aa r X i v : . [ m a t h . N T ] S e p Category Of C -MotivesOver Finite Fields Esmail Arasteh Rad and Urs HartlOctober 2, 2020
Abstract
In this article we introduce and study a motivic category in the arithmetic of func-tion fields, namely the category of motives over an algebraic closure L of a finite fieldwith coefficients in a global function field over this finite field. It is semi-simple, non-neutral Tannakian and possesses all the expected fiber functors. This category gener-alizes the previous construction due to Anderson and is more relevant for applicationsto the theory of G -Shtukas, such as formulating the analog of the Langlands-Rapoportconjecture over function fields. We further develop the analogy with the category ofmotives over L with coefficients in Q for which the existence of the expected fiberfunctors depends on famous unproven conjectures. Mathematics Subject Classification (2000) : 11G09
Let F q be a finite field with q elements, let C be a smooth, projective, geometrically con-nected curve over F q with function field Q = F q ( C ), and let G be a flat affine group schemeof finite type over C . The moduli stacks of global G -shtukas appear as function field analogsfor Shimura varieties; for some explanation regarding this analogy see for example [Ara,Chapter 3]. There are several approaches to study the geometry of these moduli stacks.In [AraHar14], [AraHab19, Chapter 4] and [AraHar19] the authors developed an approachwhich is based on the relation between these moduli stacks and certain moduli spaces forlocal P -shtukas. Namely, in [AraHar14] we developed the local theory of global G -shtukas.In particular, for smooth group scheme G over C with reductive connected generic fiber,we constructed Rapoport-Zink spaces for local P -shtukas, where P is the base change of G INTRODUCTION C . This construction generalizes the previous con-struction in [HV11] for constant groups G = G × F q Spec F q [[ z ]], where G is split reductiveover F q , to the ramified case. Furthermore in [AraHab19, Chapter 4] the authors use thedeformation theory of local P -shtukas, investigated in [AraHar14], to establish the analogueof Rapoport-Zink theory of local models for moduli of G -shtukas. On the other hand, usingRapoport-Zink spaces for local P -shtukas we established the uniformization theory of themoduli stacks of global G -shtukas in [AraHar19]. Based on these results we investigate theanalog of the Langlands-Rapoport conjecture over function fields in [AraHar16]. To this aim,as we will discuss in this article, one has to introduce and study the properties, of a relevantmotivic category M ot νC ( L ) over a finite field extension L of F q , as well as the associatedfundamental motivic groupoid. Here ν denotes an n-tuple ( ν , . . . , ν n ) of pairwise differentclosed points of C . In this context, our initial motivation for studying this category was thefact that it provides a tool to describe the quasi-isogeny classes of G -shtukas in terms of therepresentations of the corresponding motivic groupoid.Apart from what we mentioned above, it seems to us that the category M ot νC ( L ) deservesto be studied for its own as well. Note that it basically arises as a natural generalizationof previous constructions of the category of motives over function fields considered by var-ious authors; for example see [And], [Tae07] and [BH09]. In addition let us mention thatin another research project we aim to clarify the possible relation with the Galois gerbeintroduced by Kottwiz in [Kot].Let us now briefly explain how the category M ot νC ( L ) naturally arises in the contextof motives over function fields. In [Tae07], Taelman proposes several categories that servethe analogous role (of the Grothendieck category of Chow motives) over function fields. Tothis goal, he first considers the Anderson category of A -motives [And], for A = F q [[ z ]], thenafter extending the coefficients to rational coefficients (by tensoring up the Hom-sets with Q = Frac( A )), he formally inverts (tensor powers of) the Carlitz motive C . The resultingcategory t M ◦ together with the obvious fiber functor ω : t M ◦ → Q − vector spaces providesa tannakian category which is a candidate for the analogous motivic category over functionfields. Still one may naturally want- to consider motives which admit multiplications by a Dedekind domain which is strictlybigger than F q [ z ],- to construct a category analogous to the category (mixed) motives over a general base,and- to geometrize this category. More precisely, one may think of Shimura varieties as amoduli for motives according to the Deligne’s conception of Shimura varieties [Del71]. INTRODUCTION A .To handle the above issues, one makes Anderson A -motives completed at the place in-finity ∞ of a curve C , with A := Γ( ˙ C, O C ), in the following sense. Namely, we replace the(locally) free A L -module M by a locally free sheaf M of O C L -modules (or equivalently avector bundle over C L ). Note that one should be a bit careful with extending τ over infinity.Recall that, according to the definition of A -motives, one requires the morphism τ to haveit’s zeros along V ( J ), where J is the ideal corresponding to the graph of a (characteristic)section s : S := Spec L → ˙ C = Spec A . This in particular indicates that the morphism τ can not be defined over the whole (relative) curve C S . This is because over a projectivecurve, the balance between order of zero’s and poles of τ should be preserved. Therefore toprovide an appropriate definition, one should allow further characteristic section(s) s i . Notethat aside from making the definition more efficient, in fact, introducing several character-istics turns out to be an extremely useful tool in function fields set up, which is still absentover number fields. Note in addition that introducing further characteristic sections corre-sponds to inverting several Carlitz-Hayes motives, see [Ha-Ju], in Taelman’s construction.Regarding this, one eventually comes to the Definition 2.1 of the Q -linear category M ot νC ( L )whose objects are C -motives M = ( M , τ M ) of characteristic ν , with quasi-morphisms as itsmorphisms; see Definition 2.1 b).Note that sending a C -motive ( M , τ M ) to the generic fiber M η equips the category witha fiber functor ω : M ot νC ( L ) −→ Q L -vector spaces . Let us now briefly explain the content of this article. In Section 2.1 we construct on M ot νC ( L ) ´etale realization functors ω νQ ν when ν / ∈ ν and crystalline realization functors Γ ν when ν ∈ ν . Let ν / ∈ ν be a closed point of C away from the characcteristic places. For a C -motive M over L the ´etale realization ω νQ ν ( M ) is a vector space over the v -adic completion Q v of Q equiped with a continuous action of the Galois group Γ L = Gal( F q /L ). This allowsto formulate the analog of the Tate conjecture for this category: Theorem 1.1.
Let M and M ′ be C -motives over a finite field L . Then there are isomor-phisms QHom L ( M , M ′ ) ⊗ Q Q ν ˜ −→ Hom Q ν [Γ L ] ( ω νQ ν ( M ) , ω νQ ν ( M ′ )) of Q ν -vector spaces. Moreover if M = M ′ then the above are ring isomorphisms. INTRODUCTION
Theorem 1.2.
Let L/ F q be a finite field extension. Let M be a C -motive over L andlet π ∈ QEnd L ( M ) be its L -Frobenius. Let ν / ∈ ν be a closed point of C away from thecharaccteristic places. The following statements are equivalent(a) M ∈ M ot νC ( L ) is semi-simple.(b) E := QEnd L ( M ) is semi-simple.(c) End Q ν [Γ L ] ( ω νQ ν ( M )) is semisimple.(d) F ν := Q ν [ π ν ] is semisimple, where π ν = ω νQ ν ( π ) ∈ End Q ν ( ω νQ ν ( M )) .(e) F := Q [ π ] is semi-simple. This is Theorem 3.5 in the text.We study the endomorphism algebra QEnd L ( M ) further and in particular we proveproposition 3.9 and Proposition 3.10.As we mentioned earlier the notion of C -motives is relevant for G -Shtukas; we discussthis in Section 4. In particular we introduce the category of G - C -motives, see Definition4.1, and we explain its relation to the category of G -shtukas. In Proposition 4.5 we prove afiniteness theorem for G - C -motives that arise from G -shtukas.In section 4.3 we recall the construction of the global-local functor and furthermore, as anadditional analogy with the theory of abelian varieties, we see that for a fixed C -motive M in M ot νC ( L ), one can express the set of quasi-isogenies ϕ : M ′ → M in terms of Galoisstable sublattices in the ´etale realization of M at the place ν . See Corollary 4.7. Proposition 1.3.
Let ϕ : M ′ → M be a quasi-isogeny of C -motives in M ot νC ( L ) . Then ω νQ ( ϕ ) identifies ω ν ( M ′ ) with a Γ L -stable sublattice of ω νQ ( M ) . This gives a one to onecorrespondence between the following sets { quasi-isogenies M ′ → M in M ot νC ( L ) which are isomorphisms above ν } INTRODUCTION and { Γ L -stable sublattice Λ ν ⊆ ω νQ ( M ) which are contained in ω ν ( M ) } . Note that M ot νC ( L ) consists of mixed motives rather than pure ones, and to define thesubcategory of pure objects one has to impose purity conditions. But nevertheless, as wewill see in Section 5, a modified version of the Honda-Tate theory applies to this category.We introduce the analog W ν of the Weil pro-torus, and we discuss the Honda-Tate theoryin Section 5; see Propositions 5.1 and 5.2, in which we obtain a quasi-isogeny criterion for C -motives, and Theorem 5.3. We further observe that Theorem 1.4.
There is a bijectionthe set Σ of simple objects in M ot νC ( F q ) ←→ Γ Q \ W ν × N > / ∼ . Note that the Honda-Tate theory of shtukas is also well studied in the PhD thesisof F. R¨oting [R¨ot]. In the present article we give the proof for the fact that the mapΣ → Γ Q \ W ν × N > / ∼ is injective. This proof also appears in [R¨ot], while the main achieve-ment in [R¨ot] is the fact that this map is also surjective. We mention this in the proof ofthe above theorem.We also discuss the zeta-functions associated with C -motives and G -shtukas in section 6.The Semi-simplicity of the category of C -motives over F q is proved in Theorem 7.1. Thistheorem is similar to the semisimplicity result of Jannsen [Jan] and states the following: Theorem 1.5.
The category M ot νC ( F q ) with the fiber functor ω is a semi-simple tannakiancategory. In particular the kernel group P := P ∆ of the corresponding motivic groupoid P := Aut ⊗ (cid:0) ω |M ot νC ( F q ) (cid:1) is a pro-reductive group over Q F q . Notice that this theorem only holds for L = F q and not for a finite field L . This is aremarkable difference to the number fields case. The reason behind this comes from thefollowing elementary observation. Namely, unlike the characteristic zero case, a non semi-simple matrix may become semi-simple after raising to a relevant power. Apart from thisdifference, note in addition that the category of C -motives consists of “mixed” motives, in thesense that the eigenvalue of Frobenius endomorphism operating on the cohomology groupsmight have different absolute values; compare also [Gos][Theorem 5.6.10]. In this sense, thisresult maybe also viewed as a (partial) analog for the semi-simplicity of the subcategory of ONTENTS K -groups over finite fieldsaccording to Quillen [Qui]. Acknowledgments
The authors would like to thank the unknown referee for careful reading and many usefulcomments and explanations which led to a considerable improvement of the exposition andsimplified several proofs.The first author is grateful to Chia-Fu Yu for his continues encouragements and limitlesssupport. Moreover, he thanks Giuseppe Ancona, Arash Rastegar and Mehrdad Karimi forinspiring conversations and support.The authors acknowledge support of the DFG (German Research Foundation) in form ofSFB 878 and Germany’s Excellence Strategy EXC 2044–390685587 “Mathematics M¨unster:Dynamics–Geometry–Structure”. The first author also acknowledges the support of NCTS(National Center For Theoretical Sciences) and IPM (Institute for Research in FundamentalSciences).
