F-ISOCRYSTALS
MARCO D’ADDEZIO
Abstract. It has been proven by Serre, Larsen–Pink and Chin that on a smooth curve over a finite field, theπ0 and the neutral components of the monodromy groups of a compatible system of lisse
sheaves, satisfying certain conditions, areλ-independent. We generalize their results to compatible systems of lisse sheaves and overconvergentF-isocrystals, satisfying the same conditions, on arbitrary smooth varieties.
Studying the monodromy groups, we also prove thatι-pureF-isocrystals on abelian varieties become constant after a finite ´etale cover (see also Remark 5.1.3). We give two proofs of this fact, one of which does not employ the Langlands programme. This yields a proof of Deligne’s conjectures for lisse sheaves and overconvergentF-isocrystals on abelian varieties which does not use automorphic representations. At the same time, we recover a theorem of Tsuzuki on the constancy of the Newton polygons.
To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergentF-isocrystals. We use the Tannakian formalism to make explicit the simi-larities between the two types of “local systems”.
Contents
1. Introduction 2
1.1. Background 2
1.2. Main results 3
1.3. The structure of the article 4
1.4. Acknowledgements 4
2. Notation 4
2.1. Weil lisse sheaves 5
2.2. OverconvergentF-Isocrystals 6
3. Generalities 8
3.1. Coefficient objects 8
3.2. Monodromy groups 10
3.3. The K¨unneth formula 14
3.4. Weights 15
3.5. Deligne’s conjectures 17
3.6. Compatible systems 19
4. Independence of the monodromy groups 20
4.1. The group of connected components 20
4.2. The neutral component 21
5. Coefficient objects on abelian varieties 25
5.1. A finiteness result 25
5.2. The Newton polygons 27
References 27
1
1. Introduction
1.1. Background. Deligne has formulated in Weil II the following conjectures [Del80, Conjecture 1.2.10].
Conjecture 1.1.1. Let X0 be a normal scheme of finite type over a finite field Fq of characteristic p
andV0 an irreducible lisse Qℓ-sheaf whose determinant has finite order.
(i) V0 is pure of weight0;
(ii) There exists a number fieldE⊆Qℓ such that for every closed pointx0 inX0, the characteristic
polynomialdet(1−Fx0t,Vx)has coefficients in E, where Fx0 is the geometric Frobenius at x0;
(iii) For every primeℓ6=p, the eigenvalues of the Frobenii at closed points ofV0 are ℓ-adic units;
(iv) For a suitable fieldE(maybe larger than in (iii)) and for every finite placeλnot dividingp, there exists a lisse Eλ-sheaf compatible withV0, namely a lisseEλ-sheaf with the same characteristic
polynomials of the Frobenii at closed points asV0;
(v) Whenλdividesp, there exists some compatible crystalline object (“petits camarades cristallins”).
Conjecture (iv) is commonly called thecompanion conjecture(for lisse sheaves). Sometimes it is also stated in the following weaker form.
(iv’) If E is a number field as in (ii), for every finite place λ not dividing p, there exists a lisse Eλ-sheaf compatible withV0.
Thanks to the work of Chin in [Chi03], we know that conjectures (ii) and (iv’) for every finite ´etale cover ofX0 imply Conjecture (iv).
In [Laf02] L. Lafforgue proves the Langlands reciprocity conjecture for GLr over function fields. As
a consequence, he obtains conjectures (i), (ii), (iii) and (iv’), whenX0 is a smooth curve.
The lack of a Langlands correspondence for higher dimensional varieties (even at the level of the formulation) forced one to generalize Deligne’s conjectures, reducing geometrically to the case of curves. One of the difficulties is that one cannot rely on a Lefschetz theorem for the ´etale fundamental group in positive characteristic (see [Esn17, Lemma 5.4]). This means that one cannot in general find a curve
C0⊆X0 such that for every irreducible lisseQℓ-sheafV0 ofX0, the inverse image ofV0 onC0 remains irreducible.
Luckily, the Lefschetz theorem can be replaced by a weaker form. Rather than considering all the lisse sheaves at the same time, one can fix the lisse sheaf and find a suitable curve where the lisse sheaf remains irreducible (Theorem 3.5.4). In this way one can prove (i) and (iii) for arbitrary varieties, using Lafforgue’s result.
The conjectures (ii) and (iv’) require some more effort. The former has been obtained by Deligne in [Del12], the latter has been proven by Drinfeld forsmooth varieties in [Dri12] and it is still open in general. Following the ideas of Wiesend in [Wie06] Drinfeld proves a gluing theorem for lisse sheaves [Drinfeld,op. cit. , Theorem 2.5].
Passing to (v), Crew conjectured in [Cre92, Conjecture 4.13] that the correct p-adic analog of lisse sheaves might beoverconvergentF-isocrystals, introduced by Berthelot [Ber96a]. To endorse his conjec-ture, he proves theglobal monodromy theoremfor these isocrystals, over a smooth curve [Crew,op. cit.
, Theorem 4.9]. Many people have then worked on the direction suggested by Crew (see for example [Ked04a] and [Ked06]).
1.2. Main results. Following Kedlaya [Ked16b], we refer to lisse sheaves and overconvergentF-isocrystals as coefficient objects. LetX0 be a smooth connected variety over Fq. Suppose thatE0 is a coefficient object on X0 with all the eigenvalues of the Frobenii at closed points algebraic overQ. Thanks to the known cases of the companion conjecture, E0 sits in an E-compatible system {E0,λ}λ∈Σ, where E is a number field, Σ is a set of finite places ofE, containing all the places that do not dividep, and{E0,λ}λ∈Σ is a family of pairwise compatibleEλ-coefficient objects (as in (iv)), one for eachλ∈Σ (Theorem 3.6.3).
We use the new tools, presented above, to extend the results of λ-independence of the monodromy groups. LetFbe an algebraic closure ofFqandxanF-point ofX0. For everyλ∈Σ, letG(E0,λ, x) be the
arithmetic monodromy group ofE0,λandG(Eλ, x) itsgeometric monodromy group (see Definition 3.2.2).
We prove a generalization of a theorem of Serre and Larsen–Pink (see [Ser00] and [LP95, Proposition 2.2]).
Theorem 1.2.1 (Theorem 4.1.1). If the coefficient objects {E0,λ}λ∈Σare pure, the groups of connected
components ofG(E0,λ, x)andG(Eλ, x)are independent of λ.
To prove it we need first Proposition 3.2.10. This tells us that the groups of connected components of the arithmetic (resp. geometric) monodromy groups of overconvergentF-isocrystals are quotients of the arithmetic (resp. geometric) ´etale fundamental groups. Then the proof of Theorem 1.2.1 proceeds as in [LP95, Proposition 2.2].
Let’s assume now in addition that for everyℓ6=p, all the eigenvalues of the Frobenii at closed points of the family{E0,λ}λ∈Σareℓ-adic units1, and that for everyλ∈Σ the coefficient objectE0,λis semi-simple.
We obtain the following result, that is a generalization of [Chi04, Theorem 1.4].
Theorem 1.2.2 (Theorem 4.2.2). After possibly replacing E by a finite extension, there exists a con-nected split reductive group G0, defined over E, such that for every λ ∈ Σ, the group G0 ⊗Eλ is
isomorphic to the neutral component ofG(E0,λ, x).
Moreover, let ρ0,λ be the tautological representation of G(E0,λ, x). There exists a faithful E-linear
representation ρ0 of G0 and isomorphisms ϕ0,λ : G0⊗Eλ −∼→ G(E0,λ, x)◦ for every λ ∈ Σ such that
ρ0⊗Eλ is isomorphic toρ0,λ◦ϕ0,λ.
Chin, following the ideas in [Ser00], shows that for every lisse sheaf on a curve, there exist infinitely many closed points such that the Frobenius torus associated to the point is a maximal torus of the arithmetic monodromy group [Chi04, Lemma 6.4]. We generalize his lemma to overconvergent F -isocrystals on a smooth curve (Proposition 4.2.10). We deduce it from the case of lisse sheaves, just looking at the representation theory of the monodromy groups (Lemma 4.2.9). This result on Frobenius tori is then used to obtain Theorem 1.2.2 for curves, thanks to Chin’s argument.
Subsequently, we prove the theorem in full generality. We reduce to the case of curves using the weak Lefschetz theorem (Theorem 3.5.4). We need to show that the curve given by the theorem for one coefficient object has the same property for all the other coefficient objects in the compatible system. This is proven in Lemma 4.2.11.
In [Pal15, Theorem 8.23], P´al proves Theorem 1.2.2 for curves, using a strong ˇCebotarev theorem [ibid., Theorem 4.13], whose proof is not yet publically available. Our proof is different and it does not rely on this ˇCebotarev theorem. At the same time, Drinfeld in [Dri15] proves the independence of the entire arithmetic monodromy groups (not only the neutral component), overQℓ. See Remark 4.2.15 for other comments on the differences between our result and Drinfeld’s work.
Finally, we prove the following theorem on the finiteness of the coefficient objects defined on abelian varieties.
Theorem 1.2.3 (Theorem 5.1.1). LetX0 be an abelian variety. Every absolutely irreducible coefficient
object with finite order determinant is finite. In particular, everyι-pure coefficient object onX0 becomes
constant after passing to a finite ´etale cover.
