In characteristicp >0, this completes partly the motivic picture predicting that the G-S/T conjecture should be independent of the field of coefficients

CONJECTURES

ANNA CADORET, CHUN YIN HUI AND AKIO TAMAGAWA

Abstract. We investigate the relation between the Grothendieck-Serre/Tate (G-S/T for short) conjectures with Q`- andF`-coefficients for`0 going through their ultraproduct formulations. Our main result roughly asserts that the G-S/T conjecture withF`-coefficients for`0 always implies the G-S/T conjecture withQ`-coefficients for

`0 and that the converse implication holds at least in characteristicp >0. In characteristicp >0, this completes partly the motivic picture predicting that the G-S/T conjecture should be independent of the field of coefficients.

As a concrete application of our result, we obtain that over an arbitrary finitely generated fields of characteristic p >0, the Tate conjecture withQ`-coefficients for divisors and every`6=p is equivalent to the finiteness of the Galois-fixed part of the prime-to-ptorsion subgroup of the geometric Brauer group. This generalizes a well-known theorem of Tate over finite fields.

2020Mathematics Subject Classification. Primary: 14F20; Secondary: 20G35, 14C25.

1. Introduction

LetKbe a field of characteristicp≥0. Fix an algebraic closureK; writeπ1(K) :=π1(Spec(K),Spec(K))(=

Aut(K/K)) for the absolute Galois group of K. A variety overK (or a K-variety) means a scheme sep- arated and of finite type over K. Let SmP(K) denote the symmetric monoidal category of smooth projective varieties overK.

1.1. Conjectures for realization functors. ForX∈SmP(K), letCH^w(X) denote the Chow group of codimension w cycles (modulo rational equivalence) and CH(X) :=⊕_w≥0CH^w(X) the Z-graded Chow ring.

Let CHM(K) denote the category of Chow motives over K with Q-coefficients and SmP(K)^op → CHM(K) the canonical functor [A04, 4.1.3]; fix a Weil cohomology H : CHM(K)⊗CH → T_H with field of coefficients C_H and enriched Tannakian target category T_H - See [A04, 3.3, 4.2.5, 7.1]. For X ∈SmP(K), letGH(X) denote the Tannakian group of the Tannakian subcategory hH(X)i generated by H(X) in T_H. The following unifying conjecture is at the heart of the philosophy of pure motives.

1.1.1.Conjecture. For everyX ∈SmP(K),

(1) (Semisimplicity) H(X) is semisimple - equivalently G_H(X) is a reductive algebraic group over C_H; (2) (Fullness)The image of the cycle class map [−]_H :CH(X)⊗C_H → ⊕_w≥0H^2w(X)(w)is the subspace

of G_H(X)-invariant classes.

The most standard avatars of Conjecture1.1.1are (forK =C) the Hodge conjecture ( [H52], [A04, 7.2]) for singular cohomology with enriched Tannakian target category the category of Q-Hodge structures (so that C_H = Q) and (for K finitely generated over its prime field) the Grothendieck-Serre/Tate (G- S/T for short) conjecture ( [T65], [A04, 7.3]) for `-adic cohomology (` 6= p) with enriched Tannakian target category the category of finite-dimensional Q`-vector spaces endowed with a continuous action of π<sub>1</sub>(K) (so thatC<sub>H</sub> =Q`). The fullness part of Conjecture1.1.1for H implies the standard conjecture of Lefschetz type [A04, 5.2.4] for H. Ifp= 0 this is already enough to imply all the standard conjectures for H [A04, 5.4.2.2]. Ifp >0, combined with the semisimplicity part of Conjecture1.1.1for H, this also implies all the standard conjectures for H (except possibly the standard conjecture of Hodge type) [A04, 7.1.1.1].

In particular, Conjecture 1.1.1 for H implies that numerical and H-homological equivalences coincide so that, after modifying the commutativity constraint, the category of numerical motives becomes a semisimple Tannakian category over Q. Let Q_X be any finite field extension of Q neutralizing the Tannakian subcategoryhXi generated by the numerical motiveX in the category of numerical motives

1

(with modified commutativity constraint) [DM82, Rem. 3.10], let H : hX⊗QXi → V ectQX be a fiber functor and letG(X) denote the corresponding Tannakian group; this is a reductive group overQ_X acting faithfully on the finite-dimensional Q_X-vector space H(X). Assume Conjecture 1.1.1holds for another Weil cohomology H⁰ : CHM(K)⊗C_H⁰ → T_H⁰. Then the general formalism of Tannakian categories implies the following.

1.1.2.Conjecture. For every X ∈ SmP(K) and embedding of Q_X in C_H⁰, one has G(X)×_QX C_H⁰ ' G_H⁰(X)×_C

H0 C_H⁰ acting on H(X)⊗_QXC_H⁰ 'H⁰(X)⊗_C

H0 C_H⁰.

When K has characteristic 0, one expects QX = Q and the isomorphisms of Conjecture 1.1.2 to hold overC_H⁰. WhenK has characteristicp >0, as Serre noticed, this cannot always hold [Gr68,§1.7].

1.2. Realization functors arising from ´etale cohomology. Let L denote the set of all primes 6=p and letU denote the set of all non-principal ultrafilters onL. For`∈ LletF` denote the finite field with

`elements and Q` the completion of Q at`. Foru ∈ U let Qu (resp. Qu) denote the residue field of the maximal ideal ofF:=Q

`∈LF` (resp. Q:=Q

`∈LQ`) corresponding to u (See Section8 for details about ultraproducts).

The G-S/T conjecture is the incarnation of Conjecture1.1.1for the Weil cohomologies derived from ´etale cohomology.

1.2.1. These are built from the following cohomology groups:

- For every`∈ L,Q`-cohomology H^w(X_K,Q`) := (lim_←−

n

H^w(X_K,Z/`<sup>n</sup>))⊗<sub>Z`</sub>Q`; - For everyu∈ U, Qu-cohomology H^w(X_K,Qu) := (Q

`∈LH<sup>w</sup>(X<sub>K</sub>,F`)⊗_FQu; Qu-cohomology H^w(X_K,Qu) := (Q

`∈LH<sup>w</sup>(X<sub>K</sub>,Q`))⊗_QQu. The following diagram summarizes the relation between the various coefficients:

Q` Q

oooo bZ^oo

////F ^////

F`

Qu^oo Zb⊗Q ^////Qu

From now on, assume the base field K is finitely generated. Let C denote any of Q`, Qu, Qu and write HC(X) := H(X<sub>K</sub>, C). The Tannakian target category T<sub>HC</sub> is the category of finite-dimensional continuous C-representations ofπ1(K) (as usual, F` is equipped with the discrete topology, Q` with the

`-adic topology, F,Q with the product topology and Qu, Qu with the quotient topology of the product topology on F, Q). For X ∈ SmP(K) the group G_HC(X) is the Zariski-closure of the image of π₁(K) acting on H_C(X).