Contents C -Motives . . . . . . . . . . . . . . . . . . . 152.1.3 ´Etale realization of C -Motives and Tate Conjecture . . . . . . . . . . 16 G -motives and functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 G -Shtukas and functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Quasi-Isogenies and Galois Stable Sublattices . . . . . . . . . . . . . . . . . 28 ONTENTS Throughout this article we denote by F q a finite field with q elements and characteristic p , F q an algebraic closure of F q , L a ring containing F q , F a finite field containing F q , C a smooth projective geometrically irreducible curve over F q , η the generic point of C , ν a closed point of C , also called a place of C , ν = ( ν , . . . , ν n ) a tuple of n pairwise different places ν i on C ,˙ C := C r ν the open subscheme of C , Q := κ ( η ) = F q ( C ) the function field of C , F ν := κ ( ν ) the residue field at the place ν on C , A := Γ( ˙ C, O C ) the ring of regular functions outside ν , A L the ring A ⊗ F q L , Q L the ring of fractions Frac( A L ), A ν the completion of the stalk O C,ν at the place ν , Q ν := Frac( A ν ) its fraction field,ˆ A := F [[ z ]] the ring of formal power series in z with coefficients in F , b Q := Frac( ˆ A ) its fraction field, A ν the ring of integral adeles of C outside ν , A νQ := A ν ⊗ O C Q the ring of adeles of C outside ν , D R := Spec R [[ z ]] the spectrum of the ring of formal power series in z with coeffi-cients in an F -algebra R , ONTENTS D R := Spf R [[ z ]] the formal spectrum of R [[ z ]] with respect to the z -adic topology.For a formal scheme b S we denote by N ilp b S the category of schemes over b S on which anideal of definition of b S is locally nilpotent. We equip N ilp b S with the ´etale topology. We alsodenote by A ν the completion of the local ring O C n ,ν of C n at the closed point ν , N ilp A ν := N ilp Spf A ν the category of schemes over C n on which the ideal defining the closedpoint ν ∈ C n is locally nilpotent, N ilp F [[ ζ ]] := N ilp ˆ D the category of D -schemes S for which the image of z in O S is locallynilpotent. We denote the image of z by ζ since we need to distinguishit from z ∈ O D . G a flat affine group scheme of finite type over C , P ν := G × C Spec A ν , the base change of G to Spec A ν , P ν := G × C Spec Q ν , the generic fiber of P ν over Spec Q ν , P a flat affine group scheme of finite type over D = Spec F [[ z ]], P the generic fiber of P over Spec F (( z )).Let S be an F q -scheme and consider an n -tuple s := ( s i ) i ∈ C n ( S ). We denote by Γ s theunion S i Γ s i of the graphs Γ s i ⊆ C S := C × F q S .We denote by σ S : S → S the F q -Frobenius endomorphism which acts as the identity on thepoints of S and as the q -power map on the structure sheaf. We set C S := C × Spec F q S , and σ := id C × σ S . Ch ∼ ,d ( − , Q ) the group of cycles of dimension d modulo the equivalence relation ∼ ,with coefficients in Q . Here ∼ stands for an adequate equivalence relation,e.g. rational, algebraic, homological and numerical equivalence relations.Let H be a sheaf of groups (for the ´etale topology) on a scheme X . In this article a ( right ) H -torsor (also called an H -bundle ) on X is a sheaf G for the fppf topology on X togetherwith a (right) action of the sheaf H such that G is isomorphic to H on a fppf covering of X .Here H is viewed as an H -torsor by right multiplication. CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS Definition 1.6. (a) Let R be a ring and X ⊆ R be a subset. We denote by C R ( X ) the centralizer of X in R , i.e. C R ( X ) := { a ∈ R ; a · x = x · a ∀ x ∈ X } . The subring C R ( C R ( X )) is called bicommutant of X in R .(b) Let M be an R -module. Set R M := im ( R → End( M, +) , a ( a : m a.m ))(c) We denote by Z ( R ) the centralizer of R in R , i.e. Z ( R ) := C R ( R ).(d) We denote by Bicom R ( M ) := C End( M, +) (End R ( M )) Remark 1.7. (a) [Bou58, Chapitre 8, §
1, n o
2, Proposition 3]. Let K be a field and R and R ′ be two K -algebras. Let S ⊆ R and S ′ ⊆ R ′ be sub-K-algebras. Then we have C R ⊗ K R ′ ( S ⊗ K S ′ ) = C R ( S ) ⊗ K C R ′ ( S ′ )in particular Z ( R ⊗ K R ′ ) = Z ( R ) ⊗ K Z ( R ′ ).(b) [Bou58, Chapitre 8, §
4, n o
2, Corollaire 1]. Let M be a semi-simple R -module whichis finitely generated as an End R ( M )-module. Then Bicom R ( M ) = R M . In this section we present the basic definitions of the category of C -motives.Let ν = ( ν , . . . , ν n ) be an n -tuple of closed points of C and let A := Γ( ˙ C, O C ) be thecoordinate ring of the open subscheme ˙ C := C r { ν , . . . , ν n } . Definition 2.1. (a) Let S be a scheme over F q . A C -motive M with characteristic ν over S is a tuple ( M , τ M ) consisting of(i) a locally free sheaf M of O C S -modules of finite rank,(ii) an isomorphism τ M : σ ∗ ˙ M → ˙ M where ˙ M denotes the pullback of M under theinclusion ˙ C S → C S , and σ = id × σ S where σ S : S → S is the absolute Frobeniusmorphism over F q .(b) A morphism M → N is a homomorphism f : M → N of sheaves on C S which fitsinto the following commutative diagram CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS σ ∗ ˙ M τ M | ˙ CS −−−−→ ˙ M σ ∗ f y y f σ ∗ ˙ N τ N | ˙ CS −−−→ ˙ N . The set of morphisms is denoted Hom S ( M , N ).The set of quasi-morphisms QHom( M , N ) consists of the equivalence classes of thecommutative diagrams σ ∗ ˙ M τ M −−−→ ˙ M σ ∗ f y y f σ ∗ ˙ N ⊗ O ( D S ) τ N −−−→ ˙ N ⊗ O ( D S ) , where D is a divisor on C and D S := D × F q S , and two such diagrams for divisors D and D ′ are called equivalent provided that the corresponding diagrams agree when wetensor with O ( D S + D ′ S ).Note that when S = Spec L , for a field L , one can equivalently say that the set ofquasi-morphisms QHom( M , N ) is given by the following commutative diagrams σ ∗ M η τ M ,η −−−→ M ησ ∗ f y y f σ ∗ N η τ N ,η −−−→ N η . Here M η denotes the pull back of M under η × F q S → C S .(c) A quasi-isogeny between M and N is a morphism in QHom( M , N ) which admits aninverse. See also [Har19, Theorem 5.12] and Proposition 2.5 below.(d) We denote by M ot νC ( S ) the Q -linear category whose objects are C -motives of char-acteristic ν as above, with quasi-morphisms as its morphisms. We further denote by M ot νC ( S ) ◦ the category obtained by restricting the set of morphisms to quasi-isogenies.When S = Spec L we simply use the notation M ot νC ( L ). CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS Remark 2.2.
Let F ν be the residue field of the point ν ∈ C n . If L is a field extension of F ν ,then we can modify the above definition by replacing (a)(ii) as follows. Let s i : Spec L → C be composition of the natural morphism Spec L → Spec F ν ֒ → C n followed by the projectiononto the i -th component. It factors as s i : Spec L → Spec F ν i ֒ → C . We may replace (a)(ii)by (ii)’ an isomorphism τ M : σ ∗ M| C L r ∪ i Γ si → M| C L r ∪ i Γ si outside the graphs Γ s i of themorphisms s i : Spec L → C , where σ = id × σ L and σ L : L → L is the absolute Frobeniusmorphism over F q .We always have ˙ C L ⊆ C L r ∪ i Γ s i but this inclusion is strict when F ν = F q . So the differ-ence between conditions (ii) and (ii)’ is that in (ii) it is required that τ M is an isomorphismon the larger set C L r ∪ i Γ s i . Condition (ii)’ is more natural and better compatible with thetheory of global G -shtukas, see Section 4 below. Note that one can choose an embedding F ν → L , when L is large enough, then the obvious functor from the category of C -motivesover L defined with ( ii ) ′ to the category of C -motives over L defined with ( ii ) is fully faithfulwith morphisms given by QHom( M , N ). Proposition 2.3.
The obvious functor M ot νC ( L ) ′ → M ot νC ( L ) from the category M ot νC ( L ) ′ of C -motives over L defined with ( ii ) ′ to the category of C -motives over L defined with ( ii ) is fully faithful. With morphisms given by QHom L ( M , N ) it is an equivalence of categories,but with morphisms given by Hom L ( M , N ) it is not.Proof. Full faithfulness follows from the definitions. Now let
M ∈ M ot νC ( L ). We are goingto define a M ′ ∈ M ot νC ( L ) ′ which is isomorphic to M in M ot νC ( L ). For this purpose fix apoint ν ∈ ν and let A ν be the completion of the local ring O C,ν and let Q ν be the fractionfield of A ν . Since the point ν splits in C L into deg( ν )-many L -valued points, the completedtensor products A ν b ⊗ F q L = Y i ∈ Z / (deg ν ) ( A ν b ⊗ F q L ) / a i and Q ν b ⊗ F q L = Y i ∈ Z / (deg ν ) ( Q ν b ⊗ F q L ) / a i split correspondingly, where a i = h α ⊗ − ⊗ α q i : α ∈ F ν i . The map σ sends a i to a i +1 ,and hence τ M yields isomorphisms σ ∗ (cid:0) ˙ M ⊗ O ˙ CL ( Q ν b ⊗ F q L ) / a i (cid:1) = ( σ ∗ ˙ M ) ⊗ O ˙ CL ( Q ν b ⊗ F q L ) / a i +1 ∼ −→ ˙ M ⊗ O ˙ CL ( Q ν b ⊗ F q L ) / a i +1 for all i ∈ Z / (deg ν ). Since ˙ C L together with ` ν ∈ ν Spec A ν b ⊗ F q L is an fpqc-covering of C L we may construct M ′ from M by modifying it in the components M ⊗ O CL ( A ν b ⊗ F q L ) / a i for CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS i = 0 at all ν ∈ ν according to M ′ ⊗ O CL ( A ν b ⊗ F q L ) / a i := τ i M (cid:16) σ i ∗ (cid:0) M ⊗ O CL ( A ν b ⊗ F q L ) / a (cid:1)(cid:17) ⊂ ˙ M ⊗ O ˙ CL ( Q ν b ⊗ F q L ) / a i . Then by construction M ′ := ( M ′ , τ M ′ ) with τ M ′ := τ M satisfies τ M ′ : ( σ ∗ ˙ M ) ⊗ O ˙ CL ( Q ν b ⊗ F q L ) / a i ∼ −→ ˙ M ⊗ O ˙ CL ( Q ν b ⊗ F q L ) / a i for all i = 0, and so M ′ belongs to M ot νC ( L ) ′ . Since the restrictions of M and M ′ to ˙ C L coincide, we have M ∼ = M ′ in M ot νC ( L ) as desired. Proposition 2.4.
