Construction of torsion cohomology classes for KHT Shimura varieties
Pascal Boyer



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Construction of torsion cohomology classes]Construction of torsion cohomology classes for KHT Shimura varieties






boyer@math.univ-paris13.fr
Université Paris 13, Sorbonne Paris Cité 

LAGA, CNRS, UMR 7539
 
F-93430, Villetaneuse (France) 

PerCoLaTor: ANR-14-CE25







Let _K(G,μ) be a  Shimura variety of KHT type,  as introduced in <cit.>, associated 
to some similitude group G/ and a open compact subgroup K of G(). For any irreducible
algebraic _l-representation ξ of G, let V_ξ be the _l-local system 
on _K(G,μ). From <cit.>, we know that if we allow the local component
K_l of K to be small enough, then there must exists some non trivial cohomology
classes with coefficient in V_ξ. The aim of this paper is then to construct explicitly such torsion
classes with the control of K_l. As an application we obtain the construction of some new automorphic
congruences between tempered and non tempered automorphic representations of the same
weight and same level at l.






11F70, 11F80, 11F85, 11G18, 20C08










[
    Boyer Pascal
    December 30, 2023
=====================



headings arabic













§ 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 E(K)). 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 conjecture is to consider some
automorphic tempered representation Π of a reductive group G/ 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 <cit.> 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
_ξ indexed by irreducible algebraic representations ξ, called weight, of G().
For example, see corollary 2.9 of <cit.>, to each non trivial torsion cohomology class of level I,
we can associate an infinite collection of non isomorphic weakly congruent irreducible automorphic 
representations of the same weight and level but each of them being tempered.
In section 3 of <cit.>, we obtained automorphic congruences between tempered and
non tempered automorphic representations but with distinct weights.
In <cit.>, 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 <cit.>, is to say that, whatever 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 _ξ 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 existence of non trivial
cohomology classes with coefficients in _ξ, without playing with the level at l. 

As in previous work, we compute the cohomology groups of the Shimura variety (G,μ)
through the vanishing cycles spectral sequence, i.e. as the cohomology of the special
fiber of (G,μ) at some place v not dividing l, 
with coefficients in _ξ⊗Ψ_v where Ψ_v
designates the perverse sheaf of vanishing cycles at v. In <cit.> we explained
how to, using the Newton stratification of the special fiber of (G,μ), construct a
_l-filtration of Ψ_v which graduates are some intermediate extensions of
the _l-Harris-Taylor local systems constructed in <cit.>. These local systems are
indexed by irreducible _l-cuspidal entire representations π_v of the linear group of
rank g ≤ d, where d is the dimension of (G,μ): among the data is some lattice of 
the Steinberg representation _t(π_v) with tg ≤ d.
We then have a spectral sequence E_1^p,q computing the H^p+q(_K(G,μ),_ξ):
up to translation we may suppose that E_1^p,q=0 for all p<0.


  * The first idea to construct torsion could be to find some non trivial torsion classes in the E_1-page, i.e.
in the cohomology of the Harris-Taylor perverse sheaves. For example in <cit.> proposition 4.5.1,
we prove that if the modulo l reduction of such π_v 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_1 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_2 page by finding a map d_1^p,q with

    E_1^p,q⊗__l_l 
    [rr]^d_1^p,q⊗__l_l@->>[d]        E_1^p+1,q⊗__l_l 
    
    Q [rr]^∼       Q' @^(->[u]

such that the _l-lattices of Q and Q' respectively induced by E_1^p,q
and E_1^p+1,q, are not isomorphic.

The idea to realize this last point, is to use the main result of <cit.> where we describe
some of the lattices of the _t(π_v) which appears as some data of the Harris-Taylor perverse
sheaves as graduates of the filtration of stratification of Ψ_v. 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 _t(π_v) is a representation of
GL_d(F_v), then this persitence of non degeneracy is also true for E_1^0,0 while all the
E_1^p,1-p for p>0 containing _t(π_v) 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_2-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, corresponds 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.











§ COHOMOLOGY OF HARRIS-TAYLOR PERVERSE SHEAVES





 §.§ Shimura varieties of Kottwitz-Harris-Taylor type




Let F=F^+ E be a CM field where E/ is quadratic imaginary and F^+/
totally real with a fixed real embedding τ:F^+ ↪. For a place v of F,
we will denote


  * F_v the completion of F at v,


  * _v the ring of integers of F_v,


  * ϖ_v a uniformizer,


  * q_v the cardinal of the residual field κ(v)=_v/(ϖ_v).

Let B be a division algebra with center F, of dimension d^2 such that at every place x of F,
either B_x is split or a local division algebra and suppose B provided with an involution of
second kind * such that *_|F is the complexe conjugation. For any
β∈ B^*=-1, denote ♯_β the involution x ↦ x^♯_β=β x^*
β^-1 and G/ the group of similitudes, denoted G_τ in <cit.>, defined for every
-algebra R by 

    G(R)  ≃{ (λ,g) ∈ R^×× (B^op⊗_ R)^× such that  
    gg^♯_β=λ}

with B^op=B ⊗_F,c F. 
If x is a place of  split x=yy^c in E then 
thm1

    G(_x) ≃ (B_y^op)^××_x^×≃_x^××∏_z_i (B_z_i^op)^×,

where, identifying places of F^+ over x with places of F over y,
x=∏_i z_i in F^+.

Convention: for x=yy^c a place of  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 
G(F_z) in place of the factor (B_z^op)^× in the formula (<ref>).


In <cit.>, the author justify the existence of some G like before such that moreover


  * if x is a place of  non split in E then G(_x) is quasi split;


  * the invariants of G() are (1,d-1) for the embedding τ and (0,d) for the others.


As in  <cit.> bottom of page 90, a compact open subgroup U of G(^) is said 
small enough
if there exists a place x such that the projection from U^v to G(_x) does not contain any 
element of finite order except identity.


Denote  the set of open compact subgroup small enough of G(^).
For I ∈, write X_I,η⟶ F the associated
Shimura variety of Kottwitz-Harris-Taylor type.



 
Define  the set of  places v of F such that p_v:=v_|≠ l is split in E and
B_v^×≃ GL_d(F_v).  For each I ∈, write
(I) the subset of  of places which doesn't divide the level I.




