aa r X i v : . [ m a t h . N T ] O c t Pure Anderson Motives over Finite Fields
Matthias Bornhofen, Urs Hartl ∗ October 30, 2008
Abstract
In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in thearithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as theappropriate analogues of abelian varieties, G. Anderson introduced pure t -motives. In this articlewe study the arithmetic of the latter. We investigate which pure t -motives are semisimple, that is,isogenous to direct sums of simple ones. We give examples for pure t -motives which are not semi-simple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphismalgebra, but also this fails over infinite fields. Still over finite fields we study the Zeta function andthe endomorphism rings of pure t -motives and criteria for the existence of isogenies. We obtainanswers which are similar to Tate’s famous results for abelian varieties. Mathematics Subject Classification (2000) : 11G09, (13A35, 16K20)
Introduction
In the last decades the Arithmetic of Function Fields has acquired great impetus caused by Drin-feld’s [Dr1, Dr2] invention of the concepts of elliptic modules (today called
Drinfeld modules ) and elliptic sheaves in the 1970s. Both are analogues of elliptic curves. The latter live in the Arithmeticof Number Fields, like their higher dimensional generalizations abelian varieties. In [BH1, Ha1] weclaimed that pure Anderson motives (a slight generalization of the pure t -motives introduced by An-derson [An1]) and abelian τ -sheaves should be viewed as the appropriate analogues for abelian varietiesand higher dimensional generalizations of elliptic sheaves or modules. We want to further support thisclaim in the present article by developing the theory of pure Anderson motives over finite fields.To give the definition of pure Anderson motives let C be a connected smooth projective curve over F q , let ∞ ∈ C ( F q ) be a fixed point, and let A = Γ( C r {∞} , O C ). For a field L ⊃ F q let σ ∗ be theendomorphism of A L := A ⊗ F q L sending a ⊗ b to a ⊗ b q for a ∈ A and b ∈ L . Let c ∗ : A → L be an F q -homomorphism and let J = ( a ⊗ − ⊗ c ∗ ( a ) : a ∈ A ) ⊂ A L . A pure Anderson motive M = ( M, τ ) of rank r , dimension d and characteristic c ∗ consists of a locally free A L -module M of rank r and an A L -homomorphism τ : σ ∗ M := M ⊗ A L ,σ ∗ A L → M with dim L coker τ = d and J d · coker τ = 0, suchthat M possesses an extension to a locally free sheaf M on C × F q L on which τ l : ( σ ∗ ) l M → M ( k · ∞ )is an isomorphism near ∞ for some positive integers k and l . The last condition is the purity condition.The ratio kl equals dr and is called the weight of M . Anderson’s definition of pure t -motives [An1]is recovered by setting C = P F q and A = F q [ t ]. In the first two sections we recall the definition ofmorphisms and isogenies between pure Anderson motives as well as some facts from [BH1]. Also foran isogeny f between pure Anderson motives we define the degree of f as an ideal of A (2.8) whichannihilates coker f (2.10). If M is a semisimple (see below) pure Anderson motive over a finite field, thedegree of any isogeny f : M → M is a principal ideal and has a canonical generator (7.3). In particular f has a canonical dual. ∗ The second author acknowledges support of the Deutsche Forschungsgemeinschaft in form of DFG-grant HA3006/2-1and SFB 478
ONTENTS semisimple , that is, isogenousto a direct sum of simple pure Anderson motives. A pure Anderson motive is called simple if it hasno non-trivial factor motives. This question is the analogue of the classical theorem of Poincar´e-Weilon the semisimplicity of abelian varieties. By giving a counterexample (Example 6.1) we demonstratethat the answer to this question is negative in general. On the positive side we show that every pureAnderson motive over a finite base field becomes semisimple after a field extension whose degree is apower of q (6.15), and then stays semisimple after any further field extension (6.16). Let Q be thefunction field of C . Then the endomorphism Q -algebra QEnd( M ) := End( M ) ⊗ A Q of a semisimplepure Anderson motive is semisimple (2.7) and over a finite field also the converse is true (6.11). Thisis false however over an infinite field (6.13).Like for abelian varieties the behavior of a pure Anderson motive M over a finite field is controlledby its Frobenius endomorphism π (defined in 5.2). If M is semisimple we determine the dimension andthe local Hasse invariants of its endomorphism Q -algebra QEnd( M ) in terms of π (6.5, 9.1). We definea Zeta function Z M for a pure Anderson motive M (Definition 7.6) and we show that it satisfies theRiemann hypothesis (7.8), and has an expression in terms of the degrees deg(1 − π i ) for all i if M issemisimple (7.7). We prove the following isogeny criterion. Theorem 8.1.
Let M and M ′ be semisimple pure Anderson motives over a finite field and let π ,respectively π ′ , be their Frobenius endomorphisms. Then the following are equivalent:1. M and M ′ are isogenous.2. The characteristic polynomials of π and π ′ acting on the v -adic Tate modules of M , respectively M ′ , coincide for some (any) place v ∈ Spec A .3. There exists an isomorphism of Q -algebras QEnd( M ) ∼ = QEnd( M ′ ) mapping π to π ′ .4. Z M = Z M ′ . In the last section we sketch a few results for the question, which orders of QEnd( M ) occur asthe endomorphism rings of pure Anderson motives (10.7, 10.11). There is a relation between thebreaking up of the isogeny class of a semisimple pure Anderson motive into isomorphism classes, andthe arithmetic of QEnd( M ). We indicate this by treating the case of pure Anderson motives definedover the minimal field F q . In this case QEnd( M ) is commutative (10.11). Many of our results parallelTate’s celebrated article [Tat] on abelian varieties over finite fields. To prove them, a major tool are theTate modules and local shtuka attached to pure Anderson motives, which we recall in Sections 4 and3, and the analogue [Tag, Tam] of Tate’s conjecture on endomorphisms. These local structures behavelike in the classical case of abelian varieties, local shtuka playing the role of the p -divisible groups of theabelian varieties. The only difference is that p -divisible groups are only useful for abelian varieties incharacteristic p , whereas the local shtuka at any place of Q are important for the investigation of abelian τ -sheaves and pure Anderson motives. One of the aims of this article is to demonstrate the utility oflocal shtuka. For instance we apply them in the computation of the Hasse invariants of QEnd( M ) inTheorem 9.1. We also used them in [BH1] to reprove the standard fact that the set of morphismsbetween two pure Anderson motives is a projective A -module (1.3). Scattered in the text are severalinteresting examples displaying various phenomena (6.1, 6.13, 9.4, 9.5). Note that there is a two in oneversion [BH2] of the present article and [BH1] on the arXiv. Contents
Introduction 11 Pure Anderson Motives and Abelian τ -Sheaves 4 ONTENTS Q -Algebra 2510 Kernel Ideals for Pure Anderson Motives 31References 34 Notation
In this article we denote by F q the finite field with q elements and characteristic p , C a smooth projective geometrically irreducible curve over F q , ∞ ∈ C ( F q ) a fixed F q -rational point on C , A = Γ( C r {∞} , O C ) the ring of regular functions on C outside ∞ , Q = F q ( C ) = Quot( A ) the function field of C , Q v the completion of Q at the place v ∈ C , A v the ring of integers in Q v . For v = ∞ it is the completion of A at v . F v the residue field of A v . In particular F ∞ ∼ = F q .For a field L containing F q we write C L = C × Spec F q Spec L , A L = A ⊗ F q L , Q L = Q ⊗ F q L , A v,L = A v b ⊗ F q L for the completion of O C L at the closed subscheme v × Spec L , Q v,L = A v,L [ v ]. Note that this is not a field if F v ∩ L ) F q ,Frob q : L → L for the q -Frobenius endomorphism mapping x to x q , σ = id C × Spec(Frob q ) for the endomorphism of C L which acts as the identity on the points and on O C and as the q -Frobenius on L , σ ∗ for the endomorphisms induced by σ on all the above rings. For instance σ ∗ ( a ⊗ b ) = a ⊗ b q for a ∈ A and b ∈ L . σ ∗ M = M ⊗ A L ,σ ∗ A L for an A L -module M and similarly for the other rings.For a divisor D on C we denote by O C L ( D ) the invertible sheaf on C L whose sections ϕ have divisor( ϕ ) ≥ − D . For a coherent sheaf F on C L we set F ( D ) := F ⊗ O CL O C L ( D ). This notation applies inparticular to the divisor D = n · ∞ for n ∈ Z .We will fix the further notation π, F, E, µ π , π v , F v , E v , and χ v in formula (6.1) on page 15. PURE ANDERSON MOTIVES AND ABELIAN τ -SHEAVES τ -Sheaves Pure Anderson motives were introduced by G. Anderson [An1] under the name pure t -motives in thecase where A = F p [ t ]. They were further studied in [BH1]. To give their definition let L be a fieldextension of F q and fix an F q -homomorphism c ∗ : A → L . Let J ⊂ A L be the ideal generated by a ⊗ − ⊗ c ∗ ( a ) for all a ∈ A . Definition 1.1 (pure Anderson motives) . A pure Anderson motive M = ( M, τ ) of rank r , dimension d ,and characteristic c ∗ over L consists of a locally free A L -module M of rank r and an injective A L -modulehomomorphism τ : σ ∗ M → M such that1. the cokernel of τ is an L -vector space of dimension d and annihilated by J d , and2. M extends to a locally free sheaf M of rank r on C L such that for some positive integers k, l the map τ l := τ ◦ σ ∗ ( τ ) ◦ . . . ◦ ( σ ∗ ) l − ( τ ) : ( σ ∗ ) l M → M induces an isomorphism ( σ ∗ ) l M ∞ →M ( k · ∞ ) ∞ of the stalks at ∞ .We call ε := ker c ∗ ∈ Spec A the characteristic point of M and we say that M has finite charac-teristic (respectively generic characteristic ) if ε is a closed (respectively the generic) point. The ratio wt( M, τ ) := kl equals dr and is called the weight of ( M, τ ) ; see [BH1, Proposition 1.2]. Definition 1.2. (Compare [PT, 4.5])1. A morphism f : ( M, τ ) → ( M ′ , τ ′ ) between Anderson motives of the same characteristic c ∗ is amorphism f : M → M ′ of A L -modules which satisfies f ◦ τ = τ ′ ◦ σ ∗ ( f ) .2. If f : M → M ′ is surjective, M ′ is called a factor motive of M .3. A morphism f : M → M ′ is called an isogeny if f is injective with torsion cokernel.4. An isogeny is called separable (respectively purely inseparable ) if the induced morphism τ : σ ∗ coker f → coker f is an isomorphism (respectively is nilpotent , that is, if for some n themorphism τ ◦ σ ∗ τ ◦ . . . ◦ ( σ ∗ ) n τ is zero).We denote the set of morphisms between M and M ′ by Hom(
M , M ′ ) . It is an A -module. If M and M ′ are pure Anderson motives of different weights then Hom( M , M ′ ) = { } by [BH1,Corollary 3.5]. This justifies the terminology pure . The following fact is well known. A proof can befound for instance in [BH1, Theorem 9.5]. Theorem 1.3.
Let M and M ′ be pure Anderson motives over an arbitrary field L . Then Hom(
M , M ′ ) is a projective A -module of rank ≤ rr ′ . The minimal polynomial of every endomorphism of a pureAnderson motive M lies in A [ x ] . Corollary 1.4. ([BH1, Corollary 5.4]) Let f : M → M ′ be an isogeny between pure Anderson motives.Then1. there exists an element a ∈ A which annihilates coker f ,2. there exists a dual isogeny f ∨ : M ′ → M such that f ◦ f ∨ = a · id M ′ and f ∨ ◦ f = a · id M . Next we come to the notion of abelian ( τ -)sheaves. It was introduced in [Ha1] in order to constructmoduli spaces for pure Anderson motives. We briefly recall the results from [BH1] on the relationbetween pure Anderson motives and abelian τ -sheaves. Although our primary interest is on pureAnderson motives we present abelian τ -sheaves here because they can have characteristic point ε = ∞ ∈ C in contrast to pure Anderson motives, and many results for the later extend to this more generalsituation. Moreover, some results are proved most naturally via the use of abelian τ -sheaves (e.g. 9.1and 7.3 below). The fact that ε = ∞ is allowed for abelian τ -sheaves was crucial for the uniformizationof the moduli spaces of pure Anderson motives in [Ha1] and the derived consequences on analyticuniformization of pure Anderson motives. Let L ⊃ F q be a field and fix a morphism c : Spec L → C .Let J be the ideal sheaf on C L of the graph of c . Let r and d be non-negative integers. PURE ANDERSON MOTIVES AND ABELIAN τ -SHEAVES Definition 1.5 (Abelian τ -sheaf) . An abelian τ -sheaf F = ( F i , Π i , τ i ) of rank r , dimension d andcharacteristic c over L is a collection of locally free sheaves F i on C L of rank r together with injectivemorphisms Π i , τ i of O C L -modules ( i ∈ Z ) of the form · · · −−−−→ F i − Π i − −−−−→ F i Π i −−−−→ F i +1 Π i +1 −−−−→ · · · x τ i − x τ i − x τ i · · · −−−−→ σ ∗ F i − σ ∗ Π i − −−−−−→ σ ∗ F i − σ ∗ Π i − −−−−−→ σ ∗ F i σ ∗ Π i −−−−→ · · · subject to the following conditions:1. the above diagram is commutative,2. there exist integers k, l > with ld = kr such that the morphism Π i + l − ◦ · · · ◦ Π i identifies F i with the subsheaf F i + l ( − k · ∞ ) of F i + l for all i ∈ Z ,3. coker Π i is an L -vector space of dimension d for all i ∈ Z ,4. coker τ i is an L -vector space of dimension d and annihilated by J d for all i ∈ Z .We call ε := c (Spec L ) ∈ C the characteristic point and say that F has finite (respectively generic ) characteristic if ε is a closed (respectively the generic) point. If r = 0 we call wt( F ) := dr the weight of F .Remark.
1. If F is an abelian τ -sheaf and D a divisor on C , then F ( D ) := ( F i ( D ) , Π i ⊗ , τ i ⊗
1) is anabelian τ -sheaf of the same rank and dimension as F .2. Let F be an abelian τ -sheaf and let n ∈ Z . We denote by F [ n ] := ( F i + n , Π i + n , τ i + n ) the n -shifted abelian τ -sheaf of F whose collection of F ’s, Π ’s and τ ’s is just shifted by n . Definition 1.6. A morphism f between two abelian τ -sheaves F = ( F i , Π i , τ i ) and F ′ = ( F ′ i , Π ′ i , τ ′ i ) of the same characteristic c : Spec L → C is a collection of morphisms f i : F i → F ′ i ( i ∈ Z ) whichcommute with the Π ’s and the τ ’s, that is, f i +1 ◦ Π i = Π ′ i ◦ f i and f i +1 ◦ τ i = τ ′ i ◦ σ ∗ f i . We denote theset of morphisms between F and F ′ by Hom( F , F ′ ) . It is an F q -vector space. For example, the collection of morphisms ( Π i ) : F → F [ 1 ] defines a morphism between the abelian τ -sheaves F and F [ 1 ]. Definition 1.7.
Let F and F ′ be abelian τ -sheaves and let f ∈ Hom( F , F ′ ) be a morphism. Then f is called injective (respectively surjective , respectively an isomorphism ), if f i is injective (respectivelysurjective, respectively bijective) for all i ∈ Z . We call F an abelian factor τ -sheaf of F ′ , if there is asurjective morphism from F ′ onto F . If F = ( F i , Π i , τ i ) is an abelian τ -sheaf of rank r , dimension d , and characteristic c : Spec L → C with ε = im c = ∞ then M ( F ) := ( M, τ ) := (cid:16) Γ( C L r {∞} , F ) , Π − ◦ τ (cid:17) (1.1)is a pure Anderson motive of the same rank and dimension and of characteristic c ∗ : A → L . Converselywe have the following result. Proposition 1.8. ([BH1, Theorem 3.1]) Let ( M, τ ) be a pure Anderson motive of rank r , dimension d , and characteristic c ∗ : A → L over L . Then ( M, τ ) = M ( F ) for an abelian τ -sheaf F over L ofsame rank and dimension with characteristic c := Spec c ∗ : Spec L → Spec A ⊂ C . One can even findthe abelian τ -sheaf F with k, l relatively prime. ISOGENIES AND QUASI-ISOGENIES We recall the basic facts about isogenies from [BH1].