Let L be a field over F q . The category M ot νC ( L ) is a Q -linear rigid abeliantensor category. It further admits a fiber functor over the function field Q L of the curve C L .Proof. Let f : M → N ( D ) be a representative for a morphism in QHom( M , N ). Thenker f := (ker f, τ M | σ ∗ ker f ) and im f := (im f, τ N | σ ∗ im f ). Consider the cokernel F := coker f : M → N ( D ) as a coherent sheaf of O C L -modules. The torsion subsheaf T has finite supportand F / T is a torsion free sheaf. The morphism τ N induces a morphism τ F : σ ∗ F → F .We define coker f := ( F / T , τ F / T ) and coim f := ker( N → coker f ). Moreover this is clearthat the natural morphism im f → coim f is a quasi-isogeny and therefore an isomorphismin M ot νC ( L ).The tensor product of two C -motives M := ( M , τ M ) and N := ( N , τ N ) is the C -motive M ⊗ N consisting of the locally free sheaf of O C L -modules M ⊗ O CL N and the isomorphism τ M⊗N := τ M ⊗ τ N . The unit object for the tensor product is := ( O C L , id), and preciselywe have QEnd( ) = Q . One can easily see that this category has an internal Hom object.Namely we define H := Hom( M , N ) to be the object with H := Hom O CL ( M , N ) as theunderlying locally free sheaf and τ H is given by sending h ∈ H to τ N ◦ h ◦ τ − M . This furtherdefines the functor ˇ( − ) := Hom( − , ) : M ot νC ( L ) → M ot νC ( L ) , which sends M to its dual ˇ M . Finally sending a C -motive M := ( M , τ M ) to the genericfiber M η of the underlying locally free sheaf M equips the category with a fiber functor ω ( − ) : M ot νC ( L ) → Q L − Vector spaces . Proposition 2.5.
Let M and N be C -motives, over a field L . Assume that L is finite over F q of degree s . The following are equivalent CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS (a) f ∈ Hom( M , N ) is a quasi-isogeny.(b) There is a non-zero element a ∈ A and ˇ f ∈ Hom( N , M ) with ˇ f ◦ f = a. id M and f ◦ ˇ f = a. id N .Proof. The proof is similar to [BH11, Corollary 5.4] and [Tae09, Proposition 3.1.2]. See also[Har19, Corollary 5.15].
In analogy with the theory of motives, the category of C -motives also admits crystalline and´etale realizations. In this section we recall the definition of the corresponding realizationcategories and functors. We further prove the analog of the Tate conjecture for the category C -motives over a finite field. The results are similar to [BH11, §§
8, 9].
Fix a place ν ∈ C and a uniformizer z := z ν ∈ A ν = ˆ O C,ν . It yields canonical isomorphisms A ν = F ν [[ z ]] and Q ν = F ν (( z )). Let L/ F ν be a field extension. Let ˆ σ ν be the endomorphismof L [[ z ]] = A ν b ⊗ F ν L which is the F ν -Frobenius b b F ν = b q deg ν on L and fixes z . Also let L (( z )) = Q ν b ⊗ F ν L denote the fraction field of L [[ z ]]. Definition 2.6.
Keep the above notation.(a) A ˆ σ ν - crystal ˆ M (resp. ˆ σ ν - iso-crystal ) of rank r over L is a tuple ( ˆ M , ˆ τ ) (resp. ( ˙ˆ M, ˆ τ )) consisting of the following data(i) a free L [[ z ]]-modulue ˆ M (resp. an L (( z ))-vector space ˙ˆ M ) of rank r ,(ii) an isomorphism ˆ τ : ˆ σ ∗ ν ˆ M [1 /z ] → ˆ M [1 /z ] (resp. ˆ τ : ˆ σ ∗ ν ˙ˆ M → ˙ˆ M ), where ˆ M [1 /z ] :=ˆ M ⊗ L [[ z ]] L (( z )).A ˆ σ ν -crystal is ´etale if the isomorphism ˆ τ : ˆ σ ∗ ν ˆ M [1 /z ] → ˆ M [1 /z ] comes from an iso-morphism ˆ τ : ˆ σ ∗ ν ˆ M → ˆ M .(b) A quasi-morphism (resp. morphism ) between ˆ σ ν -crystals ˆ M := ( ˆ M , ˆ τ ) and ˆ M ′ :=( ˆ M ′ , ˆ τ ′ ) is a morphism f : ˆ M [1 /z ] → ˆ M ′ [1 /z ] (resp. f : ˆ M → ˆ M ′ ) such that f ◦ ˆ τ = CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS τ ′ ◦ ˆ σ ∗ ν f . We denote by ˆ σ ν - CrystIso ( L ) (resp. ˆ σ ν - Cryst ( L )) the Q ν -linear (resp. A ν -linear) category of ˆ σ ν -crystals together with quasi-morphisms (resp. morphisms) asits morphisms. We denote by ´Et ˆ σ -CrystIso ( L ) (resp. ´Et ˆ σ -Cryst ( L )) the full Q ν -linear (resp. A ν -linear) subcategory of ˆ σ - CrystIso ( L ) (resp. ˆ σ - Cryst ( L )) consistingof ´etale ˆ σ -crystals.(c) A quasi-morphism between ˆ σ ν -iso-crystals ( ˙ˆ M, ˆ τ ) and ( ˙ˆ M ′ , ˆ τ ′ ) is a morphism f : ˙ˆ M → ˙ˆ M ′ such that f ◦ ˆ τ = ˆ τ ′ ◦ ˆ σ ∗ ν f . We denote by ˆ σ ν - IsoCryst ( L ) the Q ν -linear categoryof ˆ σ ν -iso-crystals together with quasi-morphisms as its morphisms. Definition 2.7.
The first ´etale cohomology realization of an ´etale ˆ σ ν -crystal ˆ M is the Γ L := Gal ( L sep /L )-module of ˆ τ -invariantsH ( ˆ M , A ν ) := (cid:16) ˆ M ⊗ L [[ z ]] L sep [[ z ]] (cid:17) ˆ τ := (cid:8) m ∈ ˆ M ⊗ L [[ z ]] L sep [[ z ]] : m = ˆ τ (ˆ σ ∗ ν m ) (cid:9) . We set H i ´et ( ˆ M , A ν ) := ∧ i H ( ˆ M , A ν ) and H i ´et ( ˆ M , Q ν ) := H i ´et ( ˆ M , A ν ) ⊗ A ν Q ν . We recall the following crucial result.
Theorem 2.8.
We have the following statements for an ´etale ˆ σ ν -crystal ˆ M .(a) H ´et ( ˆ M , A ν ) is a free A ν -module of rank equal to rk ˆ M and the following natural mor-phism H ´et ( ˆ M , A ν ) ⊗ A ν L sep [[ z ]] → ˆ M ⊗ L [[ z ]] L sep [[ z ]] is a Γ L × ˆ τ -equivariant isomorphism of L sep [[ z ]] -modules, where on the left hand side Γ L -acts diagonally and ˆ τ acts as id ⊗ ˆ σ ∗ ν and on the right hand side Γ L only acts on L sep [[ z ]] and ˆ τ acts as m ⊗ f ˆ τ (ˆ σ ∗ ν m ) ⊗ ˆ σ ∗ ν ( f ) . In particular H ´et ( ˆ M , A ν ) is a free A ν -module of rank rk ˆ M .(b) The first ´etale cohomology functor H ´et ( − , A ν ) : ´Et ˆ σ ν -Cryst ( L ) → A ν [Γ L ] -modulesis a fully faithful embedding of the category of ´Et ˆ σ ν -Cryst ( L ) into the category of A ν [Γ L ] -modules. One can recover ˆ M from its ´etale realization by taking Galois invari-ants ˆ M := (cid:16) H ´et ( ˆ M , A ν ) ⊗ A ν L sep [[ z ]] (cid:17) Γ L . CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS Proof.
See [T-W, Proposition 6.1] and [Har11, Proposition 1.3.7 and A.4.5]. Compare alsoto [BH11, Proposition 8.4].
Proposition 2.9.
The first ´etale cohomology functor H ´et ( − , A ν ) : ´Et ˆ σ ν -Cryst ( L ) → A ν [Γ L ] − modulesis exact.Proof. Given a short exact sequence0 → ˆ M ′′ → ˆ M → ˆ M ′ → ´Et ˆ σ ν -Cryst (L), we obtain the following exact sequence0 → H ( ˆ M ′′ , A ν ) ⊗ A ν L sep [[ z ]] → H ( ˆ M , A ν ) ⊗ A ν L sep [[ z ]] → H ( ˆ M ′ , A ν ) ⊗ A ν L sep [[ z ]] → , according to Theorem 2.8 a) and faithfully flatness of L [[ z ]] → L sep [[ z ]]. By faithfully flatnessof A ν → A ν,L sep := L sep [[ z ]], the above short exact sequence descends and yields the followingshort exact sequence0 → H ( ˆ M ′′ , A ν ) → H ( ˆ M , A ν ) → H ( ˆ M ′ , A ν ) → . C -Motives Let ν ∈ C be a place, let L/ F q be a field extension, and consider the ring A ν,L := A ν b ⊗ F q L =( F ν ⊗ F q L )[[ z ]]. Let ℓ = { α ∈ L ; α F ν = α } be the intersection of F ν and L , i.e. thoseelements α ∈ L such that α q deg ν = α . Let s denote the degree of the field extension ℓ/ F q .Then ℓ = q s . The scheme Spec( A ν,L ) is the union of its connected components. Wewrite Spec A ν,L = ∐ i ∈ Z /s V ( a ˜ ν i ). The connected component V ( a ˜ ν i ) := Spec( A ν,L ) / a ˜ ν i =Spec A ν b ⊗ ℓ L corresponds to the ideal a ˜ ν i = ( a ⊗ − ⊗ a q i : a ∈ ℓ ) and σ cyclically permutesthese components and σ s leaves each of the components V ( a ˜ ν i ) stable. Thus the set ofconnected components is endowed with a free Z /s := Gal( ℓ/ F q )-action. Here ˜ ν i stands forthe places of C ℓ lying above ν ∈ C . Note that when L contains F ν , then ℓ = F ν and s = deg ν . CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS Definition 2.10.
