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 the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we 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 semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the Zeta function and the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers 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 Drinfeld’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 Arithmetic of Number Fields, like their higher dimensional generalizations abelian varieties. In [BH1, Ha1] we claimed that pure Anderson motives (a slight generalization of the pure t-motives introduced by Anderson [An1]) and abelian τ-sheaves should be viewed as the appropriate analogues for abelian varieties and higher dimensional generalizations of elliptic sheaves or modules. We want to further support this claim 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 ∖ {∞}, O_{C}). For a field L ⊃ 𝔽_{q} let σ^{∗} be the endomorphism 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 ⊗ 1 − 1 ⊗ c^{∗}(a) : a ∈ A) ⊂ A_{L}. A pure Anderson motive \underline{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, such that 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 \frac{k}{l} equals \frac{d}{r} and is called the weight of \underline{M}. Anderson’s definition of pure t-motives [An1] is recovered by setting C = ℙ_{𝔽_{q}}^{1} and A = 𝔽_{q}[t]. In the first two sections we recall the definition of morphisms and isogenies between pure Anderson motives as well as some facts from [BH1]. Also for an isogeny f between pure Anderson motives we define the degree of f as an ideal of A (2.8) which annihilates coker f (2.10). If \underline{M} is a semisimple (see below) pure Anderson motive over a finite field, the degree of any isogeny f : \underline{M} → \underline{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 HA3006/2-1 and SFB 478

Next we address the question whether every pure Anderson motive is semisimple, that is, isogenous to a direct sum of simple pure Anderson motives. A pure Anderson motive is called simple if it has no non-trivial factor motives. This question is the analogue of the classical theorem of Poincaré-Weil on the semisimplicity of abelian varieties. By giving a counterexample (Example 6.1) we demonstrate that the answer to this question is negative in general. On the positive side we show that every pure Anderson motive over a finite base field becomes semisimple after a field extension whose degree is a power of q (6.15), and then stays semisimple after any further field extension (6.16). Let Q be the function field of C. Then the endomorphism Q-algebra QEnd(\underline{M}) := End(\underline{M}) ⊗_{A} Q of a semisimple pure Anderson motive is semisimple (2.7) and over a finite field also the converse is true (6.11). This is false however over an infinite field (6.13).

Like for abelian varieties the behavior of a pure Anderson motive \underline{M} over a finite field is controlled by its Frobenius endomorphism π (defined in 5.2). If \underline{M} is semisimple we determine the dimension and the local Hasse invariants of its endomorphism Q-algebra QEnd(\underline{M}) in terms of π (6.5, 9.1). We define a Zeta function Z_{\underline{M}} for a pure Anderson motive \underline{M} (Definition 7.6) and we show that it satisfies the Riemann hypothesis (7.8), and has an expression in terms of the degrees deg(1 − π^{i}) for all i if \underline{M} is semisimple (7.7). We prove the following isogeny criterion.

Theorem 8.1. Let \underline{M} and \underline{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. \underline{M} and \underline{M}′ are isogenous.

2. The characteristic polynomials of π and π′ acting on the v-adic Tate modules of \underline{M}, respectively \underline{M}′, coincide for some (any) place v ∈ Spec A.

3. There exists an isomorphism of Q-algebras QEnd(\underline{M}) ≅ QEnd(\underline{M}′) mapping π to π′.

4. Z_{\underline{M}} = Z_{\underline{M}}′.

In the last section we sketch a few results for the question, which orders of QEnd(\underline{M}) occur as the endomorphism rings of pure Anderson motives (10.7, 10.11). There is a relation between the breaking up of the isogeny class of a semisimple pure Anderson motive into isomorphism classes, and the arithmetic of QEnd(\underline{M}). We indicate this by treating the case of pure Anderson motives defined over the minimal field 𝔽_{q}. In this case QEnd(\underline{M}) is commutative (10.11). Many of our results parallel Tate’s celebrated article [Tat] on abelian varieties over finite fields. To prove them, a major tool are the Tate modules and local shtuka attached to pure Anderson motives, which we recall in Sections 4 and 3, and the analogue [Tag, Tam] of Tate’s conjecture on endomorphisms. These local structures behave like in the classical case of abelian varieties, local shtuka playing the role of the p-divisible groups of the abelian varieties. The only difference is that p-divisible groups are only useful for abelian varieties in characteristic 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 of local shtuka. For instance we apply them in the computation of the Hasse invariants of QEnd(\underline{M}) in Theorem 9.1. We also used them in [BH1] to reprove the standard fact that the set of morphisms between two pure Anderson motives is a projective A-module (1.3). Scattered in the text are several interesting examples displaying various phenomena (6.1, 6.13, 9.4, 9.5). Note that there is a two in one version [BH2] of the present article and [BH1] on the arXiv.

Contents

Introduction 1

1 Pure Anderson Motives and Abelian τ-Sheaves 4 CONTENTS 3

2 Isogenies and Quasi-Isogenies 6 3 Local Shtuka 8 4 Tate Modules 12 5 The Frobenius Endomorphism 13 6 The Poincaré-Weil Theorem 14 7 Zeta Functions and Reduced Norms 21 8 A Quasi-Isogeny Criterion 23 9 The Endomorphism Q-Algebra 25 10 Kernel Ideals for Pure Anderson Motives 31 References 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 ∖ {∞}, 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} ˆ⊗_{𝔽_{q}}L  for the completion of O_{C_{L}} at the closed subscheme v × Spec L,
Q_{v,L} = A_{v,L}[\frac{1}{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_{C_{L}}} O_{C_{L}}(D). This notation applies in particular 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.

1 PURE ANDERSON MOTIVES AND ABELIAN τ-SHEAVES 4

1 Pure Anderson Motives and Abelian τ-Sheaves

Pure Anderson motives were introduced by G. Anderson [An1] under the name pure t-motives in the case where A = 𝔽_{p}[t]. They were further studied in [BH1]. To give their definition let L be a field extension of 𝔽_{q} and fix an 𝔽_{q}-homomorphism c^{∗} : A → L. Let J ⊂ A_{L} be the ideal generated by a ⊗ 1 − 1 ⊗ c^{∗}(a) for all a ∈ A.

Definition 1.1 (pure Anderson motives). A pure Anderson motive \underline{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}-module homomorphism τ : σ^{∗}M → M such that

1. the cokernel of τ is an L-vector space of dimension d and annihilated by J^{d}, and

2. 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−1}(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 \underline{M} and we say that \underline{M} has finite characteristic (respectively generic characteristic) if ε is a closed (respectively the generic) point. The ratio wt(M, τ) := \frac{k}{l} equals \frac{d}{r} 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 a morphism f : M → M′ of A_{L}-modules which satisfies f ∘ τ = τ′ ∘ σ^{∗}(f).

2. If f : \underline{M} → \underline{M}′ is surjective, \underline{M}′ is called a factor motive of \underline{M}.

3. A morphism f : \underline{M} → \underline{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 the morphism τ ∘ σ^{∗}τ ∘ … ∘ (σ^{∗})^{n}τ is zero).

We denote the set of morphisms between \underline{M} and \underline{M}′ by Hom(\underline{M}, \underline{M}′). It is an A-module.

If \underline{M} and \underline{M}′ are pure Anderson motives of different weights then Hom(\underline{M}, \underline{M}′) = \{0\} by [BH1, Corollary 3.5]. This justifies the terminology pure. The following fact is well known. A proof can be found for instance in [BH1, Theorem 9.5].

Theorem 1.3. Let \underline{M} and \underline{M}′ be pure Anderson motives over an arbitrary field L. Then Hom(\underline{M}, \underline{M}′) is a projective A-module of rank ≤ rr′. The minimal polynomial of every endomorphism of a pure Anderson motive \underline{M} lies in A[x].

Corollary 1.4. ([BH1, Corollary 5.4]) Let f : \underline{M} → \underline{M}′ be an isogeny between pure Anderson motives. Then

1. there exists an element a ∈ A which annihilates coker f,

2. there exists a dual isogeny f^{∨} : \underline{M}′ → \underline{M} such that f ∘ f^{∨} = a · id_{\underline{M}}′ and f^{∨} ∘ f = a · id_{\underline{M}} .

Next we come to the notion of abelian (τ-)sheaves. It was introduced in [Ha1] in order to construct moduli spaces for pure Anderson motives. We briefly recall the results from [BH1] on the relation between pure Anderson motives and abelian τ-sheaves. Although our primary interest is on pure Anderson 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 general situation. Moreover, some results are proved most naturally via the use of abelian τ-sheaves (e.g. 9.1 and 7.3 below). The fact that ε = ∞ is allowed for abelian τ-sheaves was crucial for the uniformization of the moduli spaces of pure Anderson motives in [Ha1] and the derived consequences on analytic uniformization 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.

Definition 1.5 (Abelian τ-sheaf). An abelian τ-sheaf \underline{F} = (F_{i}, Π_{i}, τ_{i}) of rank r, dimension d and characteristic c over L is a collection of locally free sheaves F_{i} on C_{L} of rank r together with injective morphisms Π_{i}, τ_{i} of O_{C_{L}}-modules (i ∈ ℤ) of the form

⋯ ⟶ F_{i−1} ⟶^{Π_{i−1}} F_{i}  ⟶^{Π_{i}} F_{i+1} ⟶^{Π_{i+1}} ⋯
↑^{τ_{i−2}}  ↑^{τ_{i−1}}  ↑^{τ_{i}}
⋯ ⟶ σ^{∗}F_{i−2} ⟶^{σ^{∗}Π_{i−2}} σ^{∗}F_{i−1} ⟶^{σ^{∗}Π_{i−1}} σ^{∗}F_{i} ⟶^{σ^{∗}Π_{i}} ⋯

subject to the following conditions:

1. the above diagram is commutative,

2. there exist integers k, l > 0 with ld = kr such that the morphism Π_{i+l−1} ∘ ⋯ ∘ 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 \underline{F} has finite (respectively generic) characteristic if ε is a closed (respectively the generic) point. If r ≠ 0 we call wt(\underline{F}) := \frac{d}{r} the weight of \underline{F}.

Remark. 1. If \underline{F} is an abelian τ-sheaf and D a divisor on C, then \underline{F}(D) := (F_{i}(D), Π_{i}⊗ 1, τ_{i}⊗ 1) is an abelian τ-sheaf of the same rank and dimension as \underline{F}.

2. Let \underline{F} be an abelian τ-sheaf and let n ∈ ℤ. We denote by \underline{F}[ n] := (F_{i+n}, Π_{i+n}, τ_{i+n}) the n-shifted abelian τ-sheaf of \underline{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 \underline{F} = (F_{i}, Π_{i}, τ_{i}) and \underline{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) which commute 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 the set of morphisms between \underline{F} and \underline{F}′ by Hom(\underline{F}, \underline{F}′). It is an 𝔽_{q}-vector space.

For example, the collection of morphisms (Π_{i}) : \underline{F} → \underline{F}[ 1 ] defines a morphism between the abelian τ-sheaves \underline{F} and \underline{F}[ 1 ].

Definition 1.7. Let \underline{F} and \underline{F}′ be abelian τ-sheaves and let f ∈ Hom(\underline{F}, \underline{F}′) be a morphism. Then f is called injective (respectively surjective, respectively an isomorphism), if f_{i} is injective (respectively surjective, respectively bijective) for all i ∈ ℤ. We call \underline{F} an abelian factor τ-sheaf of \underline{F}′, if there is a surjective morphism from \underline{F}′ onto \underline{F}.

If \underline{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

\underline{M}(\underline{F}) := (M, τ) := (Γ(C_{L} ∖ \{∞\}, F_{0}) , Π_{0}^{−1} ∘ τ_{0})  (1.1)

is a pure Anderson motive of the same rank and dimension and of characteristic c^{∗} : A → L. Conversely we 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) = \underline{M}(\underline{F}) for an abelian τ-sheaf \underline{F} over L of same rank and dimension with characteristic c := Spec c^{∗} : Spec L → Spec A ⊂ C. One can even find the abelian τ-sheaf \underline{F} with k, l relatively prime.

2 Isogenies and Quasi-Isogenies

We recall the basic facts about isogenies from [BH1].

Proposition 2.1. ([BH1, Proposition 5.1]) Let f : \underline{F} → \underline{F}′ be a morphism between two abelian τsheaves \underline{F} = (F_{i}, Π_{i}, τ_{i}) and \underline{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 \underline{F} and \underline{F}′ have the same rank and dimension,

3. \underline{F} and \underline{F}′ have the same weight and the fiber f_{i,η} at the generic point η of C_{L} is an isomorphism for some (any) i ∈ ℤ.

Definition 2.2. 1. A morphism f : \underline{F} → \underline{F}′ satisfying the equivalent conditions of Proposition 2.1

is called an isogeny. We denote the set of isogenies between \underline{F} and \underline{F}′ by Isog(\underline{F}, \underline{F}′).

2. An isogeny f : \underline{F} → \underline{F}′ is called separable (respectively purely inseparable) if for all i the induced morphism τ_{i} : σ^{∗} coker f_{i} → coker f_{i+1} is an isomorphism (respectively is nilpotent, that is, τ_{i} ∘ σ^{∗}τ_{i−1} ∘ … ∘ (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-)morphisms to have “poles” we get rings which are related to the endomorphism rings of the associated pure Anderson motives. We make the following:

Definition 2.3 (Quasi-morphism and quasi-isogeny). Let \underline{F} and \underline{F}′ be abelian τ-sheaves.

1. A quasi-morphism f between \underline{F} and \underline{F}′ is a morphism f ∈ Hom(\underline{F}, \underline{F}′(D)) for some effective divisor D on C.

2. A quasi-isogeny f between \underline{F} and \underline{F}′ is an isogeny f ∈ Isog(\underline{F}, \underline{F}′(D)) for some effective divisor D on C.

If D_{1} ≤ D_{2} the composition with the inclusion isogeny \underline{F}′(D_{1}) ⊂ \underline{F}′(D_{2}) defines an injection Hom(\underline{F}, \underline{F}′(D_{1})) ↪ Hom(\underline{F}, \underline{F}′(D_{2})). This yields an equivalence relation for quasi-morphisms and quasi-isogenies. We let QHom(\underline{F}, \underline{F}′) and QIsog(\underline{F}, \underline{F}′) be the set of quasi-morphisms, respectively quasi-isogenies, between \underline{F} and \underline{F}′ modulo this equivalence relation. We write QEnd(\underline{F}) := QHom(\underline{F}, \underline{F}) and QIsog(\underline{F}) := QIsog(\underline{F}, \underline{F}).

The Q-vector spaces QHom(\underline{F}, \underline{F}′) and QEnd(\underline{F}) are finite dimensional, and QIsog(\underline{F}) is the group of units in the Q-algebra QEnd(\underline{F}), see [BH1, Propositions 6.5 and 9.4].

Two abelian τ-sheaves \underline{F} and \underline{F}′ are called quasi-isogenous (notation: \underline{F} ≈ \underline{F}′), if there exists a quasi-isogeny between \underline{F} and \underline{F}′. The relation ≈ is an equivalence relation. If \underline{F} ≈ \underline{F}′, then the Q-algebras QEnd(\underline{F}) and QEnd(\underline{F}′) are isomorphic, and QHom(\underline{F}, \underline{F}′) is free of rank 1 both as a left module over QEnd(\underline{F}′) and as a right module over QEnd(\underline{F}).

Proposition 2.4. ([BH1, Proposition 6.10]) Let \underline{F} and \underline{F}′ be two abelian τ-sheaves of characteristic ε ≠ ∞ and let \underline{M}(\underline{F}) and \underline{M}(\underline{F}′) be their associated pure Anderson motives. Then there is a canonical isomorphism of Q-vector spaces

QHom(\underline{F}, \underline{F}′) = Hom(\underline{M}(\underline{F}), \underline{M}(\underline{F}′)) ⊗_{A} Q .