1.2.2. The G-S/T conjecture. For an integer w≥0 andX∈SmP(K), consider the following assertions¹. (S,C, ^w₂,X) The action ofπ1(K) on H^w(X_K, C) is semisimple.

(wS,C,w,X) The inclusion H^2w(X_K, C(w))^π1(K),→H^2w(X_K, C(w)) splitsπ₁(K)-equivariantly.

(wS’, C,w,X) The canonical morphismc_w: H^2w(X_K, C(w))^π1(K)→H^2w(X_K, C(w))_π1(K) induced by the identity is an isomorphism.

(F,C,w,X) The cycle map [−] :CH^w(X)⊗C→H^2w(X_K, C(w))^π1(K) is surjective.

(sF,C,w,X) The cycle map [−] :CH^w(X_K)⊗C→ lim_−→

K0/Kfinite

H^2w(X_K, C(w))^π1(K0) is surjective.

Apart from sF, the above assertions also make sense with C replaced by F`, ` ∈ L; we will use the corresponding notation.

With this notation, the classical ( [T66], where it is only formulated for C =Q`) G-S/T conjecture (=

Conjecture 1.1.1) for C asserts that (S, C, ^w₂, X) and (F, C, w, X) hold for every X ∈ SmP(K) and

1S stands for ‘semisimplicity’, wS for ‘weak semisimplicity’, F for ‘Fullness’ and sF for ‘stabilized Fullness’.

integerw≥0.

1.2.3. Known results. The G-S/T conjecture is widely open. If p >0 and K is finite (resp. p >0, resp.

p= 0), Tate [T66] (resp. Zarhin [Z75], [Z77], Mori [Mo77], resp. Faltings [FW84]) proved (S,F`, ¹₂,X),

`0 and (S, Q`, ¹₂,X) forX arbitrary and (F, F`, 1, X), (S, F`, ^w₂,X), `0 and (F,Q`, 1, X), (S, Q`, <sup>w</sup><sub>2</sub>,X) for X an abelian variety. Their proofs for F`,` 0 mimic their proofs for Q`; they do not deduce one of the statements from the other.

By works of several authors ( [N83], [NO85], [Ma14], [Cha13], [MP15], [KMP16], [MP20], [I18]), (F,Q`, w,X), (S,Q`, ^w₂,X) are now established for X a K3 surface. For K3 surfaces, (F, F`,w,X),`0 and (S,F`, <sup>w</sup><sub>2</sub>,X),`0 hold as well. This is due to Skorobogatov-Zarhin ifp≥3 ( [SkZ15]), Ito ifp= 2 [I18]

and Skorobogatov-Zarhin ( [SkZ08]). To our knowledge, these are the only instances where (F,F`,w,X),

`0 and (S,F`, ^w₂,X),`0 are deduced directly from (F,Q`,w,X),`0 and (S,Q`, ^w₂,X) (and not by mimicking or adjusting the proof for Q`-coefficients to F`-coefficients). The arguments of these authors, however, rely on specific features of K3 surfaces, in particular the Kuga-Satake construction². Eventually, formal arguments allow to deduce a few other cases from the above ones - Seee.g.[T94, Thm.

5.2].

1.3. When p = 0 and K is embedded into C, the existence of comparison isomorphisms between ´etale and singular cohomologies (See e.g. [A04, 3.4.2]) implies that H_Q†-homological equivalence is indepen- dent of † ∈ L ∪ U, which ensures that Conjecture 1.1.2 for the H_Q†, † ∈ L ∪ U and Conjecture 1.1.1 for one single H_Q†, † ∈ L ∪ U imply Conjecture 1.1.1 for every H_Q†, † ∈ L ∪ U. But, unfortunately, very little is known about Conjecture 1.1.2 when p = 0. In contrast, when p > 0, and modulo the semisimplicity part of Conjecture 1.1.1, Conjecture 1.1.2essentially boils down to the Langlands corre- spondence [L02], [Chi04], [CZ21]. However, in this case, the lack of comparison isomorphisms between the H_Q†, † ∈ L ∪ U makes it unclear whether Conjecture 1.1.1 for one single H_Q†, † ∈ L ∪ U implies Conjecture1.1.1for every H_Q†,† ∈ L ∪ U.

Letu∈ U. The aim of this note is to study a related but easier version of the above problem, namely to relate Conjecture1.1.1 (in our case, the G-S/T conjecture) for H<sub>Q`</sub>, `∈S for some S ∈u, for H_Qu and for H_Qu. One motivation is to give conceptual and completely general (i.e. working for arbitrary smooth projective varieties) proofs of results like the above mentioned results of Skorobogatov-Zarhin and Ito for K3 surfaces. Another motivation is that we may hope that some new cases of the G-S/T conjecture could be proved more easily forQu-coefficients and then transferred toQu- henceQ`-coefficients.

Assumep >0. LetCdenote any ofQ`,QuorQu. For any integerw≥0,vandX∈SmP(K), letG_C(X) denote the Zariski closure of the image ofπ1(K) acting on H^w(X_K, C(v)). Before considering Conjecture 1.1.1, we prove the following variant of Conjecture1.1.2for the group of connected components.

1.3.1.Theorem. For every X∈SmP(K) the kernel of the canonical map π₁(K)→π₀(G_C(X)) is inde- pendent of C=Q`, Qu, Qu.

When GC(X) is connected for one of (equivalently every) C = Q`, Qu, Qu, one says that X has con- nected monodromy in degrees (w, v). Under the connected monodromy assumption in degrees (2w, w), H^2w(X_K, C(w))^π1(K) = H^2w(X_K, C(w))^π1(K0) for every finite field extension K⁰/K and the G-S/T con- jecture forX and X⁰ :=X×_KK⁰ become equivalent - See Lemma 4.2.

Our second main result is the following statements.

1.3.2.Proposition. For every X ∈SmP(K), equidimensional of dimension d, and u ∈ U, the following hold.

(1) (F,Qu,d,X²) + (S, Qu, ^w₂,X) =⇒ (S,Qu, ^w₂,X);

(2) (F,Qu,d,X²) + (F,Qu,w,X) + (wS,Qu,w,X) =⇒(wS, Qu,w,X).

2The restrictionp≥3 in [SkZ15] is related to the fact that the Kuga-Satake construction was not available forp= 2 at the time of [SkZ15]. This missing ingredient was developed by Kim and Madapusi Pera in [KMP16]. Building on [KMP16]

and the method of [SkZ15], Ito extended Skorobogatov-Zarhin’s result to thep= 2 case.

1.3.3.Theorem. Assume p > 0. For every X ∈SmP(K), equidimensional of dimension d, and u ∈ U, the following hold.