In the above situation define the category ´Et ( σ, ν ) -CrystIso ( L ) as thecategory whose objects consist of tuples ( ˆ M , ˆ τ ), where ˆ M is a locally free A ν,L -module and τ : σ ∗ ˆ M → ˆ M is an isomorphism. One defines quasi-morphisms in a similar way as indefinition 2.6 and considers them as the set of morphisms of this category. Similarly onemay define the categories ´Et ( σ, ν ) -Cryst ( L ) and ( σ, ν ) -IsoCryst ( L ). Proposition 2.11.
Let L/ F ν be a field extension, and let ˜ ν i be as above. The reductionmodulo a ˜ ν i induces equivalences of categories, which are independent of i ∈ Z / deg νRed : ´Et ( σ, ν ) -Cryst ( L ) ˜ −→ ´Et ˆ σ ν -Cryst ( L ) Red : ´Et ( σ, ν ) -CrystIso ( L ) ˜ −→ ´Et ˆ σ ν -CrystIso ( L ) Red : ( σ, ν ) -IsoCryst ( L ) ˜ −→ ˆ σ ν -IsoCryst ( L )ˆ M := ( ˆ M , ˆ τ ) Red ˆ M := ( ˆ M / a ˜ ν i , ˆ τ deg ν ) Proof.
This is [BH11, Proposition 8.5].
Definition 2.12.
Let ν ∈ C be a place and let L/ F ν be a field extension. For M := ( M , τ M )in M ot νC ( L ), let ˆ M denote the pull back M ⊗ O CL A ν,L . Sending M to ( ˆ M , τ M ⊗
1) andfurther to ( ˆ
M / a ˜ ν i , ( τ M ⊗ deg ν ) defines a functor which is independent of the choice of i ∈ Z / deg ν Γ ν ( − ) : M ot νC ( L ) −→ ( σ, ν ) -IsoCryst ( L ) −→ ˆ σ ν -IsoCryst ( L ) . We call this the crystalline realization functor at ν . If ν does not lie in the characteristicplaces ν the functor produces ´etale crystalsΓ ν ( − ) : M ot νC ( L ) −→ ´Et ( σ, ν ) -Cryst ( L ) −→ ´Et ˆ σ ν -Cryst ( L ) . Remark 2.13.
The above construction at ν ∈ ν is apparently more canonical when wework with the alternative definition of M ot νC ( L ), explained in Remark 2.2. As we will see inSection 4, this is in fact the case when we work with ( G -) C -motives arising from ( G -)shtukas.This is because there is a preferred canonical section Spec L → Spec F ν → C n given by thecharacteristic sections (also called legs) of the ( G -)shtuka. C -Motives and Tate Conjecture When ν does not lie in the characteristic places ν , assigning the Γ L -module H i ´et ( ˆ M , A ν ) tothe ´etale crystal ˆ M := Γ ν ( M ), defines a functor CATEGORY OF C-MOTIVES AND REALIZATION FUNCTORS i ´et ( − , A ν ) : M ot νC ( L ) −→ A ν [Γ L ] -modules , M 7−→ H i ´et (Γ ν ( M ) , A ν ) . Tensoring up with Q ν we similarly defineH i ´et ( − , Q ν ) : M ot νC ( L ) −→ Q ν [Γ L ] -modules , M 7−→ H i ´et (Γ ν ( M ) , A ν ) ⊗ A ν Q ν . We call the above functor the i’th ´etale realization functor with coefficients in A ν (resp. Q ν ). We also use the notation ω ν ( − ) (resp. ω νQ ν ( − )) for the first cohomology functorH ( − , A ν ) (resp. H ( − , Q ν )). Proposition 2.14.
The functor ω νQ ν ( − ) is exact.Proof. This follows from exactness of Γ ν ( − ) and Proposition 2.9. Proposition 2.15.
Let M and M ′ be in M ot νC ( L ) , and let ν be a place on C different fromall the characteristic places ν i . Then the obvious morphism QHom( M , M ′ ) ⊗ Q Q ν −→ Hom Q ν [Γ L ]( ω νQν ( M ) ,ω νQν ( M ′ )) is injective. In particular dim Q QHom( M , M ′ ) ≤ rk M · rk M ′ .Proof. Clearly we have an embedding of QHom( M , M ′ ) ⊗ Q Q ν in Hom Q L ( M η , M ′ η ) ⊗ Q Q ν .The latter sits inside Hom Q ν,L (Γ ν ( M ) , Γ ν ( M ′ )) and is compatible with respect to τ and τ ′ ,as well as Γ ν ( τ ) and Γ ν ( τ ′ ). Here Q ν,L denotes the total ring of fractions of A ν,L . Thereforewe have an embeddingQHom( M , M ′ ) ⊗ Q Q ν ֒ → Hom
Cris (Γ ν ( M ) , Γ ν ( M ′ )) ⊗ A ν Q ν The latter equals Hom Q ν (Γ L ) ( ω νQ ν ( M ) , ω νQ ν ( M ′ )), see Theorem 2.8.The following Theorem can be regarded as an analog for Tate conjecture. Theorem 2.16.
Let M and M ′ be C -motives over a finite field L . Let ν be a closed pointof C away from the characteristic places ν i . Then there are isomorphisms Hom L ( M , M ′ ) ⊗ A A ν ˜ −→ Hom A ν [Γ L ] ( ω ν ( M ) , ω ν ( M ′ )) (resp. QHom L ( M , M ′ ) ⊗ Q Q ν ˜ −→ Hom Q ν [Γ L ] ( ω νQ ν ( M ) , ω νQ ν ( M ′ ))) of A ν -modules (resp. Q ν -vector spaces). Moreover if M = M ′ then the above are ringisomorphisms. THE ENDOMORPHISM RING OVER FINITE FIELDS Proof.
Without loss of generality we may assume that F ν = F q . For this first observe thatwe can replace F q by ℓ (see subsection 2.1.2), in order to assume that the coefficient ring( F ν ⊗ ℓ L )[[ z ]] of local shtukas is the power series ring (and not the product of power seriesrings); see Proposition 2.11. Then one can proceed by replacing F ν ⊗ ℓ L by L . Now we claimthat there is an isomorphismHom L ( M , M ′ ) ⊗ A A ν ˜ −→ Hom L (Γ ν ( M ) , Γ ν ( M ′ ))To see this, consider the following exact sequence0 / / Hom L ( M , M ′ ) / / Hom A L ( M , M ′ ) / / Hom A ( M , M ′ ) f ✤ / / f ◦ τ M − τ M ′ ◦ f. Let us set ˆ M := Γ ν ( M ) and ˆ M ′ := Γ ν ( M ′ ). As A → A ν is flat, we get the following exactsequence0 / / Hom L ( M , M ′ ) ⊗ A A ν / / Hom A ν,L ( ˆ M , ˆ M ′ ) / / Hom A ν ( ˆ M , ˆ M ′ ) f ✤ / / f ◦ ˆ τ ˆ M − ˆ τ ˆ M ′ ◦ f, and thus Hom L ( M , M ′ ) ⊗ A A ν ∼ = Hom L ( ˆ M , ˆ M ′ ). Note that we here use the fact that A L ⊗ A A ν = A ν,L . This is only true when L is a finite field, because A ν,L is ν -adicallycomplete and the left hand side is not in general. It remains to show thatHom L ( ˆ M , ˆ M ′ ) = Hom A v [Γ L ] ( ω ν ( M ) , ω ν ( M ′ )) . But this follows from Theorem 2.8 b).
Throughout this section we assume that the field L is a finite field extension of F q . Definition 3.1.
Let L/ F q be a field extension of degree e and let M := ( M , τ ) be a C -motivein M ot νC ( L ). The Frobenius quasi-isogeny π := π M ∈ QEnd( M ) is given by π := τ ◦ ( σ ∗ ) τ · · · ◦ ( σ ∗ ) e − τ : M = ( σ ∗ ) e M → M
THE ENDOMORPHISM RING OVER FINITE FIELDS Proposition 3.2.
Let M := ( M , τ ) be a C -motive over a finite field L = F q e and let ν bea place on C distinct from characteristic places ν i . Then(a) The action of the topological generator Frob L : x x L of Γ L on ω νQ ν ( M ) equals theaction of π − ν . Here π ν denotes ω νQ ν ( π ) .(b) The image of the continuous morphism A ν [Γ L ] → End A ν ( ω νQ ν ( M )) equals A ν [ π ν ] .Proof. a) Let ( ˆ M , ˆ τ ) be the crystal associated with M under Γ ν ( − ). Recall that ˆ M can berecovered from it’s ´etale realization H ( ˆ M , A ν ):ˆ M ∼ = (cid:16) H ( ˆ M , A v ) ⊗ A v L sep [[ z ]] (cid:17) Γ L , see Theorem 2.8, and ˆ τ can be recovered as 1 ⊗ ˆ σ . Since ϕ L = ˆ σ e and Γ L acts diagonally onH ( ˆ M , A ν ), we have ˆ σ e ⊗ ˆ σ e = id on ˆ M and thus we see that π ν = ϕ − L .b) Follows formally from the continuity of the action of Γ on ω ν ( M ). Namely, sincethe target of Γ → End A ν ( ω ν ( M )) is finitely generated over A ν , the A ν -submodule A ν [ π ν ]is closed. Since ϕ − L generates a dense subgroup of Γ L , the image of Γ L is contained in theclosed submodule A ν [ π ν ].Fix a C -motive M over L and a place ν different from characteristic places ν i . Consider F := Q [ π ] ⊆ E := QEnd( M ). Note that F is finitely generated as a Q -module by Propo-sition 2.15 and write F := Q [ x ] /µ π , where µ π is the minimal polynomial of π over Q . Notefurther that according to Theorem 2.16 we have E ⊗ Q Q ν ∼ = E ν := End Q ν [Γ L ] ( ω νQ ν ( M )) andobserve that under this isomorphism F ⊗ Q Q ν gets identified with image F ν of Q ν [Γ L ] in E ν .Let χ ν denote the characteristic polynomial corresponding to π ν := π ⊗ E ν . Remark 3.3.