If \underline{M} and \underline{M}′ are pure Anderson motives, then the elements of Hom(\underline{M}, \underline{M}′) ⊗_{A} Q which admit an inverse in Hom(\underline{M}′, \underline{M}) ⊗_{A} Q are called quasi-isogenies. With this definition we can reformulate Propositions 1.8 and 2.4 as follows.

Corollary 2.5. Let ε ≠ ∞. Then the functor \underline{F} ↦ \underline{M}(\underline{F}) defines an equivalence of categories between

1. the category with abelian τ-sheaves as objects and with QHom(\underline{F}, \underline{F}′) as the set of morphisms, 2. and the category with pure Anderson motives as objects and with Hom(\underline{M}, \underline{M}′) ⊗_{A} Q as the set of

morphisms.

We call these the quasi-isogeny categories of abelian τ-sheaves of characteristic different from ∞ and of pure Anderson motives, respectively.

Definition 2.6. Let \underline{F} be an abelian τ-sheaf.

1. \underline{F} is called simple, if \underline{F} ≠ \underline{0} and \underline{F} has no abelian factor τ-sheaves other than \underline{0} and \underline{F}.

2. \underline{F} is called semisimple, if \underline{F} admits, up to quasi-isogeny, a decomposition into a direct sum \underline{F} ≈ \underline{F}_{1} ⊕ ⋯ ⊕ \underline{F}_{n} of simple abelian τ-sheaves \underline{F}_{j} (1 ≤ j ≤ n).

We make the same definition for a pure Anderson motive.

Remark. 1. Let \underline{F} be an abelian τ-sheaf with characteristic different from ∞. Then \underline{F} is (semi-)simple if and only if the pure Anderson motive \underline{M}(\underline{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 a\underline{M} ⊂ \underline{M} is a proper sub-motive for any a ∈ A ∖ A^{×}. This shows that pure Anderson motives behave dually to abelian varieties. Namely an abelian variety is called simple if it has no non-trivial abelian subvarieties.

Theorem 2.7. ([BH1, Theorem 7.8]) Let \underline{F} be an abelian τ-sheaf of characteristic different from ∞.

1. If \underline{F} is simple, then QEnd(\underline{F}) is a division algebra over Q.

2. If \underline{F} is semisimple with decomposition \underline{F} ≈ \underline{F}_{1} ⊕ ⋯ ⊕ \underline{F}_{n} up to quasi-isogeny into simple abelian τ-sheaves \underline{F}_{j}, then QEnd(\underline{F}) decomposes into a finite direct sum of full matrix algebras over the division algebras QEnd(\underline{F}_{j}) over Q.

In the following we want to define the degree of an isogeny which should be an ideal of A since in the function field case we have substituted A for ℤ. Let f : \underline{M} → \underline{M}′ be an isogeny between pure Anderson motives. Then the A_{L}-module coker f is a finite L-vector space equipped with a morphism of A_{L}-modules τ′ : σ^{∗} coker f → coker f. Since coker f is annihilated by an element of A it decomposes by the Chinese remainder theorem

(coker f, τ′) = ⨁_{v∈supp(coker f)}(coker f, τ′) ⊗_{A} A_{v} =: ⨁_{v∈supp(coker f)} \underline{K}_{v} .

If v ≠ ε the morphism τ′ on \underline{K}_{v} is an isomorphism and so Lang’s theorem implies that

(\underline{K}_{v} ⊗_{L} L^{sep})^{τ} ⊗_{𝔽_{q}} L^{sep} ⟶^{∼} \underline{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}}(\underline{K}_{v} ⊗_{L} L^{sep})^{τ} = dim_{𝔽_{q}}(\underline{K}_{v} ⊗_{L} L^{sep})^{τ} = dim_{L^{sep}}(\underline{K}_{v} ⊗_{L} L^{sep}) = dim_{L} \underline{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_{0} := (b ⊗ 1 − 1 ⊗ c^{∗}(b) : b ∈ 𝔽_{ε}) ⊂ A_{ε,L} .

If we set n := [𝔽_{ε} : 𝔽_{q}] and 𝔞_{i} := (σ^{∗})^{i}𝔞_{0} = (b ⊗ 1 − 1 ⊗ c^{∗}(b)^{q^{i}} : b ∈ 𝔽_{ε}), then we can decompose A_{ε,L} = ⨁_{i∈ℤ/nℤ} A_{ε,L}/𝔞_{i} and τ is an isomorphism

σ^{∗}(\underline{K}_{ε}/𝔞_{i−1}\underline{K}_{ε}) ⟶^{∼} \underline{K}_{ε}/𝔞_{i}

for i ≠ 0 since τ is an isomorphism on \underline{M} and \underline{M}′ outside the graph of c^{∗}. (This argument will be used again in Proposition 3.8.) In particular

[𝔽_{ε} : 𝔽_{q}] · dim_{L}(\underline{K}_{ε}/𝔞_{0}\underline{K}_{ε}) = dim_{L} \underline{K}_{ε} .

Definition 2.8. We assign to the isogeny f the ideal

deg(f) := ∏_{v∈supp(coker f)} v^{(dim_{L} \underline{K}_{v})/[𝔽_{v}:𝔽_{q}]} = ε^{dim_{L}(\underline{K}_{ε}/𝔞_{0}\underline{K}_{ε})} · ∏_{v≠ε} v^{dim_{𝔽v}(\underline{K}_{v}⊗_{L}L^{sep})^{τ}}

of A and call it the degree of f. We call ε^{dim_{L}(\underline{K}_{ε}/𝔞_{0}\underline{K}_{ε})} the inseparability degree of f and ∏_{v≠ε} v^{dim_{𝔽v}(\underline{K}_{v}⊗_{L}L^{sep})^{τ}} the separability degree of f.

Remark. The separability degree of f is the Euler-Poincaré characteristic EP (⨁_{v≠ε} \underline{K}_{v} ⊗_{L} L^{sep})^{τ}; see Gekeler [Gek, 3.9] or Pink and Traulsen [PT, 4.6]. Recall that the Euler-Poincaré characteristic of a finite torsion A-module is the ideal of A defined by requiring that EP is multiplicative in short exact sequences, and that EP (A/v) := v for any maximal ideal v of A.

Lemma 2.9. 1. If f : \underline{M} → \underline{M}′ and g : \underline{M}′ → \underline{M}′′ are isogenies then deg(gf) = deg(f) · deg(g). 2. dim_{𝔽_{q}} A/ deg(f) = dim_{L} coker f.

Proof. 1 is immediate from the short exact sequence

\xymatrix{0 \ar[r] & coker f \ar[r]^{g} & coker(gf) \ar[r] & coker g \ar[r] & 0}

and 2 is obvious.

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 \underline{K}_{ε}/𝔞_{0}\underline{K}_{ε}. In particular a^{dim_{L}(\underline{K}_{ε}/𝔞_{0}\underline{K}_{ε})} annihilates \underline{K}_{ε}/𝔞_{0}\underline{K}_{ε}, and hence also \underline{K}_{ε}.

If v ≠ ε and a is a uniformizer at v, we obtain analogously that a^{dim_{𝔽v}(\underline{K}_{v}⊗_{L}L^{sep})^{τ}} annihilates the 𝔽_{v}-vector space (\underline{K}_{v} ⊗_{L} L^{sep})^{τ} and therefore also the L-vector space \underline{K}_{v}.

Proposition 2.11. Let f : \underline{M} → \underline{M}′ be an isogeny such that deg(f) = aA is principal (for example this is the case if C = ℙ^{1} and A = 𝔽_{q}[t]). Then there is a uniquely determined dual isogeny f^{∨} : \underline{M}′ → \underline{M} (depending on a), which satisfies f ∘ f^{∨} = a · id_{\underline{M}}′ and f^{∨} ∘ f = a · id_{\underline{M}} .

Proof. Since deg(f) annihilates coker f the proposition is immediate.

In Theorem 7.3 we will see that any isogeny f ∈ End(\underline{M}) of a semisimple pure Anderson motive over a finite field satisfies the assumption that deg(f) is principal.

3 Local Shtuka

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 next section. The local (iso-)shtuka is the analogue of the Dieudonné module of the p-divisible group attached to an abelian variety. Note however one fundamental difference. While the Dieudonné module exists only if p equals the characteristic of the base field, there is no such restriction in our theory here. And in fact 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 Dieudonné 𝔽_{q}[[z]]-modules (respectively Dieudonné 𝔽_{q}((z))-modules). They are studied in [An2, Ha2, Lau]. Over a field their definition takes the 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 \underline{M̂} = (M̂, ϕ) consisting of a free A_{v,L}-module M̂ of rank r and an injective A_{v,L}-module homomorphism ϕ : σ^{∗}M̂ → M̂.

A local σ-isoshtuka at v of rank r over L is a pair \underline{N̂} = (N̂, ϕ) consisting of a free Q_{v,L}-module N̂ of rank r and an isomorphism ϕ : σ^{∗}N̂ ⟶^{∼} N̂ of Q_{v,L}-modules.

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} is a field. For applications to pure Anderson motives this is not a problem since we may reduce to this case by Propositions 3.5 and 3.8 below.

Definition 3.3. A local shtuka \underline{M̂} = (M̂, ϕ) is called étale if ϕ is an isomorphism. The Tate module of an étale local σ-shtuka \underline{M̂} at v is the G := Gal(L^{sep}/L)-module of ϕ-invariants

T_{v}\underline{M̂} := (\underline{M̂} ⊗_{A_{v,L}} A_{v,L^{sep}})^{ϕ} .

The rational Tate module of \underline{M̂} is the G-module

V_{v}\underline{M̂} := T_{v}\underline{M̂} ⊗_{A_{v}} Q_{v} .

It follows from [TW, Proposition 6.1] that T_{v}\underline{M̂} is a free A_{v}-module of the same rank than \underline{M̂} and that the natural morphism

T_{v}\underline{M̂} ⊗_{A_{v}} A_{v,L^{sep}} ⟶^{∼} \underline{M̂} ⊗_{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 both factors and ϕ is id ⊗σ^{∗}. Since (L^{sep})^{G} = L we obtain:

Proposition 3.4. Let \underline{M̂} and \underline{M̂}′ be étale local σ-shtuka at v over L. Then

1. \underline{M̂} = (T_{v}\underline{M̂} ⊗_{A_{v}} A_{v,L^{sep}})^{G}, the Galois invariants,

2. Hom_{A_{v,L}[ϕ]}(\underline{M̂}, \underline{M̂}′) ⟶^{∼} Hom_{A_{v}[G]}(T_{v}\underline{M̂}, T_{v}\underline{M̂}′) , f ↦ T_{v}f is an isomorphism.

In particular the Tate module functor yields an equivalence of the category of étale local shtuka at v over L with the category of A_{v}[G]-modules, which are finite free over A_{v}.

Proof. 1 and 2 are immediate. Hence clearly the Tate module functor is fully faithful. That it is an equivalence follows analogously to [Kat, Proposition 4.1.1].

If the residue field 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 = ∏_{Gal(𝔽_{q^{f}} /𝔽_{q})}𝔽_{v} ⊗_{𝔽_{q^{f}}} L = ∏_{i∈ℤ/fℤ}𝔽_{v} ⊗_{𝔽_{q}} L / (b ⊗ 1 − 1 ⊗ 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 depends on 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}) generated by {b ⊗ 1 − 1 ⊗ b^{q^{i}} : b ∈ 𝔽_{q^{f}} }. Then

A_{v,L} = ∏_{Gal(𝔽_{q^{f}} /𝔽_{q})}A_{v} ˆ⊗_{𝔽_{q^{f}}}L = ∏_{i∈ℤ/fℤ}A_{v,L} / 𝔞_{i}

and similarly for Q_{v}. Note that the factors in this decomposition and the ideals 𝔞_{i} correspond precisely to the places v_{i} of C_{𝔽_{q^{f}}} lying above v.

Proposition 3.5. Fix an i. The reduction modulo 𝔞_{i} induces equivalences of categories

1. (N̂, ϕ) ⟼ (N̂/𝔞_{i}N̂ , ϕ^{f} : (σ^{∗})^{f} N̂/𝔞_{i}N̂ → N̂/𝔞_{i}N̂)

between local σ-isoshtuka at v over L and local σ^{f}-isoshtuka at v_{i} over L of the same rank.

2. (M̂, ϕ) ⟼ (M̂/𝔞_{i}M̂ , ϕ^{f} : (σ^{∗})^{f} M̂/𝔞_{i}M̂ → M̂/𝔞_{i}M̂)

between étale local σ-shtuka at v over L and étale local σ^{f}-shtuka at v_{i} over L preserving Tate modules

T_{v}(M̂, ϕ) ⟶^{∼} T_{v_{i}}(M̂/𝔞_{i}M̂, ϕ^{f}) .

Proof. Since σ^{∗}𝔞_{i} = 𝔞_{i+1} the isomorphism ϕ yields isomorphisms σ^{∗}(N̂/𝔞_{i}N̂) → N̂/𝔞_{i+1}N̂ and similarly for M̂. These allow to reconstruct the other factors from (N̂/𝔞_{i}N̂, ϕ^{f}), and likewise for ˆ \underline{M}. The isomorphism between the Tate modules follows from the observation that an element (x_{j})_{j∈ℤ/fℤ} 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 étale local σ^{f}-shtuka at v_{i} is that it is a free module over the integral domain A_{v,L}/𝔞_{i} = A_{v} ˆ⊗_{𝔽_{q^{f}}}L, and similarly for local σ^{f}-isoshtuka. So the results from [An2, Ha1, Ha2, Lau] apply.

Now let \underline{F} be an abelian τ-sheaf and v ∈ C an arbitrary place of Q. We define the local σ-isoshtuka of \underline{F} at v as

\underline{N}_{v}(\underline{F}) := (F_{0} ⊗_{O_{C_{L}}} Q_{v,L} , Π_{0}^{−1} ∘ τ_{0}) .

If v ≠ ∞ we define the local σ-shtuka of \underline{F} at v as

\underline{M}_{v}(\underline{F}) := (F_{0} ⊗_{O_{C_{L}}} A_{v,L} , Π_{0}^{−1} ∘ τ_{0}) .

Likewise if \underline{M} is a pure Anderson motive over L and v ∈ Spec A we define the local σ-(iso-)shtuka of \underline{M} at v as

\underline{M}_{v}(\underline{M}) := \underline{M} ⊗_{A_{L}} A_{v,L}  respectively  \underline{N}_{v}(\underline{M}) := \underline{M} ⊗_{A_{L}} Q_{v,L} .

These local (iso-)shtuka all have rank r. The local shtuka are étale if v ≠ ε. Note that \underline{N}_{∞}(\underline{F}) does not contain a local σ-shtuka if ε ≠ ∞, since then it is isoclinic of slope − wt(\underline{F}) < 0.

However, if 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 M̂_{i} := F_{i} ⊗_{O_{C_{L}}} A_{∞,L}. Recall the integers k, l from Definition 1.5/2 and set Π̃ := Π_{l−1} ∘ ⋯ ∘ Π_{0}. We define the big local σ-shtuka of \underline{F} at ∞ as

\underline{M̃}_{∞}(\underline{F}) := M̂_{0} ⊕ ⋯ ⊕ M̂_{l−1}  with  ϕ := ( 0  0 Π̃^{−1} ∘ z^{k}τ_{l−1} τ_{0}  0 0 0 0 τ_{l−2} 0  )  (3.1)

We also define the big local σ-isoshtuka of \underline{F} at ∞ as

\underline{Ñ}_{∞}(\underline{F}) := \underline{M̃}_{∞}(\underline{F}) ⊗_{A_{∞,L}} Q_{∞,L} .

Both have rank rl and depend on the choice of k, l and z. If ε ≠ ∞ then \underline{M̃}_{∞}(\underline{F}) is étale. Note that \underline{M̃}_{∞}(\underline{F}) and \underline{Ñ}_{∞}(\underline{F}) were used in [Ha1] to construct the uniformization at ∞ of the moduli spaces of abelian τ-sheaves.

