL

p F

q and fix t ě 1 such that tg ď d. Thanks to Igusa varieties, Harris and Taylor constructed a local system on X

s,1h

L

l

pπ

rts

q

“

eπv

à

i“1

L

l

pρ

q

where pπ

rts

q

v,h

“ À

i“1

ρ

with ρ

irreductible. The Hecke action of P

pF

q is then given through its quotient GL

ˆ Z . These local systems have stable Z

-lattices and we will write simply L p π

r t s

q

for any Z

-stable lattice that we don’t want to specify.

2.2.3. Notations. — For Π

any representation of GL

and Ξ :

1

2

Z ÝÑ Z

defined by Ξp

q “ q

, we introduce

HT Ą

p π

, Π

q : “ L p π

r t s

q

b Π

b Ξ

and its induced version

HT Ą pπ

, Π

q :“

´

Lpπ

rts

q

b Π

b Ξ

¯

ˆ

GL

pF

q, where the unipotent radical of P

pF

q acts trivially and the action of

pg

,

ˆ g

˚ 0 g

˙

, σ

q P Gp A

q ˆ P

pF

q ˆ W

is given

– by the action of g

on Π

and degpσ

q P Z on Ξ

, and

– the action of pg

, g

, valpdet g

q´deg σ

q P Gp A

qˆGL

pF

qˆ Z on L

l

pπ

rts

q

b Ξ

. We also introduce

HT pπ

, Π

q

:“ HT Ą pπ

, Π

q

rd ´ tgs,

and the perverse sheaf

P pt, π

q

:“ j

h,!˚

HT pπ

, St

pπ

qq

b L pπ

q,

and their induced version, HT p π

, Π

q and P p t, π

q , where L

is the local Langlands correspondence.

Remark: recall that π

is said inertially equivalent to π

if there exists a character ζ : Z ÝÑ Q

such that π

» π

b pζ ˝ val ˝ detq. Note, cf. [2]

2.1.4, that P pt, π

q depends only on the inertial class of π

and P pt, π

q “ e

P pt, π

q

where P pt, π

q is an irreducible perverse sheaf. When we want to speak of the Q

-versions we will add it on the notations.

Recall that the modulo l reduction of an irreducible Q

-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 of supercuspidal F

-representation

% of length in the following set t 1, m p % q , lm p % q , l

m p % q , ¨ ¨ ¨ u .

2.2.1. Definition. — Let i P Z be greater than ´1. We say that π

is of %-type i when the supercuspidal support of its modulo l reduction is a Zelevinsky segment associated to % of length 1 for i “ ´1 and mp%ql

otherwise.

2.2.4. Notation. — We denote Scusp

Fl

p g q the set of inertial equiva- lence classes of F

-supercuspidal representations of GL

pF

q. For % P Scusp

l

pgq, we then denote

– g

p % q “ g and for i ě 0, g

p % q “ m p % q l

g

p % q ;

– Scusp

p%q the set of inertial equivalence classes of irreducible cuspi- dal Q

-representations of %-type i.

2.3. Cohomology groups over Q

. — Let σ

: E ã Ñ Q

be a fixed embedding and write Φ the set of embeddings σ : F ã Ñ Q

whose re- striction to E equals σ

. There exists, [11] p.97, then an explicit bi- jection between irreducible algebraic representations ξ of G over Q

and pd ` 1q-uple `

a

, pÝ Ñ a

q

˘ where a

P Z and for all σ P Φ, we have Ý

Ñ a

“ p a

ď ¨ ¨ ¨ ď a

q . We then denote V

the associated Z

-local

system on X

Recall that an irreducible automorphic representation Π is said ξ-cohomological if there exists an integer i such that

H

`

p Lie G pRqq b

C , U, Π

b ξ

˘

‰ p 0 q ,

where U is a maximal open compact subgroup modulo the center of Gp R q.

2.3.1. Definition. — (cf. [13]) For Π an automorphic irreducible rep- resentation ξ-cohomological of Gp A q, then, see for example lemma 3.2 of [5], there exists an integer s called the degeneracy depth of Π, such that through Jacquel-Langlands correspondence and base change, its associated representation of GL

p A

q is isobaric of the following form

µ| det |

‘ µ| det |

‘ ¨ ¨ ¨ ‘ µ| det |

where µ is an irreducible cuspidal representation of GL

p A

q.