In the sequel, v will denote a place of F in . For such a place v 
the scheme X_I,η has a projective model X_I,v over _v
with special fiber X_I,s_v. For I going through , the projective system (X_I,v)_I∈ 
is naturally equipped with an action of G(^) × such that 
w_v in the Weil group W_v of F_v acts by - (w_v) ∈,
where =∘^-1 and ^-1:W_v^ab≃ F_v^× is Artin's isomorphism
which sends geometric Frobenius to uniformizers.

 (see <cit.> 1.3)
For I ∈, the Newton stratification of the geometric special fiber X_I,s̅_v is denoted

    X_I,s̅_v=:X^≥ 1_I,s̅_v⊃ X^≥ 2_I,s̅_v⊃⋯⊃ 
    X^≥ d_I,s̅_v

where X^=h_I,s̅_v:=X^≥ h_I,s̅_v - X^≥ h+1_I,s̅_v is an affine 
scheme[see for example <cit.>], smooth of pure dimension d-h built up by the geometric 
points whose connected part of its Barsotti-Tate group is of rank h.
For each 1 ≤ h <d, write

    i_h:X^≥ h_I,s̅_v↪ X^≥ 1_I,s̅_v,   
    j^≥ h: X^=h_I,s̅_v↪ X^≥ h_I,s̅_v,

and j^=h=i_h ∘ j^≥ h.












For 1 ≤ h < d, the Newton stratum X_,s̅^=h is geometrically induced
under the action of the parabolic subgroup P_h,d-h(_v) in the sense where there
exists a closed subscheme X_I,s̅,1_h^=h stabilized by the Hecke 
action of P_h,d-h(_v) and such that

    X_,s̅^=h≃ X_,s̅,1_h^=h×_P_h,d-h(_v) GL_d(_v).

For a ∈ GL_d(F_v)/P_h,d-h(F_v), we denote X_I,s̅,a^=h the image of
X_,s̅,1_h^=h by a and X_I,s̅,a^≥ h its closure in 
X_I,s̅^≥ h: they are stable under 
P_a(F_v):=aP_h,d-h(F_v)a^-1.





 §.§ 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_v of GL_d(F_v) and n ∈1/2, set 
π_v { n }:= π_v ⊗ q_v^-n ∘. Recall that
the normalized induction of two representations π_v,1 and π_v,2 
of respectively GL_n_1(F_v) and GL_n_2(F_v) is

    π_1 ×π_2:=_P_n_1,n_1+n_2(F_v)^GL_n_1+n_2(F_v)π_v,1{n_2/2}⊗π_v,2{-n_1/2}.

A representation
π_v of GL_d(F_v) is called cuspidal (resp. supercuspidal)
if it's not a subspace (resp. subquotient) of a proper parabolic induced representation.


  (see <cit.> 9 and <cit.> 1.4)
Let g be a divisor of d=sg and π_v an irreducible cuspidal 
_l-representation of GL_g(F_v). Then the normalized induced representation 

    π_v{1-s/2}×π_v {3-s/2}×⋯×π_v {s-1/2}
 
holds a unique irreducible quotient (resp. subspace) denoted _s(π_v) (resp.
_s(π_v)); it's a generalized Steinberg (resp. Speh) representation.



The local Jacquet-Langlands correspondance is a bijection between irreducible essentially square
integrable representations of GL_d(F_v), i.e. representations of the type _s(π_v)
for π_v cuspidal, and irreducible representations of D_v,d^× where
D_v,d is the central division algebra over F_v with invariant 1/d and with maximal
order _v,d.


We will denote π_v[s]_D the irreductible representation of D_v,d^× associated 
to _s(π_v^∨) by the local Jacquet-Langlands correspondance.



Let π_v be an irreducible cuspidal _l-representation of GL_g(F_v) and
fix t ≥ 1 such that tg ≤ d. Thanks to Igusa varieties, Harris and Taylor
constructed a local system on  X^=tg_,s̅,1_h

    __l(π_v[t]_D)_1_h=⊕_i=1^e_π_v__l(ρ_v,i)_1_h

where (π_v[t]_D)_|_v,h^×=⊕_i=1^e_π_vρ_v,i 
with ρ_v,i irreductible. The Hecke action of P_tg,d-tg(F_v) is then given
through its quotient GL_d-tg×. These local systems have stable
_l-lattices and we will write simply (π_v[t]_D)_1_h
for any _l-stable lattice that we don't want to specify.


 For Π_t any representation of GL_tg and 
Ξ:1/2⟶_l^× defined by 
Ξ(1/2)=q^1/2, we introduce

    HT_1(π_v,Π_t):=(π_v[t]_D)_1_h⊗Π_t ⊗Ξ^tg-d/2

and its induced version

    HT(π_v,Π_t):= ( (π_v[t]_D)_1_h⊗Π_t ⊗Ξ^tg-d/2) ×_P_tg,d-tg(F_v) GL_d(F_v),

where the unipotent radical of P_tg,d-tg(F_v) acts trivially and the action of

    (g^,v, ( [  g_v^c      *;      0 g_v^et ] ),σ_v) 
    ∈ G(^,v) × P_tg,d-tg(F_v) × W_v
 
is given


  * by the action of g_v^c on Π_t and 
(σ_v) ∈ on Ξ^tg-d/2, and


  * the action of (g^,v,g_v^et,( g_v^c)-σ_v)
∈ G(^,v) × GL_d-tg(F_v) × on __l
(π_v[t]_D)_1_h⊗Ξ^tg-d/2.

We also introduce

    HT(π_v,Π_t)_1_h:=HT(π_v,Π_t)_1_h[d-tg],

and the perverse sheaf

    P(t,π_v)_1_h:=j^=tg_1_h,!* HT(π_v,_t(π_v))_1_h⊗(π_v),

and their induced version, HT(π_v,Π_t) and P(t,π_v), where 

^∨ is the local Langlands correspondence.


recall that π'_v is said inertially equivalent to π_v 
if there exists a character ζ: ⟶_l^× such that
π'_v ≃π_v ⊗ (ζ∘∘).
Note, cf. <cit.> 2.1.4, that P(t,π_v) depends only on the inertial class of π_v and

    P(t,π_v)=e_π_v(t,π_v)
 
where (t,π_v) is an irreducible perverse sheaf. When we want to speak of
the _l-versions we will add it on the notations.



Recall that the modulo l reduction of an irreducible _l-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 _l-representation ϱ of length in the following set
{ 1, m(ϱ),l m(ϱ),l^2 m(ϱ),⋯}.

 
Let i ∈ be greater than -1.
We say that π_v is of ϱ-type i when the supercuspidal support of its modulo
l reduction is a Zelevinsky segment associated to ϱ of length 1 for i=-1 and 
m(ϱ)l^i otherwise. 