Proposition 2.1. ([BH1, Proposition 5.1]) Let f : F → F ′ be a morphism between two abelian τ -sheaves F = ( F i , Π i , τ i ) and F ′ = ( F ′ i , Π ′ i , τ ′ i ) . Then the following assertions are equivalent:1. f is injective and the support of all coker f i is contained in D × Spec L for a finite closed subscheme D ⊂ C ,2. f is injective and F and F ′ have the same rank and dimension,3. F and F ′ have the same weight and the fiber f i,η at the generic point η of C L is an isomorphismfor some (any) i ∈ Z . Definition 2.2.
1. A morphism f : F → F ′ satisfying the equivalent conditions of Proposition 2.1is called an isogeny . We denote the set of isogenies between F and F ′ by Isog( F , F ′ ) .2. An isogeny f : F → F ′ is called separable (respectively purely inseparable ) if for all i theinduced morphism τ i : σ ∗ coker f i → coker f i +1 is an isomorphism (respectively is nilpotent , thatis, τ i ◦ σ ∗ τ i − ◦ . . . ◦ ( σ ∗ ) n τ i − n = 0 for some n ). The endomorphism rings of abelian τ -sheaves are finite rings. But if we allow the (endo-)morphismsto have “poles” we get rings which are related to the endomorphism rings of the associated pureAnderson motives. We make the following: Definition 2.3 (Quasi-morphism and quasi-isogeny) . Let F and F ′ be abelian τ -sheaves.1. A quasi-morphism f between F and F ′ is a morphism f ∈ Hom( F , F ′ ( D )) for some effectivedivisor D on C .2. A quasi-isogeny f between F and F ′ is an isogeny f ∈ Isog( F , F ′ ( D )) for some effective divisor D on C . If D ≤ D the composition with the inclusion isogeny F ′ ( D ) ⊂ F ′ ( D ) defines an injec-tion Hom( F , F ′ ( D )) ֒ → Hom( F , F ′ ( D )). This yields an equivalence relation for quasi-morphismsand quasi-isogenies. We let QHom( F , F ′ ) and QIsog( F , F ′ ) be the set of quasi-morphisms, respec-tively quasi-isogenies, between F and F ′ modulo this equivalence relation. We write QEnd( F ) :=QHom( F , F ) and QIsog( F ) := QIsog( F , F ).The Q -vector spaces QHom( F , F ′ ) and QEnd( F ) are finite dimensional, and QIsog( F ) is the groupof units in the Q -algebra QEnd( F ), see [BH1, Propositions 6.5 and 9.4].Two abelian τ -sheaves F and F ′ are called quasi-isogenous (notation: F ≈ F ′ ), if there existsa quasi-isogeny between F and F ′ . The relation ≈ is an equivalence relation. If F ≈ F ′ , then the Q -algebras QEnd( F ) and QEnd( F ′ ) are isomorphic, and QHom( F , F ′ ) is free of rank 1 both as a leftmodule over QEnd( F ′ ) and as a right module over QEnd( F ). Proposition 2.4. ([BH1, Proposition 6.10]) Let F and F ′ be two abelian τ -sheaves of characteristic ε = ∞ and let M ( F ) and M ( F ′ ) be their associated pure Anderson motives. Then there is a canonicalisomorphism of Q -vector spaces QHom( F , F ′ ) = Hom( M ( F ) , M ( F ′ )) ⊗ A Q . If M and M ′ are pure Anderson motives, then the elements of Hom( M , M ′ ) ⊗ A Q which admitan inverse in Hom( M ′ , M ) ⊗ A Q are called quasi-isogenies . With this definition we can reformulatePropositions 1.8 and 2.4 as follows. Corollary 2.5.
Let ε = ∞ . Then the functor F 7→ M ( F ) defines an equivalence of categories between1. the category with abelian τ -sheaves as objects and with QHom( F , F ′ ) as the set of morphisms, ISOGENIES AND QUASI-ISOGENIES
2. and the category with pure Anderson motives as objects and with
Hom(
M , M ′ ) ⊗ A Q as the set ofmorphisms.We call these the quasi-isogeny categories of abelian τ -sheaves of characteristic different from ∞ andof pure Anderson motives, respectively. Definition 2.6.
Let F be an abelian τ -sheaf.1. F is called simple , if F 6 = 0 and F has no abelian factor τ -sheaves other than and F .2. F is called semisimple , if F admits, up to quasi-isogeny, a decomposition into a direct sum F ≈ F ⊕ · · · ⊕ F n of simple abelian τ -sheaves F j (1 ≤ j ≤ n ) .We make the same definition for a pure Anderson motive.Remark.
1. Let F be an abelian τ -sheaf with characteristic different from ∞ . Then F is (semi-)simpleif and only if the pure Anderson motive M ( F ) is (semi-)simple by [BH1, Proposition 7.3].2. It is not sensible to try defining simple pure Anderson motives via sub-motives, since for example aM ⊂ M is a proper sub-motive for any a ∈ A r A × . This shows that pure Anderson motives behavedually to abelian varieties. Namely an abelian variety is called simple if it has no non-trivial abeliansubvarieties. Theorem 2.7. ([BH1, Theorem 7.8]) Let F be an abelian τ -sheaf of characteristic different from ∞ .1. If F is simple, then QEnd( F ) is a division algebra over Q .2. If F is semisimple with decomposition F ≈ F ⊕ · · · ⊕ F n up to quasi-isogeny into simple abelian τ -sheaves F j , then QEnd( F ) decomposes into a finite direct sum of full matrix algebras over thedivision algebras QEnd( F j ) over Q . In the following we want to define the degree of an isogeny which should be an ideal of A sincein the function field case we have substituted A for Z . Let f : M → M ′ be an isogeny between pureAnderson motives. Then the A L -module coker f is a finite L -vector space equipped with a morphismof A L -modules τ ′ : σ ∗ coker f → coker f . Since coker f is annihilated by an element of A it decomposesby the Chinese remainder theorem(coker f, τ ′ ) = M v ∈ supp(coker f ) (coker f, τ ′ ) ⊗ A A v =: M v ∈ supp(coker f ) K v . If v = ε the morphism τ ′ on K v is an isomorphism and so Lang’s theorem implies that( K v ⊗ L L sep ) τ ⊗ F q L sep ∼ −→ K v ⊗ L L sep is an isomorphism; see for instance [An1, Lemma 1.8.2]. In particular[ F v : F q ] · dim F v ( K v ⊗ L L sep ) τ = dim F q ( K v ⊗ L L sep ) τ = dim L sep ( K v ⊗ L L sep ) = dim L K v . On the other hand if the characteristic is finite and v = ε , the characteristic morphism c ∗ : A → L yields F ε ⊂ L and determines the distinguished prime ideal a := ( b ⊗ − ⊗ c ∗ ( b ) : b ∈ F ε ) ⊂ A ε,L . If we set n := [ F ε : F q ] and a i := ( σ ∗ ) i a = ( b ⊗ − ⊗ c ∗ ( b ) q i : b ∈ F ε ), then we can decompose A ε,L = L i ∈ Z /n Z A ε,L / a i and τ is an isomorphism σ ∗ ( K ε / a i − K ε ) ∼ −→ K ε / a i for i = 0 since τ is an isomorphism on M and M ′ outside the graph of c ∗ . (This argument will be usedagain in Proposition 3.8.) In particular[ F ε : F q ] · dim L ( K ε / a K ε ) = dim L K ε . LOCAL SHTUKA Definition 2.8.
We assign to the isogeny f the ideal deg( f ) := Y v ∈ supp(coker f ) v (dim L K v ) / [ F v : F q ] = ε dim L ( K ε / a K ε ) · Y v = ε v dim F v ( K v ⊗ L L sep ) τ of A and call it the degree of f . We call ε dim L ( K ε / a K ε ) the inseparability degree of f and Q v = ε v dim F v ( K v ⊗ L L sep ) τ the separability degree of f .Remark. The separability degree of f is the Euler-Poincar´e characteristic EP (cid:0)L v = ε K v ⊗ L L sep (cid:1) τ ; seeGekeler [Gek, 3.9] or Pink and Traulsen [PT, 4.6]. Recall that the Euler-Poincar´e characteristic of afinite torsion A -module is the ideal of A defined by requiring that EP is multiplicative in short exactsequences, and that EP ( A/v ) := v for any maximal ideal v of A . Lemma 2.9.
1. If f : M → M ′ and g : M ′ → M ′′ are isogenies then deg( gf ) = deg( f ) · deg( g ) .2. dim F q A/ deg( f ) = dim L coker f .Proof. / / coker f g / / coker( gf ) / / coker g / / Proposition 2.10.
The ideal deg( f ) annihilates coker f .Proof. If v = ε and a is a uniformizer at ε , then multiplication with a is nilpotent on the L -vector space K ε / a K ε . In particular a dim L ( K ε / a K ε ) annihilates K ε / a K ε , and hence also K ε .If v = ε and a is a uniformizer at v , we obtain analogously that a dim F v ( K v ⊗ L L sep ) τ annihilates the F v -vector space ( K v ⊗ L L sep ) τ and therefore also the L -vector space K v . Proposition 2.11.
Let f : M → M ′ be an isogeny such that deg( f ) = aA is principal (for example thisis the case if C = P and A = F q [ t ] ). Then there is a uniquely determined dual isogeny f ∨ : M ′ → M (depending on a ), which satisfies f ◦ f ∨ = a · id M ′ and f ∨ ◦ f = a · id M .Proof. Since deg( f ) annihilates coker f the proposition is immediate.In Theorem 7.3 we will see that any isogeny f ∈ End( M ) of a semisimple pure Anderson motiveover a finite field satisfies the assumption that deg( f ) is principal. There are mainly two local structures which one can attach to pure Anderson motives and abelian τ -sheaves, namely the local (iso-)shtuka and the Tate module . We treat the Tate module in the nextsection. The local (iso-)shtuka is the analogue of the Dieudonn´e module of the p -divisible group attachedto an abelian variety. Note however one fundamental difference. While the Dieudonn´e module existsonly if p equals the characteristic of the base field, there is no such restriction in our theory here. And infact this would even allow to dispense with Tate modules at all and only work with local (iso-)shtuka.Local (iso-)shtuka were introduced in [Ha1] under the name Dieudonn´e F q [[ z ]] -modules (respectively Dieudonn´e F q (( z )) -modules ). They are studied in [An2, Ha2, Lau]. Over a field their definition takesthe following form. Definition 3.1.
Let v ∈ C be a place of Q and let L ⊃ F q be a field. An (effective) local σ -shtuka at v of rank r over L is a pair ˆ M = ( ˆ M , φ ) consisting of a free A v,L -module ˆ M of rank r and an injective A v,L -module homomorphism φ : σ ∗ ˆ M → ˆ M .A local σ -isoshtuka at v of rank r over L is a pair ˆ N = ( ˆ N , φ ) consisting of a free Q v,L -module ˆ N of rank r and an isomorphism φ : σ ∗ ˆ N ∼ −→ ˆ N of Q v,L -modules. LOCAL SHTUKA Remark 3.2.
Note that so far in the literature [An2, Ha1, Ha2, Lau] it is always assumed that A v has residue field F q , the fixed field of σ on L . So in particular A v,L is an integral domain and Q v,L isa field. For applications to pure Anderson motives this is not a problem since we may reduce to thiscase by Propositions 3.5 and 3.8 below. Definition 3.3.
A local shtuka ˆ M = ( ˆ M , φ ) is called ´etale if φ is an isomorphism. The Tate module of an ´etale local σ -shtuka ˆ M at v is the G := Gal( L sep /L ) -module of φ -invariants T v ˆ M := (cid:0) ˆ M ⊗ A v,L A v,L sep (cid:1) φ . The rational Tate module of ˆ M is the G -module V v ˆ M := T v ˆ M ⊗ A v Q v . It follows from [TW, Proposition 6.1] that T v ˆ M is a free A v -module of the same rank than ˆ M andthat the natural morphism T v ˆ M ⊗ A v A v,L sep ∼ −→ ˆ M ⊗ A v,L A v,L sep is a G - and φ -equivariant isomorphism of A v,L sep -modules, where on the left module G acts on bothfactors and φ is id ⊗ σ ∗ . Since ( L sep ) G = L we obtain: Proposition 3.4.
Let ˆ M and ˆ M ′ be ´etale local σ -shtuka at v over L . Then1. ˆ M = ( T v ˆ M ⊗ A v A v,L sep ) G , the Galois invariants,2. Hom A v,L [ φ ] ( ˆ M , ˆ M ′ ) ∼ −→ Hom A v [ G ] ( T v ˆ M , T v ˆ M ′ ) , f T v f is an isomorphism.In particular the Tate module functor yields an equivalence of the category of ´etale local shtuka at v over L with the category of A v [ G ] -modules, which are finite free over A v .Proof. F v of v is larger than F q one has to be a bit careful with local (iso-)shtuka since Q v,L is then in general not a field. Namely let F v = q n and let F q f := { α ∈ L : α q n = α } be the“intersection” of F v with L . Then F v ⊗ F q L = Y Gal( F qf / F q ) F v ⊗ F qf L = Y i ∈ Z /f Z F v ⊗ F q L / ( b ⊗ − ⊗ b q i : b ∈ F q f )and σ ∗ transports the i -th factor to the ( i + 1)-th factor. (Of course, the indexing of the factors dependson a choice of embeddings F q f ⊂ F v and F q f ⊂ L .) Denote by a i the ideal of A v,L (or Q v,L ) generatedby { b ⊗ − ⊗ b q i : b ∈ F q f } . Then A v,L = Y Gal( F qf / F q ) A v b ⊗ F qf L = Y i ∈ Z /f Z A v,L / a i and similarly for Q v . Note that the factors in this decomposition and the ideals a i correspond preciselyto the places v i of C F qf lying above v . Proposition 3.5.
Fix an i . The reduction modulo a i induces equivalences of categories1. ( ˆ N , φ ) (cid:0) ˆ N / a i ˆ N , φ f : ( σ ∗ ) f ˆ N / a i ˆ N → ˆ N / a i ˆ N (cid:1) between local σ -isoshtuka at v over L and local σ f -isoshtuka at v i over L of the same rank. LOCAL SHTUKA ( ˆ M , φ ) (cid:0) ˆ M / a i ˆ M , φ f : ( σ ∗ ) f ˆ M / a i ˆ M → ˆ M / a i ˆ M (cid:1) between ´etale local σ -shtuka at v over L and ´etale local σ f -shtuka at v i over L preserving Tatemodules T v ( ˆ M , φ ) ∼ −→ T v i ( ˆ M / a i ˆ M , φ f ) . Proof.
Since σ ∗ a i = a i +1 the isomorphism φ yields isomorphisms σ ∗ ( ˆ N / a i ˆ N ) → ˆ N / a i +1 ˆ N and similarlyfor ˆ M . These allow to reconstruct the other factors from ( ˆ N / a i ˆ N , φ f ), and likewise for ˆ M . Theisomorphism between the Tate modules follows from the observation that an element ( x j ) j ∈ Z /f Z is φ -invariant if and only if x j +1 = φ ( σ ∗ x j ) for all j and x i = φ f (( σ ∗ ) f x i ). Remark.