Remark: for a place v such that G p F

q » GL

p F

q in the sense of our previous convention, the local component Π

of Π at v is isomorphic to some Speh

p π

q where π

is an irreducible non degenerate representation, s ě 1 is an integer and Speh

p π

q is the Langlands quotients of the parabolic induced representation π

t

u ˆ π

t

u ˆ ¨ ¨ ¨ ˆ π

t

u. In terms of the Langlands correspondence, Speh

p π

q corresponds to σ ‘ σp1q ‘ ¨ ¨ ¨ ‘ σps ´ 1q where σ is the representation of Galp F ¯ {F q associated to π

by the local Langlands correspondence.

2.3.2. Notation. — For π

an irreducible cuspidal Q

-representation of GL

p F

q and t ě 1 such that tg ď d, let denote

H

pπ

, t, ξq :“ lim

ÝÑ IPIv

H

pX

v,1tg

, V

b j

tg

P pπ

, tq

q and the induced version

H

pπ

, tq :“ lim

IPIv

H

pX

, V

bj

P pπ

, tqq » H

pπ

, tqˆ

GL

pF

q,

as well as

H

pπ

, tq :“ lim

ÝÑ IPIv

H

pX

v,1tg

, V

b P pπ

, tq

q and

H

pπ

, tq

:“ lim

ÝÑ IPIv

H

pX

, V

bP pπ

, tqq » H

pπ

, tqˆ

GL

pF

q,

where I

is the set of I P I such that I

is of the form K

pnq for some

n ě 1.

In this section we only consider the Q

-cohomology groups and we recall the computations of [3]. For Π

an irreducible representation of G pA

q , denote r H

p π

, t, ξ qst Π

u the associated isotypic component.

We will use similar notations with the cohomology groups introduced above. Consider now a fixed irreducible cuspidal representation π

of GL

pF

q.

2.3.3. Proposition. — (cf. [5] §3.2 and 3.3) Let Π be an irreducible automorphic representation of Gp A q which is ξ-cohomological and with degeneracy depth s ě 1.

– If s “ 1 then rH

pπ

, t, ξqstΠ

u and rH

pπ

, t, ξqstΠ

u are all zero for i ‰ 0. For i “ 0, if rH

pπ

, t, ξqstΠ

u ‰ p0q (resp.

rH

pπ

, t, ξqstΠ

u ‰ p0q) then Π

is of the following shape St

pr π

q ˆ Π

where π r

is inertially equivalent to π

and k ď t (resp.

k “ t).

– For s ě 1, and Π

of the following shape Speh

p π

ˆ π

q , π

any irreducible representation of GL

s

p F

q , then r H

p π

, t, ξ qst Π

u (resp. r H

p π

, t, ξ qst Π

u ) is non zero if and only if i “ s ´ 1 and t ě s (resp. t “ s and i ” s ´ 1 mod 2 with | i | ď s ´ 1).

Remark: In [5], we give the complete description of these cohomology groups.

3. Torsion in the cohomology of KHT-Shimura varieties As explained in the introduction, to construct non trivial torsion co- homology classes with coefficients in L

, we use a spectral sequence E

obtained from a filtration of stratification of the perverse sheaf of nearby cycles Ψ

, whose E

terms are given by the cohomology groups of Harris- Taylor perverse sheaves. The argument takes place in two steps:

– the construction of non trivial torsion classes in the E

page, cf.

§3.3;

– we then have to prove that these previous torsion classes remain in the E

.

To be able to do the second point we need to have some informal infor-

mation about the torsion classes in the cohomology of the Harris-Taylor

perverse sheaves: it’s the aim of the next section.

3.1. Torsion for Harris-Taylor perverse sheaves. — From now on, we fix an irreducible supercuspidal F

-representation % and all the irreducible cuspidal Q

-representation π

considered will be of type %. In [4], using the adjunction maps Id ÝÑ j

j

, we construct a filtration called of stratification in loc. cit.