(1) (wS,Qu,w,X) =⇒ (wS,Qu,w,X);

(2) (S, Qu, ^w₂,X) =⇒(S, Qu, ^w₂,X);

(3) (F,Qu,i,X),i=w, d−w+ (wS, Qu,w,X) =⇒ (F,Qu,i,X),i=w, d−w (+ (wS, Qu,w,X)).

Proposition1.3.2and Theorem 1.3.3imply formally (See Lemma4.1) the following.

1.3.4.Corollary. For every X∈SmP(K), equidimensional of dimension d, (1) (F,F`,d,X<sup>2</sup>) + (S, F`, ^w₂,X),`0 =⇒ (S,Q`, ^w₂,X),`0;

(2) (F,F`,d,X<sup>2</sup>) + (F,F`,w,X) + (wS, F`,w,X), `0 =⇒(wS, Q`,w,X), `0.

Assume p >0. Then

(3) (wS,Q`,w,X),`0 =⇒ (wS,F`,w,X),`0;

(4) (S, Q`, <sup>w</sup><sub>2</sub>,X),`0 =⇒ (S,F`, <sup>w</sup><sub>2</sub>,X), `0;

(5) (F,Q`,i,X), i=w, d−w + (wS, Q`,w,X), ` 0 =⇒ (F,F`,i,X),i=w, d−w (+ (wS,F`,w, X)), `0.

1.3.5. For divisors, Theorem 1.3.3, Corollary 1.3.4 (3)-(5) yield [T94, Prop. 5.1] that for every X ∈ SmP(K),

(1) (F,Qu, 1, X) =⇒ (F,Qu, 1,X) + (wS,Qu, 1,X);

(2) (F,Q`, 1,X),`0 =⇒ (F,F`, 1,X) + (wS,F`, 1,X),`0.

In particular, for X an abelian variety or a K3 surface one can directly deduce (F, F`, 1,X) + (wS,F`, 1, X), ` 0 from (F, Q`, 1, X) (See Subsection 1.2.3) without resorting to any specific arithmetico- geometric features ofX as in [SkZ15] or [I18].

1.3.6.Remark. The implication (F, F`, w, X) =⇒ (F, Q`, w, X) always holds for ` 0 (hence the implication (F, Qu, w,X) =⇒ (F, Qu,w,X)). This follows from Nakayama’s lemma and the fact that H<sup>2w</sup>(X<sub>K</sub>,Z`) is torsion-free for `0 ( [G83] - See Fact 2.2). More precisely, we have the commutative diagram

CH^w(X) ^//

**

H^2w(X_K,Z`(w))^π1(K)

H^2w(X_K,F`(w))<sup>π1(K)oo</sup> H<sup>2w</sup>(X<sub>K</sub>,Z`(w))^π1(K)⊗F`,

where the bottom arrow is injective for`0. So if the left vertical arrow is surjective, the bottom arrow is an isomorphism hence the diagonal arrow is surjective.

1.4. Divisors and finiteness of Brauer groups. Let X ∈ SmP(K) with connected monodromy in degrees (2,1). Then (F, Q`, 1, X) is equivalent to the finiteness of the `-primary π1(K)-invariant part Br(X_K)^π1(K)[`^∞] of the Brauer group of X_K (e.g.[CCh20, Prop. 2.1.1] and the references therein). One has the following strengthening.

Corollary. Assume p >0. Then for every X∈SmP(K) the following assertions are equivalent (1) (F,Q`, 1,X),`6=p;

(2) Br(X_K)^π1(K)[p⁰]is finite,

(where Br(X_K)^π1(K)[p⁰] denotes the prime-to-p part of Br(X_K)^π1(K)).

When K is finite, Corollary 1.4 was proved by Tate [T94, Prop. 4.3]. In this setting, it is even known that (F,Q`, 1,X) is independent of`6=pand implies that Br(X) is finite (See the references in the proof of [T94, Prop. 4.3]). The delicate implication is (1)⇒(2), which requires Corollary1.3.4(3). Establishing (2) whenX is a K3 surface was the main motivation of Skorobogatov-Zarhin and Ito in [SkZ15], [I18].

1.5. The proof of Theorem 1.3.1 is carried out in Section 3, the proof of Proposition 1.3.2 in Section 5 and the proof of Theorem 1.3.3in Section6. The proof of Proposition1.3.2is formal; this is why it also holds for p = 0. The proofs of Theorem 1.3.1 and Theorem 1.3.3 rely on deeper arithmetico-geometric inputs which, for the convenience of the reader, are summarized in Section2; the assumption that p >0 is crucial. Eventually, the proof of Corollary 1.4 is carried out in Section 7. In Section 8, we gathered basic properties of ultraproducts of fields.

Acknowledgments: The first author was partly funded by the ANR project ECOVA, ANR-15-CE40- 0002-01, the CNRS-JSPS project ASPIC and is supported by the Institut Universitaire de France. This project was initiated while the first and second authors were visiting the third author at RIMS; they want to thank RIMS for providing remarkable working conditions. The third author was partly supported by JSPS KAKENHI Grant Numbers 15H03609, 20H01796.

2. Etale cohomology´

LetKbe a finitely generated field of characteristicp≥0 and letX ∈SmP(K). Letkdenote the algebraic closure of the prime field of K inK.

2.1. Convention. In several places, we will fix a smooth projective model f :X → S ofX →Spec(K) with S a smooth separated and geometrically connected scheme over k with generic point η and set of closed points |S|. In particular, for every geometric point s over a point s ∈ S, locally constant constructibleZ`-sheafF (`6=p) and up to choosing ´etale paths fromstoη, one gets canonical equivariant isomorphisms

(R^∗f∗F(v))_s ^{' //}(R^∗f∗F(v))_η H^w(X_η,F) H^w(X_K,F)

π1(s, s)

00 //π1(S, s)

;;

' //π1(S, η)

;;

π1(η, η) =π1(K).

<<

oooo

Whenp >0 (so that kis a finite field) ands∈ |S|, letϕs∈π1(s) denote the geometric Frobenius, which we identify with its image (well-defined up to conjugacy if we ignore base points, which we will do most of the time) inπ₁(S, s) ˜→π₁(S, η).

Assumep >0. Fix integersw≥0,v. The following are consequences of the theory of Frobenius weights developed by Deligne in [D80].

2.2.Fact.

(1) ( [G83])TheZ`-local systemsR<sup>w</sup>f∗Z`(v) are torsion-free (of finite constant rank) for `(6=p)0. In particular, for every geometric point son S, (R<sup>∗</sup>f∗Z`(v))_s⊗F`→(R˜ <sup>w</sup>f∗F`(v))_s, `(6=p)0;

(2) ( [CHT17, Thm. 1.3]) H⁰(S_k, R^wf∗Z`(v))⊗F`→H˜ ⁰(S_k, R^wf∗F`(v)), `(6=p)0.