The advantage of the ´etale local σ f -shtuka at v i is that it is a free module over the integraldomain A v,L / a i = A v b ⊗ F qf L , and similarly for local σ f -isoshtuka. So the results from [An2, Ha1, Ha2,Lau] apply.Now let F be an abelian τ -sheaf and v ∈ C an arbitrary place of Q . We define the local σ -isoshtukaof F at v as N v ( F ) := (cid:16) F ⊗ O CL Q v,L , Π − ◦ τ (cid:17) . If v = ∞ we define the local σ -shtuka of F at v as M v ( F ) := (cid:16) F ⊗ O CL A v,L , Π − ◦ τ (cid:17) . Likewise if M is a pure Anderson motive over L and v ∈ Spec A we define the local σ -(iso-)shtuka of M at v as M v ( M ) := M ⊗ A L A v,L respectively N v ( M ) := M ⊗ A L Q v,L . These local (iso-)shtuka all have rank r . The local shtuka are ´etale if v = ε . Note that N ∞ ( F ) doesnot contain a local σ -shtuka if ε = ∞ , since then it is isoclinic of slope − wt( F ) < v = ∞ the periodicity condition allows to define a different local (iso-)shtuka at ∞ which is of slope ≥
0. Namely, choose a uniformizer z on C at ∞ and set ˆ M i := F i ⊗ O CL A ∞ ,L . Recallthe integers k, l from Definition 1.5/2 and set e Π := Π l − ◦ · · · ◦ Π . We define the big local σ -shtuka of F at ∞ as e M ∞ ( F ) := ˆ M ⊕ · · · ⊕ ˆ M l − with φ := e Π − ◦ z k τ l − τ
000 0 τ l − (3.1)We also define the big local σ -isoshtuka of F at ∞ as e N ∞ ( F ) := e M ∞ ( F ) ⊗ A ∞ ,L Q ∞ ,L . Both have rank rl and depend on the choice of k, l and z . If ε = ∞ then e M ∞ ( F ) is ´etale. Note that e M ∞ ( F ) and e N ∞ ( F ) were used in [Ha1] to construct the uniformization at ∞ of the moduli spaces ofabelian τ -sheaves.The big local (iso-)shtuka at ∞ , e M ∞ ( F ) and e N ∞ ( F ) are always equipped with the endomorphisms Π := e Π − ◦ z k Π l − Π
000 0 Π l − , Λ( λ ) := λ · id M λ q · id M λ q l − · id M l − (3.2) LOCAL SHTUKA λ ∈ F q l ∩ L . They satisfy the relations Π l = z k and Π ◦ Λ( λ q ) = Λ( λ ) ◦ Π . We let ∆ ∞ be “the”central division algebra over Q ∞ of rank l with Hasse invariant − kl , or explicitly∆ ∞ := F q l (( z ))[ Π ] / ( Π l − z k , λz − zλ, Πλ q − λΠ for all λ ∈ F q l ) . (3.3)If F q l ⊂ L we identify ∆ ∞ with a subalgebra of End Q ∞ ,L [ φ ] (cid:0) e N ∞ ( F ) (cid:1) by mapping λ ∈ F q l ⊂ ∆ ∞ toΛ( λ ).The following two results were proved in [BH1, Theorems 8.6 and 8.7]. Theorem 3.6.
Let F and F ′ be abelian τ -sheaves of the same weight over a finite field L and let v bean arbitrary place of Q .1. Then there is a canonical isomorphism of Q v -vector spaces QHom( F , F ′ ) ⊗ Q Q v ∼ −→ Hom Q v,L [ φ ] (cid:0) N v ( F ) , N v ( F ′ ) (cid:1) .
2. If v = ∞ choose an l which satisfies 1.5/2 for both F and F ′ and assume F q l ⊂ L . Then there isa canonical isomorphism of Q ∞ -vector spaces QHom( F , F ′ ) ⊗ Q Q ∞ ∼ −→ Hom ∆ ∞ b ⊗ F q L [ φ ] (cid:0) e N ∞ ( F ) , e N ∞ ( F ′ ) (cid:1) . Theorem 3.7.
Let M and M ′ be pure Anderson motives over a finite field L and let v ∈ Spec A be anarbitrary maximal ideal. Then Hom(
M , M ′ ) ⊗ A A v ∼ −→ Hom A v,L [ φ ] ( M v ( M ) , M v ( M ′ )) . Let now the characteristic be finite and v = ε be the characteristic point. Consider a pure Andersonmotive M of characteristic c , its local σ -shtuka M ε ( M ) = ( ˆ M , φ ) at ε and the decomposition of thelater described before Proposition 3.5 M ε ( M ) = Y i ∈ Z /f Z M ε ( M ) / a i M ε ( M ) . From the morphism c : Spec L → Spec F ε ⊂ C we see that F ε ⊂ L , f = [ F ε : F q ] and that there is adistinguished place v of C F ε above v = ε = ∞ , namely the image of c × c : Spec L → C × Spec F ε .Then φ has no cokernel on M ε ( M ) / a i M ε ( M ) for i = 0 and the reasoning of Proposition 3.5 yields Proposition 3.8.
The reduction modulo a induces an equivalence of categories M ε ( M ) (cid:0) M ε ( M ) / a M ε ( M ) , φ f (cid:1) between the local σ -shtuka at ε associated with pure Anderson motives of characteristic c and the local σ f -shtuka at v associated with pure Anderson motives of characteristic c . The same is true for abelian τ -sheaves.Remark. Now the fixed field of σ f on L equals F ε , the residue field of A ε . Also M ε ( F ) / a M ε ( F )is a module over the integral domain A ε b ⊗ F ε L . So again [An2, Ha1, Ha2, Lau] apply to (cid:0) M ε ( F ) / a M ε ( F ) , φ f (cid:1) . Proposition 3.9.
Let M be a pure Anderson motive over L and let ˆ M ′ ε be a local σ f -subshtukaof M ε ( M ) / a M ε ( M ) of the same rank. Then there is a pure Anderson motive M ′ and an isogeny f : M ′ → M with M ε ( f ) (cid:0) M ε ( M ′ ) / a M ε ( M ′ ) (cid:1) = ˆ M ′ ε . The same is true for abelian τ -sheaves. TATE MODULES Proof.
Extend ˆ M ′ ε to the local σ -subshtuka L i ∈ Z /f Z φ i (cid:0) ( σ ∗ ) i ˆ M ′ ε (cid:1) of M ε ( M ) and consider K := M ε ( M ) / M i ∈ Z /f Z φ i (cid:0) ( σ ∗ ) i ˆ M ′ ε (cid:1) . The induced morphism φ K : σ ∗ K → K has its kernel and cokernel supported on the graph of c .Set M ′ = ( M ′ , τ ′ ) := (cid:0) ker( M → K ) , τ | M ′ (cid:1) . Then M ′ is a pure Anderson motive with the requiredproperties by [BH1, Proposition 1.6].There is a corresponding result at the places v = ε which is stated in Proposition 4.4. Definition 4.1. If F is an abelian τ -sheaf over L , respectively M a pure Anderson motive over L and v ∈ C (respectively v ∈ Spec A ) is a place of Q different from the characteristic point ε , we define T v F := T v ( M v ( F )) and V v F := V v ( M v ( F )) for v = ∞ ,T ∞ F := T ∞ ( e M ∞ ( F )) and V ∞ F := V ∞ ( e M ∞ ( F )) for v = ∞ 6 = ε, respectively T v M := T v ( M v ( M )) and V v M := V v ( M v ( M )) . We call T v F (respectively V v F ) the (rational) v -adic Tate module of F . If v = ∞ they both depend onthe choice of k, l , and z ; see page 10. By [TW, Proposition 6.1], T v F (and V v F ) are free A v -modules (respectively Q v -vector spaces) ofrank r for v = ∞ and rl for v = ∞ , which carry a continuous G = Gal( L sep /L )-action.Also the Tate modules T ∞ F and V ∞ F are always equipped with the endomorphisms Π and Λ( λ )for λ ∈ F q l ∩ L from (3.2). And if F q l ⊂ L we identify the algebra ∆ ∞ from (3.3) with a subalgebra ofEnd Q ∞ ( V ∞ F ) by mapping λ ∈ F q l to Λ( λ ). Remark.
Our functor T v is covariant. In the literature usually the A v -dual of our T v M is called the v -adic Tate module of M . With that convention the Tate module functor is contravariant on Andersonmotives but covariant on Drinfeld modules and Anderson’s abelian t -modules [An1] (which both giverise to Anderson motives). Similarly the classical Tate module functor on abelian varieties is covariant.We chose our non-standard convention here solely to avoid perpetual dualizations. This agrees alsowith the remark after Definition 2.6 that abelian τ -sheaves behave dually to abelian varieties.The following analogues of the Tate conjecture for abelian varieties are due to Taguchi [Tag] andTamagawa [Tam, § Theorem 4.2.
Let M and M ′ be pure Anderson motives over a finitely generated field L and let G := Gal( L sep /L ) . Let ε = v ∈ Spec A be a maximal ideal. Then the Tate conjecture holds: Hom(
M , M ′ ) ⊗ A A v ∼ = Hom A v [ G ] ( T v M , T v M ′ ) . Theorem 4.3. ([BH1, Theorem 9.9]) Let F and F ′ be abelian τ -sheaves over a finitely generated field L and let G := Gal( L sep /L ) . Let v ∈ C be a place different from the characteristic point ε .1. If v = ∞ assume ε = ∞ or wt( F ) = wt( F ′ ) . Then QHom( F , F ′ ) ⊗ Q Q v ∼ = Hom Q v [ G ] ( V v F , V v F ′ ) .
2. If v = ∞ choose an integer l which satisfies 1.5/2 for both F and F ′ and assume F q l ⊂ L . Then QHom( F , F ′ ) ⊗ Q Q ∞ ∼ = Hom ∆ ∞ [ G ] ( V ∞ F , V ∞ F ′ ) . THE FROBENIUS ENDOMORPHISM
Proposition 4.4. ([BH1, Proposition 9.11])1. Let f : M ′ → M be an isogeny between pure Anderson motives then T v f ( T v M ′ ) is a G -stablelattice in V v M contained in T v M .2. Conversely if M is a pure Anderson motive and Λ v is a G -stable lattice in V v M contained in T v M , then there exists a pure Anderson motive M ′ and a separable isogeny f : M ′ → M with T v f ( T v M ′ ) = Λ v . Proposition 4.5.
Let F ′ be an abelian factor τ -sheaf of F . Then V v F ′ is a G -factor space of V v F .The same holds if M ′ is a factor motive of a pure Anderson motive M .Proof. Let f ∈ Hom( F , F ′ ) be surjective and let ˆ M and ˆ M ′ be the (big, if v = ∞ ) local σ -shtukaof F , respectively F ′ , at v . Then the induced morphism M v ( f ) ∈ Hom( ˆ
M , ˆ M ′ ) is surjective andˆ M ′′ := ker M v ( f ) is also a local σ -shtuka at v . We get an exact sequence of local σ -shtuka which wetensor with A v,L sep yielding0 −−−−→ ˆ M ′′ ⊗ A v,L A v,L sep −−−−→ ˆ M ⊗ A v,L A v,L sep M v ( f ) −−−−→ ˆ M ′ ⊗ A v,L A v,L sep −−−−→ . The Tate module functor is left exact, because considering the morphism of A v,L sep -modules1 − τ : ˆ M ⊗ A v,L A v,L sep −→ ˆ M ⊗ A v,L A v,L sep we have by definition T v ˆ M = ker(1 − τ ), and the desired left exactness follows from the snake lemma.After tensoring with ⊗ A v Q v we get0 −−−−→ V v ˆ M ′′ −−−−→ V v ˆ M V v f −−−−→ V v ˆ M ′ . Counting the dimensions of these Q v -vector spaces, we finally also get right exactness, as desired. Suppose that the characteristic is finite, that is, the characteristic point ε is a closed point of C withfinite residue field F ε , and the map c : Spec L → C factors through the finite field ε = Spec F ε . Definition 5.1 ( s -Frobenius on abelian τ -sheaves) . Let F be an abelian τ -sheaf with finite characteristicpoint ε = Spec F ε and let s = q e be a power of the cardinality of F ε . We define the s -Frobenius on F by π := ( π i ) : ( σ ∗ ) e F → F [ e ] , π i := τ i + e − ◦ · · · ◦ ( σ ∗ ) e − τ i : ( σ ∗ ) e F i → F i + e . Clearly π is an isogeny. Observe that F ε ⊂ F s implies that ( σ ∗ ) e F has the same characteristic as F . Similarly if ε ∈ Spec A is a closed point we define Definition 5.2 ( s -Frobenius on pure Anderson motives) . Let M be a pure Anderson motive with finitecharacteristic point ε = Spec F ε and let s = q e be a power of the cardinality of F ε . We define the s -Frobenius isogeny on M by π := τ ◦ . . . ◦ ( σ ∗ ) e − τ : ( σ ∗ ) e M → M .
Remark 5.3.
Classically for (abelian) varieties X over a field K of characteristic p one defines theFrobenius morphism X → φ ∗ X where φ is the p -Frobenius on K . There p equals the cardinality ofthe “characteristic field” im( Z → K ) = F p . In view of the dual behavior of abelian τ -sheaves and pureAnderson motives our definition is a perfect analogue since here we consider the s -Frobenius for s beingthe cardinality of (a power of) the “characteristic field” im( c ∗ : A → L ) = F ε . THE POINCAR ´E-WEIL THEOREM L = F s to be a finite field with s = q e ( e ∈ N ). Let F s denote a fixed algebraicclosure of F s and set G = Gal( F s / F s ). It is topologically generated by Frob s : x x s . The followingresults for the Frobenius endomorphism of τ -modules can be found in Taguchi and Wan [TW, § Proposition 5.4.
Let M be a pure Anderson motive over F s of rank r and let ε = v ∈ Spec A be amaximal ideal.1. The generator Frob s of G acts on T v M like ( T v π ) − .2. Let Ψ : A v [ G ] → End A v ( T v M ) denote the continuous morphism of A v -modules which is inducedby the action of G on T v M . Then im Ψ = A v [ T v π ] .Proof. Remark.
The inversion of T v π in the first statement results from the dual definition of our Tate module. Proposition 5.5.
Let F be an abelian τ -sheaf over L = F s with s = q e and let π be its s -Frobenius.Then ( σ ∗ ) e F = F . Let v ∈ C be a place different from ∞ and from the characteristic point ε .1. The s -Frobenius π can be considered as a quasi-isogeny of F .2. The generator Frob s of G acts on T v F like ( T v π ) − .3. The image of the continuous morphism of Q v -vector spaces Q v [ G ] → End Q v ( V v F ) is Q v [ V v π ] .4. M ( π ) coincides with the s -Frobenius on the pure Anderson motive M ( F ) from definition 5.2.Proof.
1. Due to the periodicity condition, we have F [ e ] ⊂ F ( nk · ∞ ) for a sufficiently large n ∈ N ,since F i + e ⊂ F i + nl = F i ( nk · ∞ ) for e ≤ nl . Thus π ∈ Hom( F , F ( nk · ∞ )), and therefore π ∈ QEnd( F ).By 2.1, we have π ∈ QIsog( F ).2 and 3 again follow from [TW, Ch. 6] and the continuity of Ψ; see [BH2, Proposition 2.29] for moredetails.4 follows from the definition of π and the commutation of the Π ’s and the τ ’s. In this section we study the analogue for pure Anderson motives and abelian τ -sheaves of the Poincar´e-Weil theorem. Originally, this theorem states that every abelian variety is semisimple, that is, isogenousto a product of simple abelian varieties, see [Lan, Corollary of Theorem II.1/6]. Unfortunately, wecannot expect a full analogue of this statement for abelian τ -sheaves or pure Anderson motives as ournext example illustrates. On the positive side we show that every abelian τ -sheaf or pure Andersonmotive over a finite field becomes semisimple after a finite base field extension. Example 6.1.