The big local (iso-)shtuka at ∞, \underline{M̃}_{∞}(\underline{F}) and \underline{Ñ}_{∞}(\underline{F}) are always equipped with the endomorphisms

Π := ( 0  0 Π̃^{−1} ∘ z^{k}Π_{l−1} Π_{0}  0 0 0 0 Π_{l−2} 0  ) , Λ(λ) := (λ · id_{M_{0}}λ^{q} · id_{M_{1}} λ^{q^{l−1}} · id_{M_{l−1}})  (3.2)

for all λ ∈ 𝔽_{q^{l}} ∩ L. They satisfy the relations Π^{l} = z^{k} and Π ∘ Λ(λ^{q}) = Λ(λ) ∘ Π. We let Δ_{∞} be “the” central division algebra over Q_{∞} of rank l with Hasse invariant −\frac{k}{l}, or explicitly

Δ_{∞} := 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 Δ_{∞} with a subalgebra of End_{Q_{∞,L}[ϕ]}(\underline{Ñ}_{∞}(\underline{F})) by mapping λ ∈ 𝔽_{q^{l}} ⊂ Δ_{∞} to Λ(λ).

The following two results were proved in [BH1, Theorems 8.6 and 8.7].

Theorem 3.6. Let \underline{F} and \underline{F}′ be abelian τ-sheaves of the same weight over a finite field L and let v be an arbitrary place of Q.

1. Then there is a canonical isomorphism of Q_{v}-vector spaces

QHom(\underline{F}, \underline{F}′) ⊗_{Q} Q_{v} ⟶^{∼} Hom_{Q_{v,L}[ϕ]}(\underline{N}_{v}(\underline{F}), \underline{N}_{v}(\underline{F}′)) .

2. If v = ∞ choose an l which satisfies 1.5/2 for both \underline{F} and \underline{F}′ and assume 𝔽_{q^{l}} ⊂ L. Then there is a canonical isomorphism of Q_{∞}-vector spaces

QHom(\underline{F}, \underline{F}′) ⊗_{Q} Q_{∞} ⟶^{∼} Hom_{Δ_{∞} ˆ⊗_{𝔽q}L[ϕ]}(\underline{Ñ}_{∞}(\underline{F}), \underline{Ñ}_{∞}(\underline{F}′)) .

Theorem 3.7. Let \underline{M} and \underline{M}′ be pure Anderson motives over a finite field L and let v ∈ Spec A be an arbitrary maximal ideal. Then

Hom(\underline{M}, \underline{M}′) ⊗_{A} A_{v} ⟶^{∼} Hom_{A_{v,L}[ϕ]}(\underline{M}_{v}(\underline{M}), \underline{M}_{v}(\underline{M}′)) .

Let now the characteristic be finite and v = ε be the characteristic point. Consider a pure Anderson motive \underline{M} of characteristic c, its local σ-shtuka \underline{M}_{ε}(\underline{M}) = (M̂, ϕ) at ε and the decomposition of the later described before Proposition 3.5

\underline{M}_{ε}(\underline{M}) = ∏_{i∈ℤ/fℤ} \underline{M}_{ε}(\underline{M})/𝔞_{i}\underline{M}_{ε}(\underline{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 a distinguished place v_{0} 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 \underline{M}_{ε}(\underline{M})/𝔞_{i}\underline{M}_{ε}(\underline{M}) for i ≠ 0 and the reasoning of Proposition 3.5 yields

Proposition 3.8. The reduction modulo 𝔞_{0} induces an equivalence of categories

\underline{M}_{ε}(\underline{M}) ⟼ (\underline{M}_{ε}(\underline{M})/𝔞_{0}\underline{M}_{ε}(\underline{M}) , ϕ^{f})

between the local σ-shtuka at ε associated with pure Anderson motives of characteristic c and the local σ^{f}-shtuka at v_{0} 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 \underline{M}_{ε}(\underline{F})/𝔞_{0}\underline{M}_{ε}(\underline{F}) is a module over the integral domain A_{ε} ˆ⊗_{𝔽_{ε}}L. So again [An2, Ha1, Ha2, Lau] apply to (\underline{M}_{ε}(\underline{F})/𝔞_{0}\underline{M}_{ε}(\underline{F}), ϕ^{f}).

Proposition 3.9. Let \underline{M} be a pure Anderson motive over L and let \underline{M̂}′_{ε} be a local σ^{f}-subshtuka of \underline{M}_{ε}(\underline{M})/𝔞_{0}\underline{M}_{ε}(\underline{M}) of the same rank. Then there is a pure Anderson motive \underline{M}′ and an isogeny f : \underline{M}′ → \underline{M} with \underline{M}_{ε}(f)(\underline{M}_{ε}(\underline{M}′)/𝔞_{0}\underline{M}_{ε}(\underline{M}′)) = \underline{M̂}′_{ε}. The same is true for abelian τ-sheaves.

Proof. Extend \underline{M̂}′_{ε} to the local σ-subshtuka ⨁_{i∈ℤ/fℤ} ϕ^{i}((σ^{∗})^{i}\underline{M̂}′_{ε}) of \underline{M}_{ε}(\underline{M}) and consider

\underline{K} := \underline{M}_{ε}(\underline{M}) / ⨁_{i∈ℤ/fℤ}ϕ^{i}((σ^{∗})^{i}\underline{M̂}′_{ε}) .

The induced morphism ϕ_{K} : σ^{∗}K → K has its kernel and cokernel supported on the graph of c. Set \underline{M}′ = (M′, τ′) := (ker(\underline{M} → \underline{K}), τ|_{M′}). Then \underline{M}′ is a pure Anderson motive with the required properties by [BH1, Proposition 1.6].

There is a corresponding result at the places v ≠ ε which is stated in Proposition 4.4.

4 Tate Modules

Definition 4.1. If \underline{F} is an abelian τ-sheaf over L, respectively \underline{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}\underline{F} := T_{v}(\underline{M}_{v}(\underline{F})) and V_{v}\underline{F} := V_{v}(\underline{M}_{v}(\underline{F})) for v ≠ ∞,
T_{∞}\underline{F} := T_{∞}(\underline{M̃}_{∞}(\underline{F})) and V_{∞}\underline{F} := V_{∞}(\underline{M̃}_{∞}(\underline{F})) for v = ∞ ≠ ε,
respectively T_{v}\underline{M} := T_{v}(\underline{M}_{v}(\underline{M})) and V_{v}\underline{M} := V_{v}(\underline{M}_{v}(\underline{M})) .

We call T_{v}\underline{F} (respectively V_{v}\underline{F}) the (rational) v-adic Tate module of \underline{F}. If v = ∞ they both depend on the choice of k, l, and z; see page 10.

By [TW, Proposition 6.1], T_{v}\underline{F} (and V_{v}\underline{F}) are free A_{v}-modules (respectively Q_{v}-vector spaces) of rank r for v ≠ ∞ and rl for v = ∞, which carry a continuous G = Gal(L^{sep}/L)-action.

Also the Tate modules T_{∞}\underline{F} and V_{∞}\underline{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 Δ_{∞} from (3.3) with a subalgebra of End_{Q_{∞}}(V_{∞}\underline{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}\underline{M} is called the v-adic Tate module of \underline{M}. With that convention the Tate module functor is contravariant on Anderson motives but covariant on Drinfeld modules and Anderson’s abelian t-modules [An1] (which both give rise 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 also with 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] and Tamagawa [Tam, §2].

Theorem 4.2. Let \underline{M} and \underline{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(\underline{M}, \underline{M}′) ⊗_{A} A_{v} ≅ Hom_{A_{v}[ G]}(T_{v}\underline{M}, T_{v}\underline{M}′) .

Theorem 4.3. ([BH1, Theorem 9.9]) Let \underline{F} and \underline{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(\underline{F}) = wt(\underline{F}′). Then

QHom(\underline{F}, \underline{F}′) ⊗_{Q} Q_{v} ≅ Hom_{Q_{v}[ G]}(V_{v}\underline{F}, V_{v}\underline{F}′) .

2. If v = ∞ choose an integer l which satisfies 1.5/2 for both \underline{F} and \underline{F}′ and assume 𝔽_{q^{l}} ⊂ L. Then

QHom(\underline{F}, \underline{F}′) ⊗_{Q} Q_{∞} ≅ Hom_{Δ_{∞}[G]}(V_{∞}\underline{F}, V_{∞}\underline{F}′) .

As expected, there is the following relation between Tate modules and isogenies.

Proposition 4.4. ([BH1, Proposition 9.11])

1. Let f : \underline{M}′ → \underline{M} be an isogeny between pure Anderson motives then T_{v}f(T_{v}\underline{M}′) is a G-stable lattice in V_{v}\underline{M} contained in T_{v}\underline{M}.

2. Conversely if \underline{M} is a pure Anderson motive and Λ_{v} is a G-stable lattice in V_{v}\underline{M} contained in T_{v}\underline{M}, then there exists a pure Anderson motive \underline{M}′ and a separable isogeny f : \underline{M}′ → \underline{M} with T_{v}f(T_{v}\underline{M}′) = Λ_{v}.

Proposition 4.5. Let \underline{F}′ be an abelian factor τ-sheaf of \underline{F}. Then V_{v}\underline{F}′ is a G-factor space of V_{v}\underline{F}. The same holds if \underline{M}′ is a factor motive of a pure Anderson motive \underline{M}.

Proof. Let f ∈ Hom(\underline{F}, \underline{F}′) be surjective and let \underline{M̂} and \underline{M̂}′ be the (big, if v = ∞) local σ-shtuka of \underline{F}, respectively \underline{F}′, at v. Then the induced morphism \underline{M}_{v}(f) ∈ Hom(\underline{M̂}, \underline{M̂}′) is surjective and \underline{M̂}′′ := ker \underline{M}_{v}(f) is also a local σ-shtuka at v. We get an exact sequence of local σ-shtuka which we tensor with A_{v,L^{sep}} yielding

0 ⟶ \underline{M̂}′′ ⊗_{A_{v,L}} A_{v,L^{sep}} ⟶ \underline{M̂} ⊗_{A_{v,L}} A_{v,L^{sep}} ⟶^{\underline{M}_{v}(f)} \underline{M̂}′ ⊗_{A_{v,L}} A_{v,L^{sep}} ⟶ 0 .

The Tate module functor is left exact, because considering the morphism of A_{v,L^{sep}}-modules

1 − τ : \underline{M̂} ⊗_{A_{v,L}} A_{v,L^{sep}} ⟶ \underline{M̂} ⊗_{A_{v,L}} A_{v,L^{sep}}

we have by definition T_{v}\underline{M̂} = ker(1 − τ), and the desired left exactness follows from the snake lemma. After tensoring with ⊗_{A_{v}}Q_{v} we get

0 ⟶ V_{v}\underline{M̂}′′ ⟶ V_{v}\underline{M̂} ⟶^{V_{v}f} V_{v}\underline{M̂}′ .

Counting the dimensions of these Q_{v}-vector spaces, we finally also get right exactness, as desired.

5 The Frobenius Endomorphism

Suppose that the characteristic is finite, that is, the characteristic point ε is a closed point of C with finite 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 \underline{F} be an abelian τ-sheaf with finite characteristic point ε = Spec 𝔽_{ε} and let s = q^{e} be a power of the cardinality of 𝔽_{ε}. We define the s-Frobenius on \underline{F} by

π := (π_{i}) : (σ^{∗})^{e}\underline{F} → \underline{F}[ e], π_{i} := τ_{i+e−1} ∘ ⋯ ∘ (σ^{∗})^{e−1}τ_{i} : (σ^{∗})^{e}F_{i} → F_{i+e} .

Clearly π is an isogeny. Observe that 𝔽_{ε} ⊂ 𝔽_{s} implies that (σ^{∗})^{e}\underline{F} has the same characteristic as \underline{F}.

Similarly if ε ∈ Spec A is a closed point we define

Definition 5.2 (s-Frobenius on pure Anderson motives). Let \underline{M} be a pure Anderson motive with finite characteristic 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 \underline{M} by

π := τ ∘ … ∘ (σ^{∗})^{e−1}τ : (σ^{∗})^{e}\underline{M} → \underline{M} .

Remark 5.3. Classically for (abelian) varieties X over a field K of characteristic p one defines the Frobenius morphism X → ϕ^{∗}X where ϕ is the p-Frobenius on K. There p equals the cardinality of the “characteristic field” im(ℤ → K) = 𝔽_{p}. In view of the dual behavior of abelian τ-sheaves and pure Anderson motives our definition is a perfect analogue since here we consider the s-Frobenius for s being the cardinality of (a power of) the “characteristic field” im(c^{∗} : A → L) = 𝔽_{ε}.

Now we suppose L = 𝔽_{s} to be a finite field with s = q^{e} (e ∈ ℕ). Let \overline{𝔽}_{s} denote a fixed algebraic closure of 𝔽_{s} and set G = Gal(\overline{𝔽}_{s}/𝔽_{s}). It is topologically generated by Frob_{s} : x ↦ x^{s}. The following results for the Frobenius endomorphism of τ-modules can be found in Taguchi and Wan [TW, §6].

Proposition 5.4. Let \underline{M} be a pure Anderson motive over 𝔽_{s} of rank r and let ε ≠ v ∈ Spec A be a maximal ideal.

1. The generator Frob_{s} of G acts on T_{v}\underline{M} like (T_{v}π)^{−1}.

2. Let Ψ : A_{v}[ G] → End_{A_{v}}(T_{v}\underline{M}) denote the continuous morphism of A_{v}-modules which is induced by the action of G on T_{v}\underline{M}. Then im Ψ = A_{v}[ T_{v}π].

Proof. 1 was proved in [TW, Ch. 6] and 2 follows from the continuity of Ψ.

Remark. The inversion of T_{v}π in the first statement results from the dual definition of our Tate module.

Proposition 5.5. Let \underline{F} be an abelian τ-sheaf over L = 𝔽_{s} with s = q^{e} and let π be its s-Frobenius. Then (σ^{∗})^{e}\underline{F} = \underline{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 \underline{F}.

2. The generator Frob_{s} of G acts on T_{v}\underline{F} like (T_{v}π)^{−1}.

3. The image of the continuous morphism of Q_{v}-vector spaces Q_{v}[ G] → End_{Q_{v}}(V_{v}\underline{F}) is Q_{v}[ V_{v}π].

4. \underline{M}(π) coincides with the s-Frobenius on the pure Anderson motive \underline{M}(\underline{F}) from definition 5.2.

Proof. 1. Due to the periodicity condition, we have \underline{F}[ e] ⊂ \underline{F}(nk · ∞) for a sufficiently large n ∈ ℕ, since F_{i+e} ⊂ F_{i+nl} = F_{i}(nk ·∞) for e ≤ nl. Thus π ∈ Hom(\underline{F}, \underline{F}(nk ·∞)), and therefore π ∈ QEnd(\underline{F}). By 2.1, we have π ∈ QIsog(\underline{F}). 2 and 3 again follow from [TW, Ch. 6] and the continuity of Ψ; see [BH2, Proposition 2.29] for more details. 4 follows from the definition of π and the commutation of the Π’s and the τ’s.

6 The Poincaré-Weil Theorem

In this section we study the analogue for pure Anderson motives and abelian τ-sheaves of the Poincaré-Weil theorem. Originally, this theorem states that every abelian variety is semisimple, that is, isogenous to a product of simple abelian varieties, see [Lan, Corollary of Theorem II.1/6]. Unfortunately, we cannot expect a full analogue of this statement for abelian τ-sheaves or pure Anderson motives as our next example illustrates. On the positive side we show that every abelian τ-sheaf or pure Anderson motive over a finite field becomes semisimple after a finite base field extension.