We propose two proofs. The first one uses the K¨unneth formula (Proposition 3.3.3), an Eckmann– Hilton argument and the global monodromy theorem (Theorem 3.4.3). This proof does not rely on
the Langlands programme. The second proof is a consequence of the known cases of the companion conjecture and the global monodromy theorem for lisse sheaves. Using this method one could actually prove a more general statement (see Remark 5.1.3).
As a consequence, we give a proof of Deligne’s conjectures for coefficient objects that does not use automorphic representations (Corollary 5.1.2). We also obtain in Corollary 5.2.2 an alternative proof of a theorem of Tsuzuki on the constancy of the Newton polygons ofF-isocrystals on abelian varieties [Tsu17, Theorem 3.7].
1.3. The structure of the article. We define in§3.1 the categories of coefficient objects andgeometric
coefficient objects. We recall some definitions related to the characteristic polynomials of the Frobenii at closed points and the (weak) ˇCebotarev theorem for coefficient objects (Theorem 3.1.11).
In§3.2 we define the arithmetic and the geometric monodromy groups of coefficient objects, using the Tannakian formalism. We also introduce two Tannakian fundamental groups, one classifying coefficient objects and the other geometric coefficient objects and we present a fundamental exact sequence for coefficient objects (Proposition 3.2.4). Then we show that the groups of connected components of these fundamental groups are isomorphic to the arithmetic and the geometric ´etale fundamental group (Proposition 3.2.10).
In the§3.3 we prove the K¨unneth formula for the fundamental group classifying geometric coefficient objects for projective connected varieties with a rational point.
In§3.4 we recollect some theorems from Weil II that are now proven for coefficient objects of both types.
We present in§3.5 the state of Deligne’s conjectures and in§3.6 we give the definition of compatible systems of lisse sheaves and overconvergent F-isocrystals and we present a strong formulation of the companion conjecture in Theorem 3.6.3.
In §4 we start investigating the properties of λ-independence of the monodromy groups varying in a compatible system of coefficient objects. We start by proving in §4.1 the λ-independence of the groups of connected components, generalizing the theorem of Serre and Larsen–Pink to overconvergent
F-isocrystals. In §4.2 we extend the theory of Frobenius tori to overconvergent F-isocrystals and we prove Theorem 4.2.2.
In§5 we focus on coefficient objects on abelian varieties. We give the two proofs of Theorem 5.1.1, we prove Deligne’s conjectures for abelian varieties and we recover Tsuzuki’s theorem in Corollary 5.2.2.
1.4. Acknowledgements. I am grateful to my advisors H´el`ene Esnault and Vasudevan Srinivas for the helpful discussions. I thank Abe Tomoyuki for having kindly answered my questions on isocrystals and for all valuable comments on a draft of this text. The article greatly benefited from many hours of conversation I had with Emiliano Ambrosi. I also thank Gregorio Baldi for some important discussions we had on coefficient objects defined on abelian varieties. Finally, I would like to thank all my colleagues at Freie Universit¨at for all their support.
2. Notation
2.0.1. We fix a prime number pand a positive power q. Let Fq be a field with q elements andF an
algebraic closure ofFq. For every positive integerswe denote byFqs the subfield ofFwithqselements. Ifk is a field we will say that a separated scheme of finite type over k is a variety over k. Acurve
will be a one dimensional variety. We denote byX0 andY0 smooth connected varieties over some finite fieldk. Ifkis not specified, it is assumed to beFq. In this case we denote byX andY the extensions of
scalarsX0⊗FqFandY0⊗FqF, overF. In general, we denote with a subscript0objects and morphisms defined over Fq and the suppression of the subscript will mean the extension toF. We denote by kX0 (resp. kY0) the algebraic closure of Fq in Γ(X0,OX0) (resp. in Γ(Y0,OY0)). Sometimes we will also consider X0 and Y0 as varieties over kX0 and kY0 (without changing the scheme, just the structural morphism). We denote by |X0| the set of closed points of X0. If x0 ∈ |X0| the degree of x0 will be deg(x0) := [κ(x0) :Fq].
(Y0, y)→(X0, x) is a morphism of varietiesY0→X0 sendingy to x. On a pointed variety the F-point induces an identificationkX0=Fqs for somes∈Z>0.
2.0.2. The letter ℓ will denote a prime number and in general we will allow ℓ to be equal to p. We fix an algebraic closure Q of Q. For every number field E we denote by |E|ℓ the set of finite places
of E dividingℓ. At the same time we define |E|6=p :=Sℓ6=p|E|ℓ and |E| := Sℓ|E|ℓ. We choose in a
compatible way, for every number fieldE⊆Qand everyλ∈ |E|, a completion ofE byλ, denotedEλ.
For every primeℓ we denote byQℓthe union of all the Eλ, for every number fieldE⊆Qandλ∈ |E|ℓ.
If Kis a field of characteristic 0, an elementa∈Kis said to be an algebraic number if it is algebraic overQ. Ifais an algebraic number we will say that it isp-plain2if it is anℓ-adic unit for everyℓ6=p. 2.0.3. Let K be a field. We denote by VecK the category of finite dimensional K-vector spaces. A Tannakian categoryoverKwill be a rigid abelian symmetric⊗-categoryCtogether with an isomorphism End(✶)≃ Ksuch that there exists a faithful exact K-linear ⊗-functor ω : C → VecL, for some field extensionK⊆L. We will call such a functor afiber functor ofCoverL. If in additionCadmits a fiber functor overKitself, we say thatCis a neutral Tannakian category.
For every Tannakian categoryCoverK, we say that an object inCis atrivial objectif it is isomorphic to✶⊕nfor somen∈N. A Tannakian subcategory ofCis a strictly full abelian subcategory, closed under
⊗, duals, subobjects, and quotients. IfV is an object of C, we denote by hVithe smallest Tannakian subcategory ofC containingV.
2.0.4. Ifω is a fiber functor ofC, over an extension L, the affine group scheme Aut⊗(ω) will be the
Tannakian group of C with respect toω. For every object V ∈C, a fiber functorω ofC induces, by restriction, a fiber functor for the Tannakian categoryhVi, that we denote again byω. We also say that the Tannakian group ofhViwith respect to ω is themonodromy group ofV (with respect toω). If the monodromy group ofV is finite, we say thatV is afinite object.
2.0.5. For every affine group scheme G, we denote by π0(G) the group of connected components of
G and G◦ will be the connected component of G containing the neutral element, called the neutral
component of G.
2.1. Weil lisse sheaves. We mainly use the notations and conventions for lisse sheaves in [Del80]. 2.1.1. If x is a geometric point of X0 we denote by π´et1(X0, x) and π´1et(X, x) the ´etale fundamental groups of X0 and X respectively. If kis a finite extension of Fq andk is an algebraic closure of k we
call thegeometric Frobenius of k (with respect tok) the inverse of the automorphism ofk that raises each element to its q[k:Fq]-th power. We denote by F the geometric Frobenius of F
q, with respect to F. For every n∈Z>0 we denote by W(F/Fqn) the Weil group of Fqn (it is generated by Fn). At the same time, we denote by W(X0, x) be the Weil group of X0. If x′0 is a closed point of X0, we write
Fx′
0 ⊆W(X0, x) for the conjugacy class of the possible images of all the geometric Frobenii ofx
′
0 and we call the elements inFx′
0 the Frobenii at x
′
0.
2.1.2. For every ℓ 6=pwe have a category LS(X,Qℓ) of lisse Qℓ-sheaves defined over X, that is the 2-colimit of the categoriesLS(X, Eλ) oflisse Eλ-sheaves, whereEλvaries among the finite extensions of QℓinQℓ. IfX0is not geometrically connected overFq then these categories are not Tannakian. Ifxis a geometric point ofX0, we denote by X(x)the connected component ofX containingx. The categories
LS(X(x), E
λ) are then always neutral Tannakian categories.
If V is a lisse Qℓ-sheaf on X, an n-th Frobenius structure on V is an action of W(F/Fqn) on the pair (X,V) such thatW(F/Fqn) acts on X =X0⊗F via the natural action on F. An n-th Frobenius structure is equivalent to the datum of an isomorphism (Fn)∗V −∼→ V. The category of lisseE
λ-sheaves
equipped with a 1-st Frobenius structure will be the category of Weil lisse Eλ-sheaves of X0, denoted byWeil(X0, Eλ). The categoriesWeil(X0, Eλ) are Tannakian. We will often refer to Weil lisse sheaves
simply aslisse sheaves of X0.
For every geometric pointxofX0 and everyEλwe define a functor
Ψx,Eλ :Weil(X0, Eλ)→LS(X, Eλ)→LS(X (x), E
λ)
where the first functor forgets the Frobenius structure and the second one is the inverse image functor with respect to the open immersionX(x)֒→X. IfV
0is a Weil lisse sheaf, we remove the subscript0 to indicate the lisse sheaf Ψx,Eλ(V0).