2.3.Fact.

(1) ( [D80, 3.4.1 (iii)])R^wf∗Q`(v)|_S

k is a semisimple Q`-local system onS<sub>k</sub>,`6=p;

(2) ( [CHT17, Thm. 1.1]) R^wf∗F`(v)|_S

k is a semisimple F`-local system on S<sub>k</sub> for `(6=p)0.

2.4.Fact.

(1) ( [D80, Cor. 3.2.9]) For every closed point s ∈ |S| the characteristic polynomial P_s := det(IdT − ϕ_s|(R^wf∗Q`(v))<sub>s</sub>) of the geometric Frobenius ϕ<sub>s</sub>∈π<sub>1</sub>(s) is inZ[1/p][T], independent of`(6=p);

(2) (e.g.[LaP95, (proof of) Prop. 2.1])The characteristic polynomialP := det(IdT−ϕ|H⁰(S_k, R^wf∗Q`(v))) of the geometric Frobenius ϕ∈π<sub>1</sub>(k) is in Z[1/p][T]and independent of `.

From Fact 2.2 (1), Fact 2.4 (1) implies that, for `(6= p) 0, the reduction modulo ` of P_s ∈ Z[1/p][T] coincides with the characteristic polynomial Ps,F` := det(IdT −ϕs|(R<sup>w</sup>f∗F`(v))s)∈F`[T]. In turn, this implies that P_s ∈ Z[1/p][T] coincides with the characteristic polynomial det(IdT −ϕ_s|H^w(X_s,Qu(v))), u ∈ U. (It also directly follows from Fact 2.2 (1) that Ps ∈ Z[1/p][T] coincides with the characteristic polynomial det(IdT−ϕ_s|H^w(X_s,Qu(v))),u∈ U).

From Fact 2.2 (2), Fact 2.4 (2) implies that, for `(6= p) 0, the reduction modulo ` of P ∈Z[1/p][T] coincides with the characteristic polynomialP<sub>F`</sub> := det(IdT−ϕ|H<sup>0</sup>(S<sub>k</sub>, R<sup>w</sup>f∗F`(v)))∈F`[T].

2.5. Let Π (resp. Π) denote the image ofπ₁(S_k) (resp. π₁(S)) acting on Q

`∈L(R<sup>w</sup>f∗F`(v))_s. Then, Fact. ( [CT19,§3.1]) Π (hence Π) is a topologically finitely generated profinite group.

Let Π_Qu denote the image of π₁(S) acting on H^w(X_η,Qu(v))). Fact 2.5 has the following (non-trivial!) consequence

Corollary. For every finite index subgroup Π⁰

Qu ⊂Π_Qu there exists a connected ´etale cover S⁰ → S such thatΠ⁰

Qu coincides with the image ofπ₁(S⁰) acting on H^w(X_η,Qu(v))).

Proof. From Fact2.5, Π is a topologically finitely generated profinite group. As the inverse image Π⁰⊂Π of Π⁰

Q^u in Π is again of finite index it follows from [NS07a], [NS07b] that Π⁰ is automatically open in Π

hence corresponds to a connected ´etale coverS⁰ → S.

The fact that Π is topologically finitely generated also ensures (Lemma8.4.2) H^w(X_η,Qu(v))^π1(Sk)= (Y

`∈L

H^w(X_η,F`(v))^π1(Sk))⊗Qu= (Y

`∈L

H⁰(S_η, R^wf∗F`(v))⊗Qu

so that, from Fact 2.2 (2) and Fact 2.4 (2), P ∈ Q[T] coincides with the characteristic polynomial det(IdT−ϕ|H^w(X_s,Qu(v))^π1(Sk)), u∈ U.

(From Fact 2.4(2) and [B96, 6.3.1, 6.3.2], similar results hold forQu-coefficients).

3. Proof of Theorem 1.3.1

LetK be a finitely generated field of characteristic p >0 and let X∈SmP(K). We retain the notation of2.1. ForC =Qu,Q`,Qu, or F`set H_C := H^w(X_K, C(v)) and, forC=Qu,Q` orQu, letG_C ⊂GL(H_C) denote the Zariski-closure of the image ΠC of π1(K) acting on HC. It is enough to prove that for every

`∈ Land u∈ U,

G_Qu is connected⇔ G_Q` is connected ⇔G_Qu is connected.

We prove the assertion for Qu andQ`. The proof forQu and Q` is easier.

3.1. Characteristic polynomial map. Let V be a finite-dimensional vector space over a field Q of characteristic 0 and letG⊂GL(V) be a closed algebraic subgroup. Letr denote theQ-dimension of V and χ: GL_V →Gm,Q×A^r−1_Q the characteristic polynomial map.

Lemma. For everyg∈G(Q),χ(gG^◦)⊂Gm,Q×A^r−1_Q is a closed subvariety andχ(gG^◦) =χ(G^◦) if and only ifg∈G^◦. In particular,G=G^◦ if and only if χ(G) =χ(G^◦).

Proof. Since the characteristic polynomial mapχ: GL(V)→Gm,Q×A^r−1_Q factors through the maximal reductive quotient G G^red and since π0(G) = π0(G^red) (as the kernel of G G^red is the unipotent radical ofG, which is connected), one may assumeGis reductive. The assertion forGreductive is [LaP92,

4.9, 4.10].

3.2. For every closed point s ∈ |S|, let Ps ∈ Z[T] denote the characteristic polynomial of the geomet- ric Frobenius ϕs ∈ π1(s) (See 2.4 (1)). Write Ψ ⊂ (Gm ×A^r−1)(Z) for the set of all Ps, s ∈ |S| and Ψ^zar⊂Gm,Z×A^r−1_Z for the Zariski-closure of Ψ in Gm,Z×A^r−1_Z .

Lemma. One has(Ψ^zar)_{Q`</sub>=χ(G<sub>Q`}), (Ψ^zar)_Qu=χ(G_Qu).

Proof. Write C :=Q`,Qu and let Φ_C ⊂Π_C denote the union of the Π_C-conjugacy classes of the images of the geometric Frobenii ϕs attached to closed points s ∈ |S| (so that Ψ = χ(ΦC)). Since χ(GC) ⊂ Gm,C×A^r−1_C is Zariski-closed (by Lemma 3.1), the inclusion (Ψ^zar)_C ⊂χ(G_C) always holds.

For C = Q`, by the `-adic Cebotarev density theorem Φ<sub>Q`</sub> is `-adically-dense in Π_Q` - hence, since the

Zariski topology is coarser than the`-adic topology, Φ<sub>Q`</sub> is Zariski-dense inG_Q` and the assertion follows.

