A criterion for good reduction of Drinfeld modules and Anderson motives in terms of local shtukas

U. Hartl, S. Hüsken

Abstract For an Anderson A-motive over a discretely valued field whose residue field has A-characteristic ε, we prove a criterion for good reduction in terms of its associated local shtuka at ε. This yields a criterion for good reduction of Drinfeld modules. Our criterion is the function-field analog of Grothendieck’s [SGA 7, Proposition IX.5.13] and de Jong’s [dJ98, 2.5] criterion for good reduction of an abelian variety over a discretely valued field with residue characteristic p in terms of its associated p-divisible group. Mathematics Subject Classification (2000): 11G09, (14L05)

Contents

1 Introduction 1

2 The base rings 2

3 Frobenius modules 4

4 Anderson motives 5

5 Local shtukas and analytic Anderson motives 7

6 The reduction criterion for Anderson motives 12

References 14

1 Introduction

We fix a finite field 𝔽 with r elements and characteristic p. Let C be a smooth projective and geometrically irreducible curve over 𝔽 with function field Q = 𝔽(C). Let ∞ ∈ C be a closed point and let A = Γ(C ∖ {∞}, O_{C}) be the 𝔽-algebra of those rational functions on C which are regular outside ∞. For every 𝔽-algebra R we let σ be the endomorphism of A_{R} := A ⊗_{𝔽} R given by σ := id_{A} ⊗ Frob_{r,R} : a ⊗ b ↦ a ⊗ b^{r} for a ∈ A and b ∈ R.

Let o_{L} be a complete discrete valuation ring containing 𝔽, with fraction field L, uniformizing parameter π, maximal ideal 𝔪_{L} = (π) and residue field ℓ = o_{L}/𝔪_{L}. We assume that ℓ is a finite field extension of ℓ^{p}. This is equivalent to saying that ℓ has a finite p-basis over ℓ^{p} in the sense of [Bou81, § V.13, Definition 1]. It holds for example if ℓ is perfect, or if ℓ is a finitely generated field. Since every Anderson A-motive over L can be defined over a finitely generated subfield of L our restriction on ℓ is not serious. Let c^{∗}: A → o_{L} be a homomorphism of 𝔽-algebras such that the kernel of the composition A → o_{L} ↠ ℓ is a maximal ideal ε in A. We say that the residue field ℓ has finite A-characteristic ε. We do not assume that c^{∗}: A → o_{L} is injective. So L can have either generic A-characteristic kerc^{∗} = (0) or finite A-characteristic kerc^{∗} = ε.

In the following we will consider various A_{o_{L}} -algebras. In all of them we consider the ideal generated by {a ⊗ 1 − 1 ⊗ c^{∗}(a): a ∈ A} ⊆ A_{o_{L}}. By abuse of notation we denote all these ideals by 𝔍.

By an Anderson A-motive over L we mean a pair \underline{M} = (M, F_{M}) consisting of a locally free A_{L}-module M of finite rank, and an injective A_{L}-homomorphism F_{M} : σ^{∗}M → M where σ^{∗}M := M ⊗_{A_{L} ,σ} A_{L}, such that coker(F_{M}) is a finite dimensional L-vector space and is annihilated by a power of 𝔍. We say that \underline{M} has good reduction over o_{L} if there exists a locally free A_{o_{L}} -module M and an injective A_{o_{L}} -homomorphism F_{M} : σ^{∗}M → M such that (M, F_{M} ) ⊗_{A_{o_{L}}} A_{L} ≅ \underline{M} and coker(F_{M} ) is a finite free o_{L}-module which is annihilated by a power of 𝔍. We call \underline{M} = (M, F_{M} ) a good model of \underline{M}. In particular if \underline{M} = \underline{M}(φ) is the Anderson A-motive associated with a Drinfeld A-module φ over L, then \underline{M} has good reduction if and only if φ has good reduction; see Proposition 4.10.