Let M and M ′ be in M ot νC ( L ) and let V ν := ω νQ ν ( M ) and V ′ ν := ω νQ ν ( M ′ )at a place ν ∈ C , different from characteristic places ν i . Furthermore, assume that thecorresponding Frobenius endomprphisms π ν ∈ End Q ν [Γ L ] ( V ν ) and π ′ ν ∈ End Q ν [Γ L ] ( V ′ ν ) aresemisimple. Consider the decomposition χ ν = Q µ µ m µ (resp. χ ′ ν = Q µ µ m ′ µ ) of the charac-teristic polynomial χ ν (resp. χ ′ ν ) to its irreducible factors and set K µ := Q ν [ x ] /µ . There aredecompositions V ν ∼ = L µ ( K µ ) m µ and V ′ ν ∼ = L µ ( K µ ) m ′ µ , and thereforeHom Q ν [Γ L ] ( V ν , V ′ ν ) ∼ = M i M at m ′ µ × m µ ( K µ ) . THE ENDOMORPHISM RING OVER FINITE FIELDS Definition 3.4. A C -motive M 6 = 0 in M ot νC ( L ) is called simple , if it has no non-trivialquotient in M ot νC ( L ). Furthermore M is called semi-simple , if it admits a decompositioninto a direct sum of simple ones.The following theorem generalizes Proposition 6.8 and Theorem 6.11 of [BH09] and asin [Jan], is a key observation for the proof of the semi-simplicity of the category M ot νC ( F q ).The proof given bellow is a slight modification of the proof given in [BH09]. Theorem 3.5.
Consider the following diagram M ot νC ( L ) ◦ End Qν [Γ L ] ( ω νQν ( − )) / / QEnd( − ) / / Q -Algebras −⊗ Q ν (cid:15) (cid:15) Q ν -Algebras . The following statements are equivalent(a)
M ∈ M ot νC ( L ) is semi-simple.(b) E := QEnd( M ) is semi-simple.(c) End Q ν [Γ L ] ( ω νQ ν ( M )) is semisimple.(d) F ν := Q ν [ π ν ] is semisimple.(e) F is semi-simple.Proof. (a) ⇒ (b) follows from the fact that for a simple C -motive M the endomorphismalgebra E = QEnd( M ) is a division algebra. (b) ⇒ (c) follows from [Bou58, Corollaire7.6/4] and the fact that Q ν /Q is separable. (c) ⇒ (d) follows from [Bou58, Corollaire deProposition 6.4/9]. As Q ν /Q is separable and F ν = F ⊗ Q ν , one can argue that ( d ) and ( e ) areequivalent by [Bou58, Corollaire 7.6/4]. Also if π ν is semi-simple then End Q ν [Γ L ] ( ω νQ ν ( M )) = L µ M at m µ × m µ ( K µ ), see Remark 3.3. Thus ( d ) implies ( c ). Again we argue that ( c ) implies( b ) by [Bou58, Corollaire 7.6/4].It remains to justify ( b ) ⇒ ( a ). First observe that we can reduce to the this statement thatif QEnd( M ) is a division algebra, then M is simple. To see this, suppose QEnd( M ) = ⊕ mj =1 M at r j × r j ( E j ) be the decomposition to the matrix algebras over division algebras E j THE ENDOMORPHISM RING OVER FINITE FIELDS Q . Let { e j,i j } ≤ i j ≤ r j be the corresponding set of idempotents with P e j,i j = id r j ∈ M at r j × r j ( E j ) and e j,i j QEnd( M ) e j,i j = E j . Now consider the quasi-isogeny P i,j e j,i j : M →⊕ i,j M i j , where M i j := im e j,i j and observe that QEnd( M i j ) = e j,i j QEnd( M ) e j,i j = E j .Now assume that E := QEnd( M ) is a division algebra. We show that M has no non-trivialquotient. Let M ′ be a quotient of M under f : M → M ′ . Since E is a division algebra, it isenough to show that there is an element g ∈ QHom( M ′ , M ) such that g ◦ f = 0. Accordingto Proposition 2.14, the realization functor ω νQ ν ( − ) is exact, and thus by applying ω νQ ν ( − )we get a surjection ω νQ ν ( f ) : ω νQ ν ( M ) ։ ω νQ ν ( M ′ ) . Note that as we have seen above F ν is semi-simple and thus ω νQ ν ( M ) is a finite semi-simple F ν -algebra. This implies that there exist f ′ ν ∈ Hom Q ν [Γ L ] ( ω νQ ν ( M ′ ) , ω νQ ν ( M ))such that f ν ◦ f ′ ν = id ω νQν ( M ′ ) . For sufficiently large integer n we have h ν := z nν · f ′ ν ∈ Hom( ω ν ( M ′ ) , ω ν ( M )) ∼ = Hom L ( M ′ , M ) ⊗ A A ν . Here z ν is a uniformizer of A ν . Now take g ∈ Hom L ( M ′ , M ) such that g ≡ h ν ( mod ν m )for sufficiently large m > n . We claim that g ◦ f ∈ E × . Namely, if g ◦ f = 0 then since f issurjective we also have f ◦ g = 0 in QEnd( M ′ ). This implies that z nν · id ω νQν ( M ) = z nν · f ν ◦ f ′ ν = f ν ◦ h ν ≡ f ◦ g = 0 ( mod ν m ) , which is a contradiction. Hence we argue that f is also injective.Let K be a field and f, g ∈ K [ x ]. Consider the factorizations f = Q µ n µ and g = Q µ m µ ,where µ runs over irreducible polynomials in K [ x ]. Set r K ( f, g ) := Q µ m µ .n µ . deg µ . Proposition 3.6.
Let M be a C -motive over a finite field L = F q e with semi-simple Frobe-nius π . Then F = Q [ π ] is the center Z ( E ) of the semi-simple Q -algebra E = QEnd ℓ ( M ) .Proof. Since F ν is semisimple, the F ν -module ω ν ( M ) is semisimple; see Theorem 3.5. As ω ν ( M ) is finitely generated module over E ν which itself is finite dimensional over Q ν . There-fore we have Bicom F ν ( ω ν ( M )) = F ν ; see Remark 1.7(b). Hence Z ( E ν ) = E ν ∩ F ν = F ν and F ⊗ Q Q ν = F ν = Z ( E ⊗ Q Q ν ) = Z ( E ) ⊗ Q Q ν , see Remark 1.7(a). We conclude thatdim Q F = dim Q Z ( E ). Note that F ⊆ Z ( E ), since for every f ∈ E we have f ◦ τ M = τ M ◦ f and thus f ◦ π = π ◦ f . THE ENDOMORPHISM RING OVER FINITE FIELDS Proposition 3.7.
Let M and M ′ be C -Motives over L and let V ν := ω νQ ν ( M ) and V ′ ν := ω νQ ν ( M ′ ) , at a place ν ∈ C , different from characteristic places ν i . Assume further that π ν ∈ End Q ν [Γ] ( V ν ) (resp. π ′ ν ∈ End Q ν [Γ] ( V ν ′ ) ) is semisimple. Then the dimension of QHom( M , M ′ ) as a Q -vector space equals r Q ν ( χ ν , χ ′ ν ) .Proof. Consider the decomposition χ ν = Q µ µ m µ (resp. χ ′ ν = Q µ µ m ′ µ ) of the characteristicpolynomial χ ν (resp. χ ′ ν ) to the irreducible factors and set K µ := Q ν [ x ] /µ . Then decompose V ν ∼ = L µ ( K µ ) m µ and V ′ ν ∼ = L µ ( K µ ) m ′ µ . We getHom Q ν [Γ L ] ( V ν , V ′ ν ) ∼ = M i M at m ′ µ × m µ ( K µ ) . Now the assertion follows from Theorem 2.16.
Definition 3.8.
We say that M has complex multiplication if QEnd( M ) contains a com-mutative, semi-simple Q -algebra of dimension rk M . Proposition 3.9.
Let M be a C -motive of rank r over L with Frobenius endomorphism π .Set E := QEnd( M ) . Assume that F = Q [ π ] is a field and let h := [ F : Q ] = deg µ π . Then(a) h | r and dim Q E = r /h and dim F E = r /h .(b) For any place ν of Q different from characteristic places ν i we have E ⊗ Q ν ∼ = M at r/h × r/h ( F ⊗ Q Q ν ) and χ ν = ( µ π ) r/h , independent of ν .In particular if M is CM then F = E = QEnd( M ) is commutative.Proof. Since F is a field, π ν is semi-simple, see Theorem 3.5. Therefore the minimal polyno-mial µ π ν equals Q µ µ , with pairwise different monic irreducible polynomials µ ∈ Q ν [ x ].The characteristic polynomial χ ν then equals Q µ µ m µ . We have E ν ∼ = Q µ E µ , where E µ = M at m µ × m µ ( K µ ) and K µ = Q ν [ x ] /µ , see Remark 3.3. We get Q ν [ π ν ] = F ν = F ⊗ Q Q ν ։ Q ν [ x ] / ( µ ) = K µ and the surjection E ν ⊗ F ν K µ = ( E ⊗ Q Q ν ) ⊗ F ν K µ = E ⊗ F ( F ⊗ Q Q ν ) ⊗ F ν K µ = E ⊗ F K µ ։ E µ In particular m µ ≤ dim F E . Thusdim F E · [ F : Q ] = dim Q E = dim Q ν E ν = P µ m µ · deg x µ ≤ (dim F E ) · P µ deg x µ = (dim F E ) · deg x µ π ν = [ F : Q ] · dim F E THE ENDOMORPHISM RING OVER FINITE FIELDS m µ = dim F E for every µ and r = deg x χ ν = X µ m µ deg x µ = p dim F E · X µ deg x µ = p dim F E · [ F : Q ] . Therefore r = m µ · h and dim F E = r /h . Now to see part (b) write E ν ∼ = ⊕ µ M at r/h × r/h ( K µ ) ∼ = M at r/h × r/h ( ⊕ µ K µ ) ∼ = M at r/h × r/h ( F ν ).Let us state the following proposition, regarding the two extreme cases for F ⊆ E . Proposition 3.10.