2.1.3. If we fix a geometric point x of X0, there exists an equivalence between the category of Weil lisse Qℓ-sheaves over X0 and the finite-dimensional continuous Qℓ-representations of the Weil group
W(X0, x). The equivalence sends a Weil lisse sheaf V0 to the representation ofW(X0, x) on the stalk
V0,x. For an arbitrary variety X0 we call ´etale lisse sheaf any Weil lisse sheaf such that, for every connected component of X0, the associated representation of the Weil group factors through the ´etale fundamental group.
2.1.4. Notation as in§2.1.3. IfV0 is a Weil lisseEλ-sheaf, for every closed pointx′0∈ |X0|the Frobenii atx′
0 act onV0,x. We call these automorphisms ofV0,x, by a slightly abuse of notation,the Frobenii at
x′
0and we denote the set byFx′
0. Even if these automorphisms are a priori different, their characteristic polynomials do not change. Hence, it make sense to definePx′
0(V0, t) := det(1−tFx0′|V0,x)∈Eλ[t]. This will be the(Frobenius) characteristic polynomial ofV0 atx′0.
For every natural numbern, a lisseQℓ-sheaf is said to bepure ofweight n, if for every closed pointx′
0 ofX0, the eigenvalues of any element inFx′
0 are algebraic and all the conjugates have complex absolute value (#κ(x′
0))n/2. Ifι:Qℓ ∼
−→Candwis a real number, we say that a lisse sheaf isι-pure of ι-weight wif for every closed pointx′
0ofX0the eigenvalues ofFx′
0, after applyingι, have complex absolute value (#κ(x0))w/2. Moreover, we say that a lisseQℓ-sheaf ismixed (resp. ι-mixed) if it admits a filtration of
lisseQℓ-sheaf with pure (resp. ι-pure) successive quotients.
2.2. OverconvergentF-Isocrystals.
2.2.1. Letk be a perfect field. We denote by W(k) the ring ofp-typical Witt vectors overk and by
K(k) its fraction field. For everys∈Z>0, we denote byZqs the ring of Witt vectors overFqs andQqs its fraction field. We suppose chosen compatible morphismsQqs→Qp.
LetX0 be a smooth variety over k, we denote byIsoc†(X0/K(k)) the category of Berthelot’s
over-convergent isocrystals of X0 over K(k). See [Ber96a] for a precise definition and [Cre87] or [Ked16a] for a shorter presentation. We will also need, in our article, the interpretation of this category via the theory ofarithmetic D-modules presented in [Abe13]. The category Isoc†(X0/K(k)) is aK(k)-linear rigid abelian⊗-category, with unit objectOX†, that we will denote byK(k)X0. The endomorphism ring ofK(k)X0 is isomorphic toK(k)
s, wheresis the number of connected components of X.
We will recall now the notation for the extension of scalars and the Frobenius structure of overcon-vergent isocrystals. We mainly refer to [Abe13,§1.4].
2.2.2. For every finite extension K(k) ֒→ K we denote by Isoc†(X0/K(k))K the category of K -linear overconvergent isocrystals of X0 over K(k), namely the category of pairs (M, γ), where M ∈
Isoc†(X0/K(k)) and γ : K→End(M) is a morphism of (noncommutative) K(k)-algebras, called the
K-structure. The morphisms in Isoc†(X0/K(k))K are morphisms of overconvergent isocrystals over K(k), commuting with theK-structure. We will often omitγ in the notation.
For a pair of objects ((M, γ),(M′, γ′)), the tensor product inIsoc†(X
0/K(k))Kis defined in the
fol-lowing way. We start by considering the tensor product of the two isocrystalsM⊗M′inIsoc†(X
0/K(k)). On this objectKacts viaγ⊗id and id⊗γ′ at the same time. We defineN as the greatest quotient of
M ⊗ M′ such that the twoK-structures agree. Then we defineδas the uniqueK-structure induced on
2.2.3. IfK⊆Lare finite extensions ofK(k) and{α1, . . . , αd}is a basis ofLoverKwe define a functor
(−)⊗KL: Isoc†(X0/K(k))K→Isoc†(X0/K(k))L. An object (M, γ)∈Isoc†(X0/K(k))K is sent toLdi=1Mi, δ
, where{Mi}1≤i≤d are copies ofMand
δis defined as follows. We callιithe inclusions ofMi in the direct sum. For everyα∈Land 1≤i≤d,
we writeα·αi=Pdj=1aijαj, whereaij∈K. The restriction ofδ(α) toMiis defined asPdj=1ιj◦γ(aij).
Different choices of a basis ofLoverKinduce functors that are canonically isomorphic.
For every finite field k, and for every finite extension K(k) ⊆ K we choose as a unit object of
Isoc†(X0/K(k))K, the object KX0 := K(k)X0 ⊗K. We have endowed Isoc
†(X
0/K(kX0))K with a structure of a K-linear rigid abelian ⊗-category. When X0 is geometrically connected over k, the endomorphism ring ofKX0 is equal toK.
2.2.4. We want define now the inverse image functor for overconvergent isocrystals withK-structure. Suppose given a commutative diagram
X0 Y0
Spec(Fqs) Spec(Fqs′)
f0
that we will denote byf0:X0/Fqs→Y0/Fqs′.
We have ana¨ıve inverse image functorf0+ :Isoc†(Y0/Qqs′)Q
qs →Isoc
†(X
0/Qqs) sending (M, γ) to (f0+M, f0+γ), wheref0+Mis the inverse image ofMas isocrystal overQqs′ andf0+γis theQqs-structure, given by the composition ofγwith the morphism End(M)→End(f0+M), induced by the inverse image functor. The functorf0+ does not commute in general with the tensor, thus we need to “normalize” it. The isocrystalf0+Mis endowed with twoQqs-structures. One isf0+γ, the second is the structuralQqs -structure, as an object inIsoc†(X0/Qqs). We definef0∗(M, γ) as the greatest quotient off0+(M, γ) such that the twoQqs-structures agree. We equip it with the unique inducedQqs-structure. For every finite extension Qqs ⊆ K, this construction extends to a functor f0∗ : Isoc†(Y0/Q
qs′)K → Isoc†(X0/Qqs)K. This will be the inverse image functor we will use.
2.2.5. We denote by F : X0 →X0 the q-power Frobenius3. Let Kbe a finite extension for Qq. For
every M ∈Isoc†(X0/Qq)K and every n ∈Z>0, an isomorphism between (Fn)∗M and M will be an
n-th Frobenius structure of M. We denote by F-Isoc†(X0/Qq)K the category of overconvergent F -isocrystals with K-structure, namely the category of pairs (M,Φ) where M ∈ Isoc†(X0/Qq)K and Φ
is a 1-st Frobenius structure of M, called theFrobenius structure of the F-isocrystal. The morphisms inF-Isoc†(X0/Qq)K are the morphisms inIsoc†(X0/Qq)Kthat commute with the Frobenius structure.
For every positive integern, the isomorphism
Φn:= Φ◦F∗Φ◦ · · · ◦(Fn−1)∗Φ
will be then-th Frobenius structureof (M,Φ). The categoryF-Isoc†(X0/Qq)Kis aK-linear rigid abelian ⊗-category. In this case,F-Isoc†(X0/Qq)K has ring of endomorphisms isomorphic toK.
2.2.6. We extend the functor of the extension of scalars, imposing (M,Φ)⊗KL:= (M ⊗KL,Φ⊗KidL),
where Φ⊗KidLgoes fromF∗M ⊗KL=F∗(M ⊗KL) to M ⊗KL.
Letf0:X0→Y0be a morphism, for every extensionQq ⊆K, the functorf0∗defined for the isocrystals extends to a functor
f0∗:F-Isoc†(Y0/Qq)K→F-Isoc†(X0/Qq)K
that sends (M,Φ) to (f∗
0M, f0∗(Φ)).
3The letter F will denote two different types of Frobenius endomorphisms, depending if we are working with lisse
If (X0, x) is a smooth connected pointed variety, geometrically connected over Fqs and K is an VecK as a rigid abelian ⊗-category. At the same time if k ⊆ k′ is a finite extension of finite fields, F-Isoc†(Spec(k′)/K(k))
K is equivalent to the category of (finite dimensional)K-vector spaces endowed
with an automorphism.
2.2.8. Let (X0, x) be a smooth pointed variety, geometrically connected over Fqs. LetEλ be a finite extension of Qqs. The categoryIsoc†(X0/Qqs)E
λ admits a fiber functor over some finite extension of
Eλ. Assume that deg(x0) =n. Let i0:x0/Fqn ֒→X0/Fqs the immersion of the closed pointx0 in X0
is a fiber functor, as proven in [Cre92, Lemma 1.8]. This means that for every finite extensionEλofQqs, the category Isoc†(X0/Qqs)E the(linearized geometric) FrobeniusofM0atx′0. It corresponds to a linear automorphism of aK-vector space. This will be the(linearized geometric) Frobenius ofM0 atx′0. The characteristic polynomial
Px′
0(M0, t) := det(1−tFx′0|ie
∗
0(M0)) will be the(Frobenius) characteristic polynomial ofM0 atx′0.
In analogy with lisse sheaves, we say that overconvergent F-isocrystals are pure, ι-pure, mixed or
ι-mixed, if they respect the analogous conditions on the eigenvalues of the Frobenii at closed points.