Anderson A-motives are function-field analogs of abelian varieties. For an abelian variety A over a discretely valued field K with residue field of characteristic p there are criteria for good reduction in terms of local data. For a prime number l ≠ p the criterion of Néron-Ogg-Shavarevich [ST68, §1, Theorem 1] states that A has good reduction if and only if the l-adic Tate module T_{l}A of A is unramified as a Gal(K^{alg}/K)-representation. At the prime p the criterion of Grothendieck [SGA 7, Proposition IX.5.13] (for char(K) = 0), respectively de Jong [dJ98, 2.5] (for char(K) = p) states that A has good reduction if and only if the Barsotti-Tate group A[p^{∞} ] has good reduction.

These criteria have function-field analogs for Anderson A-motives. The analog of the Néron-Ogg-Shavarevich-criterion was proved by Gardeyn [Gar02, Theorem 1.1]. In this article we simultaneously prove the analog of Grothendieck’s and de Jong’s criterion. Here the function-field analogs of Barsotti-Tate groups are local shtukas [Har11, § 2.1] which are defined as follows. Let A_{o_{L} ,(ε,π)} be the (ε, π)-adic completion of A_{o_{L}}. An (effective) local shtuka at ε over o_{L} is a pair \underline{M̂} = (M̂, FM̂) consisting of a finite free A_{o_{L} ,(ε,π)}- module M̂ and an injective A_{o_{L} ,(ε,π)}-homomorphism FM̂ : σ^{∗}M̂ → M̂ such that coker(FM̂) is a finite free o_{L}-module and is annihilated by a power of 𝔍. The local shtuka associated with a good model \underline{M} of an Anderson A-motive is \underline{M̂}(\underline{M} ) := \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,(ε,π)}. Strictly speaking effective local shtukas are the function field analogs of the F -crystals of Barsotti-Tate groups. The analogs of the latter are called ε-divisible local Anderson-modules and their category is equivalent to the category of effective local shtukas; see [HS15] for more details. Our analog of Grothendieck’s and de Jong’s reduction criterion is now the following

Corollary 6.6. Let \underline{M} be an Anderson A-motive over L. Then \underline{M} has good reduction over o_{L} if and only if there is an effective local shtuka \underline{M̂} at ε over o_{L} and an isomorphism \underline{M} ⊗_{A_{L}} A_{o_{L} ,(ε,π)}[1/π] ≅ \underline{M̂} ⊗_{A_{o_{L} ,(ε,π)}} A_{o_{L} ,(ε,π)}[1/π].

(In the body of the text we prove a slightly stronger statement.) This applies in particular if \underline{M} is the Anderson A-motive associated with a Drinfeld module φ over L to give a criterion for good reduction of φ in terms of its associated local shtuka. The reformulation of this criterion in terms of the ε-divisible local Anderson-module of φ is given in [HS15].

Acknowledgements. We would like to thank the anonymous referee for his careful reading and for asking an interesting question which lead to the answer given in Remark 5.4. We also thank the Deutsche Forschungsgemeinschaft for supporting this research in form of SFB 878.

2 The base rings

Let o_{L} be an equi-characteristic complete discrete valuation ring containing the finite field 𝔽, with quotient field L = Frac(o_{L}) and residue field ℓ = o_{L}/𝔪_{L}, where 𝔪_{L} ⊆ o_{L} is the maximal ideal of o_{L}. We assume that ℓ is a finite field extension of ℓ^{p} := {b^{p}: b ∈ ℓ}. We fix a uniformizer π = π_{L} of o_{L} and sometimes identify o_{L} with ℓ⟦π⟧. Let v = v_{π} = ord_{π}(·) be the discrete valuation on L normalized by v(π) = 1.