 We denote __l(g) the set of inertial equivalence classes of
_l-supercuspidal representations of GL_g(F_v). For 
ϱ∈__l(g), we then denote 


  * g_-1(ϱ)=g and for i ≥ 0, g_i(ϱ)=m(ϱ)l^i g_-1(ϱ);


  * _i(ϱ) the set of inertial equivalence
classes of irreducible cuspidal _l-representations of ϱ-type i.









 §.§ Cohomology groups over _l



Let σ_0:E ↪_l be a fixed embedding and write Φ the set of embeddings 
σ:F ↪_l whose restriction to E equals σ_0.
There exists, <cit.> p.97, then an explicit bijection between irreducible algebraic representations 
ξ of G over _l and (d+1)-uple
( a_0, (a_σ)_σ∈Φ )
where a_0 ∈ and for all σ∈Φ, we have a_σ=
(a_σ,1≤⋯≤ a_σ,d ). 
We then denote V_ξ the associated _l-local system on X_ 
Recall that an irreducible automorphic representation Π is said ξ-cohomological if there exists
an integer i such that

    H^i  ( (  G()) ⊗_,U,Π_⊗ξ^∨ ) ≠ (0),

where U is a maximal open compact subgroup modulo the center of G().


  (cf. <cit.>)
For Π an automorphic irreducible representation ξ-cohomological of G(),
then, see for example lemma 3.2 of <cit.>,  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_d(_)
is isobaric of the following form

    μ |  |^1-s/2⊞μ |  |^3-s/2⊞⋯⊞μ |  |^s-1/2

where μ is an irreducible cuspidal representation of GL_d/s(_).


for a place v such that G(F_v) ≃ GL_d(F_v) in the sense of our previous convention,
the local component Π_v of Π at v is isomorphic to some _s(π_v) where π_v
is an irreducible non degenerate representation, s ≥ 1 is an integer and _s(π_v)
is the Langlands quotients of the parabolic induced representation π_v {1-s/2}×π_v {3-s/2}×⋯×π_v {s-1/2}. In terms of the Langlands
correspondence, _s(π_v) corresponds to σ⊕σ(1) ⊕⋯⊕σ(s-1) 
where σ is the representation of (F̅/F) associated to π_v by the local Langlands 
correspondence.


For π_v an irreducible cuspidal _l-representation of GL_g(F_v) and t ≥ 1
such that tg ≤ d, let denote

    H^i_!,1(π_v,t,ξ):=lim_⟶I ∈_v
    H^i_c(X_I,s̅_v,1_tg^=tg,V_ξ⊗
    j^=tg,*_1_tg(π_v,t)_1_tg)

and the induced version

    H^i_!(π_v,t):=lim_⟶I ∈_v
    H^i_c(X_I,s̅_v^≥ tg,V_ξ⊗
    j^=tg,*(π_v,t)) ≃ H^i_!,1(π_v,t)×_P_tg,d-tg(F_v) GL_d(F_v),

as well as

    H^i_!*,1(π_v,t):=lim_⟶I ∈_v
    H^i(X_I,s̅_v,1_tg^≥ tg,V_ξ⊗(π_v,t)_1_tg)

and

    H^i(π_v,t)_!*:=lim_⟶I ∈_v
    H^i(X_I,s̅_v^≥ tg,V_ξ⊗(π_v,t)) ≃ 
    H^i_!*,1(π_v,t)×_P_tg,d-tg(F_v) GL_d(F_v),

where _v is the set of I ∈ such that I_v is of the form K_v(n) for some n ≥ 1.


In this section we only consider the _l-cohomology groups and we recall the computations
of <cit.>. For Π^,v an irreducible representation of G(^,v), 
denote [H^i_!(π_v,t,ξ) ] {Π^,v} the associated isotypic component. We will use similar
notations with the cohomology groups introduced above.
Consider now a fixed irreducible cuspidal representation π_v of GL_g(F_v).




  (cf. <cit.> 3.2 and 3.3)
Let Π be an irreducible automorphic representation of G() which is ξ-cohomological and
with degeneracy depth s ≥ 1.


  * If s=1 then [H^i_!(π_v,t,ξ) ] {Π^,v} and
[H^i_!,*(π_v,t,ξ) ] {Π^,v} are all zero for i ≠ 0.
For i=0, if [H^i_!(π_v,t,ξ) ] {Π^,v}≠ (0) (resp.
[H^i_!*(π_v,t,ξ) ] {Π^,v}≠ (0)) then Π_v is of the following shape
_k(π_v) ×Π'_v where π_v is inertially equivalent to π_v
and k ≤ t (resp. k=t).


  * For s ≥ 1, and Π_v of the following shape _s(π_v ×π'_v),
π'_v any irreducible representation of GL_d-sg/s(F_v), then
[H^i_!(π_v,t,ξ) ] {Π^,v} (resp. [H^i_!*(π_v,t,ξ) ] {Π^,v}) is non zero if and only
if i=s-1 and t ≥ s (resp. t=s and i ≡ s-1  2 with |i| ≤ s-1).



In <cit.>, we give the complete description of these cohomology groups.





§ TORSION IN THE COHOMOLOGY OF KHT-SHIMURA VARIETIES


As explained in the introduction, to construct non trivial torsion cohomology classes with coefficients
in _ξ, we use a spectral sequence E_1^p,q obtained from a filtration of stratification
of the perverse sheaf of nearby cycles Ψ_, whose E_1 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_2 page, cf. <ref>;


  * 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 information about the
torsion classes in the cohomology of the Harris-Taylor perverse sheaves: it's the aim of the next
section.





 §.§ Torsion for Harris-Taylor perverse sheaves



From now on, we fix an irreducible supercuspidal _l-representation ϱ and all the
irreducible cuspidal _l-representation π_v considered will be of type ϱ.
In <cit.>, using the adjunction maps ⟶ j^=h_* j^=h,*, we construct
a filtration called of stratification in loc. cit.

    0=^-d_*(π_v,Π_t) ⊂^1-d_*(π_v,Π_t) ⊂⋯⊂^0_*(π_v,Π_t)=j^=tg_! HT(π_v, Π_t),

with free gradutates ^-r_*(π_v,Π_t):=^-r_*(π_v,Π_t)/^-r-1_*(π_v,Π_t) which are trival
except for r=kg-1 with t ≤ k ≤ s and then verifying

    p  j^= kg_!* HT (π_v,Π_t ×_k-t(π_v)) ⊗Ξ^(t-k)/2
    ^1-kg_*(π_v,Π_t) 
    p+  j^= kg_!* 
    HT (π_v,Π_t ×_k-t(π_v)) ⊗Ξ^(t-k)/2,

where we recall that _+ means a bimorphism, i.e. both a mono and a epi-morphism,
whose cokernel has support in X^≥ (kg+1_,s̅.
In <cit.>, we in fact proved that each of these graduates are isomorphic to the p-intermediate
extensions.