Let C = P F q , C r {∞} = Spec F q [ t ] and ζ := c ∗ (1 /t ) ∈ F q × . We construct an abelian τ -sheaf F over L = F q with r = d = 2 which is not semisimple. Let∆ = (cid:16)
10 01 (cid:17) + (cid:16) αγ βδ (cid:17) · t with α, β, γ, δ ∈ F q . To obtain characteristic c we need det ∆ = (1 − ζt ) , and thus we require theconditions α + δ = − ζ and αδ − βγ = ζ . We set F i := O C L ( i · ∞ ) ⊕ , we let Π i be the naturalinclusion, and we let τ i := ∆. Then F is an abelian τ -sheaf with r = d = 2 and k = l = 1. Theassociated pure Anderson motive is M = ( L [ t ] ⊕ , ∆).We see that F is not simple. If ∆ = (cid:0) − ζt − ζt (cid:1) then F is semisimple as a direct sum of two simpleabelian τ -sheaves. Otherwise, if ∆ = (cid:0) − ζt − ζt (cid:1) which is the case for example if β = 0, consider˜∆ := (cid:16) βδ + ζ (cid:17) − · ∆ · σ ∗ (cid:16) βδ + ζ (cid:17) = (cid:16) − ζt t − ζt (cid:17) THE POINCAR ´E-WEIL THEOREM τ -sheaf with e F i = O C L ( i · ∞ ) ⊕ and ˜ τ i = ˜∆ which is isomorphic to F . There is anexact sequence 0 −−−−→ F ′ ϕ −−−−→ e F ψ −−−−→ F ′′ −−−−→ τ ′ = 1 − ζt ˜ τ τ ′′ = 1 − ζt with ϕ : 1 (cid:0) (cid:1) and ψ : (cid:0) xy (cid:1) y where F ′ = F ′′ is the abelian τ -sheaf with F ′ i = O C L ( i · ∞ ) and τ ′ i = 1 − ζt . If e F were semisimple, then there would be a quasi-morphism ω : F ′′ → e F with ψ ◦ ω = id F ′′ ,hence ω : y (cid:0) e (cid:1) · y for some e ∈ F q ( t ). Thus, a necessary condition for the semisimplicity of F is(1 − ζt ) · σ ∗ ( y ) · (cid:0) e (cid:1) = (cid:16) − ζt t − ζt (cid:17) · (cid:0) σ ∗ ( e )1 (cid:1) · σ ∗ ( y )which is equivalent to the condition e − σ ∗ ( e ) = t − ζt . But this cannot be true since e − σ ∗ ( e ) = 0, thus F is not semisimple. However, this last formula issatisfied if e = λ · t − ζt for λ ∈ F q q with λ q − λ = −
1. That means that after field extension F q ( λ ) / F q we get F ∼ = F ′ ⊕ and we have QEnd( F ) = M (QEnd( F ′ )) = M ( Q ). Note that this phenomenongenerally appears, and we will state and prove it in Theorem 6.15.From now on we fix a place v ∈ Spec A which is different from the characteristic point ε of c .For a morphism f ∈ QHom( F , F ′ ) between two abelian τ -sheaves F and F ′ we denote its image V v f ∈ Hom Q v [ G ] ( V v F , V v F ′ ) just by f v . If F is defined over F s this applies in particular to the s -Frobenius endomorphism π of F (Definition 5.1).Let F be an abelian τ -sheaf over the finite field L = F s . We set E := QEnd( F ) ∋ π E v := End Q v [ G ] ( V v F ) ∋ π v F := Q [ π ] ⊂ E F v := im( Q v [ G ] → End Q v ( V v F )) (6.1)with Q v [ G ] → End Q v ( V v F ) induced by the action of G on V v F . Clearly, we have F ⊂ E and F v ⊂ E v by Proposition 5.5/3. By [BH1, Proposition 9.4], we know that dim Q E < ∞ . Thus π is algebraicover Q . We denote its minimal polynomial by µ π ∈ Q [ x ], and the characteristic polynomial of theendomorphism π v of V v F by χ v ∈ Q v [ x ]. If ε = ∞ , Theorem 1.3 shows that π is integral over A , µ π ∈ A [ x ]. The zeroes of π in Spec A [ π ] all lie above ε because π is an isomorphism locally at all v = ε ;compare with [BH1, Remark 5.5].Due to the Tate conjecture, our situation can be represented by the following diagram where wewant to fit the missing bottom right arrow with an isomorphism. E / / E ⊗ Q Q v ∼ / / E v F / / O O F ⊗ Q Q v ∼ / / ___ O O F v . O O Lemma 6.2.
The natural morphism between F ⊗ Q Q v and F v is an isomorphism.Proof. Consider the isomorphism ψ : E ⊗ Q Q v ∼ = E v ⊂ End Q v ( V v F ) and set ϕ := ψ | F ⊗ Q Q v . Then ϕ isinjective and maps into F v . Since im ϕ = Q v [ π v ], the surjectivity follows from Proposition 5.5.To evaluate the dimension of E we need the following notation. Definition 6.3.
Let K be a field. Let f, g ∈ K [ x ] be two polynomials and let f = Y µ ∈ K [ x ]irred. µ m ( µ ) , g = Y µ ∈ K [ x ]irred. µ n ( µ ) THE POINCAR ´E-WEIL THEOREM be their respective factorizations in powers of irreducible polynomials. Then we define the integer r K ( f, g ) := Y µ ∈ K [ x ]irred. m ( µ ) · n ( µ ) · deg µ . Remark.
In contrast to characteristic zero, we have for char( K ) = 0 in general different values of theinteger r K for different ground fields K . Namely, if K ⊂ L then r K ( f, g ) ≤ r L ( f, g ) with equality ifand only if all irreducible µ ∈ K [ x ] which are contained both in f and in g have no multiple factorsin L [ x ]. This is satisfied for example if the greatest common divisor of f and g has only separableirreducible factors, or if L is separable over K . See 9.4 below for an example where r K ( f, g ) < r L ( f, g ).Before we discuss semisimplicity criteria in 6.8 – 6.16, let us compute the dimension of QHom( F , F ′ ). Lemma 6.4.
Let v be a place of Q different from ε and ∞ . Let F and F ′ be abelian τ -sheaves over F s and assume that π v and π ′ v are semisimple. Factor their characteristic polynomials χ v = µ m · . . . · µ m n n and χ ′ v = µ m ′ · . . . · µ m ′ n n with distinct monic irreducible polynomials µ , . . . , µ n ∈ Q v [ x ] and m i , m ′ i ∈ N .Then1. Hom Q v [ G ] ( V v F , V v F ′ ) ∼ = n M i =1 M m ′ i × m i (cid:0) Q v [ x ] / ( µ i ) (cid:1) as Q v -vector spaces,2. End Q v [ G ] ( V v F ) ∼ = n M i =1 M m i (cid:0) Q v [ x ] / ( µ i ) (cid:1) as Q v -algebras, and3. dim Q v Hom Q v [ G ] ( V v F , V v F ′ ) = r Q v ( χ v , χ ′ v ) .Proof. Clearly 2 and 3 are consequences of 1 which we now prove. Since π v and π ′ v are semisimple, wehave the following decomposition of Q v [ G ]-modules V v F ∼ = n M i =1 ( Q v [ x ] / ( µ i )) ⊕ m i , V v F ′ ∼ = n M i =1 ( Q v [ x ] / ( µ i )) ⊕ m ′ i where Q v [ x ] / ( µ i ) =: K i are fields. Obviously, we only have non-zero Q v [ G ]-morphisms K i → K j if i = j , since otherwise µ i ( π ) = 0 in K j . Since π v operates on K ⊕ m i i as multiplication by the scalar x ,the lemma follows. Theorem 6.5.
Let F and F ′ be abelian τ -sheaves of the same weight over F s and assume that theendomorphisms π v and π ′ v of V v F and V v F ′ are semisimple at a place v = ε, ∞ of Q . Let χ v and χ ′ v be their characteristic polynomials. Then dim Q QHom( F , F ′ ) = r Q v ( χ v , χ ′ v ) . Proof.
This follows from the lemma and the Tate conjecture, Theorem 4.3.
Corollary 6.6.
Let F be an abelian τ -sheaf of rank r over F s with Frobenius endomorphism π and let µ π be the minimal polynomial of π . Assume that F = Q [ x ] / ( µ π ) is a field and set h := [ F : Q ] = deg µ π .Then1. h | r and dim Q QEnd( F ) = r h and dim F QEnd( F ) = r h .2. For any place v of Q different from ε and ∞ we have QEnd( F ) ⊗ Q Q v ∼ = M r/h ( F ⊗ Q Q v ) and χ v = ( µ π ) r/h independent of v .Proof. Since F is a field, π v is semisimple by 6.8 below. So general facts of linear algebra imply that µ π = µ · . . . · µ n with pairwise different irreducible monic polynomials µ i ∈ Q v [ x ] and χ v = µ m · . . . · µ m n n with m i ≥
1. We set K i = Q v [ x ] / ( µ i ) and use the notation from (6.1). By Lemma 6.4 the semisimple Q v -algebra E v decomposes E v ∼ = L ni =1 E i into the simple constituents E i = M m i ( K i ). By [Bou, THE POINCAR ´E-WEIL THEOREM E i = E v · e i where e i are the central idempotents with K i = F v · e i . Thus there are epimorphisms of K i -vector spacesQEnd( F ) ⊗ F K i = E v ⊗ F v K i −→→ E i . This shows that m i ≤ dim F E . So by Lemma 6.4[ F : Q ] · dim F E = dim Q v E v = n X i =1 m i deg µ i ≤≤ dim F E · n X i =1 deg µ i = dim F E · deg µ π = [ F : Q ] · dim F E .
Therefore m i = dim F E for all i . Since r = deg χ v = P i m i deg µ i = √ dim F E · [ F : Q ]. We find r = m i h and dim F E = r h , proving 1. For 2 we use that E v ∼ = M i M r/h (cid:0) Q v [ x ] / ( µ i ) (cid:1) = M r/h (cid:0)M i Q v [ x ] / ( µ i ) (cid:1) = M r/h (cid:0) Q v [ x ] / ( µ π ) (cid:1) . Next we investigate when π v is semisimple. Remark 6.7.
Notice that the completion Q v is separable over Q . Namely, in terms of [EGA, IV.7.8.1–3], we can state that O C,v is an excellent ring. Thus the formal fibers of b O C,v −→ O
C,v and therefore Q v = b O C,v ⊗ O C,v Q −→ Q are geometrically regular. This means that Q v ⊗ Q K is regular for every finitefield extension K over Q . Since ”regular” implies ”reduced”, we conclude that Q v is separable over Q . Proposition 6.8.
In the notation of (6.1) the following statements are equivalent:1. π is semisimple.2. F is semisimple.3. F ⊗ Q Q v ∼ = F v is semisimple.4. π v is semisimple.5. E ⊗ Q Q v ∼ = E v is semisimple.6. E is semisimple.Proof.
1. and 2. are equivalent by definition. So we show the equivalences from 2. to 6.Let F be semisimple. Since Q v is separable over Q , we conclude that F ⊗ Q Q v ∼ = Q v [ π v ] is semisimpleby [Bou, Corollaire 7.6/4]. Hence π v is semisimple by definition, and we showed in Lemma 6.4/2 thatthen E v ∼ = E ⊗ Q Q v is semisimple. Again by [Bou, Corollaire 7.6/4] this implies that E is semisimple.Since F ⊂ Z ( E ) is a finite dimensional Q -subalgebra of the center of E , we conclude by [Bou, Corollairede Proposition 6.4/9] that F is semisimple, and our proof is complete. Remark.
If more generally F is defined over a finitely generated field, then one cannot consider π , π v ,nor F . Nevertheless 5 and 6 remain equivalent and are still implied by 3 due to the following well-knownlemma. Namely E v is the commutant of F v in End Q v ( V v F ). We thank O. Gabber for mentioning thisfact to us and we include its proof for lack of reference. Lemma 6.9.
Let B be a central simple algebra of finite dimension over a field K and let F be asemisimple K -subalgebra of B . Then the commutant of F in B is semisimple.Proof. Let F = L i F i be the decomposition into simple constituents and let e i be the correspondingcentral idempotents, that is, F i = F e i . Consider B i = e i Be i which is again central simple over K by[Bou, Corollaire 6.4/4], since if I ⊂ B i is a non-zero two sided ideal then BIB contains 1 and so I contains the unit e i of B i . By [Bou, Th´eor`eme 10.2/2] the commutant E i of F i in B i is simple. Clearlythe commutant E of F in B satisfies E i = e i Ee i = Ee i and E = L i E i proving the lemma. THE POINCAR ´E-WEIL THEOREM Corollary 6.10.
Let F be an abelian τ -sheaf over F s of rank r with semisimple Frobenius endomor-phism π . Then the algebra F = Q ( π ) is the center of the semisimple algebra E = QEnd( F ) .Proof. Since F v is semisimple, we know by [Bou, Proposition 5.1/1] that the F v -module V v F is semi-simple. The commutant of F v in End Q v ( V v F ) is E v by definition. Trivially V v F is of finite type over E v . Thus, by the theorem of bicommutation [Bou, Corollaire 4.2/1], the commutant of E v in End( V v F )is again F v . We conclude Z ( E v ) = E v ∩ F v = F v and we have F ⊗ Q Q v = F v = Z ( E v ) = Z ( E ) ⊗ Q Q v by[Bou, Corollaire de Proposition 1.2/3]. Considering the dimensions, we obtain dim Q F = dim Q Z ( E ).Since F ⊂ Z ( E ) and the dimensions are finite, we finish by F = Z ( E ). Theorem 6.11.
Let F be an abelian τ -sheaf over a finite field L .1. If QEnd( F ) is a division algebra over Q then F is simple. If in addition ε = ∞ then bothstatements are equivalent.2. If the characteristic point ε is different from ∞ then F is semisimple if and only if QEnd( F ) issemisimple.Proof.
1. Let QEnd( F ) = E be a division algebra and let f ∈ Hom( F , F ′ ) be the morphismonto a non-zero factor sheaf F ′ of F . We show that f is an isomorphism. We know by 4.5 that f v ∈ Hom Q v [ G ] ( V v F , V v F ′ ) is surjective. By the semisimplicity of E and Proposition 6.8, F v is semi-simple, and therefore V v F is a finitely generated semisimple F v -module. Thus we get a morphism g v ∈ Hom Q v [ G ] ( V v F ′ , V v F ) with f v ◦ g v = id V v F ′ . Consider the integral Tate modules T v F and T v F ′ .We can find some n ∈ N such that v n g v ∈ Hom A v [ G ] ( T v F ′ , T v F ) ∼ = Hom( M ( F ′ ) , M ( F )) ⊗ A A v and we choose g ∈ Hom( M ( F ′ ) , M ( F )) ⊂ QHom( F ′ , F ) with g ≡ v n g v modulo v m for a sufficientlylarge m > n . If g ◦ f = 0 in E , then f ◦ g ◦ f = 0, and therefore f ◦ g = 0 in QEnd( F ′ ) due to thesurjectivity of f . This would imply v n · id V v F ′ = v n ( f v ◦ g v ) = f v ◦ ( v n g v ) ≡ f ◦ g = 0 (modulo v m )which is a contradiction. Thus g ◦ f = 0 is invertible in E , and therefore f is injective. By that, f givesthe desired isomorphism between F ′ and F . The second assertion follows from Theorem 2.7.2. We already saw one direction in Theorem 2.7/2. So let now QEnd( F ) be semisimple and letQEnd( F ) = m M j =1 M λ j ( E j )be the decomposition into full matrix algebras M λ j ( E j ) over division algebras E j over Q (1 ≤ j ≤ m ).For each j we find λ j distinct idempotents e j, , . . . , e j,λ j ∈ M λ j ( E j ) such that e j,α · QEnd( F ) · e j,α = E j for all 1 ≤ α ≤ λ j with P λ j α =1 e j,α = 1 in M λ j ( E j ). Let e , . . . , e n denote all these idempotents, n = P mj =1 λ j , and choose a divisor D on C such that e i ∈ Hom( F , F ( D )) for all 1 ≤ i ≤ n . Then P ni =1 e i = id F in QEnd( F ) and therefore F P i e i −−−−→ n M i =1 im e i ⊂ F ( D ) . The image F i := im e i is an abelian τ -sheaf by [BH1, Proposition 4.2] because ε = ∞ . Since P i e i isinjective it is an isogeny by 2.1. Since QEnd( F i ) = e i · QEnd( F ) · e i is a division algebra, F i is a simpleabelian τ -sheaf by 1. Thus F ≈ F ⊕ · · · ⊕ F n gives the decomposition into a direct sum of simpleabelian τ -sheaves F i as desired. THE POINCAR ´E-WEIL THEOREM Remark 6.12.