Example 6.1. Let C = ℙ_{𝔽_{q}}^{1}, C ∖ {∞} = Spec 𝔽_{q}[ t] and ζ := c^{∗}(1/t) ∈ 𝔽_{q}^{×}. We construct an abelian τ-sheaf \underline{F} over L = 𝔽_{q} with r = d = 2 which is not semisimple. Let

Δ = ({1 \atop 0} {0 \atop 1}) + ({α \atop γ} {β \atop δ} ) · t

with α, β, γ, δ ∈ 𝔽_{q}. To obtain characteristic c we need det Δ = (1 − ζt)^{2}, and thus we require the conditions α + δ = −2ζ and αδ − βγ = ζ^{2}. We set F_{i} := O_{C_{L}}(i · ∞)^{⊕2}, we let Π_{i} be the natural inclusion, and we let τ_{i} := Δ. Then \underline{F} is an abelian τ-sheaf with r = d = 2 and k = l = 1. The associated pure Anderson motive is \underline{M} = (L[t]^{⊕2}, Δ).

We see that \underline{F} is not simple. If Δ = ({1−ζt \atop 0} {0 \atop 1−ζt}) then \underline{F} is semisimple as a direct sum of two simple abelian τ-sheaves. Otherwise, if Δ ≠ ({1−ζt \atop 0} {0 \atop 1−ζt}) which is the case for example if β ≠ 0, consider

Δ̃ := ({β \atop δ+ζ} {0 \atop 1})^{−1}· Δ · σ^{∗}({β \atop δ+ζ} {0 \atop 1}) = ({1−ζt \atop 0} {t \atop 1−ζt})

and the abelian τ-sheaf with F̃_{i} = O_{C_{L}}(i · ∞)^{⊕2} and τ̃_{i} = Δ̃ which is isomorphic to \underline{F}. There is an exact sequence

0 ⟶ \underline{F}′ ⟶^{φ} ˜ \underline{F} ⟶^{ψ} \underline{F}′′ ⟶ 0_{τ′ = 1−ζt  τ̃  τ′′ = 1−ζt}

with φ : 1 ↦ \binom{1}{0} and ψ : \binom{x}{y} ↦ y where \underline{F}′ = \underline{F}′′ is the abelian τ-sheaf with F′_{i} = O_{C_{L}}(i · ∞) and τ′_{i} = 1−ζt. If ˜ \underline{F} were semisimple, then there would be a quasi-morphism ω : \underline{F}′′ → ˜ \underline{F} with ψ∘ω = id_{\underline{F}}′′, hence ω : y ↦ \binom{e}{1} · y for some e ∈ 𝔽_{q}(t). Thus, a necessary condition for the semisimplicity of \underline{F} is

(1 − ζt) · σ^{∗}(y) · \binom{e}{1} = ({1−ζt \atop 0} {t \atop 1−ζt}) · \binom{σ^{∗}(e)}{1} · σ^{∗}(y)

which is equivalent to the condition

e − σ^{∗}(e) = \frac{t}{1 − ζt} .

But this cannot be true since e − σ^{∗}(e) = 0, thus \underline{F} is not semisimple. However, this last formula is satisfied if e = λ · \frac{t}{1−ζt} for λ ∈ 𝔽_{q^{q}} with λ^{q} − λ = −1. That means that after field extension 𝔽_{q}(λ) / 𝔽_{q} we get \underline{F} ≅ \underline{F}′ ^{⊕2} and we have QEnd(\underline{F}) = M_{2}(QEnd(\underline{F}′)) = M_{2}(Q). Note that this phenomenon generally 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(\underline{F}, \underline{F}′) between two abelian τ-sheaves \underline{F} and \underline{F}′ we denote its image V_{v}f ∈ Hom_{Q_{v}[ G]}(V_{v}\underline{F}, V_{v}\underline{F}′) just by f_{v}. If \underline{F} is defined over 𝔽_{s} this applies in particular to the s-Frobenius endomorphism π of \underline{F} (Definition 5.1).

Let \underline{F} be an abelian τ-sheaf over the finite field L = 𝔽_{s}. We set

E := QEnd(\underline{F}) ∋ π  E_{v} := End_{Q_{v}[ G]}(V_{v}\underline{F}) ∋ π_{v}
F := Q[ π] ⊂ E  F_{v} := im(Q_{v}[ G] → End_{Q_{v}}(V_{v}\underline{F})) (6.1)

with Q_{v}[ G] → End_{Q_{v}}(V_{v}\underline{F}) induced by the action of G on V_{v}\underline{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 algebraic over Q. We denote its minimal polynomial by μ_{π} ∈ Q[x], and the characteristic polynomial of the endomorphism π_{v} of V_{v}\underline{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 we want to fit the missing bottom right arrow with an isomorphism.

\xymatrix{E \ar[r] & E ⊗_{Q} Q_{v} \ar[r]^{∼} & E_{v} \\ F \ar[r] \ar[u] & F ⊗_{Q} Q_{v} \ar@{-->}[r]^{∼} \ar[u] & F_{v} . \ar[u]}

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}\underline{F}) and set φ := ψ|_{F ⊗_{Q}Q_{v}}. Then φ is injective 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 = ∏_{{μ∈K[ x] \atop irred.}}μ^{m(μ)}, g = ∏_{{μ∈K[ x] \atop irred.}}μ^{n(μ)}

be their respective factorizations in powers of irreducible polynomials. Then we define the integer

r_{K}(f, g) := ∏_{{μ∈K[ x] \atop irred.}}m(μ) · n(μ) · deg μ .

Remark. In contrast to characteristic zero, we have for char(K) ≠ 0 in general different values of the integer r_{K} for different ground fields K. Namely, if K ⊂ L then r_{K}(f, g) ≤ r_{L}(f, g) with equality if and only if all irreducible μ ∈ K[ x] which are contained both in f and in g have no multiple factors in L[ x]. This is satisfied for example if the greatest common divisor of f and g has only separable irreducible 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(\underline{F}, \underline{F}′).

Lemma 6.4. Let v be a place of Q different from ε and ∞. Let \underline{F} and \underline{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_{1}^{m_{1}} · . . . · μ_{n}^{m_{n}} and χ′_{v} = μ_{1}^{m′_{1}}·. . .·μ_{n}^{m′_{n}} with distinct monic irreducible polynomials μ_{1}, …, μ_{n} ∈ Q_{v}[ x] and m_{i}, m′_{i} ∈ ℕ_{0}. Then

1. Hom_{Q_{v}[G]}(V_{v}\underline{F}, V_{v}\underline{F}′) ≅ ⨁_{i=1}^{n} M_{m′_{i}×m_{i}}(Q_{v}[x]/(μ_{i})) as Q_{v}-vector spaces,

2. End_{Q_{v}[G]}(V_{v}\underline{F}) ≅ ⨁_{i=1}^{n} M_{m_{i}}(Q_{v}[x]/(μ_{i})) as Q_{v}-algebras, and

3. dim_{Q_{v}} Hom_{Q_{v}[ G]}(V_{v}\underline{F}, V_{v}\underline{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, we have the following decomposition of Q_{v}[G]-modules

V_{v}\underline{F} ≅ ⨁_{i=1}^{n} (Q_{v}[ x]/(μ_{i}))^{⊕m_{i}}, V_{v}\underline{F}′ ≅ ⨁_{i=1}^{n} (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_{i}^{⊕m_{i}} as multiplication by the scalar x, the lemma follows.

Theorem 6.5. Let \underline{F} and \underline{F}′ be abelian τ-sheaves of the same weight over 𝔽_{s} and assume that the endomorphisms π_{v} and π′_{v} of V_{v}\underline{F} and V_{v}\underline{F}′ are semisimple at a place v ≠ ε, ∞ of Q. Let χ_{v} and χ′_{v} be their characteristic polynomials. Then

dim_{Q} QHom(\underline{F}, \underline{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 \underline{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}. Then

1. h|r and dim_{Q} QEnd(\underline{F}) = \frac{r^{2}}{h} and dim_{F} QEnd(\underline{F}) = \frac{r^{2}}{h^{2}} .

2. For any place v of Q different from ε and ∞ we have QEnd(\underline{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_{1}·. . .·μ_{n} with pairwise different irreducible monic polynomials μ_{i} ∈ Q_{v}[x] and χ_{v} = μ_{1}^{m_{1}} ·. . .·μ_{n}^{m_{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} ≅ ⨁_{i=1}^{n} E_{i} into the simple constituents E_{i} = M_{m_{i}}(K_{i}). By [Bou, Théorème 5.3/1 and Proposition 5.4/12], 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 spaces

QEnd(\underline{F}) ⊗_{F} K_{i} = E_{v} ⊗_{F_{v}} K_{i} ⟶ E_{i} .

This shows that m_{i}^{2} ≤ dim_{F} E. So by Lemma 6.4

[F : Q] · dim_{F} E = dim_{Q_{v}} E_{v} = ∑_{i=1}^{n} m_{i}^{2} deg μ_{i} ≤
≤ dim_{F} E · ∑_{i=1}^{n} deg μ_{i} = dim_{F} E · deg μ_{π} = [F : Q] · dim_{F} E .

Therefore m_{i}^{2} = dim_{F} E for all i. Since r = deg χ_{v} = ∑_{i} m_{i} deg μ_{i} = \sqrt{dim_{F} E} · [F : Q]. We find r = m_{i}h and dim_{F} E = \frac{r^{2}}{h^{2}} , proving 1. For 2 we use that

E_{v} ≅ ⨁_{i} M_{r/h}(Q_{v}[x]/(μ_{i})) = M_{r/h}(⨁_{i} Q_{v}[x]/(μ_{i})) = M_{r/h}(Q_{v}[x]/(μ_{π})) .

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 Ô_{C,v} ⟶ O_{C,v} and therefore Q_{v} = Ô_{C,v}⊗_{O_{C,v}}Q ⟶ Q are geometrically regular. This means that Q_{v} ⊗_{Q} K is regular for every finite field 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 semisimple by [Bou, Corollaire 7.6/4]. Hence π_{v} is semisimple by definition, and we showed in Lemma 6.4/2 that then 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, Corollaire de Proposition 6.4/9] that F is semisimple, and our proof is complete.

Remark. If more generally \underline{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-known lemma. Namely E_{v} is the commutant of F_{v} in End_{Q_{v}}(V_{v}\underline{F}). We thank O. Gabber for mentioning this fact 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 a semisimple K-subalgebra of B. Then the commutant of F in B is semisimple.

Proof. Let F = ⨁_{i} F_{i} be the decomposition into simple constituents and let e_{i} be the corresponding central 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, Théorème 10.2/2] the commutant E_{i} of F_{i} in B_{i} is simple. Clearly the commutant E of F in B satisfies E_{i} = e_{i}Ee_{i} = Ee_{i} and E = ⨁_{i} E_{i} proving the lemma.

Corollary 6.10. Let \underline{F} be an abelian τ-sheaf over 𝔽_{s} of rank r with semisimple Frobenius endomorphism π. Then the algebra F = Q(π) is the center of the semisimple algebra E = QEnd(\underline{F}).

Proof. Since F_{v} is semisimple, we know by [Bou, Proposition 5.1/1] that the F_{v}-module V_{v}\underline{F} is semisimple. The commutant of F_{v} in End_{Q_{v}}(V_{v}\underline{F}) is E_{v} by definition. Trivially V_{v}\underline{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}\underline{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 \underline{F} be an abelian τ-sheaf over a finite field L.

1. If QEnd(\underline{F}) is a division algebra over Q then \underline{F} is simple. If in addition ε ≠ ∞ then both statements are equivalent.

2. If the characteristic point ε is different from ∞ then \underline{F} is semisimple if and only if QEnd(\underline{F}) is semisimple.

Proof. 1. Let QEnd(\underline{F}) = E be a division algebra and let f ∈ Hom(\underline{F}, \underline{F}′) be the morphism onto a non-zero factor sheaf \underline{F}′ of \underline{F}. We show that f is an isomorphism. We know by 4.5 that f_{v} ∈ Hom_{Q_{v}[ G]}(V_{v}\underline{F}, V_{v}\underline{F}′) is surjective. By the semisimplicity of E and Proposition 6.8, F_{v} is semisimple, and therefore V_{v}\underline{F} is a finitely generated semisimple F_{v}-module. Thus we get a morphism g_{v} ∈ Hom_{Q_{v}[G]}(V_{v}\underline{F}′, V_{v}\underline{F}) with f_{v} ∘ g_{v} = id_{V_{v}\underline{F}}′. Consider the integral Tate modules T_{v}\underline{F} and T_{v}\underline{F}′. We can find some n ∈ ℕ such that

v^{n}g_{v} ∈ Hom_{A_{v}[ G]}(T_{v}\underline{F}′, T_{v}\underline{F}) ≅ Hom(\underline{M}(\underline{F}′), \underline{M}(\underline{F})) ⊗_{A} A_{v}

and we choose g ∈ Hom(\underline{M}(\underline{F}′), \underline{M}(\underline{F})) ⊂ QHom(\underline{F}′, \underline{F}) with g ≡ v^{n}g_{v} modulo v^{m} for a sufficiently large m > n. If g ∘ f = 0 in E, then f ∘ g ∘ f = 0, and therefore f ∘ g = 0 in QEnd(\underline{F}′) due to the surjectivity of f. This would imply

v^{n} · id_{V_{v}\underline{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 gives the desired isomorphism between \underline{F}′ and \underline{F}. The second assertion follows from Theorem 2.7.

2. We already saw one direction in Theorem 2.7/2. So let now QEnd(\underline{F}) be semisimple and let

QEnd(\underline{F}) = ⨁_{j=1}^{m} 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,1}, …, e_{j,λ_{j}} ∈ M_{λ_{j}}(E_{j}) such that e_{j,α} · QEnd(\underline{F}) · e_{j,α} = E_{j} for all 1 ≤ α ≤ λ_{j} with ∑_{α=1}^{λ_{j}} e_{j,α} = 1 in M_{λ_{j}}(E_{j}). Let e_{1}, …, e_{n} denote all these idempotents, n = ∑_{j=1}^{m} λ_{j}, and choose a divisor D on C such that e_{i} ∈ Hom(\underline{F}, \underline{F}(D)) for all 1 ≤ i ≤ n. Then ∑_{i=1}^{n} e_{i} = id\underline{F} in QEnd(\underline{F}) and therefore

\underline{F} \underrightarrow{∑_{i} e_{i}} ⨁_{i=1}^{n} im e_{i} ⊂ \underline{F}(D) .

The image \underline{F}_{i} := im e_{i} is an abelian τ-sheaf by [BH1, Proposition 4.2] because ε ≠ ∞. Since ∑_{i} e_{i} is injective it is an isogeny by 2.1. Since QEnd(\underline{F}_{i}) = e_{i} · QEnd(\underline{F}) · e_{i} is a division algebra, \underline{F}_{i} is a simple abelian τ-sheaf by 1. Thus \underline{F} ≈ \underline{F}_{1} ⊕ ⋯ ⊕ \underline{F}_{n} gives the decomposition into a direct sum of simple abelian τ-sheaves \underline{F}_{i} as desired.

Remark 6.12. Unfortunately the theorem fails if L is not finite, as Example 6.13 below shows. The reason is, that then E_{v} may still be semisimple while the image F_{v} of Q_{v}[G] in End_{Q_{v}}(V_{v}\underline{F}) is not. Nevertheless, if one adds the assumption that F_{v} is semisimple, the assertions of Theorem 6.11 remain valid over an arbitrary field L. (See also the remark after Proposition 6.8.)

Example 6.13. We construct a pure Anderson motive \underline{M} over a non-finite field L which is not semisimple, but has End(\underline{M}) = A. Any associated abelian τ-sheaf \underline{F} has QEnd(\underline{F}) = Q. Let C = ℙ_{𝔽_{q}}^{1}, A = 𝔽_{q}[t] with q > 2, and L = 𝔽_{q}(α) where α is transcendental over 𝔽_{q}. Let M = A_{L}^{⊕2} and τ = ({αt \atop 0} {t \atop t}). Then \underline{M} = (M, τ) is a pure Anderson motive of rank and dimension 2. Clearly \underline{M} is not simple, since \underline{M}′ = (A_{L}, τ′ = t) is a factor motive by projecting onto the second coordinate. We will see below that \underline{M} is not even semisimple.