3. Generalities
3.1. Coefficient objects. Let (X0, x) be a smooth pointed variety overFq, geometrically connected over Fqs. Following [Ked16b] we use a notation to work with lisse sheaves and overconvergent F-isocrystals at the same time.
Definition 3.1.1 (Admissible fields). A p-adic admissible field (for X0) will be a finite extension of
Qqs. At the same time, if ℓ6=p, we will say that a finite extension ofQℓ is an ℓ-adic admissible field
(forX0). In this case we do not require any condition on the inertia degree ofλ. If Eλ is admissible,
we will also say that the placeλisadmissible.
Definition 3.1.2 (Coefficient objects). Ifℓ is a prime different from pand K is anℓ-adic admissible field, a Weil lisse K-sheaf will be a K-coefficient object. At the same time, if K is a p-adic admissible field, every object inF-Isoc†(X0/Qq)K will be aK-coefficient object. For everyK-coefficient object, the
fieldKwill be itsfield of scalars.
Letℓbe a prime, for every admissible fieldK(of both types), we uniformly denote byCoef(X0,K) the category ofK-coefficient objects. Moreover, the category ofQℓ-coefficient objects, denotedCoef(X0,Qℓ),
will be the 2-colimit of the categoriesCoef(X0, Eλ), whereEλ varies among the finite extensions ofQℓ
3.1.3. We will introduce now the category of geometric coefficient objects. This is built from the category of coefficient objects by forgetting the Frobenius structure.
Definition(Geometric coefficient objects). For everyp-adic admissible fieldK, we have defined a functor of Tannakian categories Ψx,K:F-Isoc†(X0/Qq)K→Isoc†(X0/Qqs)Kwhich forgets the Frobenius struc-ture (see§2.2.6). We denote byCoef(X(x),K) the smallest Tannakian subcategory ofIsoc†(X
0/Qqs)K containing the essential image of Ψx,K. We will say that the categoryCoef(X(x),K) is the category of geometric K-coefficient objects (with respect to x).
Whenℓ6=p, for everyℓ-adic admissible fieldK, we have also defined a functor Ψx,K:Weil(X0,K)→
LS(X(x),K) that forgets the Frobenius structure (see §2.1.2). The category of geometric K-coefficient
objects (with respect to x) will be the smallest Tannakian subcategory of LS(X(x),K) containing the essential image of Ψx,K, denoted byCoef(X(x),K).
For everyℓ, the category of geometric Qℓ-coefficient objects will be the 2-colimit of the categories of geometric Eλ-coefficient objects when Eλ varies among the admissible fields for X0 in Qℓ. It will be
denoted byCoef(X(x),Q
ℓ)
WhenX0is geometrically connected overFq we will simply writeCoef(X, Eλ) andCoef(X,Qℓ), as
the categories do not depend ofx.
Remark 3.1.4. The category Coef(X(x),Q
ℓ) is endowed by the autoequivalence F∗. In the p-adic
case see [Abe13, Remark in§1.1.3].
Lemma 3.1.5. An irreducible object in Coef(X(x), E
λ)admits an n-th Frobenius structure for some
n∈Z>0.
Proof. By definition, an irreducible objectF in Coef(X(x), E
λ) is a subobject of some geometric
coef-ficient objectE that admits a Frobenius structure. The autoequivalenceF∗ permutes the isomorphism
classes of irreducible subobjects inE. Thus there exist 0< n2< n1such that (Fn1)∗F ≃(Fn2)∗F. This means that ifn:=n2−n1, the geometric coefficient objectF can be endowed with ann-th Frobenius
structure.
Remark 3.1.6. WhenX0is geometrically connected, the categoryCoef(X,Qq) is the same category as
the one considered by Crew in [Cre92] to define the fundamental group in [ibid., end of§2.5]. A priori this category is not the same as the one considered by Abe to define, for example, the fundamental group in [Abe13, §2.4.16]. In light of the previous lemma we only know that Coef(X,Qq) is a Tannakian
subcategory of the one defined by Abe.
Definition 3.1.7. A K-coefficient object is said constant if it is geometrically trivial, i.e. if it is isomorphic, after applying Ψx,K, to a direct sum of unit objects. We denote byCoefcst(X0, Eλ) the
full subcategory ofCoef(X0, Eλ) of constant Eλ-coefficient objects. It is a Tannakian subcategory of
Coef(X0, Eλ) that does not depend onx. We define the category of constantQℓ-coefficient objects, as
the 2-colimit of the categories of constantEλ-coefficient objects.
3.1.8. For every prime ℓ, the category Coef(Spec(Fq),Qℓ) is canonically equivalent to the category
of Qℓ-vector spaces endowed with an automorphism. For every a ∈Q×ℓ we define Q(ℓa) as the rank 1 coefficient object over Spec(Fq) associated to the vector spaceQℓendowed with the multiplication bya.
Definition. LetpX0 :X0 →Spec(Fq) be the structural morphism. For every Qℓ-coefficient object E0 and everya∈Q×ℓ, we define
E0(a):=E0⊗p∗X0
Q(ℓa)
as thetwist ofE0 bya. A twist is said to bealgebraic ifais algebraic, at the same time it isp-plain if
aisp-plain (cf. §2.0.2).
Remark 3.1.9. The twist by a ∈Q×ℓ is an exact autoequivalence of the category Coef(X0, Eλ). In
3.1.10. For every Qℓ-coefficient object E0 of rank r, we can associate at every closed point x0 of X0 the (Frobenius) characteristic polynomial of E0 at x0, denoted Px0(E0, t) = 1 +a1t+. . . art
r, where
(a1, . . . , ar)∈Q r−1
ℓ ×Q
× ℓ.
Definition. For every coefficient objectE0, the(Frobenius) characteristic polynomial functionassociated toE0 will be the function of setsPE0 :|X0| →Q
r−1
ℓ ×Q
×
ℓ that sendsx0to the coefficients of Px0(E0, t).
Theorem 3.1.11( ˇCebotarev). Twoι-mixedQℓ-coefficient objects with the same characteristic polyno-mial functions have isomorphic semi-simplifications.
Proof. For ´etale lisse sheaf this is the classic ˇCebotarev’s density theorem, explained in [Ser66, Theo-rem 7]. For general lisse sheaf one can take the semi-simplification and then take suitable twists of the absolutely irreducible components, to reduce to the ´etale case, in virtue of [Del80, Proposition 1.3.14]. In this case the assumption that the coefficient object isι-mixed is not needed. For thep-adic case it is
proven in [Abe13, A.3].
Remark 3.1.12. We will see later that actually every coefficient object is ι-mixed (Corollary 3.5.6), thus the previous theorem applies to every coefficient object.
Definition 3.1.13. Letℓbe a prime number,Ka field andτ:K֒→Qℓ. We will say that a coefficient objectE0isK-rational with respect toτif the characteristic polynomials at closed points have coefficients in the image ofτ. AK-rational coefficient object will be the datum ofτ:K֒→Qℓ and aQℓ-coefficient object that isK-rational with respect toτ. We will also say that anEλ-coefficient object isE-rational
if it isE-rational with respect to the natural morphismE→Eλ→Qℓ.
We say that a coefficient object isalgebraicif it isQ-rational for one (or equivalently any)τ :Q֒→Qℓ. At the same time, a coefficient object is said p-plain if it is algebraic and all the eigenvalues at closed points arep-plain (see 2.0.2 for the notation).
Remark 3.1.14. An absolutely irreducible lisse sheaf is ´etale if and only if its determinant is anℓ-adic unit by [Del80, Proposition 1.3.14], hencep-plain lisse sheaves are always ´etale.
3.1.15. We can compare two K-rational coefficient objects with different fields of scalars looking at their characteristic polynomial functions.
Definition. Two coefficient objects E0 and F0 that are K-rational with respect to τ and τ′ respec-tively are said to be K-compatible if their characteristic polynomials at closed points are the same as polynomials inK[t], after the identifications given byτ andτ′.
3.2. Monodromy groups. We will introduce now the main characters of the article: thefundamental groups and the monodromy groups of coefficient objects. We will define them using the Tannakian formalism.
3.2.1. Let (X0, x) be a smooth pointed variety, geometrically connected overFqs. For everyℓ6=pand everyℓ-adic admissible fieldEλwe take the fiber functorωx,Eλ:Weil(X0, Eλ)→VecEλ attached tox that sends a lisse sheafV0to the stalkV0,x. At the same time, ifEλis ap-adic admissible field, in§2.2.8
we have defined a fiber functorωx,Eλ of the categoryIsoc
†
(X0/Qqs)Eλ, overEλ(x0),attached tox. When
ℓ6=pandEλ is anℓ-adic admissible field we defineEλ(x0):=Eλ. Thus we have for every admissible field
Eλ, we have a fiber functorωx,Eλ ofCoef(X, Eλ) overE (x0)
λ . We will denote with the same symbol the
fiber functor induced on Coef(X0, Eλ). As the fiber functors commute with the extension of scalars,
for everyℓ we also have a fiber functor overQℓ forQℓ-coefficient objects. We will denote itωx,Q
ℓ.