For C = Qu, as the ultraproduct topology is not Hausdorff, it is unclear whether Φ_Qu is Zariski-dense in G_Qu. Still, one can argue as follows. If χ(Π_Qu) 6⊂ (Ψ^zar)_Qu there should exist f ∈ Z[T0, . . . , Tr−1] vanishing on Ψ^zarandπ = (π<sub>`</sub>)`∈L ∈Π such thatf(χ(π))6= 0 withπ∈Π_Qu the image ofπ viaΠΠ_Qu. The conditionf(χ(π))6= 0 is equivalent to requiring that the set Sπ of all `∈ Lsuch that f◦χ(π`)6= 0 in H_{F`</sub> is in u. From Facts 2.2 (1) and 2.4 (1) there exists (a cofinite subset) S ∈u such that for every s ∈ |S| and ` ∈ S, P_{s,F`</sub> = P<sub>s</sub>mod ` while from the Cebotarev density theorem, for every ` ∈ S there existss<sub>`}∈ |S|such thatπ_{`</sub> is the image of the Frobeniusϕs atsacting on H<sub>F`}. So, for every`∈Sπ∩S, one should have f◦χ(π<sub>`}) =f(P<sub>s,F`</sub>) =f(P<sub>s</sub>) mod`= 0. This yields a contradiction sinceS_π∩S∈u(as a filter is stable by finite intersection) henceSπ∩S6=∅(by definition of a filter - See Subsection8.1).

3.3. Let{C₁, C₂}={Q`,Qu}. By symmetry, it is enough to prove π₀(G_C1) = 0 ⇒ π₀(G_C2) = 0. Assume π0(GC2) 6= 0 but π0(GC1) = 0. Since π0(GC2) is finite, there exists a connected ´etale cover S⁰ → S such that G^◦_C

2 is the Zariski closure of the image Π⁰_C

2 of π1(S⁰) acting on H_C2 (See Corollary 2.5 for C₂ = Qu; for C₂ = Q` this follows directly from the fact that the Zariski-topology is coarser than the

`-adic topology). Let Ψ⁰ ⊂(Gm×A^r−1)(Z) denote the set of allP_s⁰,s⁰ ∈ |S⁰|and let Ψ^0zar ⊂Gm,Z×A^r−1_Z denote the Zariski closure of Ψ⁰. Then, from Lemma3.2 and Lemma3.1,

χ(G^◦_C2) = (Ψ^0zar)C2 (χ(GC2) = (Ψ^zar)C2

while, asG_C1 is connected,

χ(G^◦_C1) = (Ψ^0zar)_C1 =χ(G_C1) = (Ψ^zar)_C1. This yields a contradiction. Indeed, by construction, one has

(Ψ^0zar_ )C1

ιC1

//

(Ψ^0zar_{_} )_Q

ι

(Ψ^0zar_ )C2

ιC2

oo

(Ψ^zar)_C1 ^//(Ψ^zar)_Q ^oo (Ψ^zar)_C2

SinceιC1 is surjective, descent for surjectivity shows thatιis surjective as well. But then, by base-change, ι_C2 is surjective as well.

4. Preliminary observations

Let X ∈ SmP(K). We begin by the following elementary observations, which follow from the formal properties of ultraproducts.

4.1. Lemma. For (?,??) = (S,^w₂), (wS,w), (wS’,w), (F,w) we have

(1) For every u∈ U, (?,Qu, ??, X) ⇐⇒ The set of all`∈ L such that (?, Q`, ??, X) holds is in u. In particular, (?,Q`, ??, X),`0 ⇐⇒(?,Qu, ??,X) for every ultrafilter u∈ U;

(2) For every u∈ U, (?,Qu, ??,X) ⇐⇒The set of all `∈ L such that (?,F`, ??, X) holds is in u.

In particular, (?,F`, ??, X), `0 ⇐⇒(?,Qu, ??,X) for every u∈ U.

Proof. For ? =F, see8.3.3(with P the property of being surjective) and8.4.2(which can be applied by??

(2)). For ? =S, see8.4.5 (with P the property of acting semisimply). For ? =wS, see8.4.6. For ? =wS’, see8.4.2,8.4.1and 8.3.3(with P the property of being an isomorphism).

4.2. LetC =Q`,Qu orQu and letK⁰/K be a finite field extension. WriteX⁰ :=X×_KK⁰. Then, Lemma.

(1) (S, C, ^w₂,X⁰)⇔ (S, C, ^w₂,X);

(2) (sF, C,w,X⁰) ⇔(sF, C,w,X);

(3) If K⁰/K is Galois,(F,C,w,X⁰) ⇒ (F,C,w,X).

Assume furthermore X has connected monodromy in degrees (2w, w). Then, (4) (wS, C,w,X) ⇔ (wS,C,w,X⁰);

(5) The assertions (sF, C,w,X), (sF,C,w,X⁰), (F, C,w,X), (F,C,w,X⁰) are all equivalent.

Proof. We show (3); the other assertions are purely group-theoretic and elementary. Let α∈H^2w(X_K, C(w))^π1(K)⊂H^2w(X_K, C(w))^π1(K0).

Then, from (F, C, w, X⁰), one can write α =P

1≤i≤rλi[Y_i⁰] with λi ∈ C and Y_i⁰ ∈ Z^ω(X⁰) an integral cycle. But, then,

α= 1 [K⁰ :K]

X

1≤i≤r

λ_i X

σ∈Gal(K⁰/K)

σ[Y_i⁰] = 1 [K⁰ :K]

X

1≤i≤r

λ_i[ X

σ∈Gal(K⁰/K)

σY_i⁰].

The conclusion follows from the fact that P

σ∈Gal(K⁰/K)σY_i⁰ is in Z^ω(X⁰)^Gal(K0/K)=Z^ω(X).

4.3.Lemma. Assume p >0. Then,

(1) For `6=p,(wS,Q`,w,X) ⇐⇒(wS’, Q`,w,X);

(2) For `0, (wS,F`,w,X)⇐⇒ (wS’, F`,w,X).

Proof. Let C =Q` or F`. We retain the notation of 2.1. Write H := H^2w(X, C(w)) and Π := π1(S_k), Π :=π₁(S). The implication (wS’,C,w, X) ⇒ (wS,C,w, X) is straightforward since the composition of c⁻¹_w : H_Π→H˜ ^Π with the canonical projection H H_Π provides a Π-equivariant splitting of H^Π ,→ H.

Conversely, let φ ∈ Π such that φ and Π generate Π. As Π acts semisimply on H (Fact 2.3) the canonical morphism H^Π→H_Π is an isomorphism. Assume (wS, C,w,X) and consider a Π-equivariant decomposition H = H^Π ⊕M; in particular M^Π = 0. Then it is enough to show that 0 = M_Π = (M_Π)ϕ←(M˜ ^Π)ϕ but this follows from the exact sequence

0→M^Π= (M^Π)^ϕ →M^Πϕ−1→ M^Π→(M^Π)_ϕ →0.