We assume that there is an o_{L}-valued point c ∈ C(o_{L}) such that the corresponding 𝔽-morphism c: Spec(o_{L}) → C factors via C ∖ {∞} ⊆ C. Such a datum corresponds to a homomorphism of 𝔽-algebras c^{∗}: A → o_{L} which we call the characteristic map. We further assume that the closed point V (π) ⊆ Spec(o_{L}) is mapped to a closed point ε of Spec(A) ⊆ C. The latter is the kernel of the composition A → o_{L} ↠ ℓ. So, in accordance with Drinfeld’s terminology [Dri76], we call ε the residue characteristic or residual characteristic place of Q. By continuity, the characteristic map c^{∗}: A → o_{L} factors through a morphism of complete discrete valuation rings A_{ε} → o_{L} where A_{ε} is the completion of A at the characteristic place ε. Note that A_{ε} → o_{L} is injective if c^{∗} is injective, and factors through A/ε if c^{∗} is not injective.

Remark 2.1. Since A is a Dedekind domain there is a power ε^{m} which is a principal ideal in A. We fix a generator t of ε^{m} and frequently use the finite flat monomorphism of 𝔽-algebras ι: 𝔽[z] → A, z ↦ t.

For any 𝔽-algebra R we abbreviate A_{R} := A ⊗_{𝔽} R. In particular, A_{o_{L}} ⊆ A_{L} is a noetherian integral domain, and by virtue of the equality A_{ℓ} ≅ A_{o_{L}} /πA_{o_{L}} it follows that π ∈ o_{L} is a prime element of A_{o_{L}}.

Definition 2.2. Let A_{o_{L} ,π} (resp., A_{o_{L} ,(ε,π)}) be the completion of the o_{L}-algebra A_{o_{L}} for the π-adic topology (resp., the (ε, π)-adic topology).

By Krull’s Theorem ([Bou67], III.3.2), the ring A_{o_{L}} is separated for both the π-adic and the (ε, π)-adic topology. The topological o_{L}-algebra A_{o_{L} ,π} is admissible in the sense of Raynaud, i.e. it is of topologically finite presentation and has no π-torsion. In particular, the L-algebra A_{o_{L} ,π}[1/π] is affinoid in the sense of rigid analytic geometry; see [Bos14, BL93a, BGR84].

For example if C = ℙ_{𝔽}^{1} and A = 𝔽[z] then we have A_{o_{L}} = o_{L}[z] and correspondingly A_{L} = L[z]. Let us specify that ε = z𝔽[z]. Our choice of a uniformizer π gives rise to an identification o_{L} = ℓ⟦π⟧. Consequently o_{L}⟦z⟧ = ℓ⟦π⟧⟦z⟧ = ℓ⟦π, z⟧ = A_{o_{L} ,(ε,π)}. On the other hand, the π-adic completion of o_{L}[z]

equals o_{L}⟨z⟩ := {∑_{i=0}^{∞} b_{i}z^{i}: v(b_{i}) → ∞(i → ∞)}, and since L⟨z⟩ = o_{L}⟨z⟩ ⊗_{o_{L}} L, we may view A_{o_{L} ,π}[1/π] as a

replacement, for general C, of the Tate algebra L⟨z⟩ of strictly convergent power series in one indeterminate z over L, which serves as coordinate ring for the one-dimensional affinoid unit ball in rigid analytic geometry.

There is a natural embedding A_{L} → A_{o_{L} ,π}[1/π] which, for general C, replaces the completion homomorphism L[z] → L⟨z⟩, and which itself can be regarded as a completion map with respect to the L-algebra norm-topology on the reduced affinoid L-algebra A_{o_{L} ,π}[1/π] and its restriction on A_{L}; see [Bos14, §1.4, Proposition 19]. Note that the canonical homomorphism A_{o_{L}} → A_{o_{L} ,(ε,π)} factors uniquely via A_{o_{L} ,π}, where the induced map A_{o_{L} ,π} → A_{o_{L} ,(ε,π)} identifies A_{o_{L} ,(ε,π)} with the (ε, π)A_{o_{L} ,π}-adic completion of A_{o_{L} ,π}. Since A_{o_{L} ,π} is a regular integral domain, it is (ε, π)A_{o_{L} ,π}-adically separated by Krull’s theorem and A_{o_{L} ,π} → A_{o_{L} ,(ε,π)} is injective and flat.