Let M be a C -motive over L with semisimple Frobenius endomorphism π := π M , i.e. F = Q [ π ] is a product of fields. Let ν be a place on C apart from characteristicplaces ν i . Let χ ν denotes the characteristic polynomial of π ν := ω ν ( π ) . We have the followingstatements(a) F = Q ( π ) is the center of the semisimple Q -algebra E = QEnd L ( M ) .(b) rk M ≤ [ E : Q ] := dim Q E ≤ (rk M ) (c) The following are equivalenti) E = F ii) E is commutative,iii) [ F : Q ] = rk M iv) [ E : Q ] = rk M v) χ ν is product of pairwise different irreducible polynomials in Q ν [ x ] (d) The following are equivalenti) F = Q ii) E ∼ = M at n × n ( D ) , for a division algebra D with center Z ( D ) = Q iii) [ F : Q ] = 1 iv) [ E : Q ] = (rk M ) v) χ ν = µ rk M for a linear polynomial µ ∈ Q ν [ x ] . This is the minimal polynomial µ π . RELATION TO G-SHTUKAS Proof. (a) was proved in Proposition 3.6. For (b) let χ ν = Q µ µ m µ with µ ∈ Q ν [ x ] irreduciblepairwise different. We have the decomposition E ⊗ Q Q ν ∼ = Q µ M at m µ × m µ ( Q ν [ x ] / ( µ )) andthus [ E : Q ] = dim Q E = X µ m µ · deg x µ and X µ m µ · deg x µ = deg x χ ν = dim Q ν ω νQ ( M ) = rk M . Thereforerk M = X µ m µ · deg x µ ≤ X µ m µ deg x µ = [ E : Q ] ≤ ( X µ m µ deg x µ ) = (rk M ) . (3.1)Let us prove part c). The implications i ) ⇔ ii ) follows from (a). As we have seen above[ E : Q ] = rk M if and only if m µ = 1 for all µ . This shows that iv ) ⇔ v ) ⇒ iii )We know that µ π is the minimal polynomial of π ν := ω ν ( π ) and since divides the character-istic polynomial, we argue that µ π = χ ν . Hence iii ) ⇒ v ). Now if we have iii ) then we alsohave iv ), and as F ⊆ E we thus see that E = F . Finally i ) implies that E is a product offields therefore we may conclude that i ) ⇒ v ) by Remark 3.3.It remains to prove ( d ). If F = Q , then by a) the center of E is a field, and therefor ii)follows from Artin-Wedderburn theorem. Conversely if E ∼ = M at n × n ( D ) then F = Z ( E ) = Z ( M at n × n ( D )) = Z ( D ) = Q . The equivalence [ E : Q ] = (rk M ) ⇔ χ ν = µ rk M π followsfrom 3.1. Finally, assuming v ), we see by semi-simplicity of π that µ π = µ and therefore[ F : Q ] = deg x µ π = deg x µ = 1 . On the other hand if µ = µ π is linear then χ ν = ( µ π ) rk M by Cayley-Hamilton. Let G be a flat affine group scheme of finite type over the curve C with generic fiber G . Let H ( C, G ) denote the stack whose S points parameterize G -bundles on C S := C × F q S . RELATION TO G-SHTUKAS G -motives and functoriality One may endow the category of C -motives M ot νC ( S ) with a G -structure. This leads to thefollowing definition. Definition 4.1.
Let Rep G denote the category of representations of G in finite free O C -modules V . By a G -motive (resp. G -motive ) over S we mean a tensor functor M G : Rep G →M ot νC ( S ) (resp. M G : Rep Q G → M ot νC ( S )). We say that two G -motives (resp. G -motives)are isomorphic if they are isomorphic as tensor functors. We denote the resulting categoryof G -motives (resp. G -motives) over S by G - M ot νC ( S ) (resp. G - M ot νC ( S )).Note that the construction of the category G - M ot νC ( S ) is functorial both in G and C . Namely- morphism ρ : G → G ′ (resp. G → G ′ ) induces a functor Rep G ′ → Rep G (resp.Rep G ′ → Rep G ) and this further induces a functor ρ ∗ : G - M ot νC ( S ) → G ′ - M ot νC ( S )(resp. ρ ∗ : G - M ot νC ( S ) → G ′ - M ot νC ( S )).- Suppose we have a morphism C → C ′ of smooth projective geometrically irreduciblecurves which is of degree d . The characteristic places ν on C induce an n-tuple of char-acteristic places on C ′ , which we denote by ν ′ . In addition the push forward functor f ∗ induces a functor from the category of locally free sheaves of rank r over C S to thecategory of locally free sheaves of rank r · d over C ′ S . This further induces the pushforward functor f ∗ : G - M ot νC ( S ) → G - M ot ν ′ C ′ ( S ). G -Shtukas and functoriality Let’s now discuss the geometrization of the category G - M ot νC ( S ). For this we first recall thefollowing definition of the moduli (stacks) of global G -shtukas. Definition-Remark 4.2. (a) A global G -shtuka G over an F q -scheme S is a tuple ( G , s, τ )consisting of- a G -bundle G over C S ,- an n -tuple s of (characteristic) sections and- an isomorphism τ : σ ∗ G| C S r Γ s ∼ −→ G| C S r Γ s . RELATION TO G-SHTUKAS ∇ n H ( C, G ) denote the stack whose S -points parameterizes global G -shtukasover S . Sometimes we will fix the sections s := ( s i ) i ∈ C n ( S ) and simply call G = ( G , τ )a global G -shtuka over S .(b) We let ∇ n H ( C, G )( S ) Q denote the category which has the same objects as ∇ n H ( C, G )( S ),but the set of morphisms is enlarged to quasi-isogenies of G -shtukas. A quasi-isogeny f : G → G ′ is a commutative diagram G f −−−→ G ′ τ x x τ ′ σ ∗ G σ ∗ f −−−→ σ ∗ G ′ , defined outside Γ s ∪ D × F q S for a closed subscheme D ⊆ C . Note that as a morphismof torsors f is automatically an isomorphism. We denote by QIsog S ( G , G ′ ) the set ofquasi-isogenies between G and G ′ .(c) We denote by ∇ n H ( C, G ) ν , the formal stack which is obtained by taking the formalcompletion of the stack ∇ n H ( C, G ) at a fixed n -tuple of pairwise different characteris-tic places ν = ( ν , . . . , ν n ) of C in the sense of [Har05, Definition A.12]. This means welet A ν be the completion of the local ring O C n ,ν , and we consider global G -shtukas onlyover schemes S whose characteristic morphism S → C n factors through N ilp A ν . Wesimilarly denote by ∇ n H ( C, G ) ν ( S ) Q the category over S , with morphisms enlargedto quasi-isogenies. Remark 4.3.
The category of vector bundles of rank n on C S can be identified with the cate-gory of GL n -bundles on C S . Consequently, one can identify the category ∇ n H ( C, GL n ) ν ( S ) Q with the subcategory C -motives M ot νC ( L ) ◦ of rank n .The assignment of the moduli stack H ( C, G ) to a group G , is functorial. In other wordsa morphism ρ : G → G ′ of algebraic groups gives rise to a morphism H ( C, G ) → H ( C, G ′ ) , G 7→ ρ ∗ G := G × G ,ρ G ′ In particular any representation ρ : G → GL( V ) induces a natural morphism of F q -stacks H ( C, G ) → H ( C, GL( V )) , G 7→ ρ ∗ G . (4.2) RELATION TO G-SHTUKAS G -bundle over C L is equivalent with giving a tensorfunctor f from Rep G to the category Vect C ( L ) of vector bundles over C L .Regarding the functoriality of H ( C, − ), the assignment of the moduli stack ∇ n H ( C, G )of global G -shtukas to a group G is functorial in G , i.e. a morphism G → G ′ gives rise tothe morphism ρ ∗ ( − ) : ∇ n H ( C, G ) → ∇ n H ( C, G ′ ) . Given a G -shtuka G and a representation ρ : G → GL( V ) of the group G on a Q -vectorspace, we may consider a lift ρ : G → GL( V ) and use it to push forward G to produce aGL( V )-shtuka ρ ∗ ( G ) in ∇ n H ( C, GL( V ))( L ). According to Remark 4.3, ρ ∗ ( G ) can be viewedas a C -motive in M ot νC ( L ). Note that taking a different lift ρ ′ of the representation ρ , ρ ′∗ G and ρ ∗ G are canonically isomorphic in M ot νC ( L ). This is because ρ and ρ ′ agree over anopen U ⊂ C . In particular we obtain the following pairing ∇ n H ( C, G ) ν ( L ) Q × Rep Q G → M ot νC ( L ) . (4.3) Definition 4.4.
The above pairing 4.3 induces a functor M − G : ∇ n H ( C, G ) ν ( L ) Q −→ G - M ot νC ( L ) , G 7→ M G G (4.4)We say that M G in G - M ot νC ( L ) is geometric if it arises from a G -shtuka G via the abovefunctor. Note that this functor may fail to be essentially surjective for general G .Recall that, for a perfect field F a motive M in the category DM effgm ( F sep ) of effectivegeometric motives, see [VSF] for the definition and notation, comes from a geometric motiveover some finite field extension E/F . As a consequence of the geometrization of the category G - M ot νC ( S ), one can immediately see that a similar fact also holds in the category of ( G -) C -motives. Proposition 4.5.
Let F be a field over F q . Any geometric G - C -motive M G in G - M ot νC ( F sep ) comes by base change from a G - C -motive in G - M ot νC ( E ) for a finite extension E/F .Proof.
First let us assume that M G arises from G = ( G , τ ) under (4.4) and then observe thatany G -bundle G over C F sep is a scheme of finite type over C F sep . To see this, one can take an fppf -covering U ′ → C F sep which trivializes the G -bundle G . Therefore we observe that G| U ′ ∼ = G × C U ′ is of finite type over U ′ , and since U ′ → C F sep is an fppf -cover, we may argue that G is of finite type by fppf -descent [EGA, VI 2.7.1]. As for G := ( G , τ ) in ∇ n H ( C, G )( F sep ),the morphism τ is defined between schemes of finite type, we see that G comes from a point in ∇ n H ( C, G )( E ) for some finite extension E/F . This defines the corresponding G - C -motive RELATION TO G-SHTUKAS G - M ot νC ( E ). Note that one can alternatively prove this statement, using the ind-algebraicstructure on ∇ n H ( C, G ) constructed in [AraHar19, Theorem 3.15]. Like abelian varieties we can pull back a C-motive along a morphism to the correspondinglocal iso-crystal.
Proposition 4.6.