0 “ Fil

pπ

, Π

q Ă Fil

pπ

, Π

q Ă ¨ ¨ ¨ Ă Fil

pπ

, Π

q “ j

HT pπ

, Π

q, with free gradutates gr

p π

, Π

q : “ Fil

p π

, Π

q{ Fil

p π

, Π

q which are trival except for r “ kg ´ 1 with t ď k ď s and then verifying

p

j

HT p π

, Π

Ñ Ý

ˆ St

p π

qq b Ξ

ã gr

pπ

, Π

q ã

p`

j

HT p π

, Π

Ý Ñ

ˆ St

p π

qq b Ξ

, where we recall that ã

means a bimorphism, i.e. both a mono and a epi-morphism, whose cokernel has support in X

. In [6], we in fact proved that each of these graduates are isomorphic to the p-intermediate extensions.

Meanwhile when g “ 1 and π

“ χ

is a character, then the associated Harris-Taylor local system on X

is just the trivial one Z

where the fundamental group Π

pX

q acts by its quotient Π

pX

q D

with D

acting by the character χ

. Then as X

h

is smooth over Spec F

, then this Harris-Taylor local system shifted by the dimension d ´ h, is perverse for both t-structures p and p`, in particular the two intermedi- ate extensions are equal so the previous short exact sequence is trivially true when g “ 1.

One of the main result of [6] is that this equality of perverse extensions remains true for every Harris-Taylor local systems associated to any ir- reducible cuspidal representation π

such that its modulo l reduction is still supercuspidal, that is, with definition 2.2.1, is of %-type ´1.

More generally for every π

P Scusp

p % q we prove the following long exact sequence

0 Ñ j

HT pπ

, Π

Ñ Ý ˆ Speh

pπ

qq b Ξ

ÝÑ ¨ ¨ ¨ ÝÑ j

HT pπ

, Π

Ñ Ý

ˆ Speh

pπ

qq b Ξ

ÝÑ j

HT pπ

, Π

Ý Ñ ˆπ

q b Ξ

ÝÑ j

HT p π

, Π

q ÝÑ

j

HT p π

, Π

q Ñ 0, (3.1.1)

which is equivalent to the property that the sheaf cohomology groups of

p

j

HT p π

, Π

q are torsion free.

For I P I a finite level, let T

: “ ś

qPUnrpIq

T

be the unramified Hecke algebra where Unr p I q is the union of places q where G is ramified and I

is not maximal, and where T

» Z

rX

pT

qs

for T

a split torus, W

the spherical Weyl group and X

p T

q the set of Z

-unramified characters of T

.

3.1.2. Notation. — Consider a fixed maximal ideal m of T

. For every q P UnrpIq let denote S

pqq the multi-set of modulo l Satake parameters at q associated to m. For any T

-module M , we moreover denote M

its localization at m.

Consider π P Scusp

p%q and denote g “ g

p%q and let denote H

pπ

, tq

:“ lim

ÝÑ IPIv

H

pX

, V

b

j

HT pπ

, tqq

and

H

p π

, t q

: “ lim

ÝÑ IPIv

H

p X

v

, V

b j

HT p π

, t qq

.

In [9] we proved that torsion classes arising in some cohomology group of the whole Shimura variety, can be raised in characteristic zero to some automorphic tempered representation of G pAq in the following sense.

3.1.1. Definition. — A torsion class either in H

pπ

, tq

or in H

pπ

, tq

, is said tempered ξ-cohomologial if there exists an irreducible automorphic and ξ-cohomological tempered representation Π unramified outside I and p with Π

a subquotient of lim

H

pX

, V

Ql

q

. From now on we moreover suppose that d “ g

p % q m p % q l

and we will pay attention to irreducible GL

pF

q-subquotient of either H

pπ

, tq

rls or H

pπ

, tq

rls, isomorphic to ρ

.

3.1.2. Lemma. — With the previous notations and assumptions with π

P Scusp

p%q for i ě ´1, if there exists an irreducible subquotient of H

pπ

, tq

rls (resp. H

pπ

, tq

rls), which is GL

pF

q-isotypic for ρ

, then j P t0, 1u (resp. j “ 1).