Definition(Fundamental groups). For every admissible fieldEλ, we denote byπλ1(X0, x) the Tannakian group overE(x0)
λ ofCoef(X0, Eλ) with respect toωx,Eλ. At the same time, we denote byπ
λ
1(X, x) the Tannakian group ofCoef(X(x), E
λ), with respect to the restriction ofωx,Eλ. The forgetful functor Ψx,Eλ :Coef(X0, Eλ)→Coef(X, Eλ)
induces a closed immersion πλ
1(X, x) → πλ1(X0, x). We also denote by πλ1(X0, x)cst the quotient of
πλ
3.2.2. EveryEλ-coefficient objectE0generates threeEλ-linear Tannakian categories, thearithmeticone
hE0i ⊆Coef(X0, Eλ), thegeometric onehEi ⊆Coef(X(x), Eλ) and the Tannakian category of constant
objects hE0icst ⊆ hE0i. We will consider these categories endowed with the fiber functors obtained by the restriction ofωx,Eλ.
Definition. (Monodromy groups) We denote byG(E0, x) thearithmetic monodromy groupofE0, namely the Tannakian group ofhE0i. At the same time the Tannakian group ofhEiwill be denoted byG(E, x) and it will be the geometric monodromy group of E0. Finally it will be also useful to consider the quotient G(E0, x)cst that is the Tannakian group of hE0icst. These three groups are quotients of the previous fundamental groups. The natural functorshE0icst→ hE0i → hEiinduce morphisms G(E, x)→
G(E0, x)→G(E0, x)cstcompatible with the morphisms of above.
Remark 3.2.3. WhenM0 is an overconvergentF-isocrystal, G(M, x) is the same group defined by Crew in [Cre92] and denoted DGal(M, x). This is also the same as the DGal(M, x) appearing in [AE16]. This agrees with our previous Remark 3.1.6.
We will present now what we call the fundamental exact sequence attached to a place λ that is admissible forX0. The sequence has been already essentially proven in [Pal15, Proposition 4.7].
Proposition 3.2.4. Let (X0, x)be a smooth connected pointed variety and λan admissible place for
X0. The natural morphisms previously presented give an exact sequence
(3.2.4.1) 1→π1λ(X, x)ss →π1λ(X0, x)gss →π1λ(X0, x)cst→1.
called the fundamental exact sequence. If X0 is geometrically connected over Fqs, we have a natural
isomorphism
π1λ(X0, x)cst
∼
−→π1λ(Spec(Fqs), Eλ).
In particular, the affine group scheme πλ
1(X0, x)cst is commutative and π0(πλ1(X0, x)cst) is canonically
isomorphic toGal(F/Fqs).
Proof. We can verify that it is exact thanks to [EHS07, Theorem A.1]. The point (i) comes from the fact thatCoefcst(X0, Eλ) is a Tannakian subcategory ofCoef(X0, Eλ). The condition (ii) is a consequence
of the definition of Coef(X(x), E
λ). As every semi-simple geometric coefficient object is actually a
subobject of a geometric coefficient object that is in the essential image of Ψx,Eλ, we also obtain (iii.c). By definition of the categoryCoefcst(X0, Eλ) the condition (iii.a) is satisfied.
To prove (iii.b), we notice that the endofunctor F∗ of Coef(X(x), E
λ) preserves the trivial objects.
Thus, ifE0 is a coefficient object, the maximal trivial subobjectF ⊆ E is preserved by F∗. Hence F lifts to a subobject ofE0.
For the last part, assume that X0 is geometrically connected over Fqs via the map qX0 : X0 → Spec(Fqs), induced by theF-pointx. We have a functorqX∗
0 :Coef(Spec(Fqs), Eλ)→Coef cst(X
0, Eλ).
We want to construct a quasi-inverse qX0∗ :Coef cst
(X0, Eλ)→Coef(Spec(Fqs), Eλ). For every E0 ∈
Coefcst(X0, Eλ), we take the Eλ-vector space of global sections of E, H0(X,E) := Hom(Eλ,X,E),
where Eλ,X is the unit object in Coef(X(x), Eλ). We have a canonical identification H0(X, F∗E) =
H0(X,E), thus the Frobenius structure Φ of E
0 induces an automorphism ofH0(X,E) that we denote byqX0∗(Φ). We define qX0∗(E0) as the pair (H
0(X,E), q
X0∗(Φ)) ∈Coef(Spec(Fqs), Eλ). At the same time, for a morphismf0 ofCoefcst(X0, Eλ), we define qX0∗(f0) as the morphism induced byf0on the geometric global sections. The functorqX0∗ is a quasi-inverse ofq
∗
X0, thusq
∗
X0 induces an isomorphism
πλ
1(X0, x)cst
∼
−→πλ
1(Spec(Fqs), Eλ).
Corollary 3.2.5. Let Eλ be an admissible field. For every geometrically semi-simple Eλ-coefficient
objectE0, we have a commutative diagram
1 πλ
1(X, x)ss π1λ(X0, x)gss π1λ(X0, x)cst 1
with exact rows and surjective vertical arrows. In particular, the group G(E0, x)cst is commutative. 3.2.6. We recall a classical lemma, that we will frequently use to apply the previous result to semi-simple coefficient objects.
Lemma. Every semi-simple coefficient object is geometrically semi-simple.
Proof. LetE0be a semi-simple coefficient object andF is the maximal semi-simple subobject in E. As
Fis preserved by the autoequivalenceF∗, it lifts to a subobjectF
0ofE0. At the same time, the quotient
E → E/F lifts to a quotientE0 → E0/F0. As E0 is semi-simple, there exists a splitE0/F0 → E0. This induces a splitE/F → E. By the maximality of F, the quotientE/F is zero, as we wanted.
Remark 3.2.7. We will see later, in Proposition 3.5.9, that the fundamental exact sequence (3.2.4.1) is also functorial when (X0, x) varies among smooth connected pointed varieties.
We present now two results on the π0 of πλ1(X0, x) and π1λ(X, x). The statements are fairly easy for lisse sheaves and difficult for overconvergentF-isocrystals. In the latter case the problem has been studied by Crew in [Cre92]. He obtains results for overconvergentF-isocrystals only over a smooth curve. The generalization of Theorem 1.4 [ibid.] to higher dimensional varieties in [Tsu02, Theorem 1.3.1] and [Ked11, Theorem 2.3.7] allows a generalization of Crew’s work.
In [DK16, Appendix B], Drinfeld and Kedlaya have already presented how to perform such a gen-eralization for the arithmetic fundamental group of overconvergent F-isocrystals. We will be mainly interested in the extension of their result to the geometric fundamental group.
3.2.8. Let whereH varies among the normal open subgroups ofπ´et
1(X0, x). There exists a natural fully faithful functor of Tannakian categories
Repsmooth
E(x0 ) λ
(π1´et(X0, x))→Coef(X0, Eλ)
in the case of overconvergent F-isocrystals the existence is a consequence of the ´etale descent [Ete02, Theorem 1] (see [DK16,§B.7.2]). This induces a mapπλ is the (pro-constant) profinite group scheme overE(x0)
λ associated to π´1et(X0, x). The subcategory
is sent via the functor to Coef(X0, Eλ)cst. Thus, by the fundamental exact sequence for the ´etale
fundamental group, there exists a unique morphismπλ
1(X, x)→π1´et(X, x) making
We will show that the horizontal morphisms have connected kernels. We firstly need a lemma, that is a consequence of Proposition 3.2.4.
Lemma 3.2.9. LetEλ be an admissible field forX0and letE0be an absolutely irreducibleEλ-coefficient
object with finite order determinant. Then the natural mapG(E, x)֒→G(E0, x)is an open immersion.
Proof. In virtue of Corollary 3.2.5 it is enough to show that G(E0, x)cst is finite. We also know by the corollary that G(E0, x)cstis commutative. AsG(E0, x)cst is a quotient ofG(E0, x), it is enough to show that G(E0, x)◦ is semi-simple. We can verify it after extending the scalars to Qℓ. Let Z be the center
sumχ⊕r, whereχ is a character ofZ andris the rank of E0. By the assumption on the determinant,
χhas finite order. At the same time, asZ embeds intoG(E0, x), the characterχgenerates the group of characters ofZ. This implies thatZ is a finite group, as we wanted.
Proposition 3.2.10. Let (X0, x)be a connected smooth pointed variety. For every admissible fieldEλ,
we have a commutative square
where the vertical arrow are closed immersions and the horizontal ones are isomorphisms. The square is functorial on (X0, x)when it varies among the connected smooth pointed varieties.
Proof. The diagram is constructed applying the functorπ0to (3.2.8.1), hence the functoriality. We recall that theπ0of the Tannakian group of a Tannakian category is the Tannakian group of the subcategory of finite objects. Every finite lisse Eλ-sheaf is a Eλ-representation of a finite quotient of π´et1(X0, x), hence in this case we prove thatϕ0is an isomorphism. In thep-adic case one can prove the same using [Ked11, Theorem 2.3.7] as it is explained in [DK16, Proposition B.7.6.(i)].