Unfortunately the theorem fails if L is not finite, as Example 6.13 below shows. Thereason is, that then E v may still be semisimple while the image F v of Q v [ G ] in End Q v ( V v F ) is not.Nevertheless, if one adds the assumption that F v is semisimple, the assertions of Theorem 6.11 remainvalid over an arbitrary field L . (See also the remark after Proposition 6.8.) Example 6.13.
We construct a pure Anderson motive M over a non-finite field L which is not semi-simple, but has End( M ) = A . Any associated abelian τ -sheaf F has QEnd( F ) = Q . Let C = P F q , A = F q [ t ] with q >
2, and L = F q ( α ) where α is transcendental over F q . Let M = A ⊕ L and τ = (cid:0) αt tt (cid:1) .Then M = ( M, τ ) is a pure Anderson motive of rank and dimension 2. Clearly M is not simple, since M ′ = ( A L , τ ′ = t ) is a factor motive by projecting onto the second coordinate. We will see below that M is not even semisimple.Let (cid:16) eg fh (cid:17) ∈ M ( A L ) be an endomorphism of M , that is, (cid:18) ασ ∗ e + σ ∗ gσ ∗ g ασ ∗ f + σ ∗ hσ ∗ h (cid:19) = (cid:18) αeαg e + fg + h (cid:19) . Choose β ∈ F q ( α ) alg r F q ( α ) satisfying β q − = α (for β / ∈ F q ( α ) we use q > σ ∗ g = αg implies g ∈ β · F q [ t ]. Since also g ∈ F q ( α )[ t ] we must have g = 0. Now σ ∗ e = e and σ ∗ h = h yielding e, h ∈ F q [ t ].Let γ ∈ F q ( α ) alg rF q ( β ) with γ q − γ = β and set ˜ f := βf − γ · ( e − h ). Then ασ ∗ f − f = e − σ ∗ h = e − h implies σ ∗ ˜ f − ˜ f = β q σ ∗ f − βf − ( γ q − γ )( e − h ) = β ( ασ ∗ f − f − ( e − h )) = 0. Thus ˜ f ∈ F q [ t ] and γ · ( e − h ) ∈ F q ( β )[ t ]. So we must have e = h and then βf = ˜ f ∈ F q [ t ] implies f = 0. This shows thatEnd( M ) = F q [ t ] = A .The same argument shows that M is not even semisimple. Namely, the projection M → M ′ has nosection M ′ → M , (cid:0) f (cid:1) , since there is no solution f for the equation αtσ ∗ f + t = tf .It is also not hard to compute F v for instance at the place v = ( t − z = t − β ∈ L sep with β q − = α , and consider the basis (cid:0) y/β (cid:1) , (cid:0) xy (cid:1) of the Tate module T v ( M ) with (cid:18) xy (cid:19) = ∞ X i =0 (cid:18) x i y i (cid:19) z i and x i , y i ∈ L sep , y = 0 . They are subject to the equations y = tσ ∗ y = (1 + z ) σ ∗ y and x = αtσ ∗ x + tσ ∗ y = α (1 + z ) σ ∗ x + y ,that is, y i − y qi = y qi − , and x i − αx qi = αx qi − + y i . There are elements γ and δ of G = Gal( L sep /L ) operating as γ ( y i ) = y i , γ ( x i ) = x i , γ ( β ) = β/η for an η ∈ F q × r { } , respectively as δ ( y i ) = y i , δ ( β ) = β , δ ( x i ) = x i + y i /β . With respect to our basis of T v ( M ) they correspond to matrices γ v = (cid:0) η (cid:1) and δ v = (cid:0)
10 11 (cid:1) . We conclude that F v is the Q v -algebraof upper triangular matrices. Its commutant in M ( Q v ) equals Q v · Id ∼ = End( M ) ⊗ A Q v . Remark. If q = 2 any pure Anderson motive of rank rk M = 2 on A = F q [ t ], which is not semisimple hasEnd( M ) ) A . One easily sees this by choosing a basis of M for which τ has the form (cid:16) α ( t − θ ) d ∗ β ( t − θ ) d (cid:17) with α, β, θ ∈ L . Then (cid:16) β/α (cid:17) is an endomorphism.However, we expect that also for q = 2 there are examples similar to 6.13 (of rk M ≥ F be an abelian τ -sheaf over F s and let F s ′ / F s be a finite field extension. The base extension F ⊗ F s F s ′ := ( F i ⊗ O C F s O C F s ′ , Π i ⊗ , τ i ⊗ THE POINCAR ´E-WEIL THEOREM τ -sheaf over F s ′ with π ′ = ( π ⊗ t for s ′ = s t , and we have a canonical isomorphismbetween V v F and V v F ′ .For the next result recall that an endomorphism ϕ of a finite dimensional vector space V over afield K is called absolutely semisimple if for every field extension K ′ /K the endomorphism ϕ ⊗ ∈ End K ′ ( V ⊗ K K ′ ) is semisimple. The following characterization is taken from [Bou, Proposition 9.2/4and Proposition 9.2/5]. Lemma 6.14.
Let K be a field and let V be a finite dimensional K -vector space. Let ϕ ∈ End K ( V ) be an endomorphism.1. ϕ is absolutely semisimple, if and only if there exists a perfect field extension K ′ /K such that ϕ ⊗ ∈ End K ′ ( V ⊗ K K ′ ) is semisimple.2. ϕ is absolutely semisimple, if and only if its minimal polynomial is separable. Theorem 6.15.
Let F be an abelian τ -sheaf over the finite field F s . Then there exists a finite fieldextension F s ′ / F s whose degree is a power of char F s such that F ⊗ F s F s ′ has an absolutely semisimpleFrobenius endomorphism. Thus if moreover ε = ∞ then F ⊗ F s F s ′ is semisimple.Remark. It suffices to take [ F s ′ : F s ] as the smallest power of char F s which is ≥ rk F . Proof.
Let s ′ = s t for some arbitrary t ∈ N . Let F ′ := F ⊗ F s F s ′ be the abelian τ -sheaf over F s ′ inducedby F . Let v ∈ Spec A be a place different from ε . Over Q v alg we can write π v ∈ End Q v ( V v F ) in Jordannormal form B − ( π v ⊗ B = λ ∗ λ . . .. . . ∗ λ r for B ∈ GL r ( Q v alg ) and for some λ j ∈ Q v alg , 1 ≤ j ≤ r . Thus, by a suitable choice of t ∈ N as a powerof char F q (as in the remark), we can achieve that π ′ v = ( π v ⊗ t is of the form B − π ′ v B = λ t λ t . . .0 λ tr . Since Q v alg is perfect, we conclude by 6.14/1 that π ′ v and thus π ′ is absolutely semisimple.The following corollary illustrates that, in contrast to endomorphisms of vector spaces, there is noneed of the term ”absolutely semisimple” for abelian τ -sheaves or pure Anderson motives over finitefields. Corollary 6.16.
Let F be an abelian τ -sheaf over F s of characteristic different from ∞ . If F issemisimple, then F ⊗ F s F s ′ is semisimple for every finite field extension F s ′ / F s . The same is true forpure Anderson motives.Proof. Let F be semisimple and let F s ′ / F s be a finite field extension with s ′ = s t . We set F ′ := F ⊗ F s F s ′ . By 6.11 and 6.8, we know that QEnd( F ) ⊗ Q Q v ∼ = End Q v [ π v ] ( V v F ) is semisimple. Since Q v [ π tv ] ⊂ Q v [ π v ] we conclude by [Bou, Corollaire de Proposition 6.4/9] that Q v [ π tv ] is semisimple, aswell. As V v F ′ = V v F , we have π ′ v = π tv , and therefore π ′ v is semisimple. Thus, by 6.8, QEnd( F ′ ) issemisimple and F ′ is semisimple by 6.11/2. ZETA FUNCTIONS AND REDUCED NORMS In this section we generalize Gekeler’s results [Gek] on Zeta functions for Drinfeld modules to pureAnderson motives. But let us begin by recalling a few facts about reduced norms; see for instance[Rei, § M be a semisimple pure Anderson motive over a finite field and let π be its Frobeniusendomorphism. Then F = Q ( π ) is the center of the semisimple algebra E by Corollary 6.10. Write F = L i F i and E = L i E i where the F i are fields and E i is central simple over F i . Note thatby 6.11 the pure Anderson motive M decomposes correspondingly up to isogeny M ≈ L i M i with E i = End( M i ) ⊗ A Q . We apply 6.6 to M i and obtain P i [ E i : F i ] / · [ F i : Q ] = r . Let f ∈ E and writeit as f = P i f i with f i ∈ E i . Choose for each i a splitting field K i of E i with α i : E i ⊗ F i K i ∼ −→ M n i ( K i )where n i = [ E i : F i ]. The reduced norm of f is then defined by N ( f ) := nr E/Q ( f ) := Y i N F i /Q (cid:0) det α i ( f i ⊗ (cid:1) , where N F i /Q is the usual field norm. The reduced norm is an element of Q which is independent of thechoices of K i and α i . It satisfies N ( a ) = a r for all a ∈ Q , and N ( f ) = 0 if and only if f ∈ E × , that is, f is a quasi-isogeny. If f ∈ End( M ) or more generally f is contained in a finite A -algebra then N ( f ) ∈ A since A is normal. Theorem 7.1.
Let F be a semisimple abelian τ -sheaf over a finite field L and let f ∈ QEnd( F ) bea quasi-isogeny. Then for any place v = ε, ∞ of Q we have N ( f ) = det V v f , the determinant of theendomorphism V v f ∈ End Q v ( V v F ) . For v = ∞ 6 = ε we have N ( f ) l = det V ∞ f , where l comes fromDefinition 4.1 and satisfies dim Q ∞ V ∞ F = l · rk F .Proof. Clearly, if t is a power of q then N ( f t ) = det V v f t implies N ( f ) = det V v f since 1 is the only t -th root of unity in Q v for v = ∞ , and likewise for v = ∞ . Writing V v f in Jordan canonical form over Q alg v we find as in the proof of Theorem 6.15 a power t of q such that V v f t is absolutely semisimple over Q v and hence its minimal polynomial is separable by 6.14. Then F v ( f t ) and F ( f t ) are semisimple by[Bou, Proposition 9.1/1 and Corollaire 7.7/4]. We now replace f by f t and thus assume that F ( f ) issemisimple.As is well known there is a semisimple commutative subalgebra H = L i H i of E containing F ( f ) with dim F i H i = n i and hence dim Q H = r . Then nr E/Q ( f ) equals the determinant of the Q -endomorphism ˜ f : x f x of H . The reason for this is that H i ⊗ F i K i is still semisimple and com-mutative if we choose a splitting field K i which is separable over F i . By Lemma 7.2 below H i ⊗ F i K i is isomorphic to K n i i as left H i ⊗ F i K i -modules, and this implies that nr E i /F i ( f i ) = det α i ( f i ) = det ˜ f i ,the determinant of the F i -endomorphism ˜ f i : x f i x of H i , and N ( f ) = det ˜ f the determinant of the Q -endomorphism ˜ f of H .If v = ∞ then again by Lemma 7.2, H v is H v -isomorphic to V v F and N ( f ) = det ˜ f = det V v f .If v = ∞ we embed E ⊕ l ∞ into End Q ∞ ,L [ φ ] (cid:0) e N ∞ ( F ) (cid:1) . Namely, if ( f (0) , . . . , f ( l − ) ∈ E ⊕ l ∞ , where f ( m ) = (cid:0) f ( m ) i : F i ⊗ O CL Q ∞ ,L → F i ⊗ O CL Q ∞ ,L (cid:1) , we set g ij := Π i − ◦ . . . ◦ Π j ◦ f ( i − j ) j if 0 ≤ j ≤ i ≤ l − z k Π − i ◦ . . . ◦ Π − j − ◦ f ( l + i − j ) j if 0 ≤ i < j ≤ l − . Then g ij : F j ⊗ O CL Q ∞ ,L → F i ⊗ O CL Q ∞ ,L and a straightforward computation shows that the ho-momorphism g = ( g ij ) i,j =0 ...l − commutes with φ from (3.1) on page 10, that is, g is an element ofEnd Q ∞ ,L [ φ ] (cid:0) e N ∞ ( F ) (cid:1) = End Q ∞ [ G ] ( V ∞ F ); use Proposition 3.4. Now we apply Lemma 7.2 to H ⊕ l ∞ ⊂ E ⊕ l ∞ ⊂ End Q ∞ ( V ∞ F ), and we compute N ( f ) l = (det ˜ f ) l = det Q ∞ (cid:0) H ⊕ l ∞ → H ⊕ l ∞ , h f h (cid:1) = det V ∞ f asdesired. Lemma 7.2.
Let K be a field and let H ⊂ M n ( K ) be a semisimple commutative K -algebra with dim K H = n . Then as a (left) module over itself H is isomorphic to K n . ZETA FUNCTIONS AND REDUCED NORMS Proof.
Decomposing H into a direct sum of fields L κ L κ and K n into a direct sum L λ V λ of simple H -modules, each V λ is isomorphic to an L κ ( λ ) . The injectivity of H → M n ( K ) and dim K H = n implythat H is isomorphic to L λ End L κ ( λ ) ( V λ ) and a fortiori isomorphic as left module over itself to K n . Theorem 7.3.
Let M be a semisimple pure Anderson motive of rank r over a finite field L and let f ∈ End( M ) be an isogeny. Then1. dim L coker N ( f ) = r · dim L coker f .2. The ideal deg( f ) = N ( f ) · A is principal and has a canonical generator.3. There exists a canonical dual isogeny f ∨ ∈ End( M ) satisfying f ◦ f ∨ = N ( f ) = f ∨ ◦ f .Remark.
1. This shows that N (1 − π n ) ∈ A is the analogue for pure Anderson motives of the numberof rational points X ( F q n ) = deg(1 − Frob nq ) ∈ Z on an abelian variety X over the finite field F q ; seealso Theorem 7.7 below.2. The dual isogeny satisfies ( f g ) ∨ = g ∨ f ∨ , because N ( f g ) = N ( f ) N ( g ). Note however, that wecannot expect that ( f + g ) ∨ = f ∨ + g ∨ unless r = 2 because for f = a ∈ A we have N ( a ) = a r and a ∨ = a r − . Proof.