5. Proof of Proposition1.3.2

5.1. LetX ∈ SmP(K) of dimension d. For C =Q`,Z`,F` write H_C := H^w(X_K, C) and set Π :=π₁(K).

To prove Proposition 1.3.2, one may freely replace L by a subset in u; in particular one may replace L by a cofinite subset hence assume

(5.1.1) H<sub>Z`</sub>⊗F`= H<sub>F`</sub>, `∈ L

(Fact 2.2(1) if p >0; if p= 0, this follows from comparison between singular andZ`-cohomology, using the fact that for every embeddingK ⊂C, Hsing(X(C),Z) is a finitely generatedZ-module). By K¨unneth formula and Poincar´e duality, (F,Qu,d,X²) ensures that up to replacingL by a subset in u one has

(5.1.2) End_Π(H_{Z`</sub>)⊗F` = (H_{Z`</sub>⊗H<sup>∨</sup><sub>Z`})^Π⊗F`→(H˜ <sub>F`}⊗H^∨_{F`</sub>)<sup>Π</sup>= End<sub>Π</sub>(H<sub>F`}), `∈ L.

5.2. Proof of Proposition 1.3.2 (1).

5.2.1. LetQbe a field and Γ a group. In this subsection, a Γ-module means a finite-dimensionalQ-vector space endowed with an action of Γ byQ-linear automorphisms. For a Γ-module V, let V^ss denote the Γ-semisimplification ofV.

Lemma. One has dim(End_Γ(V))≤dim(End_Γ(V^ss)) and dim(End_Γ(V)) = dim(End_Γ(V^ss))if and only if V is a semisimple Γ-module.

Proof. Let 0→A→V →B→0 be a short exact sequence of Γ-modules and W a Γ-module. Then 0−→Hom_Γ(B, W)−→Hom_Γ(V, W)−→Hom_Γ(A, W)

and

0−→Hom_Γ(W, A)−→Hom_Γ(W, V)−→Hom_Γ(W, B) are exact and hence we obtain

dim Hom_Γ(V, W)≤dim Hom_Γ(A⊕B, W) and

dim Hom_Γ(W, V)≤dim Hom_Γ(W, A⊕B).

By takingW =V in the first inequality and W =A⊕B in the second, we obtain dim End_Γ(V)≤dim End_Γ(A⊕B)

and induction implies

(∗) dim End_Γ(V)≤dim End_Γ(V^ss).

When (∗) is an equality, V is semisimple. Indeed, all the inequalities become equalities. Hence, the sequence

0−→HomΓ(A⊕B, A)−→HomΓ(A⊕B, V)−→HomΓ(A⊕B, B)−→0 and thus

0−→Hom_Γ(B, A)−→Hom_Γ(B, V)−→Hom_Γ(B, B)−→0

are exact, implying that 0→A→V →B →0 splits.

5.2.2. From (5.1.1) and (5.1.2) dim(End_Π(H_{Q`</sub>)) = dim(End<sub>Π</sub>(H<sub>F`})). On the other hand (S,Qu, ^w₂,X) ensures that up to replacing L by a subset in u one may assume (S,F`, <sup>w</sup><sub>2</sub>, X), `∈ L (See Lemma 4.1 (2)). Let T_Z` ⊂ H^ss_Q

` be any Π-stable Z`-lattice and set T_{F`</sub> := T<sub>Z`} ⊗F`. Then since T^ss

F` and H<sub>F`</sub> are semisimple Π-modules with the same traces, they are isomorphic. Hence

dim(End_Π(H^ss

Q`))≥dim(End<sub>Π</sub>(H<sub>Q`</sub>)) = dim(End_Π(H_F`))

= dim(EndΠ(T_F^ss

`))≥dim(EndΠ(T<sub>F`</sub>))≥dim(EndΠ(H^ss_Q

`)), where the first and second inequalities follow from Lemma 5.2.1 and the third inequality always holds.

As a result, one obtains

dim(EndΠ(H_{Q`</sub>)) = dim(EndΠ(H<sup>ss</sup><sub>Q`})).

The conclusion follows from the equality case in Lemma5.2.1.

5.3. Proof of Proposition 1.3.2(2).

5.3.1. Given a ringR, letIdem(R) andCIdem(R) denote respectively the idempotents and central idem- potents inR.

LetA be a Z`-algebra which, as a Z`-module, is free of finite rank.

Lemma. (Lifting idempotents) The reduction modulo-` morphism A A⊗F` restricts to a surjective map Idem(A)Idem(A⊗F`) and to a bijective map CIdem(A) ˜→CIdem(A⊗F`).

Proof. First, observe that for everya, a⁰ ∈A such that [a, a⁰] = 0 anda−a⁰∈`<sup>N</sup>A, we havea<sup>`n</sup>−a^0`n ∈

`<sup>N+n</sup>A. Indeed, writea−a<sup>0</sup> =`^Nb0 ∈`<sup>N</sup>A. Then,b0 commutes witha,a<sup>0</sup> and one has a<sup>`</sup>−a^0` = X

1≤k≤`

` k

`<sup>N k</sup>a<sup>0`−k</sup>b^k₀ =`^N+1 X

1≤k≤`

` k

`^{N k}

`<sup>N+1</sup>a<sup>0`−k</sup>b^k₀ =`^N+1b1. The conclusion follows by straightforward induction.

- Let ∈ Idem(A⊗F`) and pick any a ∈ A such that a = . By construction, a<sup>`m</sup> −a ∈ `A, m ≥ 0 hence, from the preliminary observation,

a^{`n+p</sup>−a<sup>`n} = (a^{`p</sup>)<sup>`n}−a<sup>`n</sup> ∈`ⁿ⁺¹A, n≥0 hence {a^`n}_n is a Cauchy sequence. Set e:= lim

n→∞a^{`n</sup>. By construction, for n0 we have e=a<sup>`n} = <sup>`n</sup> =. Furthermore, sincea<sup>2</sup>−a∈`A, we get, again, a^{2`n</sup>−a<sup>`n} ∈`<sup>n+1</sup>A,n≥0. Since (−)<sup>2</sup> :A→A is continuous, one getse<sup>2</sup> =e. This showsIdem(A)Idem(A⊗F`).

- Lete∈Idem(A) such that e∈CIdem(A⊗F`). Thene(A⊗F`)(1−e) = 0 forces eA(1−e)⊂`A=e`Ae⊕(1−e)`Ae⊕e`A(1−e)⊕(1−e)`A(1−e).