Meanwhile when g=1 and π_v=χ_v is a character, 
then the associated Harris-Taylor local system on X^=h_,s̅ is just the trivial one
_l where the fundamental group Π_1(X^=h_,s̅) 
acts by its quotient Π_1(X^=h_,s̅) ↠_v,h^× with
_v,h^× acting by the character χ_v. Then as X^≥ h_,s̅,1_h
is smooth over _p, 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 intermediate extensions are 
equal so the previous short exact sequence is trivially true when g=1.

One of the main result of <cit.> is that this equality of perverse extensions remains true
for every Harris-Taylor local systems associated to any irreducible cuspidal representation π_v such
that its modulo l reduction is still supercuspidal, that is, with definition <ref>, is of ϱ-type
-1.

More generally for every π_v ∈_i(ϱ) we prove the following long exact sequence
thm1

    0 → j_!^=s g HT(π_v,Π_t ×_s-t(π_v)) ⊗Ξ^s-t/2⟶⋯⟶
    
    j^=(t+2)g_! HT(π_v,Π_t×_2(π_v)) ⊗Ξ^1 
    ⟶ j_!^= (t+1)g HT(π_v,Π_t ×π_v) ⊗Ξ^1/2
    ⟶  
    j^=tg_! HT(π_v,Π_t) ⟶ p j^=tg_!* HT(π_v,Π_t)
    → 0,

which is equivalent to the property that the sheaf cohomology groups of p j^=tg_!* HT(π_v,Π_t)
are torsion free.


For I ∈ a finite level, let _I:=∏_x ∈(I)_x be the unramified Hecke algebra
where (I) is the union of places x where G is unramified and I_x maximal, and where
_x ≃_l[X^un(T_x)]^W_x for T_x a split torus,
W_x the spherical Weyl group and X^un(T_x) the set of _l-unramified characters 
of T_x. 


Consider a fixed maximal ideal 𝔪 of _I. For every q ∈(I) let denote
S_𝔪(q) the multi-set 
of modulo l Satake parameters at q associated to 𝔪. For any _I-module M,
we moreover denote M_𝔪 its localization at 𝔪.



Consider π∈_-1(ϱ) and denote g=g_-1(ϱ) and let denote

    H^i_!*(π_v,t)_𝔪:=lim_⟶I ∈_v
    H^i(X_I,s̅_v^≥ tg,V_ξ⊗ p j^=tg_!* HT(π_v,t))_𝔪

and

    H^i_!(π_v,t)_𝔪:=lim_⟶I ∈_v
    H^i(X_I,s̅_v^≥ tg,V_ξ⊗ j^=tg_! HT(π_v,t))_𝔪.

In <cit.> 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() in 
the following sense.


A torsion class either in H^i_!*(π_v,t)_𝔪 or in H^i_!(π_v,t)_𝔪,
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_→ I ∈ 
H^d-1(X_I,η̅,V_ξ,_l)_𝔪.


From now on we moreover suppose that d=g_-1(ϱ)m(ϱ)l^u and we will pay attention to
irreducible GL_d(F_v)-subquotient of either H^i_!*(π_v,t)_𝔪[l] or 
H^i_!(π_v,t)_𝔪[l], isomorphic to ρ_u.


 
With the previous notations and assumptions with π_v,i∈_i(ϱ) for i ≥ -1, 
if there exists an irreducible subquotient
of H^j_!*(π_v,i,t)_𝔪[l] (resp. H^j_!(π_v,i,t)_𝔪[l]), 
which is GL_d(F_v)-isotypic for ρ_u, then j ∈{ 0,1 } (resp. j=1).


 
Consider first the case of i=-1.
We argue by induction from t=s=m(ϱ)l^u to t=1 with both H^i_!,1(π_v,-1,t)_𝔪
and H^i_!*(π_v,-1,t)_𝔪. Concerning H^i_!*(π_v,-1,t)_𝔪, 
recall that, as π_v,-1∈_-1(ϱ) so that 

    p j^=tg_!* HT(π_v,-1,t)  ≃p+ j^=tg_!* HT(π_v,-1,t),

then we only have to consider the case i ≤ 0.
By Artin's theorem, see for example theorem 4.1.1 of <cit.>, using the affiness of X^=h_I,s̅_v, 
we know that H^i_!(π_v,-1,t)_𝔪 is zero for every i<0 and without torsion for i=0.

Note first that for t=s, then HT(π_v,-1,s) has support in dimension zero, so that 
H^i_!*(π_v,-1,s)_𝔪=H^i_!(π_v,-1,s)_𝔪 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>t_0 and consider the case of 
H^i_!*(π_v,-1,t)_𝔪 through the spectral sequence associated to the 
resolution (<ref>). Note first that concerning irreducible subquotient of the l-torsion
of the cohomology groups which are GL_d(F_v)-isomorphic to ρ_u, then we can truncate
(<ref>) to the short exact sequence of its last three terms.
thm1

    0  j_!^= (t+1)g HT(π_v,-1,Π_t ×π_v,-1) ⊗Ξ^1/2⟶
    
    j^=tg_! HT(π_v,-1,Π_t) ⟶ p j^=tg_!* HT(π_v,-1,Π_t)
    → 0.

Then considering our problem for H^i_!*(π_v,-1,t)_𝔪, there is nothing to prove for i ≤ -1
and for i=0 we conclude because over _l

    H^0(X_I,s̅_v, j_!^= (t+1)g HT(π_v,-1,Π_t ×π_v,-1) ⊗Ξ^1/2)
    ⟶ H^0(X_I,s̅_v,j^=tg_! HT(π_v,-1,Π_t))

is related to tempered automorphic ξ-cohomological representations.
We are then done with H^i_!*(π_v,-1,t)_𝔪.
The result about H^i_!(π_v,-1,t)_𝔪, then follows from the long exact sequence associated
to (<ref>).