We show now that ψ is a closed immersion. It is enough to prove it after extending the scalars to the algebraic closureQℓ. Thus, we have to prove that for every absolutely irreducible finite geometric
Qℓ-coefficient object E, there exists a finite object F0 ∈ Coef(X0,Qℓ), such that E is a subquotient
of F. By definition, there exists F′
0 ∈ Coef(X0,Qℓ) such that E is a subquotient of F′. As E is
absolutely irreducible, we can even assume F′
0 to be absolutely irreducible. In particular, there exist
g1, . . . , gn ∈ G(F0′, x)(Qℓ) such that ωx,Qℓ(F
monodromy groups as representations ofG(F′, x) are all finite, as they are conjugated to the monodromy
group ofE. ThereforeF′, being a sum of finite objects, is itself finite. By Corollary 3.4.7, we know that
F′
0 is the twist of a finite object F0. As the twist does not affect the associated geometric coefficient objects,F′ ≃ F. ThusF
0 is the coefficient object we were looking for.
Finally we prove that ϕ is an isomorphism. We notice that the category of finite objects is semi-simple, thusπ0(π1λ(X, x)) =π0(π1λ(X, x)ss) andπ0(π1λ(X0, x)) =π0(π1λ(X0, x)gss). Applying the functor
π0to the fundamental exact sequence (3.2.4.1), as we know thatψis a closed immersion, we deduce that
π0(πλ1(X, x)) is the kernel of the morphismπ0(πλ1(X0, x))→π0(π1λ(X0, x)cst). At the same time we know
(ii) There exists a choice off0such that, after extending the field of scalars ofE0, the natural maps
induce isomorphismsG(f∗
Proof. We notice that by Proposition 3.2.10 the group of connected components of the arithmetic mon-odromy group (resp. geometric monmon-odromy group) are quotients of the arithmetic ´etale fundamental group (resp. geometric ´etale fundamental group), thus (i) implies (ii).
Proposition 4.6] 4, when X0 is a curve. The result in higher dimension is obtained replacing Corol-lary 4.5 inloc. cit. by [DK16, Proposition B.7.6.(ii)] applied to overconvergentF-isocrystals defined on
X0⊗kX0FoverK(F).
Corollary 3.2.12. Let E0 be a coefficient object on (X0, x). For every finite ´etale morphism f0 : (Y0, y)→(X0, x)of pointed varieties, E0 is semi-simple (resp. geometrically semi-simple) if and only if
f∗
0E0 is semi-simple (resp. geometrically semi-simple).
3.3. The K¨unneth formula. To prove Theorem 5.1.1 without using the existence of companions we will need the K¨unneth formula for the fundamental group parameterizing geometric coefficient objects. For simplicity we will prove it just for smooth connected projective varieties admitting a rational point. The main ingredient for the K¨unneth formula is the existence of a direct image functor for smooth and proper morphisms of coefficient objects, that is a right adjoint of the inverse image functor and that satisfies the proper base change. In the case of lisse sheaves the classical direct image has the desired properties. In thep-adic case the construction is more problematic. We will use the direct image functor ofarithmeticD-modules.
3.3.1. For every smooth projective variety X0 we take the triangulated category with t-structure of
holonomic complexes D♭
hol(X/Qq), see [Abe13, Definition 1.1.1]. We have chosenX0to be projective in order to makeX0realizable (cf. loc. cit.). We also consider the category of holonomic complexes with
F-structure, denotedDhol♭ (X0/Qq), and for everyp-adic admissible fieldEλthe categories enriched with
Eλ-structure, denotedD♭hol(X/Qq)Eλ andD
♭
hol(X0/Qq)Eλ [ibid. §1.4].
For every proper smooth morphismf0:X0→Y0between geometrically connected projective smooth varieties we dispose of adjuctions (f+, f
+) and (f0+, f0+) of inverse and direct image forDhol♭ (X/Qq)Eλ defined in§1.1.3.11 and§2.4.15 of [Abe,op. cit.]. They are fully faithful functors commuting with the inverse images.
Remark 3.3.2. We notice that whenY0= Spec(Fq) andf0is the structural morphism, by the adjuction property, for everyM ∈Coef(X, Eλ), the pushf∗M ∈Coef(Spec(F), Eλ) is the vector space of global
sections ofM.
Proposition 3.3.3. Let (X0, x) and (Y0, y)be two smooth projective connected pointed varieties such
thatx0 andy0 are rational points. For every admissible fieldEλ, the projections ofX0×Y0to its factors
induce an isomorphism
π1λ(X×FY, x×y)
∼
−→πλ1(X, x)×π1λ(Y, y).
Proof. Let’s denotef0:X0×Y0→Y0 the projection to the second factor andg0:X0×y0֒→X0×Y0 the natural inclusion. The morphismg0 induces a closed immersionX0֒→X0×Y0 that we denote by the same letter. Ifϕandψare the morphisms induced bygandf on the fundamental groups classifying geometric coefficient objects we obtain a sequence
4In [Cre92,§2.1.10] there is a small typo. Crew refers to 1.3.2 in [10], but the correct reference is Proposition 1.3.2 in
We want to use [EHS07, Theorem A.1] to show that it is an exact sequence. This will imply the original statement. Let’s consider the sequence of functors of the Tannakian categories
Coef(Y, Eλ) f∗
−→Coef(X×FY, Eλ) g∗
−→Coef(X, Eλ).
The point (i) of Theorem A.1 (loc. cit.) follows from the existence of a section of f0, namely the closed immersion ofx0×Y0֒→X0×Y0. The point (ii) and (c) are consequence of the existence of a retraction forg0, given by the first projectionX0×Y0→X0.
We want to show now that (a) and (b) are satisfied. Let’s consider the commutative square
X0×Y0 Y0
Lemma 3.3.4. For every geometric Eλ-coefficient object E, the adjuction morphism f∗f∗E → E is
injective. Moreover, after applying g∗, the morphism g∗f∗f
∗E → g∗E makes g∗f∗f∗E the maximal
trivial subobject ofg∗E.
Proof. To show the injectivity of the adjuction morphism, we use the fiber functors of the Tannakian categories, associated to the rational points we are considering. Let G and H be the affine group schemes πλ
1(X ×FY, x×y) and πλ1(Y, y). We know that the functor f∗ is equivalent to the functor ResHG :RepEλ(H)→RepEλ(G), induced byψ:G→H.
As we have already proven thatψ:G→H is surjective, the induction functor IndHG :RepEλ(G)→
RepEλ(H) is well defined at the level of finite dimensional representations and it is the right adjoint of ResHG. If we takeN := Ker(ψ), the functor IndGHsendsV ∈RepE
λ(G) toV
N, the induced representation
ofH on the subspace ofV fixed byN.
By the uniqueness of the right adjoint of f∗, the counit of (f∗, f
∗) is isomorphic to the counit of
the adjunction (ResHG,IndGH), induced by ψ : G → H. If we apply the counit of (ResHG,IndGH) to a
representationV ∈RepEλ(G) we obtain the natural inclusionVN ֒→V, in particular an injective map.
As a consequence, the maps induced by the counit of the adjunction (f∗, f
∗) on geometric coefficient
objects are always injective.
We show now the second part of the statement. Asg∗commutes with the fiber functors, the morphism
g∗f∗f
∗E →g∗E is injective. We have natural isomorphisms,
g∗f∗f∗E ≃f′∗g′∗f∗E ≃f′∗f∗′g∗E,
the second one given by the proper base change. At the same time we know thatf′
∗g∗E ≃H0(X, g∗E),
in thep-adic case thanks to the Remark 3.3.2. Thus g∗f∗f
∗E ≃f′∗H0(X, g∗E) is the maximal trivial
subobject ofg∗E, as we wanted.
We now verify (a). It is enough to show that if E is anEλ-geometric coefficient object of X×FY
such that g∗E is trivial, then f∗f
∗E ≃ E. As g∗E is trivial, by Lemma 3.3.4, we know that g∗f∗f∗E
andg∗E are isomorphic, thus f∗f
∗E and E have the same rank. Therefore we know that the adjuction
map f∗f
∗E → E is an injective map between two objects of the same rank. This means that it is an
isomorphism.
To check (b) we have to show that for every geometric coefficient objectE, there existsF ⊆ E, such that g∗F is the maximal trivial subobject of g∗E. We know by the lemma thatf∗f∗E, equipped with
the adjunction morphism f∗f
∗E → E, is a subobject of E. We also know by the lemma that after
applyingg∗, the pullbackg∗f∗f
∗E becomes the maximal trivial subobject of g∗E. Thus we can always
takeF :=f∗f
∗E.
3.4.1. We recall the trace formula forQℓ-coefficient objects. For everyQℓ-coefficient objectE0, we can put together the pieces of information of all the characteristic polynomials at closed points in a formal series inQℓ[[t]]:
LX0(E0, t) := Y
x′
0∈|X0|
Px′
0(E0, t deg(x′
0))−1.
For lisse sheaves, we consider ´etale cohomology groups with compact support onX, endowed with a linear automorphism given by the action of the geometric Frobenius ofX. In thep-adic case we work with rigid cohomology with compact support onX0, and we have a Frobenius automorphism induced by theq-power absolute Frobenius. To refer to these groups simultaneously, we write simplyHi
c(X,E) for
every coefficient object and we denote the Frobenius automorphism acting on the cohomology groups byF.