Multiplying byeon the left and 1−eon the right, one getseA(1−e) =`eA(1−e) hence, by Nakayama’s lemma,eA(1−e) = 0. Similarly (1−e)Ae= 0. Hence for everya∈A,

ea=ea(e+ (1−e)) =eae= (e+ (1−e))ae=ae.

This showsCIdem(A)CIdem(A⊗F`). Let e, e<sup>0</sup> ∈CIdem(A) such that e=e<sup>0</sup> that is, e−e<sup>0</sup> ∈`A.

Since [e, e⁰] = 0, the preliminary observation shows that e−e⁰ =e^{`n</sup>−e<sup>0`n} ∈`<sup>n+1</sup>A,n≥0 hencee=e<sup>0</sup>. This showsCIdem(A) ˜→CIdem(A⊗F`).

5.3.2. From (5.1.1) one has EndΠ(H<sub>Z`</sub>)⊗F` = EndΠ(H_{F`</sub>), ` ∈ L. On the other hand, (F, Qu, w, X), (wS,Qu, w,X) ensure that up to replacing L by a subset in u, one also has (F, F`, w, X), (wS, F`,w, X), (wS’ ,F`, w,X),`∈ L(Lemma 4.1(2), Lemma 4.3(2)). Using (5.1.1) one can apply Lemma5.3.1 to A = EndΠ(H_{Z`</sub>). Write M0 := H<sup>Π</sup><sub>F`}, M1 := ker(H_{F`</sub> → H<sub>F`}Π). Then, by (wS’ , F`, w, X), one has the canonical decomposition H<sub>F`} = M₁ ⊕M₀ as Π-modules. By definition of M₀, M₁, any element in End_Π(H_{F`</sub>) stabilizes both M<sub>0</sub> andM<sub>1</sub> hence the elementse<sub>i</sub>: H<sub>F`} M_i ,→H_{F`</sub> (obtained by composing the canonical projection followed by the canonical injection), i= 1,2 are in CIdem(EndΠ(H<sub>F`})). From Lemma5.3.1,e₀, e₁ lift uniquely to ˜e₀,˜e₁ ∈CIdem(End_Π(H_{Z`</sub>)) with Id= ˜e<sub>0</sub>+ ˜e<sub>1</sub>. Let ˜M1−i:= ker(˜e<sub>i</sub>), i = 0,1. Then H<sub>Z`} = ˜M₁ ⊕M˜₀ with ˜M_i ⊗F` = M_i, i = 0,1. It remains to check that ˜M₀ = H^Π

Z`. Since ˜M0⊗F` = M0(= H^Π_{F`</sub>) = H<sup>Π</sup><sub>Z`}⊗F`, by Nakayama’s lemma, it is enough to show that H<sup>Π</sup><sub>Z`</sub> ⊂M˜0. Since H^Π_Z

` = (H^Π_Z

`∩M˜0)⊕(H^Π_Z

` ∩M˜1), this is equivalent to H^Π_Z

`∩M˜1 = 0. Let h ∈ H^Π_Z

`∩M˜1. Then h mod ` ∈ H^Π_F

` ∩M₁ = 0 that is H^Π_Z

` ∩M˜<sub>1</sub> ⊂ `H_{Z`</sub>. But as H<sub>Z`}/(H^Π

Z` ∩M˜<sub>1</sub>) ,→ (H<sub>Z`</sub>/H^Π_Z

`) ×M˜<sub>0</sub> is torsion-free (equivalently, H<sup>Π</sup><sub>Z`</sub>∩M˜1 ⊂H<sub>Z`</sub> is a Z`-direct summand), (`HZ`)∩(H^Π<sub>Z`</sub>∩M˜1) =`(H^Π_Z`∩M˜1).

As a result, H^Π_Z

`∩M˜1 =`(H^Π_Z

`∩M˜1) which, by Nakayama’s lemma, forces H^Π_Z

`∩M˜1= 0.

6. Proof of Theorem 1.3.3

LetK be a finitely generated field of characteristic p >0 and let X∈SmP(K). We retain the notation of2.1. Set Π :=π₁(S_k), Π :=π₁(S). Again, to prove Theorem1.3.3one may freely replaceLby a subset in u; in particular one may assume H⁰(X_η, R^∗f∗Z`)⊗F`→H˜ ⁰(X_η, R^∗f∗F`), ` ∈ L (Fact 2.2 (1)). From Lemma4.1, it is enough to show

(1’) For`0, (wS, Q`,w,X) =⇒ (wS,F`,w,X) (2’) For`0, (S, Q`, <sup>w</sup><sub>2</sub>,X) =⇒ (S,F`, ^w₂,X)

(3’) For`0, (F,Q`,i,X), i=w, d−w + (wS,Q`,w,X) =⇒(F,F`,i,X),i=w, d−w+ (wS, F`,w,X) 6.1. Proof of (1’). For C =Q`,F`, write H_C := H^2w(X_K, C(w)) and consider the following seemingly

weak variant of (wS,C,w,X).

(wS”, C,w,X) The inclusion H^Π_C ,→H^Π_C splits π1(k)-equivariantly.

Write H^Π_C{1} := ∪_n≥1ker((Id−ϕ)ⁿ|H^Π_C) for the generalized eigenspace attached to 1. By definition dim(H^Π_C{1}) =:δC(1) is the multiplicity of 1 in the roots of the characteristic polynomial ofϕ acting on H^Π_C. One has

(6.1.1) (wS”,C,w,X) ⇔ δ_C(1) = dim(H^Π_C) ⇔ δ_C(1)≤dim(H^Π_C)

(where the last equivalence follows from the fact thatδC(1)≥dim(H^Π_C) always holds). One also has 6.1.2Lemma. (wS,Q`,w,X)⇔ (wS”, Q`,w,X) and (wS,F`,w,X) ⇔ (wS”,F`,w,X),`0.

Proof. The implications⇒are straightforward. For the converse implications, from Fact2.3the canonical Π-equivariant morphism H^Π_C →H_CΠ is an isomorphism. So, settingN := ker(H_C →H_CΠ), one obtains

a direct sum decomposition as Π-modules H_C = H^Π_C⊕N.

6.1.3 From6.1, it is enough to show

(1”) For`0, δ<sub>Q`</sub>(1)≤dim(H^Π

Q`) =⇒ δ<sub>F`</sub>(1)≤dim(H^Π

F`).

This follows from the facts that, for`0,δ<sub>Q`</sub>(1) =δ_F`(1) (See the last paragraph of 2.5) and dim(H^Π_F

`)≤δ<sub>F`</sub>(1) =δ_Q`(1)≤dim(H^Π_Q

`)≤dim(H^Π_F

`).