Consider now the case i ≥ 0. Recall, cf. <cit.> proposition 2.3.3, that the semi-simplification of
the modulo l reduction of π_v,i[t]_D, doesn't depend of the choice of a stable lattice, and is equal to

    ∑_k=0^m(ϱ)l^i -1τ{ - m(ϱ)l^i-1/2 +k}
 
where τ is the modulo l reduction of π_v,-1[tm(ϱ)l^i]_D which is irreducible, and
τ{ n }:=τ⊗ q^-n ∘ where  is the reduced norm.
In particular for any representation Π_t of GL_tg_-1(ϱ)(F_v), we have
smfthm1

    m(ϱ)l^i  [  j_!^= tm(ϱ)l^ig_-1(ϱ) HT(π_v,-1,Π) ] =m(ϱ)l^i
    j_!^= tm(ϱ)l^ig_-1(ϱ) [  HT(π_v,-1,Π)  ]  
     = 
    j_!^= tg_i(ϱ) [  HT(π_v,i,Π)  ] =  [ 
    j_!^= tg_i(ϱ) HT(π_v,i,Π)  ] .
        Note moreover that concerning subquotient isomorphic to ρ_u,
        the only case where it can appeared in the modulo l reduction of some irreducible
        subquotient of the free quotient of H^j(X_I,s̅_v,j^=tg_! HT(π_v,-1,Π_t)) is when 
        either (t,j)=(s-1,1) or j=0.
        The result about H^j_!(π_v,i,t)_𝔪[l] 
        then follows from the previous case where i=-1 using (<ref>) and the following 
        wellknown short exact sequence
    0 → H^n(X,) ⊗__l_l ⟶ H^n(X, ) 
    ⟶ H^n+1(X,) [l] → 0,
        for any _q-scheme X and any _l-perverse free sheaf .
        
        Then the result about the cohomology of p j^=tg_!* HT(π_v,i,Π_t) follows from
        the resolution analog of (<ref>), and the case of p+ j^=tg_!* HT(π_v,i,Π_t)
        is obtained by Grothendieck-Verdier duality.
        
        
         §.§ Perverse sheaf of vanishing cycles and lattices
        
        For I ∈, let 
        
            Ψ_I,v:=RΨ_η_v,I(_l[d-1])(d-1/2)
        
        be the vanishing cycle autodual perverse sheaf on X_I,s̅.
        
        
        In <cit.>,  we gave a decomposition
    Ψ_≃⊕_g=1^d ⊕_ϱ∈__l(g)Ψ_ϱ
        with Ψ_ϱ⊗__l_l ≃⊕_π_v ∈(ϱ)Ψ_π_v where
        the irreducible constituant of 
        Ψ_π_v are exactly the
        perverse Harris-Taylor sheaves attached to π_v.
        
        Recall now from <cit.>, how to construct filtrations of a free perverse sheaf such that its
        graduates are free. Start first from an open subscheme j:U ↪ X and 
        i: F:=X \ U ↪ X, such that j is affine. We then have the following adjunction morphism
        where we use the notation L  L' (resp.   L _+ 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 [drr]^_*,L
    p+ j_! j^* L [ur]^_!,L@->>[r]|-+    p j_!*j^* L @^(->>[r]_+    p+ j_!* j^* L @^(->[r]|-+    p j_*j^* L
    
        where below is, cf. the remark following 1.3.12 de <cit.>,
        the canonical factorisation of p+ j_! j^* L ⟶p j_* j^* L
        and where the maps _!,L and _*,L are given by the adjonction property.
        
        
         (cf. lemma 2.1.2 of <cit.>)
        We introduce the filtration ^-1_U,!(L) ⊂^0_U,!(L) ⊂ L with
        
            ^0_U,!(L)=_ (_!,L)    and   ^-1_U,!(L)=
            _ ( (_!,L)_|_L ),
        
        where _L:=i_* p ^-1_libre i^*j_* j^* L is the kernel of
        _ ( p+ j_!  j^* L ↠ p j_!* j^* L  ).Remark: we have
        L/^0_U,!(L) ≃ i_* p+ i^* L and 
        p j_!* j^* L _+ ^0_U,!(L)/^-1_U,!(L), which gives, cf.
        lemma 1.3.13 of <cit.>, a commutative triangle
     p j_!* j^* L @^(->>[rr]_+ @^(->>[drr]_+ 
          ^0_U,!(L)/^-1_U,!(L) @^(->>[d]_+ 
          p+ j_!* j^* L.
    
        Consider now X equipped with a stratification
    X=X^≥ 1⊃ X^≥ 2⊃⋯⊃ X^≥ d,
        and let L ∈(X,_l).
        For 1 ≤ h <d, let denote
        X^1 ≤ h:=X^≥ 1-X^≥ h+1 and j^1 ≤ h:X^1≤ h↪
        X^≥ 1. We then define
    ^r_!(L):=_ ( p+ j^1 ≤ r_! j^1 ≤ r,*L ⟶ L ),
        which gives a filtration
    0=^0_!(L) ⊂^1_!(L) ⊂^1_!(L) ⋯⊂^d-1_!(L) 
    ⊂^d_!(L)=L.
        With these notations, the graduate ^h_!(Ψ_ϱ) verify
    j^=h,*^h_!(Ψ_ϱ) ≃{[           0   si  g ∤ h; __l(ϱ[t]_D) pour  h=tg. ].
        In particular for h=d, the perverse sheaf ^d_!(Ψ_ϱ) is a quotient of Ψ_ϱ
        and concentrated of the supersingular locus. One of the main result of <cit.>
        can be stated as follow.
        
         (cf. <cit.> proposition 4.2.4)
        There exists a filtration
        
            0=^-1-u_ϱ⊂^-u_ϱ⊂^1-u_ϱ⊂⋯^0_ϱ⊂^1_ϱ= ^d_!(Ψ_ϱ)
        
        verifying the following properties.
        
          *  The graduates ^-k_ϱ=^-k_ϱ/^-k-1_ϱ are free of ϱ-type
        k in the following sense
        
            ^-k_ϱ⊗__l_l ≃⊕_π_v ∈_k(ϱ)
            HT(π_v,_s_k(ϱ)(π_v)) ⊗_g_k(ϱ)(π_v)(1-s_k(ϱ)/2),
        
        where for i ≥ 0 recall g_k(ϱ)=g_-1m(ϱ)l^k and g_k(ϱ)s_k(ϱ)=d.
        