Proof. — Consider first the case of i “ ´1. We argue by induction from

t “ s “ m p % q l

to t “ 1 with both H

p π

, t q

and H

p π

, t q

.

Concerning H

p π

, t q

, recall that, as π

P Scusp

p % q so that

p

j

HT pπ

, tq »

j

HT pπ

, tq,

then we only have to consider the case i ď 0. By Artin’s theorem, see for example theorem 4.1.1 of [1], using the affiness of X

, we know that H

pπ

, tq

is zero for every i ă 0 and without torsion for i “ 0.

Note first that for t “ s, then HT pπ

, sq has support in dimension zero, so that H

p π

, s q

“ H

p π

, s q

is zero for i ‰ 0 and free for i “ 0, so the result is trivially true. Suppose by induction, the result true for all t ą t

and consider the case of H

p π

, t q

through the spectral sequence associated to the resolution (3.1.1). Note first that concerning irreducible subquotient of the l-torsion of the cohomology groups which are GL

pF

q-isomorphic to ρ

, then we can truncate (3.1.1) to the short exact sequence of its last three terms.

0 99K j

HT pπ

, Π

Ñ Ý

ˆ π

q b Ξ

ÝÑ

j

HT p π

, Π

q ÝÑ

j

HT p π

, Π

q Ñ 0. (3.1.3) Then considering our problem for H

pπ

, tq

, there is nothing to prove for i ď ´1 and for i “ 0 we conclude because over Q

H

pX

, j

HT pπ

, Π

Ñ Ý ˆ π

qbΞ

q ÝÑ H

pX

, j

HT pπ

, Π

qq is related to tempered automorphic ξ-cohomological representations. We are then done with H

p π

, t q

. The result about H

p π

, t q

, then follows from the long exact sequence associated to (3.1.3).

Consider now the case i ě 0. Recall, cf. [10] proposition 2.3.3, that the semi-simplification of the modulo l reduction of π

r t s

, doesn’t depend of the choice of a stable lattice, and is equal to

mp%qlⁱ´1

ÿ

k“0

τt´ m p % q l

´ 1

2 ` ku

where τ is the modulo l reduction of π

rtmp%ql

s

which is irreducible, and τ t n u : “ τ b q

where nrd is the reduced norm. In particular for any representation Π

of GL

p F

q , we have

mp%ql

F

”

j

HT pπ

, Πq ı

“ mp%ql

j

”

F HT pπ

, Πq ı

“ j

”

F HT pπ

, Πq ı

“ F

”

j

HT pπ

, Πq ı

. (3.1.3)

Note moreover that concerning subquotient isomorphic to ρ

, the only case where it can appeared in the modulo l reduction of some irreducible subquotient of the free quotient of H

p X

, j

HT p π

, Π

qq is when either pt, j q “ ps ´1, 1q or j “ 0. The result about H

pπ

, tq

rls then fol- lows from the previous case where i “ ´1 using (3.1.3) and the following wellknown short exact sequence

0 Ñ H

pX, P q b

F

ÝÑ H

pX, F P q ÝÑ H

pX, P qrls Ñ 0, for any F

-scheme X and any Z

-perverse free sheaf P .

Then the result about the cohomology of

j

HT p π

, Π

q follows from the resolution analog of (3.1.3), and the case of

j

HT pπ

, Π

q is obtained by Grothendieck-Verdier duality.

3.2. Perverse sheaf of vanishing cycles and lattices. — 3.2.1. Notation. — For I P I, let

Ψ

:“ RΨ

p Z

rd ´ 1sqp d ´ 1 2 q be the vanishing cycle autodual perverse sheaf on X

.

In [6], we gave a decomposition Ψ

»

d

à

g“1

à

%PScusp

Flpgq

Ψ

with Ψ

b

Q

» À

πvPCuspp%q

Ψ

where the irreducible constituant of Ψ

are exactly the perverse Harris-Taylor sheaves attached to π

.