6.2. Proof of (2’). This is proved in [CHT17,§11]. We give here a more elementary argument, which avoids Larsen-Pink’s theory of regular semisimple Frobenii. ForC =F`,Q`,Z`, write H_C := H^w(X_K, C).

Also, let Π_{`</sub> and Π<sub>`} denote the image of Π and Π acting on H_Z` respectively.

We begin with the following Lemma. Recall that (S,Q`, <sup>w</sup><sub>2</sub>,X), (S,Qu, <sup>w</sup><sub>2</sub>, X) hence - as this holds for everyu ∈ U (8.4.5 for P the property of acting semisimply) - (S, F`, ^w₂, X) for` 0 are insensitive to

finite field extensions of K (Lemma 4.2(1)).

Lemma. After replacingK by a finite field extension, there exists a monic polynomialP ∈Q[T]and for every `6=p a semisimple elementφ`∈Π` such that, for`0,Π` is generated byΠ` andφ`, and φ` has characteristic polynomial P.

Proof. LetG_{Z`</sub>,G<sub>Z`}denote respectively the Zariski closure of Π_{`</sub>, Π<sub>`}in GL(H_{Z`</sub>). After possibly replacing S by a connected ´etale cover, one may assumeG<sub>Q`} is connected for every`∈ L(Theorem1.3.1(2)). One may also assume S carries a k-point s∈ S(k). Let ϕ` denote the image of the geometric Frobenius ϕs

acting on H_{Z`</sub>; recall that its characteristic polynomialP<sub>s</sub> is inQ[T] and independent of`( [D80]). Write ϕ`=ϕ<sup>ss</sup><sub>`} ϕ^u<sub>`</sub> for the multiplicative Jordan decomposition ofϕ` inG<sub>Q`</sub>. There exists polynomials P<sup>ss</sup>, P<sup>u</sup> inQ[T] and independent of `such thatϕ^ss_{`</sub> =P<sup>ss</sup>(ϕ<sub>`}),ϕ^u_{`</sub> =P<sup>u</sup>(ϕ<sub>`}). LetF_Z`,F^ss

Z`,F^u

Z` denote the Zariski closure in G<sub>Z`</sub> of the subgroup generated by ϕ_{`</sub>, ϕ<sup>ss</sup><sub>`} and ϕ^u_{`</sub> respectively. Then F<sub>Z`} = F^ss

Z`F^u

Z`. Since G<sub>Q`</sub>/G_Q` is connected, abelian, reductive³, it is a torus. Hence F^u

Q` ⊂G<sub>Q`</sub>. In particular, ϕ^u<sub>`</sub> ∈G(Q`).

But, actually,ϕ^u_{`</sub> ∈G(Z`) for`0. Indeed,ϕ<sup>u</sup><sub>`} =P^u(ϕ_{`</sub>) is in End<sub>Z`}(H_{Z`</sub>) for`0 sinceP^u is inQ[T] and independent of `. Also det(ϕ<sup>u</sup><sub>`}) = 1 ∈ Z^×_{`</sub> shows that ϕ<sup>u</sup><sub>`} ∈ G(Q`)∩GL(H<sub>Z`</sub>). It only remains to check that G(Q`)∩GL(H<sub>Z`</sub>) =G(Z`). The inclusionG(Q`)∩GL(H<sub>Z`</sub>)⊃G(Z`) is straightforward. The converse inclusion is the valuative criterion of properness for the closed immersion G_{Z`</sub> ,→GL(H<sub>Z`}):

G_{Z`</sub> <sup> //</sup>GL(H<sub>Z`})

Q`

OO //Z`

OOdd .

From [CHT17, Thm. (7.3.2)], there exists an integerN ≥1 independent of`such that (ϕ<sup>u</sup><sub>`</sub>)^N ∈Π`. But, then, (ϕ<sup>ss</sup><sub>`</sub> )^N =ϕ^N_{`</sub> (ϕ<sup>u</sup><sub>`})^−N ∈Π_`; after replacingkby its degree-N field extension, we may assumeN = 1.

Then φ` =ϕ<sup>ss</sup><sub>`</sub> works.

We can now conclude the proof. The fact that φ` acts semisimply on H<sub>Q`</sub> is equivalent to the fact that the minimal polynomial Q_{`</sub> of φ<sub>`} is separable. Since P is in Q[T] and independent of `, Q := Q<sub>`</sub> is in Q[T] and independent of `as well. And since one assumes H<sub>Z`</sub> is torsion free, the minimal polynomial of φ_{`</sub> acting on H<sub>F`} is the reduction modulo-`of Qfor`0; in particular, it is again separable for`0.

This shows that φ_{`</sub> acts semisimply on H<sub>F`} for ` 0 hence that its image in GL(H<sub>F`</sub>) is of prime-to-`

order. Thus (S,F`, ^w₂,X) follows from Fact2.3 (2) and [S94a, Lem. 5(b)].

6.3. Proof of (3’). One retains the notation of Subsection 6.1. Since we may assume `0, (F, Q`,i, X),i=w, d−w + (wS,Q`,w,X) imply that the canonical morphismZ<sup>w</sup>(X)⊗Z` →H^Π

Z` is surjective ( [MiR04, Lem. 3.1]) and, in particular, that the morphismZ<sup>w</sup>(X)⊗Z`→H_Z` has torsion-free cokernel.

This and the fact that one assumes H<sub>Z`</sub> is torsion free show that the images of Z<sup>w</sup>(X)⊗C → H<sup>Π</sup><sub>C</sub> for C=Q`,F`,Z` have the same rank - say δ. As a result

(F,X,Q`,w) ⇔ δ= dim(H^Π

Q`) (F,X,F`,w) ⇔ δ= dim(H^Π

F`) Thus the conclusion follows from the implications:

δ_Q`(1) = dim(H^Π

Q`) <sup>6.1</sup>⇔(wS, Q`,w,X) ⁽¹

0)

⇒ (wS,F`,w,X) <sup>6.1</sup>⇔δ<sub>F`</sub>(1) = dim(H^Π

F`).

and the fact that for `0,δ<sub>Q`</sub>(1) =δ_F`(1) (See the last paragraph of 2.5).

7. Proof of Corollary 1.4

Assume p > 0 and let X ∈SmP(K) with dimension d. Let Br(X_K) := H²(X_K,Gm) denote the Brauer group of X_K. For a prime `6=p and integern ≥1, let Br(X<sub>K</sub>)[`ⁿ]⊂Br(X_K) denote the kernel of the multiplication-by-`ⁿ map,

T_`(Br(X_K)) := lim

←−Br(X_K)[`<sup>n</sup>], V<sub>`</sub>(Br(X_K))[`<sup>n</sup>]) :=T<sub>`</sub>(Br(X_K))[`<sup>n</sup>])⊗Q`.

3This is where we use (S,Q^`, ^w₂,X).