        
          *  For any geometric supersingular point i_z={ z }↪ X^=d_,s̅_v, then
        
            _(D_v,d^×)^0 ϖ_v^^D_v,d^× p h^0 i_z^* ^-k_ϱ⊕_π_v ∈_k(ϱ)Γ_G(π_v) ⊗Γ_DW(π_v)
        
        where Γ_G(π_v) (resp. Γ_DW(π_v)) is a stable lattice of 
        _s_k(ϱ)(π_v) (resp. of π_v[s_k(ϱ)]_D ⊗_g_k(ϱ)(π_v)(1-s_k(ϱ)/2).
        
        
          *  Let M_d(F_v) be the mirabolic subgroup of GL_d(F_v) defined by imposing the first column
        to be (1,0,⋯,0). Then if τ is an irreducible sub-M_d(F_v)-representation of
        Γ_G(π_v) ⊗__l_l, then τ is isomorphic to 
        the non degenerate representation τ_nd, unique up to isomorphism. As a consequence any
        irreducible GL_d(F_v)-representation of Γ_G(π_v) ⊗__l_l
        is isomorphic to ρ_u.
         About Γ_DW(π_v), note that for π_v ∈_-1(ϱ) as the modulo l
        reduction of π_v[s_k(ϱ)]_D ⊗_g_k(ϱ)(π_v)(1-s_k(ϱ)/2)
        is irreducible, up to isomorphism, there exists only one stable lattice which is then a tensor product
        Γ_D(π_v) ⊗Γ_W(π_v).
        
        
        In <cit.> proposition 2.3.3, we explained how, using ^-1(U,!), to construct 
        an exhaustive filtration of Ψ_ϱ
            0=^-2^d-1_!(Ψ_ϱ) ⊂^-2^d-1+1_!(Ψ_ϱ) 
            ⊂⋯⊂^0_!(Ψ_ϱ) ⊂⋯
            ⊂^2^d-1-1_!(Ψ_ϱ)=Ψ_ϱ,
        
        such that each of the graduate is over _l, 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
    _!,ϱ^0 ⊂_!,ϱ^1 ⊂⋯⊂_!,ϱ^r
        where each graduate is a Harris-Taylor perverse _l-sheaf. Note the lattices constructed
        in this way, may depend of the construction but concerning the quotient of the previous proposition,
        the statement remains true. Precisely we can manage so that ^r_!,ϱ:=^r_!,ϱ/
        ^r-1_!,ϱ verify, with the notation of the previous proposition
    _(D_v,d^×)^0 ϖ_v^^D_v,d^× p h^0 i_z^* ^r_!,ϱ≃Γ_G(π_v) ⊗Γ_D(π_v) ⊗Γ_W(π_v),
        where π_v is any fixed irreducible cuspidal representation in _-1(ϱ).
        
        
        
        
        
        
         §.§ Main result
        
        
        Start from an irreducible automorphic cuspidal representation Π of G() verifying the following
        properties:
        
          *  it is ξ-cohomological with non trivial invariant under some fixed I ∈;
        
        
          *  its degeneracy depth is equal to s>1;
        
        
          *  its local component at v is isomorphic to _s(π_v) with π_v ∈_-1(ϱ)
        and where d=g_u(ϱ) for some u ≥ 0.
         For π_v the trivial characte, the hypothesis d=g_u(ϱ) for u=0 is equivalent to ask
        that the order of q ∈_l, which is the cardinal of the residue field of F_v, 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 _I associated to Π.
        Remember that 𝔪 encodes the multiset of Satake's parameters of Π outside I.
        
        
        Under the previous hypothesis, the torsion of H^1(X_I,η̅_v,_ξ[d-1])_𝔪
        is non trivial.
        
        Consider the spectral sequence 
        
            E_1^p,q=H^p+q(X_I,s̅_v,^-p_!,ϱ)_𝔪⇒
            H^p+q(X_I,η̅_v,V_ξ)_𝔪
        
        associated to the filtration ^∙_!,ϱ of Ψ_ϱ.
        First note that over _l:
        
          * E_1^-r,r⊗__l_l has a direct factor isomorphic to 
        (Π^,v)^I ⊗_s(π_v) ⊗_g(π_v) (1-s/2);
        
        
          * d_1^-r,r⊗__l_l induces a injection from the previous
        direct factor into a direct factor of E_1^-r+1,r which, as a representation of GL_d(F_v)
        is parabolically induced from P_(s-1)g,d(F_v) to GL_d(F_v).
        
        From the last remark of the previous section,  
        _(D_v,d^×)^0 ϖ_v^^D_v,d^× p h^0 i_z^* E_1^-r,r
        as a GL_d(F_v)-representation, has a subspace isomorphic to
         Γ_G(π_v)
        where Γ_G(π_v) is a stable lattice of _s(π_v)
        such that ρ_u is the only irreducible sub-representation of 
        Γ_G(π_v) ⊗__l_l. Moreover we know that
        ρ_u can't be a subspace of parabolically induced representation.
        From these facts we conclude that the torsion of E_2^-r+1,r is non trivial and more precisely
        that ρ_u is a subquotient of E_2^-r+1,r[l].
        
        Now there are two alternatives. Either ρ_u as a subquotient of E_2^-r+1,r[l] remains a
        subquotient of E_^1[l] and we are done. Suppose by absurdity it's not the case. 
        First about the free quotient E_k,free^p,q of the E_k^p,q, 
        we know from <cit.> that:
        
          *  if ρ_u is a subquotient of E_1,free^p,q⊗__l_l
        with p+q ≠ 0, then _!,ϱ)^-p is isomorphic to some (π_v,s_i(ϱ)-1)
        with π_v ∈_i(ϱ) and then p+q=± 1;
        
        
          *  for k ≥ 2 and p+q ≠ 0, then ρ_u is never a subquotient of
        E_k,free^p,q⊗__l_l.
        
        Then there must exist (p,q) and a torsion class in (E_1,tor^p,q)_𝔪 with
        p+q=2 such that ρ_u is a subquotient of its l-torsion which contradicts lemma <ref>.
        
        
        Thanks to the main result of <cit.>, associated to this torsion cohomology class is a
         tempered irreducible automorphic representation Π' of G() which is
        ξ-cohomological of level I^l I'_l 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 _l-representation associated to 𝔪.
        
        
        
        
        plain
        
        § ABOUT SOME SELMER GROUPS
        
        
        We follow closely the presentation of Selmer groups given in  10 of Klosin's thesis.
        