Recall now from [4], how to construct filtrations of a free perverse sheaf such that its graduates are free. Start first from an open subscheme j : U ã Ñ X and i : F :“ XzU ã Ñ X, such that j is affine. We then have the following adjunction morphism where we use the notation L ã L

(resp. L ã

L

) for a bimorphism, i.e. both a monomorphism and a epimorphism (resp. such that its cokernel is of dimension strictly less than those of the support of L):

L

can˚,L

**

j

j

L

can!,L

99

`

// //

j

j

L ^

`

// //

j

j

L ^

^//

j

j

L

where below is, cf. the remark following 1.3.12 de [4], the canonical factorisation of

j

j

L ÝÑ

j

j

L and where the maps can

and can

are given by the adjonction property.

3.2.2. Notation. — (cf. lemma 2.1.2 of [4]) We introduce the filtra- tion Fil

pLq Ă Fil

pLq Ă L with

Fil

pLq “ Im

pcan

q and Fil

pLq “ Im

´

pcan

q

¯ , where P

:“ i

H

i

j

j

L is the kernel of Ker

´

j

j

L

j

j

L

¯ . Remark : we have L{ Fil

pLq » i

i

L and

j

j

L ã

Fil

pLq{ Fil

pLq, which gives, cf. lemma 1.3.13 of [4], a commutative triangle

p

j

j

L ^

`

// //

w

`

)) ))

Fil

pLq{ Fil

pLq

 _

`

p`

j

j

L.

Consider now X equipped with a stratification X “ X

Ą X

Ą ¨ ¨ ¨ Ą X

and let L P F p X, Z

q . For 1 ď h ă d, let denote X

: “ X

´ X

and j

: X

ã Ñ X

. We then define

Fil

p L q : “ Im

´

j

j

L ÝÑ L

¯ , which gives a filtration

0 “ Fil

pLq Ă Fil

pLq Ă Fil

pLq ¨ ¨ ¨ Ă Fil

pLq Ă Fil

pLq “ L.

With these notations, the graduate gr

pΨ

q verify j

gr

pΨ

q »

"

0 si g - h

L

l

p%rts

q pour h “ tg.

In particular for h “ d, the perverse sheaf gr

pΨ

q is a quotient of Ψ

and concentrated of the supersingular locus. One of the main result of [8] can be stated as follow.

3.2.1. Proposition. — (cf. [8] proposition 4.2.4) There exists a filtra- tion

0 “ Fil

Ă Fil

Ă Fil

Ă ¨ ¨ ¨ Fil

Ă Fil

“ gr

pΨ

q

verifying the following properties.

– The graduates gr

“ Fil

{ Fil

are free of %-type k in the following sense

gr

b

Q

» à

πvPScusp_kp%q

HT pπ

, St

pπ

qq b L

pπ

qp 1 ´ s

p%q 2 q, where for i ě 0 recall g

p % q “ g

m p % q l

and g

p % q s

p % q “ d.

– For any geometric supersingular point i

“ t z u ã Ñ X

v

, then ind

ˆ v,d

pD^ˆ_v,dq⁰$_v^Z

p

h

i

gr

à

πvPScusp_kp%q

Γ

pπ

q b Γ

pπ

q

where Γ

pπ

q (resp. Γ

pπ

q) is a stable lattice of St

pπ

q (resp.

of π

rs

p%qs

b L

pπ

qp

q.

– Let M

pF

q be the mirabolic subgroup of GL

pF

q defined by impos- ing the first column to be p1, 0, ¨ ¨ ¨ , 0q. Then if τ is an irreducible sub-M

pF

q-representation of Γ

pπ

q b

F

, then τ is isomorphic to the non degenerate representation τ

, unique up to isomor- phism. As a consequence any irreducible GL

pF

q-representation of Γ

pπ

q b

F

is isomorphic to ρ

.

Remark: About Γ

pπ

q, note that for π

P Scusp

p%q as the modulo l reduction of π

rs

p%qs

b L

pπ

qp

q is irreducible, up to isomor- phism, there exists only one stable lattice which is then a tensor product Γ

pπ

q b Γ

pπ

q.