Recall that there is a finite flat monomorphism of 𝔽-algebras ι: 𝔽[z] → A which identifies the indeterminate z with the generator t ∈ A of ε^{m} chosen in Remark 2.1. The o_{L}-algebra homomorphism ι ⊗ id: o_{L}[z] → A_{o_{L}}, ∑ _{ν} a_{ν}z^{ν} ↦ ∑ _{ν} t^{ν} ⊗ a_{ν}, is finite flat, so that we obtain finite flat maps

o_{L}⟨z⟩ → A_{o_{L} ,π}, L⟨z⟩ → A_{o_{L} ,π}[1/π], o_{L}⟦z⟧ → A_{o_{L} ,(t,π)}, ℓ[z] → A_{ℓ}.  (2.1)

Here the (t, π)-adic completion A_{o_{L} ,(t,π)} of A_{o_{L}} equals A_{o_{L} ,(ε,π)} since (ε, π)^{m} ⊆ (ε^{m}, π) = (t, π) in A_{o_{L}}.

Lemma 2.3. If A_{o_{L} ,ε} denotes the ε-adic completion of A_{o_{L}}, the canonical map A_{o_{L} ,ε} → A_{o_{L} ,(ε,π)} is an isomorphism.

3 Frobenius modules

The r-Frobenius Frob_{r}: o_{L} → o_{L}, x ↦ x^{r}, gives rise to an endomorphism

σ = id_{A} ⊗ Frob_{r}: A_{o_{L}} → A_{o_{L}} , a ⊗ x ↦ a ⊗ x^{r},

which extends to give a map id_{A}⊗Frob_{r,L} : A_{L} → A_{L} again denoted by σ. On the other hand, reducing mod π gives σ̄ = id_{A} ⊗ Frob_{r,ℓ}: A_{ℓ} → A_{ℓ}. The latter is a finite flat endomorphism of the Dedekind domain A_{ℓ}, because ℓ is finite over ℓ^{p}. The map σ: A_{o_{L}} → A_{o_{L}} is π-adically and (ε, π)-adically continuous and therefore extends to give endomorphisms A_{o_{L} ,π} → A_{o_{L} ,π} and A_{o_{L} ,(ε,π)} → A_{o_{L} ,(ε,π)}, again denoted by σ.

Lemma 3.1. In the commutative diagram

\xymatrix{A_{o_{L}} \ar[r] \ar[d]_{σ} & A_{o_{L} ,π} \ar[r] \ar[d]_{σ} & A_{o_{L} ,(ε,π)} \ar[d]_{σ} \\ A_{o_{L}} \ar[r] & A_{o_{L} ,π} \ar[r] & A_{o_{L} ,(ε,π)}}

both squares are cocartesian, and the vertical arrows are finite flat.

We let the proof be preceded by the following

Remark. Via the identification o_{L} = ℓ⟦π⟧, the r-Frobenius Frob_{r,o_{L}} : o_{L} → o_{L} is mirrored by the map ℓ⟦π⟧ → ℓ⟦π⟧, ∑_{ν=0}^{∞} a_{ν}π^{ν} ↦ ∑_{ν=0}^{∞} a_{ν}^{r}π^{rν}. Choosing an ℓ^{r}-basis of ℓ and lifting it to a subset W of o_{L}, this implies

(Frob_{r,o_{L}} )_{∗}o_{L} = ⨁_{i=0}^{r−1} ⨁_{w∈W} o_{L} wπ^{i}, so that Frob_{r,o_{L}} : o_{L} → o_{L} is finite flat.