        
        
         §.§ General definitions
         Let M be a (F̅/F)-module and
        𝔭 a prime ideal of _F not dividing l. We denote 
        
            H^1_nr(F_𝔭,M):= ( H_cont^1(D_𝔭,M) ⟶ 
            H_cont^1(I_𝔭,M)  ),
        
        where D_𝔭 is the decomposition group at 𝔭 and
        I_𝔭 is its inertia subgroup.
        
        
        Let B_cris denote the Fontaine's ring of l-adic periods.
        
         Let (V,ρ) be a representation of (F̅/F) defined over a finite extension 
        of _l, and let T be a stable lattice. We denote W=V/T.
        
          *  For 𝔭 a finite place of F not dividing l, we denote
        
            H^1_f(F_𝔭,V)=H^1_nr(F_𝔭,V).
        
          *  For 𝔭 dividing l, let
        
            H^1_f(F_𝔭,V):= ( H^1(D_𝔭,V) ⟶ H^1(I_𝔭,V)
            ⊗ B_cris ).
        
        We also denote H^1_f(F_𝔭,W) the image of H^1_f(F_𝔭,V).
         For 𝔭 not dividing l, as the cohomological dimension of 
        (κ_𝔭/κ_𝔭) ≃ equals 1,
        we have H^1_f(F_𝔭,W)=H^1_nr(F_𝔭,W).
        
        
        
         Let Σ_l be the set of places of F above l. For any
        Σ⊃Σ_l,  we denote G_Σ the Galois group of the
        extension F_Σ of F which is maximally unramified outside Σ.
         With the previous notations and for Σ' ⊂Σ\Σ_l,
        the Selmer group associated to (Σ,Σ',W) is
        
            _Σ(Σ',W):=  ( H^1(G_Σ,W) res⊕_𝔭∈Σ' ∪Σ_lH^1(F_𝔭,W)/H^1_f(F_𝔭,W) ).
        
        When Σ'=∅, we simply denote it _Σ(W) and for
        Σ'=Σ\Σ_l, we obtain the Selmer group of Kato which we'll denote
        H^1_f(G_Σ,W).
         For (ρ,M) a representation of G over a free R-module M, the adjoint
        representation of ρ is the R[G]-module
        
            ρ:=_R(ρ,ρ).
        
        
        Recall that for R an unitary commutative ring and for (ρ',M') and (ρ”,M”) two 
        G-representations defined over some free R-modules of finite type, we have a bijection
        
            _R[G](ρ”,ρ') = H^1(G,_R(ρ”,ρ'))
        
        which to some extension
    
    0 [r]     M' [r]^α    M [r]^β    M”[r] @/^1pc/[l]^s     0
    
        associates
    [                               ϕ:                     G ⟶_R(ρ”,ρ');                                  g ↦ (m”↦α^-1(gs(g^-1m”)-s(m”))). ]
        For G=(F̅/F) and ρ',ρ” representations with coefficients in some finite extension
        K of _l, we fix stable lattices Λ', Λ” and we denote
    V=_K(ρ”,ρ')  and  T=__K(ρ”_Λ”,ρ'_Λ')
        as well as
    W=V/T  and  W_n:={ w ∈ W: ϖ_K^n w=0 }≃ T/ϖ_K^n T.
        The bijection (<ref>) gives then, at every finite place 𝔭 of F, a natural 
        bijection
    H^1(F_𝔭,W_n)=_K[D_𝔭](W”_n,W'_n).
        Using this description, it is then possible, following <cit.> 1, to define local Selmer groups
        H^1_f(F_𝔭,W_n) as a subgroup of H^1(F_𝔭,W_n) such that under
        the natural map
    H^1(F_𝔭,W_n) ⟶ H^1(F_𝔭,W_n+1)
        H^1_f(F_𝔭,W_n) is the preimage of H^1_f(F_𝔭,W_n+1) and 
        such that the natural isomorphism lim_⟶n 
        H^1(F_𝔭,W_n) ≃ H^1(F_𝔭,W), induces a natural isomorphism
    lim_⟶n H^1(F_𝔭,W_n) ≃ H^1(F_𝔭,W).
        (cf. <cit.> prop 2.2)
        
          *  For 𝔭 not dividing l, we set
        
            H^1_f(F_𝔭,W_n):=H^1_nr(F_𝔭,W_n),
        
          *  and for 𝔭 | l, we define 
        H^1_f(F_𝔭,W_n) ⊂ H^1(F_𝔭,W_n)
        as the subset of cohomology classes corresponding to extensions
        
            0 → W'_n ⟶W_n ⟶ W”_n → 0
        
        where W_n is in the essential image of the functor 𝐕 defined 
        in section 1.1.2 of <cit.>.
         For ρ:(F/F) ⟶ GL_2(V) a Galois representation
        which is short at 𝔭 | l in the sense of <cit.> 1.1.2, then, with
        previous notations, if W_n sits into an exact sequence
    0 → W'_n ⟶W_n ⟶ W”_n → 0,
        then such extension gives rise to an element of H^1_f(F_𝔭,W_n),
        cf. formula (32) of <cit.> 2.1.
        
        
        
        
         §.§ Construction of non trivial elements
        
        
        
        Consider now an entire irreducible representation ρ':G_F ⟶ GL(V')
        which reduction modulo l, denoted ρ', is supposed to be irreducible. 
        Suppose moreover that there exists another entire irreducible representation
        ρ”:G_F ⟶ GL(V”) which the semi-simplification of its modulo l reduction is
        of the shape ρ' ⊕ (ρ' ⊗ϵ̅) where ϵ̅ 
        is the modulo l reduction of the cyclotomic character.
        
        We know then, cf. for example <cit.>, that there exists a stable lattice T” of V” 
        such that ρ”:G_F ⟶ GL(T”/ϖ T”) is of the following shape
    ρ”=  ( [     ρ'      φ;      0 ρ' ⊗ϵ̅ ] )
        and is not semi-simple. The map φ is then seen as a non trivial element of
    (ρ' ⊗ϵ̅,ρ') ≃ρ' ⊗ϵ̅^-1.
        Denoting V the vector space of ρ:=ρ' ⊗ϵ^-1, T a stable lattice and 
        W=V/T, the map φ gives then a non trivial element of H_cont^1(G_F,W_1). We have H_cont^0(G_F,W)=0.
        
        As the action of G_F on ρ=ρ' does not modify the determinant,
        there isn't non trivial subspace of ρ on which G_F acts through the character
        ϵ̅, so  H_cont^0(G_F,W_1)=0.
        
        Let then α∈ H_cont^0(G_F,W). If α≠ 0, let then n ≥ 1 be minimal such that 
        ϖ^n α=0. We then have ϖ^n-1α∈ H_cont^0(G_F,W_1) 
        which is zero thanks to the previous lemma, contradiction. 
        