Theorem (Trace formula). For every coefficient objectE0,
LX0(E0, t) = 2d
Y
i=1
det(1−F t, Hci(X,E))(−1)
i+1
.
Proof. For lisse sheaves this is the classic Grothendieck’s formula, in thep-adic case see [ES93, Th´eor`eme
6.3].
The formula is handy to understand properties shared by compatible coefficient objects. The problems one usually encounter working with this identity are related to the possible cancellations between the factors of the numerator and the denominator.
The main tool to control this phenomenon is given by the theory of weights. We will recollect now some results of Weil II that are now known for arbitrary coefficient objects and we will present themain theorem on weights (Theorem 3.4.4).
3.4.2. One of the starting point of Weil II is a finiteness result which relies on the class field theory for function fields.
Theorem. Every rank oneEλ-coefficient object is a twist of a finiteEλ-coefficient object. Hence, every
Eλ-coefficient object is a twist of aQℓ-coefficient object with determinant of finite order under⊗.
Proof. For lisse sheaves see [Del80, Proposition 1.3.4], in thep-adic case [Abe15, Lemma 6.1]. This is used, for example, to prove theglobal monodromy theorem.
Theorem 3.4.3(Grothendieck, Crew). For every coefficient objectE0, the radical subgroup5ofG(E, x)
is unipotent.
Proof. In the case of lisse sheaves it is a theorem of Grothendieck and it is proven in [Del80, Th´eor`eme 1.3.8]. In thep-adic case, Crew has proven the result whenX0 is a smooth curve [Cre92, Theorem 4.9]. We obtain the result in higher dimension replacing [ibid. , Proposition 4.6] by Proposition 3.2.11 and [ibid., Corollary 1.5] by Theorem 3.4.2.
We arrive now to the main theorem on weights. The proof for lisse sheaves has been given by Deligne in Weil II, as a generalization of his proof of the Riemann Hypothesis. Later, Kedlaya has obtained the analogous result for overconvergentF-isocrystals. We will see some consequences of this theorem that we will use frequently later on.
Theorem 3.4.4(Deligne, Kedlaya). Let X0 be a smooth variety and E0 aι-mixed coefficient object of
ι-weights ≤w. Ifαis an eigenvalue ofF acting onHn
c(X,E), then|ι(α)| ≤q(w+n)/2.
Proof. For lisse sheaves it is the main result in [Del80]. For overconvergentF-isocrystals it is proven by
Kedlaya in [Ked06].
Proposition 3.4.5 ([Laf02, Cor. VI.3], [Abe13, Prop. 4.3.3]). Let X0 be a geometrically connected
smooth variety overFq of dimensiond. For everyι-pure coefficient objectE0ofι-weightw, the dimension
of H0(X,E) is equal to the number of poles ofι(L(X
0,E0∨(d))), counted with multiplicity, with absolute
value qw/2. If we also assumeE
0 to be semi-simple the dimension of H0(X0,E0) :=H0(X,E)F is equal
to the order of the pole of L(X0,E0∨(d))at1.
Proof. By Poincar´e duality the dimension ofH0(X,E) is equal to the dimension of H2d
c (X,E∨(d)) and
the eigenvalues ofF acting onHc2d(X,E∨(d)) haveι-weight−α. At the same time, by Theorem 3.4.4,
for every 0≤i≤2dthe groupsHi
c(X,E∨(d)) haveι-weights less or equal than−α−2d+i. The first part
of the statement is then a consequence of the trace formula (Theorem 3.4.1) applied to L(X0,E0∨(d)). Indeed, by the observations on the weights, the polynomial det(1−F t, H2d
c (X,E)) is coprime with the
numerator, hence we conclude looking at its degree.
For the second part, we also use that by assumption the endomorphism F acts semi-simply on
H0(X,E). In particular, the geometric and the algebraic multiplicities of the eigenvalue 1 are the same. Therefore, thanks to Poincar´e duality, the dimension of H0(X
0,E0) is equal to the multiplicity of 1 of det(1−F t, H2d
c (X,E)). By the previous reasoning, this is the same as the order of the pole of
L(X0,E0∨(d)) at 1.
Proposition 3.4.6([Del80, Th´eor`eme 3.4.1.(iv)], [AC13a, Theorem 4.3.1]). LetE0be aι-pure coefficient
object, thenE is semi-simple.
Corollary 3.4.7. For everyι-pure coefficient objectE0, the neutral componentG(E, x)◦ is a semi-simple
algebraic group. It coincides with the derived subgroup ofG(E0, x)◦.
Proof. By Proposition 3.4.6, the algebraic group G(E, x)◦ is reductive, thus the unipotent radical is
trivial. At the same time, by Theorem 3.4.3, the unipotent radical is equal to the radical. This shows that
G(E, x)◦ is semi-simple, therefore G(E, x)◦ ⊆[G(E
0, x)◦, G(E0, x)◦]. By Propositon 3.2.4, the quotient
G(E0, x)◦/G(E, x)◦ is commutative, thus [G(E0, x)◦, G(E0, x)◦]⊆G(E, x)◦.
3.5. Deligne’s conjectures. We will rephrase now Deligne’s conjectures (Conjecture 1.1.1) for arbi-trary coefficient objects.
Conjecture 3.5.1. Let X0 be a smooth variety over Fq andE0 an absolutely irreducible Qℓ-coefficient
object whose determinant has finite order. There exists a number fieldE⊆Qℓ such that: (i) E0 is pure of weight0;
(ii) E0 isE-rational;
(iii) E0 isp-plain;
(iv’) For every prime ℓ′ (even ℓ′ = ℓ or ℓ′ = p) and for every inclusion τ : E ֒→ Q
ℓ′ there exists an absolutely irreducibleQℓ′-coefficient object,E-rational with respect toτ, that isE-compatible withE0.
Remark 3.5.2. We will postpone the discussions on the analog of the Conjecture 1.1.1.(iv), in§3.6. Thanks to the Langlands correspondence for GLr over function fields, for lisse sheaves and
overcon-vergentF-isocrystals, proven by L. Lafforgue and Abe, and the Ramanujan-Petersson conjecture, proven by L. Lafforgue, one can show the following result.
Theorem 3.5.3 ([Laf02, Th´eor`eme VII.6], [Abe13, §4.4]). If X0 is a smooth curve, Conjecture 3.5.1
holds.
The generalization of the previous theorem to higher dimensional varieties is performed via results of reduction to curves as the following one.
Theorem 3.5.4(Deligne, Drinfeld, Abe-Esnault, Kedlaya). Let(X0, x)be a smooth pointed variety over
Fq. For every coefficient object E0 over X0 there exists a connected smooth curve C0 and a morphism
Proof. For lisse shaves see [Del12, §1.5-1.8] and [Dri12, Proposition 2.17]. In the p-adic case see the (proof of) [AE16, Theorem 3.10] and [Ked16b, Corollary 4.5].
The first three conjectures are now proven, for both types of coefficient objects.
Theorem 3.5.5. (L. Lafforgue, Abe, Deligne, Drinfeld, Abe-Esnault, Kedlaya) Conjectures (i), (ii) and (iii) hold for every smooth variety.
Proof. The conjectures (i) and (iii) are consequences of Theorem 3.5.4. The conjecture (ii) is proven in [Del12, Theorem 3.1] for lisse sheaves and in [AE16, Lemma 4.1] and [Ked16b, Theorem 5.1] for
isocrystals.
Even if this theorem is stated for absolutely irreducible coefficient objects with finite order determi-nant, it has important consequences on more general coefficient objects, thanks to Theorem 3.4.2. We present here some of them.
Corollary 3.5.6. Every coefficient object is ι-mixed.
Proof. To prove the statement it is enough to show that every absolutely irreducibleQℓ-coefficient object
E0 is ι-pure. Thanks to Theorem 3.4.2, E0 is isomorphic toF0(a) where F0 is an absolutely irreducible coefficient object with finite order determinant anda∈Q×ℓ is algebraic. By Theorem 3.5.5,F0isι-pure of weight 0, thusE0isι-pure ofι-weight logq(ι(a)).
Remark 3.5.7. In thep-adic case this corollary is proven in [AE16] and [Ked16b] without Theorem 3.5.5 and it is actuallyused to prove 3.5.4.
Corollary 3.5.8. LetE0 andF0be two compatible coefficient objects. IfE0 is absolutely irreducible, the
same is true for F0. At the same time, if E is absolutely irreducible even F is absolutely irreducible.
Proof. Suppose E0 absolutely irreducible, then it is ι-pure by Corollary 3.5.6, hence the same is true for F0. At the same time we know that End(E0) is a one dimensional vector space. As End(E0) and
End(Fss
0) are semi-simple ι-pure compatible coefficient objects, by Proposition 3.4.5, End(F0ss) is one dimensional. Hence Fss
0 is absolutely irreducible, so the same is true for F0. For the second part we proceed in the same way, applying Proposition 3.4.5 toEnd(E) andEnd(F).
Corollary 3.5.9. On a smooth variety, aQℓ-coefficient object is geometrically semi-simple if and only if it is a direct sum of ι-pure Qℓ-coefficient objects. In particular, for every morphism f0 : Y0 → X0
between smooth varieties, if E0 is a geometrically semi-simple coefficient object onX0, the same is true
forf∗
0E0. Thus, the fundamental exact sequence (3.2.4.1) is functorial in(X0, x), when it varies among
smooth pointed varieties.