1. Clearly for any a ∈ A we have dim L M /aM = r · dim F q A/ ( a ) = − r ·∞ ( a ) where ∞ ( a ) denotesthe ∞ -adic valuation of a . Now let F be an abelian τ -sheaf with M = M ( F ), and let f : F → F ( n · ∞ )for some n be the isogeny induced by f . Using Theorem 7.1 we compute the dimension l · dim L coker f = nrl − dim L L l − j =0 (cid:0) F j ( n · ∞ ) /f j ( F j ) (cid:1) ∞ = nrl − dim L e M ∞ (cid:0) F ( n ·∞ ) (cid:1) / e M ∞ ( f ) (cid:0) e M ∞ ( F ) (cid:1) = nrl − dim F q (cid:0) T ∞ F ( n · ∞ ) /T ∞ f ( T ∞ F ) (cid:1) = −∞ (det V ∞ f ) = − l · ∞ (cid:0) N ( f ) (cid:1) . Here the first equality follows from the identities F j ( n · ∞ ) /f j ( F j ) = (cid:0) F j ( n · ∞ ) /f j ( F j ) (cid:1) ∞ ⊕ coker f and dim L (cid:0) F j ( n · ∞ ) /f j ( F j ) (cid:1) = deg F j ( n · ∞ ) − deg f j ( F j ) = nr . The second equality is the definitionof e M ∞ , and the third follows from the isomorphism e M ∞ ( F ) ⊗ A ∞ ,L A ∞ ,L sep ∼ = T ∞ F ⊗ A ∞ A ∞ ,L sep . Thefourth equality follows from the elementary divisor theorem. From this we obtain 1.2. Let v = ε be a maximal ideal of A . Using Theorem 7.1 we compute the v -adic valuation of N ( f ) v ( N ( f )) = v (det T v f ) = dim F v (cid:0) T v M /T v f ( T v M ) (cid:1) = dim F v (cid:0) (coker f ) v ⊗ L L sep (cid:1) τ = v (deg f ) . Again the second equality follows from the elementary divisor theorem, the third equality comes fromthe fact that the τ -invariants of the v -primary part (coker f ) v ⊗ L L sep are isomorphic to T v M /T v f ( T v M ),and the last equality is the definition of deg f . From 1 and Lemma 2.9 we obtain r · dim F q A/ deg( f ) = r · dim L coker f = dim L coker N ( f )= dim L (cid:0) ( A/N ( f )) r ⊗ F q L (cid:1) = r · dim F q A/N ( f ) . From the identity dim F q A/ a = P v [ F v : F q ] · v ( a ) for any ideal a ⊂ A we conclude ε (deg f ) = ε ( N ( f ))and therefore deg( f ) = N ( f ) · A .Finally 3 is immediate since N ( f ) annihilates coker f by Proposition 2.10. Remark 7.4.
We do not know of a proof of 1 and 2 for arbitrary pure Anderson motives which doesnot make use of the associated abelian τ -sheaf F . In the special case when M comes from a Drinfeldmodule, Gekeler [Gek, Lemma 3.1] argued that both sides of the equation in 2 are extensions to E ofthe ∞ -adic valuation on Q . But this argument fails in general, since there may be more than one suchextension as one sees from Example 9.5 below. A QUASI-ISOGENY CRITERION Corollary 7.5.
Let M be a semisimple pure Anderson motive of dimension d over a finite field L andlet π be its Frobenius endomorphism. Let v = ε be a maximal ideal of A and let χ v be the characteristicpolynomial of π v . Then1. χ v ∈ A [ x ] is independent of v and χ v ( a ) · A = det V v ( a − π ) · A = deg( a − π ) for every a ∈ A ,2. ε d · [ L : F ε ] = deg( π ) = χ v (0) · A = N ( π ) · A is principal.Proof. χ v ( a ) = N ( a − π ) = χ w ( a ) ∈ A for all a ∈ A .2 follows from the fact that coker π is supported on ε and from the equation dim L coker π = [ L : F q ] · dim L coker τ = d · [ L : F q ]. Definition 7.6.
We define the
Zeta function of a pure Anderson motive M over a finite field F s as Z M ( t ) := Y ≤ i ≤ r det(1 − t ∧ i π v ) ( − i +1 where ε = v ∈ Spec A is a maximal ideal and ∧ i π v ∈ End Q v ( ∧ i V v M ) . By 7.5/1 the Zeta function Z M ( t ) is independent of the place v and lies in Q ( t ). This also followsfrom work of B¨ockle [Boe] and Gardeyn [Gar, § Theorem 7.7. If M is semisimple and P i a i t i is the power series expansion of t ddt log Z M ( t ) , then a i = N (1 − π i ) ∈ A .Proof. By standard arguments a i = det(1 − π iv ); see [Gek, Lemma 5.6]. Now our assertion follows fromTheorem 7.1This Zeta function satisfies the Riemann hypothesis: Theorem 7.8.
In an algebraic closure of Q ∞ all eigenvalues of ∧ i π v ∈ End Q v ( ∧ i V v M ) have the sameabsolute value ( F s ) i wt( M ) .Proof. This was proved by Goss [Gos, Theorem 5.6.10] for i = 1 and follows for the remaining i bygeneral arguments of linear algebra. Similarly to the theory for abelian varieties, the characteristic polynomials of the Frobenius endomor-phisms on the associated Tate modules play an important role for the study of abelian τ -sheaves. Forexample, we can decide on quasi-isogeny of two abelian τ -sheaves F and F ′ just by considering thesecharacteristic polynomials. Theorem 8.1.
Let F and F ′ be abelian τ -sheaves over F s with respective Frobenius endomorphisms π and π ′ , and let µ π and µ π ′ be their minimal polynomials over Q . Let v ∈ C be a place differentfrom ∞ and ε . Let χ v and χ ′ v be the characteristic polynomials of π v and π ′ v , respectively, and let G := Gal( L sep /L ) . Assume in addition that ε = ∞ , or that F and F ′ have the same weight.1. Consider the following statements:1.1. F ′ is quasi-isogenous to an abelian factor τ -sheaf of F .1.2. V v F ′ is G -isomorphic to a G -factor space of V v F .1.3. χ ′ v divides χ v in Q v [ x ] . A QUASI-ISOGENY CRITERION µ π ′ divides µ π in Q [ x ] and rk F ′ ≤ rk F We have 1.1 ⇒ ⇒ ⇐ π v and π ′ v are semisimple,1.2 ⇐ ⇐ µ π is irreducible in Q [ x ] ,1.1 ⇐ ∞ .2. Consider the following statements:2.1. F and F ′ are quasi-isogenous.2.2. V v F and V v F ′ are G -isomorphic.2.3. χ v = χ ′ v .2.4. µ π = µ π ′ and rk F = rk F ′ .2.5. There is an isomorphism of Q -algebras QEnd( F ) ∼ = QEnd( F ′ ) mapping π to π ′ .2.6. There is a Q v -isomorphism QEnd( F ) ⊗ Q Q v ∼ = QEnd( F ′ ) ⊗ Q Q v mapping π v to π ′ v .2.7. If ε = ∞ also consider the statement Z M ( F ) = Z M ( F ′ ) .We have 2.1 ⇔ ⇒ ⇒ ⇔ ∞ ,2.2 ⇐ ⇐ π v and π ′ v are semisimple,2.2 ⇐ ⇐ ⇐ µ π and µ π ′ are irreducible in Q [ x ] .Proof.
1. For the implication 1 . ⇒ . F ′ can itself be considered as abelianfactor τ -sheaf of F and the implication follows from Proposition 4.5. The implication 1 . ⇒ . . ⇒ . µ π is also the minimal polynomial of π v over Q v by Lemma 6.2. ByProposition 5.5 statement 1.2 implies µ π ( π ′ v ) = 0, whence 1.4.For 1 . ⇒ . π v and π ′ v be semisimple. Let χ v = µ · . . . · µ n and χ ′ v = µ ′ · . . . · µ ′ n ′ be thefactorization in Q v [ x ] into irreducible factors and set V i := Q v [ x ] / ( µ i ) and V ′ i := Q v [ x ] / ( µ ′ i ). Then wecan decompose V v F = V ⊕ · · · ⊕ V n and V v F ′ = V ′ ⊕ · · · ⊕ V ′ n ′ . Since χ ′ v divides χ v , we can now easilyconstruct a surjective G -morphism from V v F onto V v F ′ which gives the desired result.Next if µ π is irreducible, 1.4 implies µ π ′ = µ π and 1.3 follows from Corollary 6.6. It further followsfrom Proposition 6.8 that π v and π ′ v are semisimple and this implies 1.2 by the above.For 1 . ⇒ . ε = ∞ . Let f v : V v F → V v F ′ be a surjective morphismof Q v [ G ]-modules. We may multiply f v by a suitable power of v to get a morphism f v : T v F → T v F ′ of the integral Tate modules which is not necessarily surjective, but satisfies v n T v F ′ ⊂ f v ( T v F ) fora sufficiently large n . Let M := (cid:0) Γ( C L r {∞} , F ) , Π − ◦ τ (cid:1) . This is a “ τ -module on A ” in thesense of [BH1, Definition 3.2]. If ε = ∞ then M is the pure Anderson motive M ( F ) associated with F in (1.1). Also let M ′ := (cid:0) Γ( C L r {∞} , F ′ ) , Π ′ − ◦ τ ′ (cid:1) . By [BH1, Theorem 9.8] (or Theorem 4.2if ε = ∞ ), f v lies inside Hom( M , M ′ ) ⊗ A A v , so we can approximate f v by some f ∈ Hom(
M , M ′ )with T v ( f ) ≡ f v modulo v n +1 T v M ′ . Since v n T v M ′ ⊂ f v ( T v M ) we find inside im T v ( f ) generatorsof v n T v M ′ /v n +1 T v M ′ . They generate an A v -submodule of v n T v M ′ whose rank must at least be r ′ since v n T v M ′ /v n +1 T v M ′ ∼ = ( A v /vA v ) r ′ . Thus im T v ( f ) has rank r ′ . Either by assumption or by [BH1,Corollary 3.5] if ε = ∞ , both F and F ′ have the same weight. So by [BH1, Proposition 6.10/1], f comesfrom a quasi-morphism f ∈ QHom( F , F ′ ), that is, a morphism f : F → F ′ ( D ) for a suitable divisor D . Now we finally assume that the characteristic is different from ∞ . By [BH1, Proposition 4.2], theimage im (cid:0) f : F → F ′ ( D ) (cid:1) is an abelian factor τ -sheaf of F and im f → F ′ ( D ) is an injective morphismbetween abelian τ -sheaves of the same rank and weight, hence an isogeny by Proposition 2.1.2. A large part of 2 follows from 1. We prove the rest. To show 2 . ⇒ . . ⇒ . r = dim Q v V v F = dim Q v V v F ′ = r ′ , the morphism f : F → F ′ ( D ) is an injective morphism betweenabelian τ -sheaves of the same rank and weight, hence an isogeny by Proposition 2.1. THE ENDOMORPHISM Q -ALGEBRA . ⇒ . g ∈ QIsog( F , F ′ ). Then the map QEnd( F ) → QEnd( F ′ ) sending f gf g − is an isomorphism with π ′ = gπg − . The implication 2 . ⇒ . . ⇒ . χ v yields the knowledge of det(1 − t ∧ i π v ) andthus of Z M ( F ) by linear algebra. Conversely we know from Theorem 7.8 that all zeroes of det(1 − t ∧ i π v )have absolute value s − i wt( F ) in an algebraic closure of Q ∞ . So we can recover χ v from Z M ( F ) by simplylooking at this absolute value. This proves 2 . ⇐ . π v and π ′ v are semisimple 2 . ⇒ . . ⇒ . µ π and µ π ′ are irreducible, 2.4 follows from 2.6 by Corollary 6.6 since µ π is also theminimal polynomial of π v over Q v by Lemma 6.2. Also 2.3 follows from 2.4 by Corollary 6.6 and π v and π ′ v are semisimple, so 2 . ⇒ . Q -Algebra In this section we study the structure of QEnd( F ) for a semisimple abelian τ -sheaf F over a finitefield and calculate the local Hasse invariants of QEnd( F ) as a central simple algebra over Q ( π ). Fora detailed introduction to central simple algebras, Hasse invariants and the Brauer group, we refer to[Rei, Ch. 7, §§ Theorem 9.1.
Let F be an abelian τ -sheaf over the finite field F s of rank r with semisimple Frobeniusendomorphism π , that is, Q ( π ) is semisimple. Let v ∈ C be a place different from ∞ and from thecharacteristic point ε . Let χ v be the characteristic polynomial of π v .1. The algebra F = Q ( π ) is the center of the semisimple algebra E = QEnd( F ) .2. We have r ≤ [ E : Q ] = r Q v ( χ v , χ v ) ≤ r .
3. Consider the following statements:3.1. E = F .3.2. E is commutative.3.3. [ F : Q ] = r .3.4. [ E : Q ] = r .3.5. χ v has no multiple factor in Q v [ x ] .3.6. χ v is separable.We have 3.1 ⇔ ⇔ ⇔ ⇔ ⇐ ⇒ π v is absolutely semisimple.4. Consider the following statements:4.1. F = Q .4.2. E is a central simple algebra over Q .4.3. [ E : Q ] = r .4.4. χ v is the r -th power of a linear polynomial in Q v [ x ] .4.5. χ v is purely inseparable.We have 4.1 ⇔ ⇔ ⇔ ⇒ ⇐ π v is absolutely semisimple.If 4.2 holds and moreover the characteristic point ε := c (Spec F s ) ∈ C F s is different from ∞ , E ischaracterized by inv ∞ E = wt( F ) , inv ε E = − wt( F ) and inv v E = 0 for any other place v ∈ C .5. In general the local Hasse invariants of E at the places v of F equal inv v E = − [ F v : F q ][ F s : F q ] · v ( π ) . Inparticular inv v E = (cid:26) if v ∤ ε ∞ , wt( F ) · [ F v : Q ∞ ] if v |∞ and ε = ∞ . THE ENDOMORPHISM Q -ALGEBRA (Here F v denotes the completion of F at the place v and F v is the residue field of the place v .) Remark 9.2. If ε = ∞ and F is an elliptic sheaf, that is, d = 1 and M ( F ) is the Anderson motive of aDrinfeld module, Gekeler [Gek, Theorem 2.9] has shown that there is exactly one place v of F above ε ,and exactly one place w of F above ∞ , and that inv w E = [ F : Q ] · wt( F ) and inv v E = − [ F : Q ] · wt( F ).Note that Gekeler actually computes the Hasse invariants of the endomorphism algebra of the Drinfeldmodule. So his invariants differ from ours by a minus sign, since passing from Drinfeld modules toabelian τ -sheaves is a contravariant functor, see [BS, Theorem 3.2.1]. Corollary 9.3.
Let F be an abelian τ -sheaf over the smallest possible field L = F q such that QEnd( F ) is a division algebra. Then QEnd( F ) is commutative and equals Q ( π ) .Proof. QEnd( F ) is a central division algebra over F by Theorem 9.1, which splits at all places of F by9.1/5, hence equals F . Proof ( of Theorem 9.1 ) . χ v = n Y i =1 µ m i i ∈ Q v [ x ]with distinct irreducible µ i ∈ Q v [ x ] and m i > ≤ i ≤ n . Then P ni =1 m i · deg µ i = deg χ v = r ,and by Theorem 6.5 we have [ E : Q ] = r Q v ( χ v , χ v ) = P ni =1 m i · deg µ i . The result now follows fromthe obvious inequalities r = n X i =1 m i · deg µ i (1) ≤ n X i =1 m i · deg µ i (2) ≤ n X i =1 m i · deg µ i ! = r . (9.1)3. Since F = Z ( E ), the equivalence 3 . ⇔ . m i = 1 for all 1 ≤ i ≤ s which establishes the equivalence 3 . ⇔ .