In [4] proposition 2.3.3, we explained how, using Fil

p U, ! q , to con- struct an exhaustive filtration of Ψ

0 “ Fill

pΨ

q Ă Fill

pΨ

q Ă ¨ ¨ ¨ Ă Fill

pΨ

q Ă ¨ ¨ ¨

Ă Fill

pΨ

q “ Ψ

, such that each of the graduate is over Q

, a direct sum of Harris-Taylor perverse sheaves of the same Newton stratum. It’s then possible to refine this filtration to obtain a new one

Fill

Ă Fill

Ă ¨ ¨ ¨ Ă Fill

where each graduate is a Harris-Taylor perverse Z

-sheaf. Note the lat-

tices constructed in this way, may depend of the construction but con-

cerning the quotient of the previous proposition, the statement remains

true. Precisely we can manage so that grr

: “ Fill

{ Fill

verify, with the notation of the previous proposition

ind

ˆ v,d

pD^ˆ_v,dq⁰$^Z_v

p

h

i

grr

» Γ

pπ

q b Γ

pπ

q b Γ

pπ

q,

where π

is any fixed irreducible cuspidal representation in Scusp

p%q.

3.3. Main result. — Start from an irreducible automorphic cuspidal representation Π of Gp A q verifying the following properties:

– it is ξ-cohomological with non trivial invariant under some fixed I P I;

– its degeneracy depth is equal to s ą 1;

– its local component at v is isomorphic to Speh

pπ

q with π

P Scusp

p%q and where d “ g

p%q for some u ě 0.

Remark: For π

the trivial characte, the hypothesis d “ g

p % q for u “ 0 is equivalent to ask that the order of q P F

, which is the cardinal of the residue field of F

, is equal to d. Another way to formulate this condition, is to say that the L-function of the trivial character modulo l, has a pole at s “ 1.

Denote m the maximal ideal of T

associated to Π. Remember that m encodes the multiset of Satake’s parameters of Π outside I.

3.3.1. Proposition. — Under the previous hypothesis, the torsion of H

pX

, L

rd ´ 1sq

is non trivial.

Proof. — Consider the spectral sequence

E

“ H

pX

, grr

q

ñ H

pX

, V

q

associated to the filtration Fill

of Ψ

. First note that over Q

:

– E

b

Q

has a direct factor isomorphic to pΠ

q

b St

pπ

q b L

pπ

qp

q;

– d

b

Q

induces a injection from the previous direct factor into a direct factor of E

which, as a representation of GL

pF

q is parabolically induced from P

pF

q to GL

pF

q.

From the last remark of the previous section, ind

ˆ v,d

pD^ˆ_v,dq⁰$_v^Z

p

h

i

E

as

a GL

pF

q-representation, has a subspace isomorphic to Γ

pπ

q where

Γ

pπ

q is a stable lattice of St

pπ

q such that ρ

is the only irreducible

sub-representation of Γ

pπ

q b

Q

. Moreover we know that ρ

can’t

be a subspace of parabolically induced representation. From these facts we conclude that the torsion of E

is non trivial and more precisely that ρ

is a subquotient of E

rls.

Now there are two alternatives. Either ρ

as a subquotient of E

rls remains a subquotient of E

rls and we are done. Suppose by absurdity it’s not the case. First about the free quotient E

of the E

, we know from [3] that:

– if ρ

is a subquotient of E

b

F

with p ` q ‰ 0, then grr

is isomorphic to some P pπ

, s

p%q ´ 1q with π

P Scusp

p%q and then p ` q “ ˘1;

– for k ě 2 and p ` q ‰ 0, then ρ

is never a subquotient of E

b

F

.

Then there must exist pp, qq and a torsion class in pE

q

with p `q “ 2 such that ρ

is a subquotient of its l-torsion which contradicts lemma 3.1.2.

Thanks to the main result of [9], associated to this torsion cohomology class is a tempered irreducible automorphic representation Π

of G p A q which is ξ-cohomological of level I

I

and weakly m-congruent to Π in the sense it shares the same multiset of Satake’s parameters than Π outside I. In particular for s “ 2, as in Ribet’s proof of Herbrand theorem, we should obtain a non trivial element in the Selmer group of the adjoint representation of the Galois F

-representation associated to m.

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PerCoLaTor: ANR-14-CE25