        
        From the short exact sequence
    
    0 [r]     W_1[r]     W [r]^×ϖ    W [r]     0
    
        we deduce a canonical injection
        H_cont^1(G_F,W_1) ↪ H_cont^1(G_F,W). Let Σ⊃Σ_l containing the ramification places of
        ρ'. Then φ defines a non trivial element of H^1_f(G_Σ,W).
        
        If Σ' ⊂Σ\Σ_l gathers all the places where
        ρ” is unramified, then φ clearly verifies the local conditions
        of _Σ(Σ',W) at every place 𝔭 of Σ'.
        
        Consider then
        a place w where ρ” is ramified and ρ' is unramified.
        From the short exact sequence
        
            0 → H^1(G_Σ',W) ⟶ H^1(G_Σ' ∪{ w },W) ⟶
            H^1(I_w,W),
        
        it suffices to show that H^1(I_w,W) is trivial. As ρ' is unramified at w, then
        
            H^1(I_w,W)=H^1(I_w^ab,W)=H^1((I_w^ab)^tame,W) = ((I_w^ab)^tame,W).
        
        Moreover form
        
            (I_w^ab)^tame≃lim_←n_q^n^×
        
        we deduce
        
            ((I_w^ab)^tame,W) ≃lim_→n(_q^n^×,W),
        
        which is then trivial as (_q^n^×,W) is non zero if and only if l | (q^n-1) which then
        implies that l ∤ (q^n+1-1).
        plain
        
        
        
        
        
        
        
        
        Following <cit.>, we introduce
    j^≥ h_≠ 1: X^≥ h_,s̅,1_h∖ 
    X^≥ h+1_,s̅,1_h+1↪ X^≥ h_,s̅,1_h,
        and j^=h_≠ 1:=i^h_1 ∘ j^≥ h_≠ 1.
        In <cit.> corollary 6.6, for π_v ∈_i(ϱ),
        we proved the following short exact sequence
        thm1
            0 → p j^=(t+1)g_i(ϱ)_1_tg_i(ϱ)+1,!* HT__l,
            1_tg_i(ϱ)+1 (π_v,_t+1(π_v)_|P_tg_i(ϱ),1,g_i(ϱ)-1(F_v) ) ⊗Ξ^1/2⟶
             
            j^=tg_i(ϱ)_≠ 1,!j^=tg_i(ϱ),*_≠ 1 (  p j^=tg_i(ϱ)_1_tg_i(ϱ),!* HT__l,
            1_tg_i(ϱ)(π_v,_t(π_v))
             ) 
            ⟶ p  j^=tg_i(ϱ)_≠ 1,!*j^=tg_i(ϱ),*_≠ 1 
            HT__l,1_tg_i(ϱ)(π_v,_t(π_v)) → 0,
        
        where for Π any representation of P_tg_i(ϱ)+1,g_i(ϱ)-1(F_v), we denote
    HT__l,1_tg_i(ϱ)+1 ( π_v,Π ):=
    _P_tg_i(ϱ)+1,g_i(ϱ)-1,d-(t+1)g_i(ϱ)(F_v)^P_tg_i(ϱ)+1,d-tg_i(ϱ)-1(F_v)
    HT__l,1_(t+1)g_i(ϱ) (π_v,Π)
        the local system on
    X^=(t+1)_i(ϱ)g_,s̅,1_tg_i(ϱ)+1:=
    ∐_a ∈ P_tg_i(ϱ)+1,d-tg_i(ϱ)-1(F_v)/P_(t+1)g_i(ϱ),d-(t+1)g_i(ϱ)(F_v) 
    X^=(t+1)g_i(ϱ)_,s̅,a,
        with j^=(t+1)g_i(ϱ)_1_tg_i(ϱ)+1:X^=(t+1)g_i(ϱ)_,s̅,
        1_tg_i(ϱ)+1↪
        X^≥ (t+1)g_i(ϱ)_,s̅↪ X^≥ 1_,s̅.
        
        
        
        
        
        
        
        
        
        
        follow the proof of lemma 2.3 in <cit.> using the short 
        exact sequence (<ref>).
        By Artin's theorem,
        see for example theorem 4.1.1 of <cit.>, using the affiness of X^=h_I,s̅_v, 
        we know that
    H^i  ( X_I,s̅_v, V_ξ⊗
    j^=tg_≠ 1,!j^=tg,*_≠ 1 (  p j^=tg_1_tg,!* HT__l,
    1_tg(π_v,_t(π_v)) )  )
     )
        is zero for every i<0 and without torsion for i=0, so that for i>0, we have
        thm1
            H^-i-1 ( X_I,s̅_v,V_ξ⊗ p  j^=tg_≠ 1,!*j^=tg,*_≠ 1 
            HT_1_tg(π_v,_t(π_v))  ) ≃
            
            H^-i ( X_I,s̅_v,V_ξ⊗ p j^=(t+1)g_1_tg+1,!* HT_1_tg+1 (π_v,_t+1(π_v)_|P_tg,1,g-1(F_v) ) ⊗Ξ^1/2 )
        
        and for i=0,
        thm1
            0 → H^-1 ( X_I,s̅_v,V_ξ⊗ p  j^=tg_≠ 1,!*
            j^=tg,*_≠ 1 HT_1_tg(π_v,_t(π_v))  ) ⟶
            
            H^0 ( X_I,s̅_v,V_ξ⊗ p j^=(t+1)g_1_tg+1,!* HT_1_tg+1 (π_v,_t+1(π_v)_|P_tg,1,g-1(F_v) ) ⊗Ξ^1/2 )
            ⟶
            
            H^0  ( X_I,s̅_v, V_ξ⊗ j^=tg_≠ 1,!j^=tg,*_≠ 1 (  p j^=tg_1_tg,!* HT__l,
            1_tg(π_v,_t(π_v))  )  ) ⟶⋯
        
        Thus for t=t_0, we obtain i_0 ≥ 0. Moreover as π_v is of ϱ-type -1 then,
        by <cit.> 2.4.1, p  j^=t_0g_≠ 1,!*
        j^=t_0g,*_≠ 1 HT_1_tg(π_v,_t_0(π_v)) is autodual so that i_0 ≤ 0,which
        proves the first part of the statement.
        
        About the second part of the lemma, we argue by induction on t from t_0 to 1. The result follows
        then clearly from (<ref>) and (<ref>).