5. In orderto prove 3 . ⇒ . µ v of π v over Q v . If χ v has no multiplefactor, then µ v = χ v and therefore [ F : Q ] = [ Q v ( π v ) : Q v ] = r . Next 3 . ⇒ . F ⊂ E and (dim Q v F v )(dim Q v E v ) = dim Q v End Q v ( V v F ) = r by [Bou, Th´eor`eme 10.2/2], since E v is thecommutant of F v in End Q v ( V v F ). Note that 3 . ⇒ . . ⇒ . E = F implies r ≥ [ Q v ( π v ) : Q v ] = [ F : Q ] = [ E : Q ] ≥ r . For 3 . ⇒ . χ v = µ v . 3 . ⇒ . F = Q , then E is simple with center Q , so E is a central simple algebra over Q . Since F = Z ( E ),the converse is obvious. This shows 4 . ⇔ . . We have equality in (2) of (9.1) if and only if n = 1,deg µ = 1 and m = r which establishes 4 . ⇔ . . In order to connect 4 . ⇔ . . ⇔ . χ v be a power of a linear polynomial. By [Bou, Proposition 9.1/1] the minimal polynomial of π v over Q v is linear and thus F = Q . The converse is trivial. For 4 . ⇒ . µ v is linear. 4 . ⇒ . k, l ) = 1 using Tate modules, since this is much shorter here and exhibits a different techniquethan 5. By the Tate conjecture 4.3, E ⊗ Q Q v is isomorphic to End Q v ( V v F ) ∼ = M r ( Q v ) for all places v ∈ C which are different from ε and ∞ , so the Hasse invariants of E at these places are 0. Since thesum of all Hasse invariants is 0 (modulo 1), we only need to calculate inv ∞ E .As a first step, we show that F q l is contained in F s . In our situation, π lies inside Q . Thus, by 7.8 weget s k/l = | π | ∞ = q m for some m ∈ Z as | Q ×∞ | ∞ = q Z . Since q e = s , we conclude that e · k/l = m ∈ Z and hence l | e , since k and l are assumed to be relatively prime. Therefore F q l ⊂ F q e = F s .Consider the rational Tate module V ∞ ( F ) at ∞ and the isomorphism of Q ∞ -algebras E ⊗ Q Q ∞ ∼ = End ∆ ∞ [ G ] ( V ∞ F ) = End ∆ ∞ ( V ∞ F ) THE ENDOMORPHISM Q -ALGEBRA Q ∞ ∆ ∞ = l and dim Q ∞ V ∞ F = rl , we conclude that V ∞ F is a left r/l -dimensional ∆ ∞ -vector space and hence isomorphic to ∆ r/l ∞ . Thus we have E ⊗ Q Q ∞ ∼ = End ∆ ∞ (∆ r/l ∞ ) = M r/l (End ∆ ∞ (∆ ∞ )) = M r/l (∆ op ∞ ) . Our proof now completes by inv ∞ E = inv ∆ op ∞ = − inv ∆ ∞ = kl = wt( F ) .5. We prove the general case using local (iso-)shtuka rather than Tate modules which were used in 4.Our method is inspired by Milne’s and Waterhouse’ computation for abelian varieties [WM, Theorem 8].However in the function field case this method can be used to calculate the Hasse invariant at all places,whereas in the number field case it applies only to the place which equals the characteristic of the groundfield. Let w be a place of Q and let N w := N w ( F ) be the local σ -isoshtuka of F at w . Let F w be theresidue field of w and F q f = F w ∩ F s the intersection inside an algebraic closure of F q . Let a be theideal ( b ⊗ − ⊗ b : b ∈ F q f ) of Q w ⊗ F q F s and let R := ( Q w ⊗ F q F s / a )[ T ] = Q w ⊗ F qf F s [ T ] be thenon-commutative polynomial ring with T · ( a ⊗ b ) = ( a ⊗ b q f ) · T for a ∈ Q w and b ∈ F s . Since Q w ⊗ F qf F s is a field, R is a non-commutative principal ideal domain as studied by Jacobson [Jac, Chapter 3]. Itscenter is the commutative polynomial ring Q w [ T g ] where g = [ F s : F q f ] = ef . From Theorem 3.6 andProposition 3.5 we get isomorphismsQEnd( F ) ⊗ Q Q w ∼ = End Q w ⊗ F q F s [ φ ] ( N w ) ∼ = End R ( N w / a N w )where T operates on N w / a N w as φ f .By [Jac, Theorem 3.19] the R -module N w / a N w decomposes into a finite direct sum indexed bysome set I N w / a N w ∼ = M v ∈ I N ⊕ n v v (9.2)of indecomposable R -modules N v with N v = N v ′ for v = v ′ . The annihilator of N v is a two sided idealof R generated by a central element µ v ∈ Q w [ T g ] by [Jac, § Q w [ T g ]-modules and µ v is the minimal polynomial of T g on N v by [Jac, Lemma 3.1]. Therefore the least common multiple µ of the µ v is the minimal polynomial of T g on N w / a N w . Note that T g operates on N w / a N w as the Frobenius π , hence µ = mipo π |F and F = Q ( π ) = Q [ T g ] / ( µ ), where we write mipo for the minimal polynomial. By the semisimplicity of π (and Proposition 6.8) µ has no multiple factors in Q w [ T g ]. Since the µ v are powers of irreduciblepolynomials by [Jac, Theorem 3.20] we conclude that all µ v are themselves irreducible in Q w [ T g ]. Again[Jac, Theorem 3.20] implies that µ v = µ v ′ since N v = N v ′ and µ = mipo π |F = Y v ∈ I µ v inside Q w [ T g ] . Thus F ⊗ Q Q w = Q w [ T g ] / ( µ ) = Q v ∈ I Q w [ T g ] / ( µ v ) = Q v | w F v . So I is the set of places of F dividing w and F v = Q w [ T g ] / ( µ v ) is the completion of F at v , justifying our notation. Let π v be the image of π in F v . Its minimal polynomial over Q w is µ v . This implies that E ⊗ Q Q w decomposes further E ⊗ Q Q w = M v ∈ I End R ( N ⊕ n v v ) = M v ∈ I E ⊗ F F v and E ⊗ F F v ∼ = End R ( N ⊕ n v v ). THE ENDOMORPHISM Q -ALGEBRA v above w and consider the diagram of field extensions F v F s F v g/h ggggggggggggggggggg OOOOOOO F w F si F w F s ∩ F vg/h nnnnnn F w ( F v ∩ F s ) i F s F wi h ppppppp F v ∩ F s g/h nnnnnnnn F q f = F w ∩ F s = F w ∩ ( F v ∩ F s ) h pppppp f F q Let h := [ F v ∩ F s : F q f ] = gcd([ F v : F q f ] , g ). Let i := [ F w : F q f ]. From the formulas[ F w F s : F w ] = [ F s : F q f ] = g, [ F w ( F v ∩ F s ) : F w ] = [ F v ∩ F s : F q f ] = h, [ F w F s : ( F w F s ∩ F v )] = [ F v F s : F v ] = [ F s : F v ∩ F s ] = gh , and F w ( F v ∩ F s ) ⊂ F w F s ∩ F v , we obtain F w F s ∩ F v = F w ( F v ∩ F s ) = F q fhi . Let F v,L be the compositum of Q w ⊗ F qf F s and F v inan algebraic closure of Q w . Note that F v,L is well defined since F s / F q f is Galois. Let F v,L [ T ′ ] be thenon-commutative polynomial ring with T ′ · ( a ⊗ b ) = ( a ⊗ b q fhi ) · T ′ and T ′ · x = x · T ′ for a ∈ Q w , b ∈ F s , and x ∈ F v and set ∆ v = F v,L [ T ′ ] / (cid:0) ( T ′ ) g/h − π iv (cid:1) . Observe that the commutationrules of T ′ are well defined since ( Q w ⊗ F qf F s ) ∩ F v has residue field F w F s ∩ F v = F q fhi and is unramifiedover Q w , because Q w ⊗ F qf F s is. Moreover, the extension F v,L /F v is unramified of degree [ F v F s : F v ] = gh and e T := ( T ′ ) [ F v : F q ] /fhi is its Frobenius automorphism. Since e T g/h = π [ F v : F q ] /fhv in ∆ v , our ∆ v is justthe cyclic algebra (cid:0) F v,L /F v , e T , π [ F v : F q ] /fhv (cid:1) and has Hasse invariant [ F v : F q ][ F s : F q ] · v ( π v ); compare [Rei, p. 266].We relate ∆ v to E ⊗ F F v . Firstly by [Jac, Theorem 3.20] there exists a positive integer u such that N ⊕ uv ∼ = R/Rµ v ( T g ). Therefore M u ( E ⊗ F F v ) ∼ = M u (cid:0) End R ( N ⊕ n v v ) (cid:1) = End R ( N ⊕ un v v ) = M n v (cid:0) ( R/Rµ v ( T g )) op (cid:1) . Secondly we choose integers m and n with m > mi + ng = 1. We claim that the morphism R/Rµ v ( T g ) → M h (∆ v ), which maps a ⊗ b a ⊗ b a ⊗ b q f . . . a ⊗ b q f ( h − and T π nv · T ′ ) m for a ∈ Q w and b ∈ F s , is an isomorphism of F v -algebras. It is well defined since it maps T · ( a ⊗ b ) and( a ⊗ b q f ) · T to the same element because ( T ′ ) m = ( T ′ ) /i in Gal( F v,L /F v ), and it maps T g = ( T h ) g/h to THE ENDOMORPHISM Q -ALGEBRA π ngv ( T ′ ) mg/h · Id h = π v · Id h . Since Rµ v ( T g ) ⊂ R is a maximal two sided ideal the morphism is injective.To prove surjectivity we compare the dimensions as Q w -vector spaces. We computedim F v M h (∆ v ) = h · ( gh ) = g , dim Q w ⊗ F qf F s (cid:0) R/Rµ v ( T g ) (cid:1) = g · deg µ v = g · [ F v : Q w ] , anddim Q w (cid:0) R/Rµ v ( T g ) (cid:1) = g · [ F v : Q w ] = dim Q w M h (∆ v ) . Altogether M u ( E ⊗ F F v ) ∼ = M hn v (∆ op v ) and inv v E = − inv v ∆ v = − [ F v : F q ][ F s : F q ] · v ( π v ) as claimed.It remains to convert this formula into the special form asserted for v ∤ ε ∞ or v |∞ . If v |∞ and ε = ∞ , let e v be the ramification index of F v /Q ∞ . Then we get from Theorem 7.8 the formula q e wt( F ) = | π | ∞ = q − v ( π v ) /e v , since the residue field of Q ∞ is F q . This implies as desired − [ F v : F q ][ F s : F q ] · v ( π v ) = − [ F v : F q ] · ( − e v e · wt( F )) e = wt( F ) · [ F v : Q ∞ ]Finally if w = ε, ∞ is a place of Q , the local σ -shtuka M w ( F ) at w is ´etale. So µ = mipo π |F hascoefficients in A w with constant term in A × w . Therefore v ( π v ) = 0 for all places v of F dividing w . Example 9.4.
Let C = P F q , C r {∞} = Spec F q [ t ] and L = F q . Let d be a positive integer. Let F i := O ( d ⌈ i ⌉ · ∞ ) ⊕ O ( d ⌈ i − ⌉ · ∞ ) for i ∈ Z and let τ := (cid:16) t d (cid:17) . Then F = ( F i , Π i , τ i ) is anabelian τ -sheaf of rank 2, dimension d , and characteristic ε = V ( t ) ∈ P over F q . Hence the Frobeniusendomorphism π equals τ . If d is odd then F is primitive (that means ( d, r ) = 1) and therefore simpleby [BH1, Proposition 7.4]. In particular, π is semisimple. We have µ π = χ v = x − t d = ( x − √ t d )( x + √ t d )which means that π v is not absolutely semisimple in characteristic 2. Moreover, we calculate r Q v ( χ v , χ v ) = 1 · · Q v ( √ t ) / Q v we have r Q v ( √ t ) ( χ v , χ v ) = (cid:26) · · · · · · E = F = Q ( π ) commutative and [ E : Q ] = 2 = r . Moreover, | π | ∞ = | √ t d | ∞ = q d/ and χ v is irreducible. But χ v is not separable in characteristic 2.If d = 2 n is even then the minimal polynomial of π is µ π = χ v = x − t d = ( x − t d/ )( x + t d/ ) . So π is semisimple if and only if char( F q ) = 2. In this case F is quasi-isogenous to the abelian τ -sheaf F ′ with F ′ i = O C L ( in · ∞ ) ⊕ and τ ′ i = (cid:16) − t n t n (cid:17) . The quasi-isogeny f : F ′ → F is given by f ,η = (cid:16) − t n t n (cid:17) : F ′ ,η ∼ −→ F ,η . The abelian τ -sheaf F ′ equals the direct sum F (1) ⊕ F (2) where F ( j ) i = O C L ( in · ∞ ) and τ ( j ) i = ( − j t n . Note that F (1) and F (2) are not isogenous over F q , since theequation − t n · σ ∗ ( g ) = g · t n has no solution g ∈ Q for char( F q ) = 2. Therefore Q ⊕ Q = M j =1 QEnd( F ( j ) ) ∼ = E = F = Q [ x ] / ( x − t n ) ∼ = Q ⊕ Q .
Now we consider the same abelian τ -sheaf over L = F q . This means π = τ = t d ∈ Q and therefore χ v = ( x − t d ) . Thus π is semisimple. By Theorem 9.1/4 we have F = Q ( π ) = Q and E is central THE ENDOMORPHISM Q -ALGEBRA Q with [ E : Q ] = 4 and inv ∞ E = inv ε E = d . Moreover, | π | ∞ = | t d | ∞ = q d . In thiscase, π v is absolutely semisimple. Note that if d is even and char( F q ) = 2 this is another example forTheorem 6.15.If d is odd then F is still primitive, whence simple and E is a division algebra. If d = 2 n is eventhen the abelian τ -sheaves F (1) and F (2) defined above are isomorphic F (1) ∼ −→ F (2) , λ where λ ∈ F q satisfies λ q − = −
1. Therefore M ( Q ) = M (cid:0) QEnd( F (1) ) (cid:1) ∼ = E in accordance with the Hasseinvariants just computed. Example 9.5.
We compute another example which displays other phenomena. Let C = P F q and let C r {∞} = Spec F q [ t ]. Let F i = O C L ( ⌈ i − ⌉ · ∞ ) ⊕ ⊕ O C L ( ⌈ i ⌉ · ∞ ) ⊕ , let Π i be the natural inclusion,and let τ i be given by the matrix T := a b t − b t with a, b ∈ F q r { } . Then F is an abelian τ -sheaf of rank 4 and dimension 2 with l = 2 , k = 1 and characteristic ε = V ( t ) ∈ P . One checks that the minimal polynomial of the matrix T is x − b x − at which is irreducibleover Q if char( F q ) = 2, since it has neither zeroes in F q [ t ] nor quadratic factors in Q [ x ]. If char( F q ) = 2then the minimal polynomial is a square and F is not semisimple.For L = F q and 2 ∤ q we obtain π = τ semisimple and E = F = Q ( π ) = Q [ x ] / ( x − b x − at ).For L = F q we have π = τ and the minimal polynomial of π over Q is x − b x − at , which isirreducible also in characteristic 2 since it has no zeroes in F q [ t ]. Hence π is semisimple, F is a fieldwith [ F : Q ] = 2 and [ E : F ] = 4 by Corollary 6.6. This again illustrates Theorem 6.15. We computethe decomposition of ∞ and ε in F . Decomposition of ε : Modulo t the polynomial x − b x − at has two zeroes x = b and x = 0 in F q . Soby Hensel’s lemma F ⊗ Q Q ε ∼ = F v ⊕ F v ′ splits with F v ∼ = F v ′ ∼ = Q ε and v ( π ) = 0 and v ′ ( π ) = v ′ ( at ) = 2.Thus the Hasse invariants of E are inv v E = inv v ′ E = 0. Decomposition of ∞ : Set y = π/t . Then y − b t y − a = 0. Case (a).
If 2 | q then ( y − a q/ ) − b t ( y − a q/ ) − b t a q/ = 0, that is, ∞ ramifies in F , F ⊗ Q Q ∞ = F w with w ( πt − a q/ ) = 1 and w ( t ) = 2 · ∞ ( t ) = 2. So [ F w : Q ∞ ] = 2 and inv w E = 0. Case (b).
If 2 ∤ q and √ a ∈ F q then the polynomial y − b t y − a has two zeroes y = ±√ a modulo t .So by Hensel’s lemma F ⊗ Q Q ∞ ∼ = F w ⊕ F w ′ splits with [ F w : Q ∞ ] = [ F w ′ : Q ∞ ] = 1. Thus the localHasse invariants of E are inv w E = inv w ′ E = . As was remarked in 9.2 such a distribution of theHasse invariants can occur only if d ≥ Case (c).