Fix a place ν on ˙ C . Let M be a C -motive over L . Let ˆ M denote thecrystal Γ ν ( M ) associated to M at the place ν . For a given quasi-isogeny (i.e. a morphismwhich is generically an isomorphism) ˆ f : ˆ M ′ → ˆ M of ´etale crystals, one can construct aunique C -motive M ′ with a unique quasi-isogeny f : M ′ → M such that it gives ˆ f back,after applying the functor Γ ν ( − ) . We denote M ′ by ˆ f ∗ M .Proof. Recall that for a noetherian ring A , the I -adic completion ˆ A of A , with respect to anideal I ⊂ A , is flat over A . Thus the natural maps( C r { ν } ) ⊔ Spec( ˆ O C,ν ) → C and ( C r { ν } ) L ⊔ Spec( ˆ O C,ν ˆ ⊗ L ) → C L are fpqc -covers. Let z ν be a uniformizer of ˆ O C,ν . Consider the following Cartesian diagramSpec( ˆ O C,ν ˆ ⊗ L [1 /z ν ]) j −−−→ ( C r { ν } ) L y y Spec( ˆ O C,ν ˆ ⊗ L ) −−−→ C L . By the fpqc -descent of quasi-coherent sheaves, to give a vector bundle M over C L is thesame as to give a triple ( N , ˆ M , ι ), consisting of a vector bundle N over ( C r { ν } ) L , a finiteprojective ˆ O C,ν ˆ ⊗ L -module ˆ M and an isomorphism ι : j ∗ N → ˙ˆ M , where the latter meansthe sheaf associated to the module ˆ M [1 /z ν ]. For a given vector bundle M over C L anda quasi-isogeny (i.e. a morphism which is generically isomorphism) ˆ f : ˆ M ′ → ˆ M of finiteprojective ˆ O C,ν ˆ ⊗ L -modules, the triple ( M| ( C r { ν } ) L , ˆ M ′ , ( ˆ f [1 /z ν ]) − ◦ ι ) and the map ( id , ˆ f )define a unique vector bundle M ′ and a unique quasi-isogeny f : M ′ → M such that f isisomorphism outside ν and gives ˆ f back after passing to the completion at ν . Similarly, fora C -motive M over L , the map τ of M := ( M , τ ) can be interpreted as a map betweentriples ( σ ∗ M| ( C r { ν,ν } ) L , σ ∗ ˆ M , σ ∗ ι ) → ( M| ( C r { ν,ν } ) L , ˆ M , ι ) . RELATION TO G-SHTUKAS f : ˆ M ′ → ˆ M of the category ´Et ( σ, ν ) -Cryst ( L ), the vectorbundle M ′ can be equipped with an isomorphism τ ′ : σ ∗ M ′ | ˙ C L → M ′ | ˙ C L . This yields a C -motive M ′ = ( M ′ , τ ′ ) over L , and moreover, the map f gives rise to a quasi-isogeny f : M ′ → M of C -motives. By construction the quasi-isogeny f gives ˆ f after applyingΓ ν ( − ). Now we may conclude by Proposition 2.11. Corollary 4.7.
Fix a place ν on ˙ C . Let ϕ : M ′ → M be a quasi-isogeny of C -motives in M ot νC ( L ) . Then ω νQ ( ϕ ) identifies ω ν ( M ′ ) with a Γ L -stable sublattice of ω νQ ( M ) . This givesa one to one correspondence between the following sets { quasi-isogenies M ′ → M in M ot νC ( L ) which are isomorphisms above ν } and { Γ L -stable sublattice Λ ν ⊆ ω νQ ( M ) which are contained in ω ν ( M ) } . Proof.
Note that ν does not lie in the characteristic places ν , and hence the associatedcrystals Γ ν ( M ′ ) and Γ ν ( M ) at ν are ´etale. We may view ω ν ( M ′ ) as a Γ L -stable sublatticeof ω νQ ( M ) contained in ω ν ( M ) by applying ω ν ( − ) to a given quasi-isogeny f : M ′ → M ω ν ( M ′ ) ֒ → ω ν ( M ) ⊆ ω νQ ν ( M ) . Vice versa, consider the inclusionΛ ν ⊗ A ν L sep [[ z ]] ⊆ ω ν ( M ) ⊗ A ν L sep [[ z ]] = Γ ν ( M ) ⊗ L [[ z ]] L sep [[ z ]] . Therefore we get a quasi-isogeny ˆ f from ˆ M ′ := (Λ ν ⊗ A ν L sep [[ z ]]) Γ L to ˆ M := (Γ ν ( M ) ⊗ L [[ z ]] L sep [[ z ]]) Γ L . Note that both ˆ M and ˆ M ′ are ´etale crystals. Namely, ˆ M is ´etale by Theorem 2.8,and ˆ M ′ is ´etale, because of the following reason. Since ˆ M ′ ֒ → ˆ M , we observe that the formermodule is also a finite projective A ν,L -module, moreover the Frobenius map on ˆ M ′ = (Λ ν ⊗ A ν L sep [[ z ]]) Γ L is given by id ⊗ ˆ σ ∗ ν , so it is an isomorphism. Hence, according to Proposition 4.6,we may construct the pull back M ′ := ˆ f ∗ M of M along ˆ f , which comes with a canonicalquasi-isogeny f : M ′ → M . By construction the above assignments are inverse to eachother. QUASI-ISOGENY CLASSES AND HONDA-TATE THEORY Recall that a Weil p n -number is an algebraic number π for which there exists an integer m such that ππ = p n for all Q [ π ] → C . Here π denotes the complex conjugate of π . TheHonda-Tate theory, [Hon] and [Tat66], states that sending an abelian variety A over a finitefield with q -elements to the eigenvalue of Frobenius endomorphism π A on the first ´etalecohomolgy group, gives a bijection between isogeny classes of simple abelian varieties over F q and the set of Weil p n -numbers W ( p n ) (up to conjugation).In this section we discuss the analogous picture for M ot νC ( F q ). Note that, unlike theabove case of abelian varieties, as it is mentioned earlier, C -motives are not pure. Thismeans that the eigenvalues of Frobenius endomorphism may have different valuations. So inparticular one must modify the group of Weil (q-)numbers. Proposition 5.1.
Let L/ F q be a finite field. For M and M ′ in M ot νC ( L ) , we let π := π M and π ′ := π M ′ denote the corresponding Frobenius endomorphism with minimal polynomials µ := µ π M and µ ′ := µ π M . Let χ ν and χ ′ ν denote the characteristic polynomials of π ν and π ′ ν .Consider the following statements(a) M ′ is quasi-isogenous to a quotient of M .(b) ω νQ ν ( M ′ ) is Γ L -isomorphic to a Γ L -quotient space of ω νQ ν ( M ) .(c) χ ′ ν divides χ ν in Q ν [ x ] .(d) µ ′ divides µ in Q [ x ] and rk M ≤ rk M ′ .then (a) and (b) are always equivalent and imply (c) and (d). Furthermore we have thefollowing statementsi) If π ν and π ′ ν are semi-simple then (c) also implies (b).ii) If µ is irreducible, then all the above statements are equivalent.Proof. ( a ) ⇒ ( b ) is obvious. So let us first show that ( b ) ⇒ ( a ). The main ingredient to provethis is the analog of Tate conjecture; See Theorem 2.16. Consider the quotient morphism f ν : ω νQ ν ( M ) → ω νQ ν ( M ′ ). Multiplying with a suitable power of the uniformizer z ν ∈ A ν , wemay assume that it is defined with integral coefficients f ν : ω ν ( M ) → ω ν ( M ′ ) with z Nν ω ν ( M ′ ) ⊆ f ν ( ω ν ( M )) , QUASI-ISOGENY CLASSES AND HONDA-TATE THEORY N ≫
0. By Theorem 2.16 f ν can be viewed as an element ofHom L ( M , M ′ ) ⊗ A A ν and thus induces a morphism f : M → M ′ such that ω ν ( f ) = f ν ( mod z N +1 ν ). We claimthat dim Q ν ω νQ ν ( f )( ω ν ( M )) = r ′ := rk M ′ . To see this first notice that from the aboveexplanation we have F r ′ ν ∼ = ω ν ( M ′ ) /z ν · ω ν ( M ′ ) ∼ = z Nν ω ν ( M ′ ) /z N +1 ν · ω ν ( M ′ ) ⊆ ω ν ( f )( ω ν ( M )) /z N +1 ν · ω ν ( M ′ ) , and thus one may take x , . . . , x r ′ ∈ ω ν ( f )( ω ν ( M )) such that they generate the F ν -vectorspace ω ν ( M ′ ) /z ν · ω ν ( M ′ ) . Let H := P r ′ i =1 A ν · x i ⊆ ω ν ( f )( ω ν ( M )) ⊆ ω ν ( M ′ ). As ω ν ( M ′ ) is a free module of rank r ′ , computing the image of H in ω ν ( f )( ω ν ( M )) /z N +1 ν ω ν ( M ′ ), we see that the rank of thefree module H also equals r ′ . Therefore from Q r ′ ν ∼ = H ⊗ A ν Q ν = r ′ X i =1 Q ν · x i ⊆ ω νQ ν ( f )( ω νQ ν ( M )) ⊆ ω νQ ν ( M ′ ) , we see that dim Q ν ω νQ ν ( f )( ω ν ( M )) = r ′ . Now observe that rk(im f ) = r ′ . To see this, apply ω ν ( − ) to the morphism M ։ im f ⊆ M ′ , to get a surjection ω ν ( f ) : ω ν ( M ) → ω ν (im f ) ⊆ ω ν ( M ′ ). Consequently we have r ′ = dim Q ν ω νQ ν (im f ) = rk(im f ) , and therefore im f → M ′ is a quasi-isogeny.( b ) ⇒ ( c ) and ( d ), precisely because of the following commutative diagram ω νQ ( M ) f ν −−−→ ω νQ ( M ′ ) −−−→ π ν y y π ′ ν ω νQ ( M ) f ν −−−→ ω νQ ( M ′ ) −−−→ . Now assume that π ν and π ′ ν are semi-simple with characteristic polynomials χ ν and χ ′ ν .Write χ ′ ν = Q n ′ i =1 P ′ i for irreducible polynomials P ′ i ∈ Q ν [ x ]. By semi-simplicity we maywrite ω νQ ν ( M ′ ) = ⊕ n ′ i =1 V ′ i as Q ν [ π ν ]-module, where V ′ i ∼ = Q ν [ x ] /P ′ i , see Remark 3.3. Thus QUASI-ISOGENY CLASSES AND HONDA-TATE THEORY χ ν = χ ′ ν · u ( x ) for some u ( x ) ∈ Q ν [ x ], and hence ⊕ n ′ i =1 V ′ i appears as a summand of ω νQ ν ( M ).Assume that d) holds and suppose further that µ π is irreducible, then we see that µ π = µ ′ π and F = Q [ x ] /µ π is a field. Therefore by Proposition 3.9 we have χ ′ ν = ( µ ′ π ) rk M ′ / [ F : Q ] | ( µ π ) rk M / [ F : Q ] = χ ν . Furthermore π ν and π ′ ν are semi-simple and (b) follows from (c) as in i).This proposition has the following consequence. Proposition 5.2.