Proof of Lemma 3.1. By base change the remark implies that σ = id_{A} ⊗ Frob_{r,o_{L}} : A_{o_{L}} → A_{o_{L}} is finite flat, and that A_{o_{L}} ⊗_{σ,A_{o_{L}}} A_{o_{L} ,π} is a finite flat A_{o_{L} ,π}-module and hence equals the π-adic completion of the A_{o_{L}} -module σ_{∗}A_{o_{L}}. If we let 𝔞 = σ(πA_{o_{L}})A_{o_{L}} = π^{r}A_{o_{L}} and 𝔟 = πA_{o_{L}}, we get 𝔟^{r} = 𝔞 ⊆ 𝔟. Consequently, by [Eis95, Lemma 7.14], the inverse systems (A_{o_{L}} /𝔞^{n})_{n} and (A_{o_{L}} /𝔟^{n})_{n} give the same limit, which shows that the square on the left is cocartesian, and that σ: A_{o_{L} ,π} → A_{o_{L} ,π} is finite flat. Similarly, we have σ(ε, π)A_{o_{L}} = (ε, π^{r}) ⊆ (ε, π) as well as (ε, π)^{r} ⊆ (ε, π^{r}), which proves that the displayed diagram qualifies A_{o_{L} ,(ε,π)} as tensor product A_{o_{L} ,(ε,π)} ⊗_{A_{o_{L}} ,σ} A_{o_{L}}, and that σ: A_{o_{L} ,(ε,π)} → A_{o_{L} ,(ε,π)} is finite flat.

Finally, note that the embedding of o_{L}-algebras ι ⊗ id: o_{L}[z] → A_{o_{L}} commutes with σ: A_{o_{L}} → A_{o_{L}} and the r-Frobenius lift of o_{L}[z], given by o_{L}[z] → o_{L}[z], ∑ _{ν} a_{ν}z^{ν} ↦ ∑ _{ν} a_{ν}^{r}z^{ν}. Consequently, also the embeddings from (2.1) are Frobenius-equivariant.

Let B be an o_{L}-algebra together with a ring endomorphism σ: B → B such that σ and Frob_{r,o_{L}} : o_{L} → o_{L} are compatible with the structure map o_{L} → B. For example, B could be any of the base rings considered above.

Definition 3.2. We define the category FMod(B) of Frobenius B-modules (or simply F -modules over B) as follows:

— An object of FMod(B) is a pair \underline{M} = (M, F) consisting of a B-module M which is locally free of finite rank, together with an injective B-linear map F = F_{M} : σ^{∗}M → M, where σ^{∗}M := M ⊗_{B,σ} B.

— A morphism of Frobenius B-modules (M, F_{M}) → (N, F_{N}) is a B-linear map φ: M → N between the underlying B-modules such that φ is F -equivariant, i.e. such that φ ∘ F_{M} = F_{N} ∘ σ^{∗}φ. It is called an isomorphism if φ is an isomorphism of the underlying B-modules.

Let B′ be a flat B-algebra together with a ring endomorphism σ: B′→ B′ extending the Frobenius lift of B, as explained before. Then the exact functor · ⊗_{B} B′ from B-modules to B′-modules yields a functor FMod(B) → FMod(B′). If the structure map B → B′ is, in addition, injective then the induced functor on FMod(B) is faithful since, given a map f: M → N of finite projective B-modules, restricting its image f ⊗ id: M ⊗_{B} B′→ N ⊗_{B} B′ to M gives back f. In particular, we obtain a natural commutative diagram of categories and faithful functors

\xymatrix{FMod(A_{o_{L}}) \ar[r] \ar[d] & FMod(A_{o_{L} ,π}) \ar[d] \ar[r] & FMod(A_{o_{L} ,(ε,π)}) \ar[d] \\ FMod(A_{L}) \ar[r] & FMod(A_{o_{L} ,π}[1/π]) \ar[r] & FMod(A_{o_{L} ,(ε,π)}[1/π])}

Slightly abusing notation, we agree to write \underline{M} ⊗_{B} B′for (M ⊗_{B} B′, F_{M} ⊗id_{B′}), whenever \underline{M} = (M, F_{M}).

4 Anderson motives

Let 𝔍 ⊆ A_{o_{L}} be the ideal generated by a ⊗ 1 − 1 ⊗ c^{∗}(a) for all a ∈ A. For example, if C = ℙ_{𝔽}^{1} and A = 𝔽[z], then 𝔍 = (z − ζ) ⊆ o_{L}[z] where ζ = c^{∗}(z). Note that the convention introduced in Remark 2.1 that (z) = ε^{m} implies ζ ∈ 𝔪_{L}. So ζ = 0 if c^{∗} is not injective. By abuse of notation we denote the ideal generated by 𝔍 in any A_{o_{L}} -algebra again by 𝔍. We consider the following variant of Anderson’s [And86] t-motives.