If 2 ∤ q and √ a / ∈ F q then y − b t y − a is irreducible modulo t and ∞ is inert in F , F ⊗ Q Q ∞ = F w with [ F w : Q ∞ ] = 2. Thus the Hasse invariant of E is inv w E = 0.In case (b) E is a division algebra and F is simple. In cases (a) and (c) E ∼ = M ( F ) and F isquasi-isogenous to ( F ′ ) ⊕ for an abelian τ -sheaf F ′ of rank 2, dimension 1 and QEnd( F ′ ) = F . Thissurprising result is due to the fact that F ′ , being of dimension 1, is associated with a Drinfeld moduleand thus of the form F ′ i = O C L ( ⌈ i ⌉ · ∞ ) ⊕ O C L ( ⌈ i − ⌉ · ∞ ) with τ ′ i = (cid:0) cd t (cid:1) and c, d ∈ F q . Then π ′ = ( τ ′ ) = (cid:16) c q +1 + d q tc q d ctdt (cid:17) has minimal polynomial x − ( c q +1 + ( d + d q ) t ) x + d q +1 t which must beequal to x − b x − at . This is possible only if d + d q = 0 and d q +1 = − a . So either d ∈ F q and 2 | q and we are in case (a), or d ∈ F q r F q , d q = − d , and a = d . The later implies 2 ∤ q and √ a = d / ∈ F q and we are in case (c). If we choose c = b in case (c) a quasi-isogeny f : F → ( F ′ ) ⊕ over F q is givenfor instance by d a − bd/t
00 0 − d a d/t a/t − d bd/t .
10 Kernel Ideals for Pure Anderson Motives
In this section we investigate which orders of E can arise as endomorphism rings End( M ) for pure An-derson motives M . For this purpose we define for each right ideal of the endomorphism ring End( M )an isogeny with target M and discuss its properties. This generalizes Gekeler’s results for Drinfeldmodules [Gek, §
3] and translates the theory of Waterhouse [Wat, §
3] for abelian varieties to the func-tion field case. These two sources are themselves the translation, respectively the higher dimensionalgeneralization of Deuring’s work on elliptic curves [Deu].Let M be a pure Anderson motive over L and abbreviate R := End( M ). Let I ⊂ R be a right idealwhich is an A -lattice in E := R ⊗ A Q . This is equivalent to saying that I contains an isogeny, sinceevery lattice contains some isogeny a · id M for a ∈ A and conversely the existence of an isogeny f ∈ I implies that the lattice f · f ∨ · R is contained in I . Definition 10.1.
1. Let M I be the pure Anderson sub-motive of M whose underlying A L -module is P g ∈ I im( g ) . This is indeed a pure Anderson motive, since if I = f R + . . . + f n R are arbitrarygenerators, then M I equals the image of the morphism ( f , . . . , f n ) : M ⊕ . . . ⊕ M −→ M . As I contains an isogeny, M I has the same rank as M and the natural inclusion is an isogenywhich we denote f I : M I → M .2. If I = { f ∈ R : im( f ) ⊂ M I } then I is called a kernel ideal for M . The later terminology is borrowed from Waterhouse [Wat, § { f ∈ R : im( f ) ⊂ M I } is theright ideal annihilating coker f I one should maybe use the name “cokernel ideal” instead. Proposition 10.2.
Let I ⊂ R be a right ideal which is a lattice, and consider the right ideal J := { f ∈ R : im( f ) ⊂ M I } ⊂ R containing I . Then M J = M I . In particular, J is a kernel ideal for M . Wecall J the kernel ideal for M associated with I .Proof. Obviously J is a right ideal and M J ⊂ M I by definition of J . Conversely M I ⊂ M J since I ⊂ J . Lemma 10.3.
1. For any g ∈ I , f − I ◦ g : M → M I is a morphism and g = f I ◦ ( f − I ◦ g ) .2. If I = gR is principal, g an isogeny, then f − I ◦ g : M → M I is an isomorphism and I is a kernelideal.Proof. g lies inside M I .2. Clearly f − I ◦ g is injective since g is an isogeny and surjective by construction, hence an isomorphism.To show that I is a kernel ideal let f ∈ R satisfy im( f ) ⊂ M I . Consider the diagram M h (cid:15) (cid:15) (cid:31)(cid:31)(cid:31) f − I ◦ f / / M I f I / / MM f − I ◦ g rrrrrrrrrrrr and let h := ( f − I ◦ g ) − ◦ ( f − I ◦ f ). Then f = gh ∈ I as desired. Example. If a ∈ A and I = aR , then M I = aM and coker f I = M /aM . More generally if a ⊂ A is anideal and I = a R then M I = a M and coker f I = M / a M . Proposition 10.4.
Let I ⊂ R and J ⊂ End( M I ) be right ideals which are lattices in E . Then also theproduct K := f I · J · f − I · I is a right ideal of R and a lattice in E and f − K ◦ f I ◦ f J is an isomorphismof ( M I ) J with M K ( M I ) J f J −−−→ M I f I −−→ M f K ←−−− M K . Proof. If f ∈ I and g ∈ J then the morphism f − I ◦ f : M → M I can be composed with f I ◦ g to yieldan element of R . Since I and J contain isogenies, K is a right ideal and contains an isogeny. Clearlythe images of f I ◦ f J and f K in M coincide since they equal the sum P i,j f I ◦ g j ◦ f − I ◦ f i ( M ) for setsof generators { f i } of I and { g j } of J . Theorem 10.5.
Let
I, J ⊂ End( M ) =: R be right ideals which are lattices in E := R ⊗ A Q and considerthe following assertions:1. I and J are isomorphic R -modules,2. the pure Anderson motives M I and M J are isomorphic.Then 1 implies 2 and if moreover I and J are kernel ideals, also 2 implies 1.Proof. ⇒
2. Since I and J are lattices, the R -isomorphism I → J extends to an E -isomorphism of E and is thus given by left multiplication with a unit g ∈ E × , that is, J = gI . There is an a ∈ A suchthat ag ∈ I ⊂ R . Then im( ag ) ⊂ M I , that is, f − I ◦ ag : M → M I is an isogeny.Let K be the right ideal f I · (cid:0) f − I ◦ ag ◦ f I · End( M I ) (cid:1) · f − I · I of R . We claim that M K ∼ = M ( ag ) I .Namely, M ( ag ) I ⊂ M K since agI ⊂ K . Conversely if f ∈ I , h ∈ End( M I ), and m ∈ M , then we find m ′ := f I ◦ h ◦ f − I ◦ f ( m ) ∈ M I , that is, m ′ = P i f i ( m i ) for suitable f i ∈ I and m i ∈ M . It follows that ag ( m ′ ) = P i agf i ( m i ) ∈ M ( ag ) I and therefore M ( ag ) I = M K .Applying Lemma 10.3 and Proposition 10.4 now yields an isomorphisms M I ∼ = M K = M ( ag ) I .Likewise we obtain M J ∼ = M aJ and the equality aJ = agI then implies M J ∼ = M I as desired.2 ⇒
1. Let I and J be kernel ideals and let u : M I → M J be an isomorphism. There is an a ∈ A with aM ⊂ M I . Therefore g := f J ◦ u ◦ ( f − I ◦ a ) : M → M is an isogeny.We claim that gI = aJ , that is, left multiplication by a − g is an isomorphism of I with J . Let f ∈ I ,then h := f J ◦ u ◦ ( f − I ◦ f ) ∈ R has im( h ) ⊂ M J . So h ∈ J since J is a kernel ideal, and gf = ah ∈ aJ ,since a commutes with all morphisms. Conversely let h ∈ J , then f := f I ◦ u − ◦ ( f − J ◦ h ) ∈ R hasim( f ) ⊂ M I . So f ∈ I since I is a kernel ideal, and ah = gf ∈ gI as desired. Proposition 10.6.
Let I ⊂ R be a right ideal which is a lattice in E . Then f I · End( M I ) · f − I containsthe left order O = { f ∈ E : f I ⊂ I } of I and equals it if I is a kernel ideal.Remark. Recall that End( M I ) ⊗ A Q is identified with E by mapping h ∈ End( M I ) to f I ◦ h ◦ f − I . Proof.
Let f ∈ O and g ∈ I . Then f g ∈ I and f − I ◦ f ◦ f I ◦ ( f − I ◦ g ) = f − I ◦ f g is a morphismfrom M to M I . If g varies, the images of f − I ◦ g exhaust all of M I . Hence f − I ◦ f ◦ f I is indeed anendomorphism of M I . Conversely let I be a kernel ideal and let f = f I ◦ h ◦ f − I ∈ f I · End( M I ) · f − I .If g ∈ I then f ◦ g = f I ◦ h ◦ ( f − I ◦ g ) ∈ R has im( f ◦ g ) ⊂ M I . So f g ∈ I as desired.We will now draw conclusions about the endomorphism ring R similar to Waterhouse’ results [Wat]on abelian varieties by simply translating his arguments. Theorem 10.7.
Every maximal order in E occurs as the endomorphism ring f · End( M ′ ) · f − ⊂ E of a pure Anderson motive M ′ isogenous to M via an isogeny f : M ′ → M .Proof. Let S be a maximal order of E . Then the lattice R contains aS for some a ∈ A . Consider theright ideal I = aS · R whose left order contains S . By Proposition 10.6, f I · End( M I ) · f − I contains theleft order of I . Since S is maximal we find S = f I · End( M I ) · f − I . Theorem 10.8. If E is semisimple and End( M ) is a maximal order in E , so is f I · End( M I ) · f − I forany right ideal I ⊂ R .Proof. By [Rei, Theorem 21.2] the left order of I is also maximal and then Proposition 10.6 yields theresult. L is a finite field and we set e := [ L : F q ]. Let π be the Frobeniusendomorphism of M . Proposition 10.9.
The order R in E contains π and deg( π ) /π .Proof. Clearly the isogeny π belongs to R . Let now a ∈ deg( π ). Then a annihilates coker π by 2.10and so there is an isogeny f : M → M with π ◦ f = a . The image a/π of f in E belongs to R . Proposition 10.10. If M is a semisimple pure Anderson motive over a finite field and End( M ) is amaximal order in E = End( M ) ⊗ A Q , then every right ideal I ⊂ End( M ) , which is a lattice, is a kernelideal for M , and deg( f I ) = N ( I ) := (cid:0) N ( f ) : f ∈ I (cid:1) .Proof. (cf. [Wat, Theorem 3.15]) Let f ∈ I , then f = f I ◦ f − I f and N ( f ) ∈ deg( f ) ⊂ deg( f I ) byLemma 2.9. Therefore N ( I ) ⊂ deg( f I ). Let R ′ be the left order of I . It is maximal by [Rei, Theorem21.2]. For a suitable a ∈ A the set J ′ := { x ∈ E : xI ⊂ aR } is a right ideal in R ′ and a lattice in E and satisfies J ′ · I = aR by [Rei, Theorem 22.7]. Let J := f − I J ′ f I ⊂ End( M I ) be the induced rightideal of End( M I ) = f − I R ′ f I ; see 10.6. Then coker f I ◦ f J = coker f J ′ I = coker a by Proposition 10.4.Therefore Theorem 7.3 and [Rei, 24.12 and 24.11] imply N ( a ) · A = N ( J ′ ) · N ( I ) ⊂ (deg f J )(deg f I ) = deg( a ) = N ( a ) · A .
By the above we must have N ( I ) = deg( f I ) since A is a Dedekind domain. If I were not a kernel idealits associated kernel ideal would be a larger ideal with the same norm. But this is impossible by [Rei,24.11].Like for abelian varieties there is a strong relation between the ideal theory of orders of E and theinvestigation of isomorphy classes of pure Anderson motives isogenous to M . We content ourselves withthe following result which is analogous to Waterhouse [Wat, Theorem 6.1]. The interested reader willfind many other results without much difficulty. Theorem 10.11.
Let M be a simple pure Anderson motive of rank r and dimension d over the smallestpossible field F q . Then1. End( M ) is commutative and E := End( M ) ⊗ A Q = Q ( π ) .2. All orders R in Q ( π ) containing π are endomorphism rings of pure Anderson motives isogenousto M . Any such order automatically contains N ( π ) /π = N Q ( π ) /Q ( π ) /π .3. For each such R the isomorphism classes of pure Anderson motives isogenous to M with endo-morphism ring R correspond bijectively to the isomorphism classes of A -lattices in E with order R .Proof. R be an order in Q ( π ) containing π and let v = ε be a maximal ideal of A . Since [ E : Q ] = r and E v is semisimple, there is by Lemma 7.2 an isomorphism E v ∼ −→ V v M of (left) E v -modules givenby f f ( x ) for a suitable x ∈ V v M . It identifies R v := R ⊗ A A v with a π -stable lattice Λ v = R v · x in V v M , which without loss of generality is contained in T v M . By Proposition 4.4 there is an isogeny f : M ′ → M of pure Anderson motives with T v f ( T v M ′ ) = Λ v . By Theorem 4.2 we concludeEnd( M ′ ) ⊗ A A v = End A v [ π ] (Λ v ) = R v . For v = ε note that Q ε,L = Q ε since L = F q . In particular F ε = F q . Since dim Q ε N ε ( M ) = r = [ E : Q ],Theorem 3.7 together with Lemma 7.2 show that E ε is isomorphic to N ε ( M ) as left E ε -modules. Since R contains π , the image of R ε := R ⊗ A A ε in N ε ( M ) is a local σ -subshtuka ˆ M ′ of M ε ( M ) of the samerank. (If it is not contained in M ε ( M ), multiply it with a suitable a ∈ A .) Then Proposition 3.9 yieldsan isogeny of pure Anderson motives f : M ′ → M such that M ε ( f ) (cid:0) M ε ( M ′ ) (cid:1) = ˆ M ′ andEnd( M ′ ) ⊗ A A ε = End A ε [ φ ] ( ˆ M ′ ) = R ε EFERENCES M ) at the respective place v , thisshows that we may modify M at all places to obtain a pure Anderson motive M ′ with End( M ′ ) = R .Now the last statement follows from Proposition 10.9 and Theorem 7.3.3. Let R be such an order. By what we proved in 2 there is a pure Anderson motive e M for whichall T v e M ∼ = R v and M ε ( e M ) ∼ = R ε . Let I ⊂ R be a (right) ideal which is an A -lattice in E andconsider the isogeny f I : e M I → e M . Under the above isomorphisms T v f I ( T v e M I ) ∼ = I ⊗ A A v =: I v and M ε f I ( M ε e M I ) ∼ = I ⊗ A A ε =: I ε . Conversely if f : M ′ → e M is an isogeny then M v f ( M v M ′ ) is a (left) R v -module because R = End( e M ), hence isomorphic to an R v -ideal I v . This shows that any isogeny f : M ′ → e M is of the form f I : e M I → e M .If now f ∈ R satisfies im( f ) ⊂ e M I then f ∈ I ε and f ∈ I v for all v and therefore f ∈ I . This showsthat every I is a kernel ideal for e M . By Proposition 10.6, End( e M I ) is the (left) order of I . Since everylattice with order R in E is isomorphic to an ideal of R , we have { A -lattices in E with order R } / ∼ { I ⊂ R Ideals with order R } / ∼ ∼ / / (cid:8) e M I f I −→ e M → M with End( e M I ) = R (cid:9) / ∼ and the assertion now follows from Theorem 10.5. References [An1] G. Anderson: t -Motives, Duke Math. J. (1986), 457–502.[An2] G. Anderson: On Tate Modules of Formal t -Modules, Internat. Math. Res. Notices (1993), 41–52.[BH1] M. Bornhofen, U. Hartl: Pure Anderson Motives and Abelian τ -Sheaves , Preprint on arXiv:0709.2809 ,version 1 from 09/18/2007.[BH2] M. Bornhofen, U. Hartl: Pure Anderson Motives and Abelian τ -Sheaves over Finite Fields , Preprint on arXiv:math.NT/0609733 , version 3 from 09/18/2007.[BS] A. Blum, U. Stuhler: Drinfeld Modules and Elliptic Sheaves, in: Vector Bundles on Curves: New Di-rections , S. Kumar, G. Laumon, U. Stuhler, M. S. Narasimhan, eds., pp. 110–188, Lecture Notes inMathematics 1649, Springer-Verlag, Berlin, etc. 1991.[Boe] G. B¨ockle: Global L -functions over function fields. Math. Ann. (2002), no. 4, 737–795.[Bou] N. Bourbaki:
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