Let ({e \atop g} {f \atop h}) ∈ M_{2}(A_{L}) be an endomorphism of \underline{M}, that is,

({ασ^{∗}e + σ^{∗}g \atop σ^{∗}g} {ασ^{∗}f + σ^{∗}h \atop σ^{∗}h}) = ({αe \atop αg} {e + f \atop g + h}) .

Choose β ∈ 𝔽_{q}(α)^{alg} ∖ 𝔽_{q}(α) satisfying β^{q−1} = α (for β ∉ 𝔽_{q}(α) we use q > 2). Then 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}∖𝔽_{q}(β) with γ^{q}−γ = β and set f̃:= βf −γ·(e−h). Then ασ^{∗}f −f = e−σ^{∗}h = e−h implies σ^{∗}f̃ − f̃ = β^{q}σ^{∗}f − βf − (γ^{q} − γ)(e − h) = β(ασ^{∗}f − f − (e − h)) = 0. Thus f̃ ∈ 𝔽_{q}[t] and γ · (e − h) ∈ 𝔽_{q}(β)[t]. So we must have e = h and then βf = f̃ ∈ 𝔽_{q}[t] implies f = 0. This shows that End(\underline{M}) = 𝔽_{q}[t] = A.

The same argument shows that \underline{M} is not even semisimple. Namely, the projection \underline{M} → \underline{M}′ has no section \underline{M}′ → \underline{M}, 1 ↦ \binom{f}{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 − 1). Let z = t − 1 and β ∈ L^{sep} with β^{q−1} = α, and consider the basis \binom{y/β}{0}, \binom{x}{y} of the Tate module T_{v}(\underline{M}) with

\binom{x}{y} = ∑_{i=0}^{∞} \binom{x_{i}}{y_{i}}z^{i} and x_{i}, y_{i} ∈ L^{sep} , y_{0} ≠ 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_{i}^{q} = y_{i−1}^{q} , and
x_{i} − αx_{i}^{q} = αx_{i−1}^{q} + 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}^{×} ∖ {1}, 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}(\underline{M}) they correspond to matrices γ_{v} = ({η \atop 0} {0 \atop 1}) and δ_{v} = ({1 \atop 0} {1 \atop 1}). We conclude that F_{v} is the Q_{v}-algebra of upper triangular matrices. Its commutant in M_{2}(Q_{v}) equals Q_{v} · Id_{2} ≅ End(\underline{M}) ⊗_{A} Q_{v}.

Remark. If q = 2 any pure Anderson motive of rank rk \underline{M} = 2 on A = 𝔽_{q}[t], which is not semisimple has End(\underline{M}) ⊋ A. One easily sees this by choosing a basis of \underline{M} for which τ has the form ({α(t−θ)^{d} \atop 0} {∗ \atop β(t−θ)^{d}} )

with α, β, θ ∈ L. Then ({0 \atop 0} {β/α \atop 0}) is an endomorphism.

However, we expect that also for q = 2 there are examples similar to 6.13 (of rk \underline{M} ≥ 3), although we have not tried to find one.

Let \underline{F} be an abelian τ-sheaf over 𝔽_{s} and let 𝔽_{s′}/𝔽_{s} be a finite field extension. The base extension

\underline{F} ⊗_{𝔽_{s}} 𝔽_{s′} := (F_{i} ⊗_{O_{C_{𝔽s}}}O_{C_{𝔽_{s}′}} , Π_{i} ⊗ 1, τ_{i} ⊗ 1)

is an abelian τ-sheaf over 𝔽_{s′} with π′ = (π ⊗ 1)^{t} for s′ = s^{t}, and we have a canonical isomorphism between V_{v}\underline{F} and V_{v}\underline{F}′.

For the next result recall that an endomorphism φ of a finite dimensional vector space V over a field K is called absolutely semisimple if for every field extension K′/K the endomorphism φ ⊗ 1 ∈ End_{K′}(V ⊗_{K} K′) is semisimple. The following characterization is taken from [Bou, Proposition 9.2/4 and 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 ⊗ 1 ∈ 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 \underline{F} be an abelian τ-sheaf over the finite field 𝔽_{s}. Then there exists a finite field extension 𝔽_{s′}/𝔽_{s} whose degree is a power of char 𝔽_{s} such that \underline{F} ⊗_{𝔽_{s}} 𝔽_{s′} has an absolutely semisimple Frobenius endomorphism. Thus if moreover ε ≠ ∞ then \underline{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 \underline{F}.

Proof. Let s′ = s^{t} for some arbitrary t ∈ ℕ. Let \underline{F}′ := \underline{F} ⊗_{𝔽_{s}} 𝔽_{s′} be the abelian τ-sheaf over 𝔽_{s′} induced by \underline{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}\underline{F}) in Jordan normal form

B^{−1} (π_{v} ⊗ 1) B = ( \begin{matrix}λ_{1} & ∗ & & 0 \\ & λ_{2} & ... \\ & & ... & ∗ \\ 0 & & & λ_{r} \end{matrix} )

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 power of char 𝔽_{q} (as in the remark), we can achieve that π′_{v} = (π_{v} ⊗ 1)^{t} is of the form

B^{−1} π′_{v} B = ( \begin{matrix}λ_{1}^{t} & & & & & 0 \\ & λ_{2}^{t} \\ & & . & . & . \\ 0 & & & & & λ_{r}^{t} \end{matrix} ) .

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 no need of the term ”absolutely semisimple” for abelian τ-sheaves or pure Anderson motives over finite fields.

Corollary 6.16. Let \underline{F} be an abelian τ-sheaf over 𝔽_{s} of characteristic different from ∞. If \underline{F} is semisimple, then \underline{F} ⊗_{𝔽_{s}} 𝔽_{s′} is semisimple for every finite field extension 𝔽_{s′}/𝔽_{s}. The same is true for pure Anderson motives.

Proof. Let \underline{F} be semisimple and let 𝔽_{s′}/𝔽_{s} be a finite field extension with s′ = s^{t}. We set \underline{F}′ := \underline{F} ⊗_{𝔽_{s}} 𝔽_{s′}. By 6.11 and 6.8, we know that QEnd(\underline{F}) ⊗_{Q} Q_{v} ≅ End_{Q_{v}[ π_{v} ]}(V_{v}\underline{F}) is semisimple. Since Q_{v}[ π_{v}^{t}] ⊂ Q_{v}[ π_{v}] we conclude by [Bou, Corollaire de Proposition 6.4/9] that Q_{v}[ π_{v}^{t}] is semisimple, as well. As V_{v}\underline{F}′ = V_{v}\underline{F}, we have π′_{v} = π_{v}^{t}, and therefore π′_{v} is semisimple. Thus, by 6.8, QEnd(\underline{F}′) is semisimple and \underline{F}′ is semisimple by 6.11/2.

7 Zeta Functions and Reduced Norms

In this section we generalize Gekeler’s results [Gek] on Zeta functions for Drinfeld modules to pure Anderson motives. But let us begin by recalling a few facts about reduced norms; see for instance [Rei, §9]. Let \underline{M} be a semisimple pure Anderson motive over a finite field and let π be its Frobenius endomorphism. Then F = Q(π) is the center of the semisimple algebra E by Corollary 6.10. Write F = ⨁_{i} F_{i} and E = ⨁_{i} E_{i} where the F_{i} are fields and E_{i} is central simple over F_{i}. Note that by 6.11 the pure Anderson motive \underline{M} decomposes correspondingly up to isogeny \underline{M} ≈ ⨁_{i} \underline{M}_{i} with E_{i} = End(\underline{M}_{i}) ⊗_{A} Q. We apply 6.6 to \underline{M}_{i} and obtain ∑_{i}[E_{i} : F_{i}]^{1/2} · [F_{i} : Q] = r. Let f ∈ E and write it as f = ∑_{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}^{2} = [E_{i} : F_{i}]. The reduced norm of f is then defined by

N(f) := nr_{E/Q}(f) := ∏_{i} N_{F_{i}/Q}(det α_{i}(f_{i} ⊗ 1)),

where N_{F_{i}/Q} is the usual field norm. The reduced norm is an element of Q which is independent of the choices 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(\underline{M}) or more generally f is contained in a finite A-algebra then N(f) ∈ A since A is normal.

Theorem 7.1. Let \underline{F} be a semisimple abelian τ-sheaf over a finite field L and let f ∈ QEnd(\underline{F}) be a quasi-isogeny. Then for any place v ≠ ε, ∞ of Q we have N(f) = det V_{v}f, the determinant of the endomorphism V_{v}f ∈ End_{Q_{v}}(V_{v}\underline{F}). For v = ∞ ≠ ε we have N(f)^{l} = det V_{∞}f, where l comes from Definition 4.1 and satisfies dim_{Q_{∞}} V_{∞}\underline{F} = l · rk \underline{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_{v}^{alg} 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) is semisimple.

As is well known there is a semisimple commutative subalgebra H = ⨁_{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 f̃: x ↦ fx of H. The reason for this is that H_{i} ⊗_{F_{i}} K_{i} is still semisimple and commutative 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_{i}^{n_{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 f̃_{i}, the determinant of the F_{i}-endomorphism f̃_{i} : x ↦ f_{i}x of H_{i}, and N(f) = det f̃ the determinant of the Q-endomorphism f̃ of H.

If v ≠ ∞ then again by Lemma 7.2, H_{v} is H_{v}-isomorphic to V_{v}\underline{F} and N(f) = det f̃ = det V_{v}f. If v = ∞ we embed E_{∞}^{⊕l} into End_{Q_{∞,L}[ϕ]}(\underline{Ñ}_{∞}(\underline{F})). Namely, if (f^{(0)}, …, f^{(l−1)}) ∈ E_{∞}^{⊕l}, where

f^{(m)} = (f_{i}^{(m)} : F_{i} ⊗_{O_{C_{L}}} Q_{∞,L} → F_{i} ⊗_{O_{C_{L}}} Q_{∞,L}), we set

g_{ij} := \{ \begin{matrix}Π_{i−1} ∘ … ∘ Π_{j} ∘ f_{j}^{(i−j)} & if 0 ≤ j ≤ i ≤ l − 1 \\ z^{k}Π_{i}^{−1} ∘ … ∘ Π_{j−1}^{−1} ∘ f_{j}^{(l+i−j)} & if 0 ≤ i < j ≤ l − 1 . \end{matrix}

Then g_{ij} : F_{j} ⊗_{O_{C_{L}}} Q_{∞,L} → F_{i} ⊗_{O_{C_{L}}} Q_{∞,L} and a straightforward computation shows that the homomorphism g = (g_{ij})_{i,j=0...l−1} commutes with ϕ from (3.1) on page 10, that is, g is an element of End_{Q_{∞,L}[ϕ]}(\underline{Ñ}_{∞}(\underline{F})) = End_{Q_{∞}[G]}(V_{∞}\underline{F}); use Proposition 3.4. Now we apply Lemma 7.2 to H_{∞}^{⊕l} ⊂ E_{∞}^{⊕l} ⊂ End_{Q_{∞}}(V_{∞}\underline{F}), and we compute N(f)^{l} = (det f̃)^{l} = det_{Q_{∞}}(H_{∞}^{⊕l} → H_{∞}^{⊕l}, h ↦ fh) = det V_{∞}f as desired.

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}.

Proof. Decomposing H into a direct sum of fields ⨁_{κ} L_{κ} and K^{n} into a direct sum ⨁_{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 imply that H is isomorphic to ⨁_{λ} End_{L_{κ(λ)}}(V_{λ}) and a fortiori isomorphic as left module over itself to K^{n}.

Theorem 7.3. Let \underline{M} be a semisimple pure Anderson motive of rank r over a finite field L and let f ∈ End(\underline{M}) be an isogeny. Then

1. 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(\underline{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 number of rational points X(𝔽_{q^{n}}) = deg(1 − Frob_{q}^{n}) ∈ ℤ on an abelian variety X over the finite field 𝔽_{q}; see also Theorem 7.7 below.

2. The dual isogeny satisfies (fg)^{∨} = g^{∨}f^{∨}, because N(fg) = N(f)N(g). Note however, that we cannot expect that (f + g)^{∨} = f^{∨} + g^{∨} unless r = 2 because for f = a ∈ A we have N(a) = a^{r} and a^{∨} = a^{r−1}.

Proof. 1. Clearly for any a ∈ A we have dim_{L} \underline{M}/a\underline{M} = r·dim_{𝔽_{q}} A/(a) = −r·∞(a) where ∞(a) denotes the ∞-adic valuation of a. Now let \underline{F} be an abelian τ-sheaf with \underline{M} = \underline{M}(\underline{F}), and let f : \underline{F} → \underline{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} ⨁_{j=0}^{l−1}(F_{j}(n · ∞)/f_{j}(F_{j}))_{∞}
= nrl − dim_{L} \underline{M̃}_{∞}(\underline{F}(n·∞))/\underline{M̃}_{∞}(f)(\underline{M̃}_{∞}(\underline{F}))
= nrl − dim_{𝔽_{q}}(T_{∞}\underline{F}(n · ∞)/T_{∞}f(T_{∞}\underline{F}))
= −∞(det V_{∞}f) = −l · ∞(N(f)) .

Here the first equality follows from the identities F_{j}(n · ∞)/f_{j}(F_{j}) = (F_{j}(n · ∞)/f_{j}(F_{j}))_{∞} ⊕ coker f and dim_{L}(F_{j}(n · ∞)/f_{j}(F_{j})) = deg F_{j}(n · ∞) − deg f_{j}(F_{j}) = nr. The second equality is the definition of \underline{M̃}_{∞}, and the third follows from the isomorphism \underline{M̃}_{∞}(\underline{F}) ⊗_{A_{∞,L}} A_{∞,L^{sep}} ≅ T_{∞}\underline{F} ⊗_{A_{∞}} A_{∞,L^{sep}}. The fourth 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}}(T_{v}\underline{M}/T_{v}f(T_{v}\underline{M})) = dim_{𝔽_{v}}((coker f)_{v} ⊗_{L} L^{sep})^{τ} = v(deg f) .

Again the second equality follows from the elementary divisor theorem, the third equality comes from the fact that the τ-invariants of the v-primary part (coker f)_{v}⊗_{L}L^{sep} are isomorphic to T_{v}\underline{M}/T_{v}f(T_{v}\underline{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}((A/N(f))^{r} ⊗_{𝔽_{q}} L) = r · dim_{𝔽_{q}} A/N(f) .

From the identity dim_{𝔽_{q}} A/𝔞 = ∑_{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 does not make use of the associated abelian τ-sheaf \underline{F}. In the special case when \underline{M} comes from a Drinfeld module, Gekeler [Gek, Lemma 3.1] argued that both sides of the equation in 2 are extensions to E of the ∞-adic valuation on Q. But this argument fails in general, since there may be more than one such extension as one sees from Example 9.5 below.

Corollary 7.5. Let \underline{M} be a semisimple pure Anderson motive of dimension d over a finite field L and let π be its Frobenius endomorphism. Let v ≠ ε be a maximal ideal of A and let χ_{v} be the characteristic polynomial of π_{v}. Then

1. 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. 1 is a direct consequence of Theorems 7.1 and 7.3 and the Lagrange interpolation theorem applied to the fact that 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 \underline{M} over a finite field 𝔽_{s} as

Z_{\underline{M}}(t) := ∏_{0≤i≤r} det(1 − t ∧^{i} π_{v})^{(−1)^{i+1}}

where ε ≠ v ∈ Spec A is a maximal ideal and ∧^{i}π_{v} ∈ End_{Q_{v}}(∧^{i}V_{v}\underline{M}).

By 7.5/1 the Zeta function Z_{\underline{M}}(t) is independent of the place v and lies in Q(t). This also follows from work of Böckle [Boe] and Gardeyn [Gar, §7]. The name “Zeta function” is justified by the following theorem (see also the remark after Theorem 7.3).