Proof. Thanks to Proposition 3.4.6, we already know that ι-pure coefficient objects are geometrically semi-simple, hence the same holds for a direct sum of ι-pure coefficient objects. For the converse we use that every coefficient object isι-mixed (Corollary 3.5.6). Assume thatF0andG0 areQℓ-coefficient
objects that have no commonι-weights. It is enough to show that a geometrically trivial extension of
G0 byF0 is trivial.
We recall that we have an exact sequence
0→Hom(G,F)F →Ext1(G0,F0)→Ext1(G,F)F,
whereF it is the automorphism induced by the geometric Frobenius for lisse sheaves and by the absolute Frobenius for overconvergentF-isocrystals. By hypothesis, the coefficient objectHom(G0,F0) does not contain anyι-pure subobject ofι-weight 0, hence Hom(G,F)F = 0. This implies that the right morphism
of the sequence is injective, as we wanted.
Corollary 3.5.10. For every algebraicQℓ-coefficient objectE0, there exists a number fieldE⊆Qℓ, such
that E0 isE-rational. Moreover, ifE0 is absolutely irreducible, it is isomorphic to an algebraic twist of
an absolutely irreducible coefficient object with finite order determinant.
Proof. It is enough to prove the result whenE0 is absolutely irreducible. Thanks to Theorem 3.4.2,E0 is isomorphic toF0(a), whereF0is a coefficient object with finite order determinant anda∈Q
×
ℓ. As the
determinant of E0and F0 are algebraic, also the numberais algebraic. Theorem 3.5.5 implies that F0 isE-rational for some number fieldE⊆Qℓ, thusE0isE(a)-rational. The generalization of conjecture (iv’) to higher dimensional varieties is yet incomplete. For the moment we know how to construct from a coefficient object of both types, compatible lisse sheaves. In dimension greater than 1, we do not know how to construct compatible overconvergentF-isocrystals.
Theorem 3.5.11 (L. Lafforgue, Abe, Drinfeld, Abe-Esnault, Kedlaya). Let X0 be a smooth variety
overFq andE a number field. For every absolutely irreducibleE-rational coefficient objectE0with finite
order determinant, everyℓdifferent frompand every inclusionτ:E ֒→Qℓ, there exists aQℓ-coefficient object that isE-rational with respect toτ andE-compatible with E0
Proof. Drinfeld has proven the theorem whenE0is a lisse sheaf, generalizing Lafforgue’s result [Dri12]. The proof uses a gluing theorem for lisse sheaves [ibid., Theorem 2.5]. The proof whenE0 is an overcon-vergentF-isocrystal has been given in [AE16] and later in [Ked16b]. They both use Drinfeld’s gluing theorem for lisse sheaves, once they have obtained Theorem 3.5.4 for overconvergentF-isocrystals. 3.6. Compatible systems. Thanks to Theorem 3.5.11, from a coefficient object we can construct many compatible coefficient objects with different fields of scalars.
Definition 3.6.1(Compatible systems). IfE⊆Qis a number field, anE-compatible system overX0, denotedE0, is a family{E0,λ}λ∈Σwhere:
– Σ is a set of finite places ofE containing|E|6=p;
– For everyλ∈Σ,E0,λis anE-rationalEλ-coefficient object;
– The coefficient objectsE0,λ are pairwiseE-compatible.
For every λ, the coefficient objectE0,λ will be the λ-component of the compatible system. We say
that a compatible system istrivial, geometrically trivial, pure, p-plain, irreducible, absolutely irreducible or semi-simple if eachλ-component has the respective property.
IfE⊆E′⊆Qis an inclusion of two number fields andE
0 is anE-compatible system, the compatible system obtained fromE0by extending the scalars toE′, is theE′-compatible system{E0′,λ′}λ′∈Σ′, where
Σ′ is the set of places of E′ over the places in Σ and for every place λ′ ∈ Σ′ over λ ∈ Σ, we set
E′
0,λ′ :=E0,λ⊗EλE
′ λ′.
Remark 3.6.2. The properties of an E-compatible system to be trivial, geometrically trivial, pure,
p-plain, absolutely irreducible or semi-simple are invariant under the extension of scalars.
Thanks to the work of Chin in [Chi03], one can prove a stronger formulation of Theorem 3.5.11.
Theorem 3.6.3 (L. Lafforgue, Abe, Deligne, Drinfeld, Abe-Esnault, Kedlaya, Chin). Let X0 be a
smooth variety and E0 an algebraic Qℓ-coefficient object ofX0. There exists a number field E, a finite
placeλ∈ |E| and anE-compatible systemE0, such thatE0 is aλ-component of E0.
Proof. After extending the field of scalars ofE0 and taking the semi-simplification we reduce to the case whenE0 is absolutely irreducible. In light of Corollary 3.5.10, the coefficient objectE0 is isomorphic to
F0(α), where α∈ Qℓ× is an algebraic number and F0 is an E-rational coefficient objectF0 with finite order determinant. After enlargingE we takea∈E and a place µofE such thata goes toαvia the natural inclusionE ֒→Eµ⊆Qℓ.
Whenℓ6=pwe can takeν overµ. The coefficient object F0 is E-compatible with every component of
V0.
Whenℓ6=p, we take theE-compatible system
{V0(a,λ−1)}λ∈|E|=6 p,λ6=ν∪ {E0⊗Eν}, ifℓ=p, we take the E-compatible system
{V0(a,λ−1)}λ∈|E|6=p∪ {E0⊗Eµ′} for someµ′ overµ.
Remark 3.6.4. Even if a coefficient object E0 isE-rational for some number fieldE, it could be still necessary to enlargeE to obtain theE-compatible systemE0.
For example, let’s take a connected smooth varietyX0 that admits a Galois cover with Galois group
Q, the quaternion group. We consider the natural four dimensionalQ-linear representation ofQon the quaternion algebraH.
The representationH⊗QQℓis irreducible overQℓif and only ifℓ= 2. If we takeℓ6= 2, thenH⊗QQℓ
decomposes as a direct sum of two copies of an absolutely irreducible two dimensionalQℓ-representation
Vℓ with traces in Q. The representation Vℓ corresponds to an absolutely irreducible Q-rational Qℓ
-coefficient object which does not admit any Q-compatible Q2-coefficient object. Indeed, suppose that there exists a semi-simpleQ2-coefficient object V2, that isQ-compatible withVℓ. Then V2⊕2 would be
Q-compatible withH⊗QQ2. By Theorem 3.1.11, the coefficient objectV2⊕2would actually be isomorphic toH⊗QQ2. But this is impossible, asH⊗QQ2is irreducible.
4. Independence of the monodromy groups
4.1. The group of connected components. In [Ser00] and [LP95, Proposition 2.2] Serre and Larsen– Pink have proven that on a smooth curve, for every pureQ-compatible system of ´etale lisseQℓ-sheaves,
the groups of connected components of the arithmetic and the geometric monodromy groups are inde-pendent ofℓ.
We will generalize their result to compatible pure compatible coefficient objects on varieties of ar-bitrary dimension. The main effort to adapt Larsen–Pink proof to arar-bitrary coefficient objects has already been done in§3.2. In light of Proposition 3.2.10, for every coefficient objectE0 we have functo-rial surjective morphismsψE0 :π
´ et
1(X0, x)→π0(G(E0, x)) andψE :π´et1(X, x)→π0(G(E, x)) of profinite groups.
Theorem 4.1.1. Let (X0, x) be a connected smooth pointed variety over Fq. If E0 and F0 are two
compatible pureEλ-coefficient objects of rankr, then:
(i) There exists an isomorphism ϕ0 :π0(G(E0, x))
∼
−→π0(G(F0, x))as abstract finite groups, such
thatψF0 =ϕ0◦ψE0;
(ii) The isomorphism ϕ0 restricts to an isomorphismϕ:π0(G(E, x))
∼
−→π0(G(F, x)).
Construction 4.1.2. LetE0 be a Eλ-coefficient object of rankr. We fix a basis of ωx,Eλ(E0) and we take the representation ρE0 : G(E0, x) → GLr,Eλ, associated to E0. For every Q-linear representation
θ: GLr,Q→GL(V) we denote byE0(θ) the coefficient object associated to (θ⊗QEλ)◦ρE0. Even ifE0(θ) depends on the choice of a basis, its class of isomorphism is uniquely determined.
Remark 4.1.3. We notice that for every pair of compatible coefficient objects (E0,F0) and for every representationθ of GLr,Q, the pair (E0(θ),F0(θ)) is again compatible. Moreover, for every irreducible representationθ of GLr,Q and every pure Qℓ-coefficient objectE0, thenE0(θ) is again pure, asE0(θ) is a subobject of En
0 ⊗(E0∨)m, for some n, m ∈ N. Thus, for every pair of pure semi-simple compatible coefficient objects (E0,F0) we can apply Proposition 3.4.5 to (E0(θ),F0(θ)) for everyθ. To do this we will considerX0as a variety overkX0. For everyθ, the equalities dim(H
0(X