Keep the notation from the above proposition. Consider the followingstatements(a) M is quasi-isogenus to M ′ .(b) There exists an isomorphism ω νQ ν ( M ) ˜ → ω νQ ν ( M ′ ) in Hom Q ν [Γ L ] ( ω νQ ν ( M ) , ω νQ ν ( M ′ )) .(c) χ ν = χ ′ ν .(d) µ π = µ π ′ and rk M = rk M ′ .(e) There exist an isomorphism of Q -algebras α : QEnd L ( M ) ˜ → QEnd L ( M ′ ) , with α ( π ) = π ′ .(f ) There exist an isomorphism of Q ν -algebras α ν : QEnd L ( ω ν ( M )) ˜ → QEnd L ( ω ν ( M ′ )) , with α ( π ν ) = π ′ ν .Then we have the following statementsi) ( a ) and ( b ) are equivalent and imply ( c ) , ( d ) and ( e ) . Also ( e ) implies ( f ) .ii) if π ν and π ′ ν are semisimple then we have ( a ) ⇔ ( b ) ⇔ ( c ) ⇔ ( f ) ⇔ ( e ) and ( c ) ⇒ ( d ) . QUASI-ISOGENY CLASSES AND HONDA-TATE THEORY iii) if µ π and µ π ′ are irreducible in Q [ x ] , then all the above statements are equivalent.Proof. The statements about ( a ), ( b ), ( c ) and ( d ) follow from the above Proposition 5.1.Precisely ( a ) implies ( e ). Namely, a quasi-isogeny f : M → M ′ gives the isomorphismQEnd k ( M ) ˜ → QEnd k ( M ′ )by sending g f ◦ g ◦ f − . Furthermore we have α ( π ) = f ◦ π ◦ f − = f ◦ τ M ◦ ( σ ∗ ) τ M · · · ( σ ∗ ) e − τ ◦ f − , where using f ◦ τ M = σ ∗ f ◦ τ M ′ , the later equals π ′M .Suppose π ν and π ′ ν are semisimple. Then the assertion ( f ) implies ( c ) follows from decompo-sition QEnd k ( M ) = ⊕ ni =1 M at m µ × m µ ( K µ ) with χ ν = Q µ µ m µ and K µ = Q ν [ x ] /µ ; see remark3.3.Suppose µ π and µ π ′ are irreducible, then F = Q [ x ] /µ π and F ′ = Q [ x ] /µ π ′ are fields andtherefore π ν and π ν ′ are semi-simple. As we have seen above, this implies that χ ν = χ ′ ν . Weconclude that µ π = µ π ′ by Proposition 3.9(b). -The Grothendieck Ring K ( M ot νC ( F ))Recall that the category Cor ∼ ( k, Q ) is the category whose objects are smooth projectiveschemes over k and whose morphisms are given byHom( X, Y ) = ⊕ X i Ch ∼ , dim X i ( X i × k Y, Q ) . Here X i denote the connected components of X and Ch ∼ ,d ( − , Q ) denotes the group of cy-cles of dimension d modulo the equivalence relation ∼ , with coefficients in Q . The category Ch eff ∼ ( k ) is the pseudo-abelian envelop of Cor ∼ ( k, Q ) and the category Ch ∼ ( k, Q ) is obtainedby inverting Lefschetz motive L . The Lefschetz motive L comes from canonical decompo-sition [ P k ] = [Spec k ] ⊕ L in Ch eff ∼ ( k ). When ∼ is the numerical equivalence ( ∼ = num )the category Ch ∼ ( k, Q ) is semi-simple abelian category by [Jan]. The Honda-Tate theory establishes a bijection between the set Σ Ch num ( k, Q ) of simple objects of Ch num ( k, Q ) andthe conjugacy classes in Gal( Q alg / Q ) \ W ( q ), where W ( q ) is the subgroup of Weil q-numbersin ( Q alg ) × . Here q := k .We now discuss the analogous picture over function fields. Set W ν = { α ∈ Q alg ; ν ( α ) =0 ∀ ν / ∈ ν } . Consider the free Z -module Z [Γ Q \ W ν × N > / ∼ ] generated by the equivalenceclasses in Γ Q \ W ν × N > / ∼ . Here ( α, n ) and ( β, m ) are equivalent if α m.l = β n.l for some inte-ger l ∈ N > . The operation ( α, · ( β,
1) = ( αβ,
1) induces a ring structure on Z [Γ Q \ W × N > / ∼ ]. Note that to compute ( α, n ) · ( β, m ), using the equivalence relation, one can adjust both THE ZETA-FUNCTION ∗ , · ( ∗ , α, n ) = ( α ′ ,
1) with α ′ n = α (resp. ( β, m ) = ( β ′ ,
1) with β ′ m = β ) and thus we have ( α, n ) · ( β, m ) = ( α ′ β ′ , α ′ and β ′ . Theorem 5.3.
There is a bijectionset Σ of simple objects in M ot νC ( F q ) ↔ elements of Γ Q \ W ν × N > / ∼ . Proof.
Let M := ( M , τ M ) be a simple object in M ot νC ( F q ). Suppose that it comes by basechange from a C -motive in M ot νC ( L ) for a finite extension L/ F q of degree n , see Proposi-tion 4.5. Let π := π M denote the corresponding Frobenius isogeny and let µ π denote thecorresponding minimal polynomial. Let α π be a zero of the minimal polynomial µ π . Thensending M to the pair ( α, n ) gives an assignment Σ → Γ Q \ W ν × N > / ∼ . This is one to oneby Proposition 5.1. The fact that it is onto was proved in [R¨ot, Theorem 3.12]. Remark 5.4.
Note that a sketchy proof of the one-to-one part is also given in [R¨ot, Corol-lary 4.2].
Corollary 5.5.
There is a morphism K ( M ot νC ( F q )) → Z [Γ Q \ W ν × N > / ∼ ] of rings. Recall that assigning a zeta function to a variety over a finite field L , factors through theGrothendieck ring K ( Ch num ( k, Q )) → t Z [[ t ]] . The zeta function satisfies sort of properties manifested in Weil conjectures. A crucial ob-servation to prove these conjectures was to establish a cohomology theory for schemes andexpressing the zeta function of X in terms of the action of Frobenius on the correspondingcohomology groups. THE ZETA-FUNCTION ν away from characteristic places ν i . In contrast with the above assignment, we define thezeta function associated to a C -motive M in M ot νC ( L ) by the following formula Z ( M , t ) := Y i det(1 − tπ ν | H i ´et ( M , Q ν )) ( − i +1 . According to Proposition 2.14, this assignment defines a morphism K ( M ot νC ( L )) → Q ν [ t ], which can be shown that in fact factors through Q [[ t ]] and gives Z ( − , t ) : K ( M ot νC ( L )) → tQ [[ t ]] , i.e. the definition is independent of the choice of the place ν . We further define Z ( M , t ) := Q p Z ( i ∗ p M , t ) for a motive M ∈ M ot νC ( S ) over a general base scheme S , which is of finitetype over F q . Here the product is over all closed points i p : Spec κ ( p ) → C . Let us explainthe reason behind the fact that these definitions are independent of the chosen place ν .First assume that M is simple. Then E := QEnd( M ) is a central simple algebra overthe field F := F ( π ). For a semi-simple element f ∈ E we let J denote the commutativesubalgebra of rank r := rk M containing f . Consider the norm function N : E → Q whichsends g to N K/Q (det( α ( g ))), here K is a splitting field for E and α : E ⊗ F K ˜ → M at n × n ( K )is an isomorphism. One can see the norm N ( f ), as the determinant of the Q -endomorphismof J ⊗ F K given by multiplication by f . Note that one can identify J ν := J ⊗ Q Q ν with ω νQ ν ( M ); see [BH09, Lemma 7.2]. Therefore N ( f ) = det( ω ν ( f )).The above defined norm induces N ( − ) : QEnd( M ) → Q for semi-simple M , for whichthe equality N ( f ) = det( ω ν ( f )) holds. Now, to see that this equality holds for generalelement f ∈ QEnd( M ), we write ω ν ( f ) in Jordan normal form S + N over Q algν , and take apower q N such that ( N ) K q = 0. So f q N is semi-simple and from the above arguments we seethat N ( f q N ) = det( ω ν ( f )) q N and thus N ( f ) = det( ω ν ( f )). Now for every a ∈ A we have χ ν ( a ) = det( a · Id − π ν ) = N ( a − π ) , and thus the characteristic polynomial χ ν is independent of the chosen place ν . This impliesthat the zeta function Z ( M , t ) lies in Q ( t ). Remark 6.1 (zeta function of a G -shtuka) . Let G be in ∇ n H ( C, G )( L ). Then to anyrepresentation ρ we can assign the zeta function of ρ ∗ G ∈ M ot νC ( L ), this gives[ Z ( G , t )] : R ( G ) → Q ( t ) , which assigns a rational function to a given class of representation in the Grothendieck ringof representation R ( G ). SEMI-SIMPLICITY OF THE CATEGORY OF C-MOTIVES OVER FINITE FIELDS Remark 6.2.
Assume that ν := (0 , ∞ ) for two specified places 0 and ∞ ∈ C . Let ζ denotethe image of the uniformizar of O C, in L . One say’s that M ♭ ∈ M ot νC ( F q ) is analog to M ∈ DM eff − ( F ) if Z ( M , t ) is the reduction of a lift of the zeta function Z ( M , t ) to Z [[ y , T ]],regarding the following diagram M ot νC ( F ) / / K ( M ot νC ( F )) Z ( − ,t ) / / A [[ t ]] Z [[ y , t ]] y = q (cid:15) (cid:15) y = z − ζ O O DM eff − ( F ) / / K ( DM eff − ( F )) Z ( − ,t ) / / ♦♦♦♦♦♦♦♦♦♦♦ Z [[ t ]]For example Carlitz module and M ( G m ) and supersingular Drinfeld module of rank 2 and“some” elliptic curve E are analog. Consider the category M ot νC ( F q ). This is a tannakian category with a fiber functor ω : M ot νC ( F q ) → Q. F q -vector spaces . This category admits ´etale and crystalline realizations. Note that according to Proposition4.5 we may regard the tannakian category V ν ( Q ν ) of germs of Q ν -adic representation ofGal( F q / F q ) as the ´etale realization category ω νQ ν ( − ) : M ot νC ( F q ) → V ν ( Q ν )Recall that this category consists of equivalence classes of continuous semisimple represen-tations of open subgroups U of Gal( F q / F q ) on the same finite dimensional Q ν -vector spaces V . Where we say that ρ and ρ on open subgroups U and U are equivalent if they agreeon an open subgroup of U ∩ U . Theorem 7.1.
The category M ot νC ( F q ) with the fiber functor ω , is a semi-simple tannakiancategory. In particular the kernel P := P ∆ of the corresponding motivic groupoid P :=Aut ⊗ (cid:0) ω |M ot νC ( F q ) (cid:1) is a pro-reductive group. EFERENCES Proof.
Since a given motive
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