Theorem 7.7. If \underline{M} is semisimple and ∑_{i} a_{i}t^{i} is the power series expansion of t\frac{d}{dt} log Z_{\underline{M}}(t), then a_{i} = N(1 − π^{i}) ∈ A.

Proof. By standard arguments a_{i} = det(1 − π_{v}^{i}); see [Gek, Lemma 5.6]. Now our assertion follows from Theorem 7.1

This 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}\underline{M}) have the same absolute value (#𝔽_{s})^{i wt(\underline{M})}.

Proof. This was proved by Goss [Gos, Theorem 5.6.10] for i = 1 and follows for the remaining i by general arguments of linear algebra.

8 A Quasi-Isogeny Criterion

Similarly to the theory for abelian varieties, the characteristic polynomials of the Frobenius endomorphisms on the associated Tate modules play an important role for the study of abelian τ-sheaves. For example, we can decide on quasi-isogeny of two abelian τ-sheaves \underline{F} and \underline{F}′ just by considering these characteristic polynomials.

Theorem 8.1. Let \underline{F} and \underline{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 different from ∞ 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 \underline{F} and \underline{F}′ have the same weight.

1. Consider the following statements:

1.1. \underline{F}′ is quasi-isogenous to an abelian factor τ-sheaf of \underline{F}. 1.2. V_{v}\underline{F}′ is G-isomorphic to a G-factor space of V_{v}\underline{F}. 1.3. χ′_{v} divides χ_{v} in Q_{v}[x].

1.4. μ_{π′} divides μ_{π} in Q[x] and rk \underline{F}′ ≤ rk \underline{F}

We have 1.1 ⇒ 1.2 ⇒ 1.3 and 1.4 always,
1.2 ⇐ 1.3  if π_{v} and π′_{v} are semisimple,
1.2 ⇐ 1.3 ⇐ 1.4 if μ_{π} is irreducible in Q[x],
1.1 ⇐ 1.2  if the characteristic is different from ∞.

2. Consider the following statements:

2.1. \underline{F} and \underline{F}′ are quasi-isogenous. 2.2. V_{v}\underline{F} and V_{v}\underline{F}′ are G-isomorphic. 2.3. χ_{v} = χ′_{v}. 2.4. μ_{π} = μ_{π′} and rk \underline{F} = rk \underline{F}′. 2.5. There is an isomorphism of Q-algebras QEnd(\underline{F}) ≅ QEnd(\underline{F}′) mapping π to π′. 2.6. There is a Q_{v}-isomorphism QEnd(\underline{F}) ⊗_{Q} Q_{v} ≅ QEnd(\underline{F}′) ⊗_{Q} Q_{v} mapping π_{v} to π′_{v}. 2.7. If ε ≠ ∞ also consider the statement Z_{\underline{M}(\underline{F})} = Z_{\underline{M}(\underline{F}′)}.

We have 2.1 ⇔ 2.2 ⇒ 2.3 , 2.4 , 2.5  always,
2.5 ⇒ 2.6 always,
2.3 ⇔ 2.7  if the characteristic is different from ∞,
2.2 ⇐ 2.3 ⇐ 2.6 if π_{v} and π′_{v} are semisimple,
2.2 ⇐ 2.3 ⇐ 2.4 ⇐ 2.6 if μ_{π} and μ_{π′} are irreducible in Q[x].

Proof. 1. For the implication 1.1 ⇒ 1.2 without loss of generality, \underline{F}′ can itself be considered as abelian factor τ-sheaf of \underline{F} and the implication follows from Proposition 4.5. The implication 1.2 ⇒ 1.3 is obvious.

For 1.2 ⇒ 1.4 note that μ_{π} is also the minimal polynomial of π_{v} over Q_{v} by Lemma 6.2. By Proposition 5.5 statement 1.2 implies μ_{π}(π′_{v}) = 0, whence 1.4.

For 1.3 ⇒ 1.2 let π_{v} and π′_{v} be semisimple. Let χ_{v} = μ_{1} · . . . · μ_{n} and χ′_{v} = μ′_{1} · . . . · μ′_{n′} be the factorization 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 we can decompose V_{v}\underline{F} = V_{1} ⊕ ⋯ ⊕ V_{n} and V_{v}\underline{F}′ = V′_{1} ⊕ ⋯ ⊕ V′_{n′}. Since χ′_{v} divides χ_{v}, we can now easily construct a surjective G-morphism from V_{v}\underline{F} onto V_{v}\underline{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 follows from Proposition 6.8 that π_{v} and π′_{v} are semisimple and this implies 1.2 by the above.

For 1.2 ⇒ 1.1 we first do not assume that ε ≠ ∞. Let f_{v} : V_{v}\underline{F} → V_{v}\underline{F}′ be a surjective morphism of Q_{v}[ G]-modules. We may multiply f_{v} by a suitable power of v to get a morphism f_{v} : T_{v}\underline{F} → T_{v}\underline{F}′ of the integral Tate modules which is not necessarily surjective, but satisfies v^{n}T_{v}\underline{F}′ ⊂ f_{v}(T_{v}\underline{F}) for a sufficiently large n. Let \underline{M} := (Γ(C_{L} ∖ \{∞\}, F_{0}), Π_{0}^{−1} ∘ τ). This is a “τ-module on A” in the sense of [BH1, Definition 3.2]. If ε ≠ ∞ then \underline{M} is the pure Anderson motive \underline{M}(\underline{F}) associated with \underline{F} in (1.1). Also let \underline{M}′ := (Γ(C_{L} ∖ \{∞\}, F′_{0}), Π′_{0}^{−1} ∘ τ′). By [BH1, Theorem 9.8] (or Theorem 4.2 if ε ≠ ∞), f_{v} lies inside Hom(\underline{M}, \underline{M}′) ⊗_{A} A_{v}, so we can approximate f_{v} by some f ∈ Hom(\underline{M}, \underline{M}′) with T_{v}(f) ≡ f_{v} modulo v^{n+1}T_{v}\underline{M}′. Since v^{n}T_{v}\underline{M}′ ⊂ f_{v}(T_{v}\underline{M}) we find inside im T_{v}(f) generators of v^{n}T_{v}\underline{M}′/v^{n+1}T_{v}\underline{M}′. They generate an A_{v}-submodule of v^{n}T_{v}\underline{M}′ whose rank must at least be r′ since v^{n}T_{v}\underline{M}′/v^{n+1}T_{v}\underline{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 \underline{F} and \underline{F}′ have the same weight. So by [BH1, Proposition 6.10/1], f comes from a quasi-morphism f ∈ QHom(\underline{F}, \underline{F}′), that is, a morphism f : \underline{F} → \underline{F}′(D) for a suitable divisor D. Now we finally assume that the characteristic is different from ∞. By [BH1, Proposition 4.2], the image im(f : \underline{F} → \underline{F}′(D)) is an abelian factor τ-sheaf of \underline{F} and im f → \underline{F}′(D) is an injective morphism between 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 2.2 ⇒ 2.1 without the hypothesis on the characteristic, we just replace the last argument of the proof of 1.2 ⇒ 1.1 by the following: Since r = dim_{Q_{v}} V_{v}\underline{F} = dim_{Q_{v}} V_{v}\underline{F}′ = r′, the morphism f : \underline{F} → \underline{F}′(D) is an injective morphism between abelian τ-sheaves of the same rank and weight, hence an isogeny by Proposition 2.1.

9 THE ENDOMORPHISM Q-ALGEBRA 25

For the implication 2.1 ⇒ 2.5 let g ∈ QIsog(\underline{F}, \underline{F}′). Then the map QEnd(\underline{F}) → QEnd(\underline{F}′) sending f ↦ gfg^{−1} is an isomorphism with π′ = gπg^{−1}. The implication 2.5 ⇒ 2.6 is obvious.

For the implication 2.3 ⇒ 2.7 note that knowledge of χ_{v} yields the knowledge of det(1 − t ∧^{i} π_{v}) and thus of Z_{\underline{M}(\underline{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(\underline{F})} in an algebraic closure of Q_{∞}. So we can recover χ_{v} from Z_{\underline{M}(\underline{F})} by simply looking at this absolute value. This proves 2.3 ⇐ 2.7.

Next if π_{v} and π′_{v} are semisimple 2.6 ⇒ 2.3 follows from Lemma 6.4/2, and 2.3 ⇒ 2.2 was already established in 1.

Finally if μ_{π} and μ_{π′} are irreducible, 2.4 follows from 2.6 by Corollary 6.6 since μ_{π} is also the minimal 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 2.3 ⇒ 2.2 by the above.

9 The Endomorphism Q-Algebra

In this section we study the structure of QEnd(\underline{F}) for a semisimple abelian τ-sheaf \underline{F} over a finite field and calculate the local Hasse invariants of QEnd(\underline{F}) as a central simple algebra over Q(π). For a detailed introduction to central simple algebras, Hasse invariants and the Brauer group, we refer to [Rei, Ch. 7, §§28–31].

Theorem 9.1. Let \underline{F} be an abelian τ-sheaf over the finite field 𝔽_{s} of rank r with semisimple Frobenius endomorphism π, that is, Q(π) is semisimple. Let v ∈ C be a place different from ∞ and from the characteristic 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(\underline{F}).

2. We have r ≤ [ E : Q] = r_{Q_{v}}(χ_{v}, χ_{v}) ≤ r^{2} .

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 ⇔ 3.2 ⇔ 3.3 ⇔ 3.4 ⇔ 3.5 ⇐ 3.6 always,
3.5 ⇒ 3.6 if 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^{2}.

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 ⇔ 4.2 ⇔ 4.3 ⇔ 4.4 ⇒ 4.5 always,
4.4 ⇐ 4.5 if 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 is characterized by inv_{∞} E = wt(\underline{F}), inv_{ε} E = − wt(\underline{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 = −\frac{[𝔽_{v}:𝔽_{q}]}{[𝔽_{s}:𝔽_{q}]} · v(π). In particular

inv_{v} E = \{ \begin{matrix}0 & if v ∤ ε∞ , \\ wt(\underline{F}) · [F_{v} : Q_{∞}] & if v|∞ and ε ≠ ∞ . \end{matrix}

(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 \underline{F} is an elliptic sheaf, that is, d = 1 and \underline{M}(\underline{F}) is the Anderson motive of a Drinfeld 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(\underline{F}) and inv_{v} E = −[F : Q]·wt(\underline{F}). Note that Gekeler actually computes the Hasse invariants of the endomorphism algebra of the Drinfeld module. So his invariants differ from ours by a minus sign, since passing from Drinfeld modules to abelian τ-sheaves is a contravariant functor, see [BS, Theorem 3.2.1].

Corollary 9.3. Let \underline{F} be an abelian τ-sheaf over the smallest possible field L = 𝔽_{q} such that QEnd(\underline{F}) is a division algebra. Then QEnd(\underline{F}) is commutative and equals Q(π).

Proof. QEnd(\underline{F}) is a central division algebra over F by Theorem 9.1, which splits at all places of F by 9.1/5, hence equals F.

Proof (of Theorem 9.1 ). 1 was already proved in Corollary 6.10.

2. Let

χ_{v} = ∏_{i=1}^{n} μ_{i}^{m_{i}} ∈ Q_{v}[ x]

with distinct irreducible μ_{i} ∈ Q_{v}[ x] and m_{i} > 0 for 1 ≤ i ≤ n. Then ∑_{i=1}^{n} 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}) = ∑_{i=1}^{n} m_{i}^{2} · deg μ_{i}. The result now follows from the obvious inequalities

r = ∑_{i=1}^{n} m_{i} · deg μ_{i} ≤^{(1)} ∑_{i=1}^{n} m_{i}^{2} · deg μ_{i} ≤^{(2)} (∑_{i=1}^{n} m_{i} · deg μ_{i})^{2} = r^{2} .  (9.1)

3. Since F = Z(E), the equivalence 3.1 ⇔ 3.2 is evident. We have equality in (1) of Equation (9.1) if and only if m_{i} = 1 for all 1 ≤ i ≤ s which establishes the equivalence 3.4 ⇔ 3.5. In order to prove 3.5 ⇒ 3.3 we consider the minimal polynomial μ_{v} of π_{v} over Q_{v}. If χ_{v} has no multiple factor, then μ_{v} = χ_{v} and therefore [ F : Q] = [ Q_{v}(π_{v}) : Q_{v}] = r. Next 3.3 ⇒ 3.1 because F ⊂ E and (dim_{Q_{v}} F_{v})(dim_{Q_{v}} E_{v}) = dim_{Q_{v}} End_{Q_{v}}(V_{v}\underline{F}) = r^{2} by [Bou, Théorème 10.2/2], since E_{v} is the commutant of F_{v} in End_{Q_{v}}(V_{v}\underline{F}). Note that 3.3 ⇒ 3.1 also follows from Lemma 7.2. Conversely 3.1 ⇒ 3.4 because E = F implies r ≥ [ Q_{v}(π_{v}) : Q_{v}] = [ F : Q] = [ E : Q] ≥ r. For 3.5 ⇒ 3.6 we use Lemma 6.14/2 as we know that χ_{v} = μ_{v}. 3.6 ⇒ 3.5 is clear.

4. If 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 4.1 ⇔ 4.2. We have equality in (2) of (9.1) if and only if n = 1, deg μ_{1} = 1 and m_{1} = r which establishes 4.3 ⇔ 4.4. In order to connect 4.1 ⇔ 4.2 with 4.3 ⇔ 4.4 let 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 4.5 ⇒ 4.4 we use again 6.14/2 to see that μ_{v} is linear. 4.4 ⇒ 4.5 is clear.

The statement about the Hasse invariants follows from 5. Nevertheless, we give a separate proof in case (k, l) = 1 using Tate modules, since this is much shorter here and exhibits a different technique than 5. By the Tate conjecture 4.3, E ⊗_{Q} Q_{v} is isomorphic to End_{Q_{v}}(V_{v}\underline{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 the sum 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 we get 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_{∞}(\underline{F}) at ∞ and the isomorphism of Q_{∞}-algebras

E ⊗_{Q} Q_{∞} ≅ End_{Δ_{∞}[G]}(V_{∞}\underline{F}) = End_{Δ_{∞}}(V_{∞}\underline{F})

from Theorem 4.3. Since dim_{Q_{∞}} Δ_{∞} = l^{2} and dim_{Q_{∞}} V_{∞}\underline{F} = rl, we conclude that V_{∞}\underline{F} is a left r/l-dimensional Δ_{∞}-vector space and hence isomorphic to Δ_{∞}^{r/l}. Thus we have

E ⊗_{Q} Q_{∞} ≅ End_{Δ_{∞}}(Δ_{∞}^{r/l}) = M_{r/l}(End_{Δ_{∞}}(Δ_{∞})) = M_{r/l}(Δ_{∞}^{op}) .

Our proof now completes by inv_{∞} E = inv Δ_{∞}^{op} = − inv Δ_{∞} = \frac{k}{l} = wt(\underline{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 ground field. Let w be a place of Q and let \underline{N}_{w} := \underline{N}_{w}(\underline{F}) be the local σ-isoshtuka of \underline{F} at w. Let 𝔽_{w} be the residue field of w and 𝔽_{q^{f}} = 𝔽_{w} ∩ 𝔽_{s} the intersection inside an algebraic closure of 𝔽_{q}. Let 𝔞_{0} be the ideal (b ⊗ 1 − 1 ⊗ b : b ∈ 𝔽_{q^{f}}) of Q_{w} ⊗_{𝔽_{q}} 𝔽_{s} and let R := (Q_{w} ⊗_{𝔽_{q}} 𝔽_{s}/𝔞_{0})[T] = Q_{w} ⊗_{𝔽_{q^{f}}} 𝔽_{s}[T] be the

non-commutative polynomial ring with T ·(a⊗b) = (a⊗b^{q^{f}})·T for a ∈ Q_{w} and b ∈ 𝔽_{s}. Since Q_{w} ⊗_{𝔽_{q^{f}}} 𝔽_{s} is a field, R is a non-commutative principal ideal domain as studied by Jacobson [Jac, Chapter 3]. Its center is the commutative polynomial ring Q_{w}[T^{g}] where g = [𝔽_{s} : 𝔽_{q^{f}}] = \frac{e}{f}. From Theorem 3.6 and Proposition 3.5 we get isomorphisms

QEnd(\underline{F}) ⊗_{Q} Q_{w} ≅ End_{Q_{w}⊗_{𝔽q}𝔽_{s}[ϕ]}(\underline{N}_{w}) ≅ End_{R}(\underline{N}_{w}/𝔞_{0}\underline{N}_{w})

where T operates on \underline{N}_{w}/𝔞_{0}\underline{N}_{w} as ϕ^{f}.

By [Jac, Theorem 3.19] the R-module \underline{N}_{w}/𝔞_{0}\underline{N}_{w} decomposes into a finite direct sum indexed by some set I

\underline{N}_{w}/𝔞_{0}\underline{N}_{w} ≅ ⨁_{v∈I} \underline{N}_{v}^{⊕n_{v}}  (9.2)

of indecomposable R-modules \underline{N}_{v} with \underline{N}_{v} ≅ \underline{N̸}_{v′} for v ≠ v′. The annihilator of \underline{N}_{v} is a two sided ideal of R generated by a central element μ_{v} ∈ Q_{w}[T^{g}] by [Jac, §3.6], which can be chosen to be monic. In particular (9.2) is an isomorphism of Q_{w}[T^{g}]-modules and μ_{v} is the minimal polynomial of T^{g} on \underline{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 \underline{N}_{w}/𝔞_{0}\underline{N}_{w}. Note that T^{g} operates on \underline{N}_{w}/𝔞_{0}\underline{N}_{w} as the Frobenius π, hence μ = mipo_{π|\underline{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 irreducible polynomials 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 \underline{N}_{v} ≅ \underline{N̸}_{v′} and

μ = mipo_{π|\underline{F}} = ∏_{v∈I} μ_{v} inside Q_{w}[T^{g}] .

Thus F ⊗_{Q} Q_{w} = Q_{w}[T^{g}]/(μ) = ∏_{v∈I} Q_{w}[T^{g}]/(μ_{v}) = ∏_{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} = ⨁_{v∈I} End_{R}(\underline{N}_{v}^{⊕n_{v}}) = ⨁_{v∈I} E ⊗_{F} F_{v}

and E ⊗_{F} F_{v} ≅ End_{R}(\underline{N}_{v}^{⊕n_{v}}).

Now fix a place v above w and consider the diagram of field extensions

[figure: 𝔽_{v}𝔽_{s}; 𝔽_{v}; g/h; 𝔽_{w}𝔽_{s}; i; 𝔽_{w}𝔽_{s} ∩ 𝔽_{v}; g/h; 𝔽_{w}(𝔽_{v} ∩ 𝔽_{s}); i; 𝔽_{s}; 𝔽_{w}; i; h; 𝔽_{v} ∩ 𝔽_{s}; g/h; 𝔽_{q^{f}} = 𝔽_{w} ∩ 𝔽_{s} = 𝔽_{w} ∩ (𝔽_{v} ∩ 𝔽_{s}); h; 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}] = \frac{g}{h}, 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_{q^{f}}} 𝔽_{s} and F_{v} in an 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 the non-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 Δ_{v} = F_{v,L}[T′]/((T′)^{g/h} − π_{v}^{i}). Observe that the commutation rules of T′ are well defined since (Q_{w} ⊗_{𝔽_{q^{f}}} 𝔽_{s}) ∩ F_{v} has residue field 𝔽_{w}𝔽_{s} ∩ 𝔽_{v} = 𝔽_{q^{fhi}} and is unramified over Q_{w}, because Q_{w} ⊗_{𝔽_{q^{f}}} 𝔽_{s} is. Moreover, the extension F_{v,L}/F_{v} is unramified of degree [𝔽_{v}𝔽_{s} : 𝔽_{v}] = \frac{g}{h}

and T̃ := (T′)^{[𝔽_{v}:𝔽_{q}]/fhi} is its Frobenius automorphism. Since T̃^{g/h} = π_{v}^{[𝔽_{v}:𝔽_{q}]/fh} in Δ_{v}, our Δ_{v} is just the cyclic algebra (F_{v,L}/F_{v}, T̃, π_{v}^{[𝔽_{v}:𝔽_{q}]/fh}) and has Hasse invariant \frac{[𝔽_{v}:𝔽_{q}]}{[𝔽_{s}:𝔽_{q}]} · v(π_{v}); compare [Rei, p. 266]. We relate Δ_{v} to E ⊗_{F} F_{v}. Firstly by [Jac, Theorem 3.20] there exists a positive integer u such that \underline{N}_{v}^{⊕u} ≅ R/Rμ_{v}(T^{g}). Therefore

M_{u}(E ⊗_{F} F_{v}) ≅ M_{u}(End_{R}(\underline{N}_{v}^{⊕n_{v}})) = End_{R}(\underline{N}_{v}^{⊕un_{v}}) = M_{n_{v}}((R/Rμ_{v}(T^{g}))^{op}) .

Secondly we choose integers m and n with m > 0 and mi + ng = 1. We claim that the morphism R/Rμ_{v}(T^{g}) → M_{h}(Δ_{v}), which maps

a ⊗ b ⟼ ( \begin{matrix}a ⊗ b \\ & a ⊗ b^{q^{f}} \\ & & . & . & . \\ & & & & & a ⊗ b^{q^{f(h−1)}} \end{matrix} )  and  T ⟼ π_{v}^{n} · ( 0  1 1 (T′)^{m}  0 )

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′)^{1/i} in Gal(F_{v,L}/F_{v}), and it maps T^{g} = (T^{h})^{g/h} to π_{v}^{ng}(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 compute

dim_{F_{v}} M_{h}(Δ_{v}) = h^{2} · (\frac{g}{h})^{2} = g^{2} ,
dim_{Q_{w}⊗_{𝔽_{q^{f}}} 𝔽_{s}}(R/Rμ_{v}(T^{g})) = g · deg μ_{v} = g · [F_{v} : Q_{w}] , and
dim_{Q_{w}}(R/Rμ_{v}(T^{g})) = g^{2} · [F_{v} : Q_{w}] = dim_{Q_{w}} M_{h}(Δ_{v}) .

Altogether M_{u}(E ⊗_{F} F_{v}) ≅ M_{hn_{v}}(Δ_{v}^{op}) and inv_{v} E = − inv_{v} Δ_{v} = −\frac{[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(\underline{F})} = |π|_{∞} = q^{−v(π_{v})/e_{v}}, since the residue field of Q_{∞} is 𝔽_{q}. This implies as desired

−\frac{[𝔽_{v} : 𝔽_{q}]}{[𝔽_{s} : 𝔽_{q}]} · v(π_{v}) = −\frac{[𝔽_{v} : 𝔽_{q}] · (−e_{v}e · wt(\underline{F}))}{e} = wt(\underline{F}) · [F_{v} : Q_{∞}]

Finally if w ≠ ε, ∞ is a place of Q, the local σ-shtuka \underline{M}_{w}(\underline{F}) at w is étale. So μ = mipo_{π|\underline{F}} has coefficients 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}}^{1}, C ∖ {∞} = Spec 𝔽_{q}[ t] and L = 𝔽_{q}. Let d be a positive integer. Let

F_{i} := O(d⌈\frac{i}{2}⌉ · ∞) ⊕ O(d⌈\frac{i−1}{2}⌉ · ∞) for i ∈ ℤ and let τ := ({0 \atop 1} {t^{d} \atop 0} ). Then \underline{F} = (F_{i}, Π_{i}, τ_{i}) is an abelian τ-sheaf of rank 2, dimension d, and characteristic ε = V (t) ∈ ℙ^{1} over 𝔽_{q}. Hence the Frobenius endomorphism π equals τ. If d is odd then \underline{F} is primitive (that means (d, r) = 1) and therefore simple by [BH1, Proposition 7.4]. In particular, π is semisimple. We have

μ_{π} = χ_{v} = x^{2} − t^{d} = (x − \sqrt{t^{d}})(x + \sqrt{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 · 1 · 2 = 2 whereas in the field extension Q_{v}(\sqrt{t}) / Q_{v} we have

r_{Q_{v}(\sqrt{t})}(χ_{v}, χ_{v}) = \{ \begin{matrix}2 · 2 · 1 = 4 & in characteristic 2, \\ 1 · 1 · 1 + 1 · 1 · 1 = 2 & in characteristic different from 2. \end{matrix}

Although the later has no further significance it illustrates the remark after Definition 6.3. By Theorem 9.1/3. we have E = F = Q(π) commutative and [ E : Q] = 2 = r. Moreover, | π |_{∞} = | \sqrt{t^{d}} |_{∞} = q^{d/2} and χ_{v} is irreducible. But χ_{v} is not separable in characteristic 2.

If d = 2n is even then the minimal polynomial of π is

μ_{π} = χ_{v} = x^{2} − t^{d} = (x − t^{d/2})(x + t^{d/2}) .

So π is semisimple if and only if char(𝔽_{q}) ≠ 2. In this case \underline{F} is quasi-isogenous to the abelian τsheaf \underline{F}′ with F′_{i} = O_{C_{L}}(in · ∞)^{⊕2} and τ′_{i} = ({−t^{n} \atop 0} {0 \atop t^{n}} ). The quasi-isogeny f : \underline{F}′ → \underline{F} is given by

f_{0,η} = ({−t^{n} \atop 1} {t^{n} \atop 1} ) : F′_{0,η} ⟶^{∼} F_{0,η}. The abelian τ-sheaf \underline{F}′ equals the direct sum \underline{F}^{(1)} ⊕ \underline{F}^{(2)} where

F_{i}^{(j)} = O_{C_{L}}(in · ∞) and τ_{i}^{(j)} = (−1)^{j}t^{n}. Note that \underline{F}^{(1)} and \underline{F}^{(2)} are not isogenous over 𝔽_{q}, since the equation −t^{n} · σ^{∗}(g) = g · t^{n} has no solution g ∈ Q for char(𝔽_{q}) ≠ 2. Therefore

Q ⊕ Q = ⨁_{j=1}^{2} QEnd(\underline{F}^{(j)}) ≅ E = F = Q[x]/(x^{2} − t^{2n}) ≅ Q ⊕ Q .

Now we consider the same abelian τ-sheaf over L = 𝔽_{q^{2}}. This means π = τ^{2} = t^{d} ∈ Q and therefore χ_{v} = (x − t^{d})^{2}. Thus π is semisimple. By Theorem 9.1/4 we have F = Q(π) = Q and E is central simple over Q with [ E : Q] = 4 and inv_{∞} E = inv_{ε} E = \frac{d}{2}. Moreover, | π |_{∞} = | t^{d} |_{∞} = q^{d}. In this case, π_{v} is absolutely semisimple. Note that if d is even and char(𝔽_{q}) = 2 this is another example for Theorem 6.15.

If d is odd then \underline{F} is still primitive, whence simple and E is a division algebra. If d = 2n is even then the abelian τ-sheaves \underline{F}^{(1)} and \underline{F}^{(2)} defined above are isomorphic \underline{F}^{(1)} ⟶^{∼} \underline{F}^{(2)}, 1 ↦ λ where λ ∈ 𝔽_{q^{2}} satisfies λ^{q−1} = −1. Therefore M_{2}(Q) = M_{2}(QEnd(\underline{F}^{(1)})) ≅ E in accordance with the Hasse invariants just computed.

Example 9.5. We compute another example which displays other phenomena. Let C = ℙ_{𝔽_{q}}^{1} and let C ∖ \{∞\} = Spec 𝔽_{q}[t]. Let F_{i} = O_{C_{L}}(⌈\frac{i−1}{2}⌉ · ∞)^{⊕2} ⊕ O_{C_{L}}(⌈\frac{i}{2}⌉ · ∞)^{⊕2}, let Π_{i} be the natural inclusion, and let τ_{i} be given by the matrix

T := ( \begin{matrix}0 & 0 & 0 & a \\ 0 & b & 1 & 0 \\ t & 0 & −b & 0 \\ 0 & t & 0 & 0 \end{matrix} ) with a, b ∈ 𝔽_{q} ∖ \{0\} .

Then \underline{F} is an abelian τ-sheaf of rank 4 and dimension 2 with l = 2, k = 1 and characteristic ε = V (t) ∈ ℙ^{1}. One checks that the minimal polynomial of the matrix T is x^{4} − b^{2}x^{2} − at^{2} which is irreducible over 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}) = 2 then the minimal polynomial is a square and \underline{F} is not semisimple.

For L = 𝔽_{q} and 2 ∤ q we obtain π = τ semisimple and E = F = Q(π) = Q[x]/(x^{4} − b^{2}x^{2} − at^{2}). For L = 𝔽_{q^{2}} we have π = τ^{2} and the minimal polynomial of π over Q is x^{2} − b^{2}x − at^{2}, which is

irreducible also in characteristic 2 since it has no zeroes in 𝔽_{q}[t]. Hence π is semisimple, F is a field with [F : Q] = 2 and [E : F] = 4 by Corollary 6.6. This again illustrates Theorem 6.15. We compute the decomposition of ∞ and ε in F. Decomposition of ε: Modulo t the polynomial x^{2} − b^{2}x − at^{2} has two zeroes x = b^{2} and x = 0 in 𝔽_{q}. So by 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}) = 2. Thus the Hasse invariants of E are inv_{v} E = inv_{v′} E = 0. Decomposition of ∞: Set y = π/t. Then y^{2} − \frac{b^{2}}{t} y − a = 0.

Case (a). If 2|q then (y − a^{q/2})^{2} − \frac{b^{2}}{t} (y − a^{q/2}) − \frac{b^{2}}{t} a^{q/2} = 0, that is, ∞ ramifies in F, F ⊗_{Q} Q_{∞} = F_{w} with w(\frac{π}{t} − a^{q/2}) = 1 and w(\frac{1}{t}) = 2 · ∞(\frac{1}{t}) = 2. So [F_{w} : Q_{∞}] = 2 and inv_{w} E = 0.

Case (b). If 2 ∤ q and \sqrt{a} ∈ 𝔽_{q} then the polynomial y^{2} − \frac{b^{2}}{t} y − a has two zeroes y = ±\sqrt{a} modulo \frac{1}{t}. So by Hensel’s lemma F ⊗_{Q} Q_{∞} ≅ F_{w} ⊕ F_{w′} splits with [F_{w} : Q_{∞}] = [F_{w′} : Q_{∞}] = 1. Thus the local Hasse invariants of E are inv_{w} E = inv_{w′} E = \frac{1}{2}. As was remarked in 9.2 such a distribution of the Hasse invariants can occur only if d ≥ 2. Case (c). If 2 ∤ q and \sqrt{a} ∉ 𝔽_{q} then y^{2} − \frac{b^{2}}{t} y − a is irreducible modulo \frac{1}{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 \underline{F} is simple. In cases (a) and (c) E ≅ M_{2}(F) and \underline{F} is quasi-isogenous to (\underline{F}′)^{⊕2} for an abelian τ-sheaf \underline{F}′ of rank 2, dimension 1 and QEnd(\underline{F}′) = F. This surprising result is due to the fact that \underline{F}′, being of dimension 1, is associated with a Drinfeld module and thus of the form F′_{i} = O_{C_{L}}(⌈\frac{i}{2}⌉ · ∞) ⊕ O_{C_{L}}(⌈\frac{i−1}{2}⌉ · ∞) with τ′_{i} = ({c \atop d} {t \atop 0}) and c, d ∈ 𝔽_{q^{2}}. Then π′ = (τ′)^{2} = ({c^{q+1}+d^{q}t \atop c^{q}d} {ct \atop dt}) has minimal polynomial x^{2} − (c^{q+1} + (d + d^{q})t)x + d^{q+1}t^{2} which must be equal to x^{2} − b^{2}x − at^{2}. 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^{2}} ∖ 𝔽_{q}, d^{q} = −d, and a = d^{2}. The later implies 2 ∤ q and \sqrt{a} = d ∉ 𝔽_{q} and we are in case (c). If we choose c = b in case (c) a quasi-isogeny f : \underline{F} → (\underline{F}′)^{⊕2} over 𝔽_{q^{2}} is given for instance by

( \begin{matrix}d & a & −bd/t & 0 \\ 0 & 0 & −d & a \\ 0 & 0 & d/t & a/t \\ 1 & −d & 0 & bd/t \end{matrix} ) .

10 Kernel Ideals for Pure Anderson Motives

In this section we investigate which orders of E can arise as endomorphism rings End(\underline{M}) for pure Anderson motives \underline{M}. For this purpose we define for each right ideal of the endomorphism ring End(\underline{M}) an isogeny with target \underline{M} and discuss its properties. This generalizes Gekeler’s results for Drinfeld modules [Gek, §3] and translates the theory of Waterhouse [Wat, §3] for abelian varieties to the function field case. These two sources are themselves the translation, respectively the higher dimensional generalization of Deuring’s work on elliptic curves [Deu].

Let \underline{M} be a pure Anderson motive over L and abbreviate R := End(\underline{M}). Let I ⊂ R be a right ideal which is an A-lattice in E := R ⊗_{A} Q. This is equivalent to saying that I contains an isogeny, since every lattice contains some isogeny a · id_{\underline{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 \underline{M}^{I} be the pure Anderson sub-motive of \underline{M} whose underlying A_{L}-module is ∑_{g∈I} im(g). This is indeed a pure Anderson motive, since if I = f_{1}R + … + f_{n}R are arbitrary generators, then \underline{M}^{I} equals the image of the morphism

(f_{1}, …, f_{n}) : \underline{M} ⊕ … ⊕ \underline{M} ⟶ \underline{M} .

As I contains an isogeny, \underline{M}^{I} has the same rank as \underline{M} and the natural inclusion is an isogeny which we denote f_{I} : \underline{M}^{I} → \underline{M}.

2. If I = \{ f ∈ R : im(f) ⊂ \underline{M}^{I} \} then I is called a kernel ideal for \underline{M}.

The later terminology is borrowed from Waterhouse [Wat, §3]. Since \{ f ∈ R : im(f) ⊂ \underline{M}^{I} \} is the right 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) ⊂ \underline{M}^{I} \} ⊂ R containing I. Then \underline{M}^{J} = \underline{M}^{I}. In particular, J is a kernel ideal for \underline{M}. We call J the kernel ideal for \underline{M} associated with I.

Proof. Obviously J is a right ideal and \underline{M}^{J} ⊂ \underline{M}^{I} by definition of J. Conversely \underline{M}^{I} ⊂ \underline{M}^{J} since I ⊂ J.

Lemma 10.3. 1. For any g ∈ I, f_{I}^{−1} ∘ g : \underline{M} → \underline{M}^{I} is a morphism and g = f_{I} ∘ (f_{I}^{−1} ∘ g).

2. If I = gR is principal, g an isogeny, then f_{I}^{−1} ∘ g : \underline{M} → \underline{M}^{I} is an isomorphism and I is a kernel ideal.

Proof. 1 is obvious since the image of g lies inside \underline{M}^{I}.

2. Clearly f_{I}^{−1}∘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) ⊂ \underline{M}^{I}. Consider the diagram

\xymatrix{\underline{M} \ar@{-->}[d]_{h} \ar[r]^{f_{I}^{−1}∘f} & \underline{M}^{I} \ar[r]^{f_{I}} & \underline{M} \\ \underline{M} \ar[ur]_{f_{I}^{−1}∘g}}

and let h := (f_{I}^{−1} ∘ g)^{−1} ∘ (f_{I}^{−1} ∘ f). Then f = gh ∈ I as desired.

Example. If a ∈ A and I = aR, then \underline{M}^{I} = a\underline{M} and coker f_{I} = \underline{M}/a\underline{M}. More generally if 𝔞 ⊂ A is an ideal and I = 𝔞R then \underline{M}^{I} = 𝔞\underline{M} and coker f_{I} = \underline{M}/𝔞\underline{M}.

Proposition 10.4. Let I ⊂ R and J ⊂ End(\underline{M}^{I}) be right ideals which are lattices in E. Then also the product K := f_{I} · J · f_{I}^{−1} · I is a right ideal of R and a lattice in E and f_{K}^{−1} ∘ f_{I} ∘ f_{J} is an isomorphism of (\underline{M}^{I})^{J} with \underline{M}^{K}

(\underline{M}^{I})^{J} ⟶^{f_{J}} \underline{M}^{I} ⟶^{f_{I}} \underline{M} ⟵^{f_{K}} \underline{M}^{K} .

Proof. If f ∈ I and g ∈ J then the morphism f_{I}^{−1} ∘ f : \underline{M} → \underline{M}^{I} can be composed with f_{I} ∘ g to yield an element of R. Since I and J contain isogenies, K is a right ideal and contains an isogeny. Clearly the images of f_{I} ∘ f_{J} and f_{K} in \underline{M} coincide since they equal the sum ∑_{i,j} f_{I} ∘ g_{j} ∘ f_{I}^{−1} ∘ f_{i}(\underline{M}) for sets of generators {f_{i}} of I and {g_{j}} of J.

Theorem 10.5. Let I, J ⊂ End(\underline{M}) =: R be right ideals which are lattices in E := R⊗_{A}Q and consider the following assertions:

1. I and J are isomorphic R-modules,

2. the pure Anderson motives \underline{M}^{I} and \underline{M}^{J} are isomorphic.

Then 1 implies 2 and if moreover I and J are kernel ideals, also 2 implies 1.

Proof. 1 ⇒ 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 such that ag ∈ I ⊂ R. Then im(ag) ⊂ \underline{M}^{I}, that is, f_{I}^{−1} ∘ ag : \underline{M} → \underline{M}^{I} is an isogeny.

Let K be the right ideal f_{I} · (f_{I}^{−1} ∘ ag ∘ f_{I} · End(\underline{M}^{I})) · f_{I}^{−1} · I of R. We claim that \underline{M}^{K} ≅ \underline{M}^{(ag)I}. Namely, \underline{M}^{(ag)I} ⊂ \underline{M}^{K} since agI ⊂ K. Conversely if f ∈ I, h ∈ End(\underline{M}^{I}), and m ∈ \underline{M}, then we find m′ := f_{I} ∘ h ∘ f_{I}^{−1} ∘ f(m) ∈ \underline{M}^{I}, that is, m′ = ∑_{i} f_{i}(m_{i}) for suitable f_{i} ∈ I and m_{i} ∈ \underline{M}. It follows that ag(m′) = ∑_{i} agf_{i}(m_{i}) ∈ \underline{M}^{(ag)I} and therefore \underline{M}^{(ag)I} = \underline{M}^{K}.

Applying Lemma 10.3 and Proposition 10.4 now yields an isomorphisms \underline{M}^{I} ≅ \underline{M}^{K} = \underline{M}^{(ag)I}. Likewise we obtain \underline{M}^{J} ≅ \underline{M}^{aJ} and the equality aJ = agI then implies \underline{M}^{J} ≅ \underline{M}^{I} as desired. 2 ⇒ 1. Let I and J be kernel ideals and let u : \underline{M}^{I} → \underline{M}^{J} be an isomorphism. There is an a ∈ A with a\underline{M} ⊂ \underline{M}^{I}. Therefore g := f_{J} ∘ u ∘ (f_{I}^{−1} ∘ a) : \underline{M} → \underline{M} is an isogeny.

We claim that gI = aJ, that is, left multiplication by a^{−1}g is an isomorphism of I with J. Let f ∈ I, then h := f_{J} ∘ u ∘ (f_{I}^{−1} ∘ f) ∈ R has im(h) ⊂ \underline{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^{−1} ∘ (f_{J}^{−1} ∘ h) ∈ R has im(f) ⊂ \underline{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(\underline{M}^{I})·f_{I}^{−1} contains the left order O = { f ∈ E : fI ⊂ I } of I and equals it if I is a kernel ideal.

Remark. Recall that End(\underline{M}^{I}) ⊗_{A} Q is identified with E by mapping h ∈ End(\underline{M}^{I}) to f_{I} ∘ h ∘ f_{I}^{−1}.

Proof. Let f ∈ O and g ∈ I. Then fg ∈ I and f_{I}^{−1} ∘ f ∘ f_{I} ∘ (f_{I}^{−1} ∘ g) = f_{I}^{−1} ∘ fg is a morphism from \underline{M} to \underline{M}^{I}. If g varies, the images of f_{I}^{−1} ∘ g exhaust all of \underline{M}^{I}. Hence f_{I}^{−1} ∘ f ∘ f_{I} is indeed an endomorphism of \underline{M}^{I}. Conversely let I be a kernel ideal and let f = f_{I} ∘ h ∘ f_{I}^{−1} ∈ f_{I} · End(\underline{M}^{I}) · f_{I}^{−1}. If g ∈ I then f ∘ g = f_{I} ∘ h ∘ (f_{I}^{−1} ∘ g) ∈ R has im(f ∘ g) ⊂ \underline{M}^{I}. So fg ∈ 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(\underline{M}′) · f^{−1} ⊂ E of a pure Anderson motive \underline{M}′ isogenous to \underline{M} via an isogeny f : \underline{M}′ → \underline{M}.

Proof. Let S be a maximal order of E. Then the lattice R contains aS for some a ∈ A. Consider the right ideal I = aS · R whose left order contains S. By Proposition 10.6, f_{I} · End(\underline{M}^{I}) · f_{I}^{−1} contains the left order of I. Since S is maximal we find S = f_{I} · End(\underline{M}^{I}) · f_{I}^{−1}.

Theorem 10.8. If E is semisimple and End(\underline{M}) is a maximal order in E, so is f_{I} · End(\underline{M}^{I}) · f_{I}^{−1} for any right ideal I ⊂ R.

Proof. By [Rei, Theorem 21.2] the left order of I is also maximal and then Proposition 10.6 yields the result.

From now on we assume that L is a finite field and we set e := [L : 𝔽_{q}]. Let π be the Frobenius endomorphism of \underline{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 2.10 and so there is an isogeny f : \underline{M} → \underline{M} with π ∘ f = a. The image a/π of f in E belongs to R.

Proposition 10.10. If \underline{M} is a semisimple pure Anderson motive over a finite field and End(\underline{M}) is a maximal order in E = End(\underline{M}) ⊗_{A} Q, then every right ideal I ⊂ End(\underline{M}), which is a lattice, is a kernel ideal for \underline{M}, and deg(f_{I}) = N(I) := (N(f) : f ∈ I).

Proof. (cf. [Wat, Theorem 3.15]) Let f ∈ I, then f = f_{I} ∘ f_{I}^{−1}f and N(f) ∈ deg(f) ⊂ deg(f_{I}) by Lemma 2.9. Therefore N(I) ⊂ deg(f_{I}). Let R′ be the left order of I. It is maximal by [Rei, Theorem 21.2]. 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}^{−1}J′f_{I} ⊂ End(\underline{M}^{I}) be the induced right ideal of End(\underline{M}^{I}) = f_{I}^{−1}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 ideal its 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 the investigation of isomorphy classes of pure Anderson motives isogenous to \underline{M}. We content ourselves with the following result which is analogous to Waterhouse [Wat, Theorem 6.1]. The interested reader will find many other results without much difficulty.

Theorem 10.11. Let \underline{M} be a simple pure Anderson motive of rank r and dimension d over the smallest possible field 𝔽_{q}. Then

1. End(\underline{M}) is commutative and E := End(\underline{M}) ⊗_{A} Q = Q(π).

2. All orders R in Q(π) containing π are endomorphism rings of pure Anderson motives isogenous to \underline{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 \underline{M} with endomorphism ring R correspond bijectively to the isomorphism classes of A-lattices in E with order R.

Proof. 1 follows from 6.11 and 9.3.

2. Let 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}\underline{M} of (left) E_{v}-modules given by f ↦ f(x) for a suitable x ∈ V_{v}\underline{M}. It identifies R_{v} := R ⊗_{A} A_{v} with a π-stable lattice Λ_{v} = R_{v} · x in V_{v}\underline{M}, which without loss of generality is contained in T_{v}\underline{M}. By Proposition 4.4 there is an isogeny f : \underline{M}′ → \underline{M} of pure Anderson motives with T_{v}f(T_{v}\underline{M}′) = Λ_{v}. By Theorem 4.2 we conclude

End(\underline{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}} \underline{N}_{ε}(\underline{M}) = r = [E : Q], Theorem 3.7 together with Lemma 7.2 show that E_{ε} is isomorphic to \underline{N}_{ε}(\underline{M}) as left E_{ε}-modules. Since R contains π, the image of R_{ε} := R ⊗_{A} A_{ε} in \underline{N}_{ε}(\underline{M}) is a local σ-subshtuka \underline{M̂}′ of \underline{M}_{ε}(\underline{M}) of the same rank. (If it is not contained in \underline{M}_{ε}(\underline{M}), multiply it with a suitable a ∈ A.) Then Proposition 3.9 yields an isogeny of pure Anderson motives f : \underline{M}′ → \underline{M} such that \underline{M}_{ε}(f)(\underline{M}_{ε}(\underline{M}′)) = \underline{M̂}′ and

End(\underline{M}′) ⊗_{A} A_{ε} = End_{A_{ε}[ϕ]}(\underline{M̂}′) = R_{ε}

REFERENCES 34

by Theorem 3.7. Since each of these operations only modifies End(\underline{M}) at the respective place v, this shows that we may modify \underline{M} at all places to obtain a pure Anderson motive \underline{M}′ with End(\underline{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 \underline{M̃} for which all T_{v}\underline{M̃} ≅ R_{v} and \underline{M}_{ε}(\underline{M̃}) ≅ R_{ε}. Let I ⊂ R be a (right) ideal which is an A-lattice in E and consider the isogeny f_{I} : \underline{M̃}^{I} → \underline{M̃}. Under the above isomorphisms T_{v}f_{I}(T_{v}\underline{M̃}^{I}) ≅ I ⊗_{A} A_{v} =: I_{v} and \underline{M}_{ε}f_{I}(\underline{M}_{ε}\underline{M̃}^{I}) ≅ I ⊗_{A} A_{ε} =: I_{ε}. Conversely if f : \underline{M}′ → \underline{M̃} is an isogeny then \underline{M}_{v}f(\underline{M}_{v}\underline{M}′) is a (left) R_{v}-module because R = End(\underline{M̃}), hence isomorphic to an R_{v}-ideal I_{v}. This shows that any isogeny f : \underline{M}′ → \underline{M̃} is of the form f_{I} : \underline{M̃}^{I} → \underline{M̃}.

If now f ∈ R satisfies im(f) ⊂ \underline{M̃}^{I} then f ∈ I_{ε} and f ∈ I_{v} for all v and therefore f ∈ I. This shows that every I is a kernel ideal for \underline{M̃}. By Proposition 10.6, End(\underline{M̃}^{I}) is the (left) order of I. Since every lattice with order R in E is isomorphic to an ideal of R, we have

[figure: \{ A-lattices in E with order R \}/_{∼}; \{ I ⊂ R Ideals with order R \}/_{∼}; ∼; \{ \underline{M̃}^{I} ⟶^{f_{I}} \underline{M̃} → \underline{M} with End(\underline{M̃}^{I}) = R \}/_{∼}]

and the assertion now follows from Theorem 10.5.

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Matthias Bornhofen Am Rösslewald 4 D – 79874 Breitnau Germany Urs Hartl Mathematisches Institut Universität Münster Einsteinstr. 62 D – 48149 Münster Germany http://www.math.uni-muenster.de/u/urs.hartl/