Definition 4.1. An Anderson A-motive over L is an object \underline{M} = (M, F_{M}) ∈ FMod(A_{L}) such that coker(F_{M}) is a finite-dimensional L-vector space and is annihilated by a power of 𝔍. A morphism of Anderson A-motives is defined as a morphism inside FMod(A_{L}).

Since Spec(A_{L}) is of finite type over L, one can consider its rigid analytification Spec(A_{L})^{an}; see [Bos14], [BGR84], [FP04]. In accordance with [BH07], we denote this rigid analytic L-space by 𝔄(∞). On the other hand, the formal completion of the o_{L}-scheme X = Spec(A_{o_{L}}) along its special fiber V (π) leads to the formal o_{L}-scheme 𝔛 = Spf(A_{o_{L} ,π}); see [EGA, I_{new}, I.10.8.3]. Its associated rigid analytic space 𝔛_{rig} ([Bos14], [FP04]) is given by the affinoid L-space 𝔄(1) := Sp(A_{o_{L} ,π}[1/π]). This space can be regarded as the unit disc of the rigid analytic space 𝔄(∞) as it corresponds to “radius of convergence 1”, hence the notation.

We study the following instance of rigid analytic τ-sheaves over A_{o_{L} ,π}[1/π], in the sense of [BH07].

Definition 4.2. An analytic Anderson A(1)-motive over L is an object \underline{M} = (M, F_{M}) ∈ FMod(A_{o_{L} ,π}[1/π]) such that coker(F_{M}) is a finite-dimensional L-vector space and is annihilated by a power of 𝔍. A morphism of analytic Anderson A(1)-motives is defined as a morphism in the category FMod(A_{o_{L} ,π}[1/π]).

Here the prefix “A(1)-” indicates that we are considering an analytic variant of Anderson A-motives over the rigid analytic “unit disc” 𝔄(1) in Spec(A_{L}).

Proposition 4.3. The natural functor FMod(A_{L}) → FMod(A_{o_{L} ,π}[1/π]) , \underline{M} ↦ \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π] restricts to a functor (Anderson A-motives over L) → (analytic Anderson A(1)-motives over L).

Definition 4.4. (a) Let \underline{M}_{L} ∈ FMod(A_{L}) be an F -module over A_{L}. A model of \underline{M}_{L} is a pair (\underline{M} , α) consisting of an object \underline{M} ∈ FMod(A_{o_{L}}) and an isomorphism α: \underline{M}_{L} ^{∼} −→ \underline{M} ⊗_{A_{o_{L}}} A_{L} inside FMod(A_{L}).

(b) Let \underline{M}_{L} ∈ FMod(A_{o_{L} ,π}[1/π]) be an F -module over A_{o_{L} ,π}[1/π]. A (formal) model of \underline{M}_{L} is a pair (\underline{M} , α) consisting of an object \underline{M} ∈ FMod(A_{o_{L} ,π}) and an isomorphism α: \underline{M}_{L} ^{∼} −→ \underline{M} ⊗_{A_{o_{L} ,π}} A_{o_{L} ,π}[1/π] inside FMod(A_{o_{L} ,π}[1/π]).

(c) In both cases a morphism of models β: (\underline{M} , α) → (\underline{M} ′, α′) is an isomorphism β: \underline{M} ^{∼} −→ \underline{M} ′of F -modules satisfying α′= β[1/π] ∘ α. In particular the sets Hom (\underline{M} , α), (\underline{M} ′, α′) contain at most one element. We will sometimes drop the α from the notation and simply speak of \underline{M} as a model of \underline{M}_{L}.

For every \underline{M} ∈ FMod(A_{o_{L}}), resp. \underline{M} ∈ FMod(A_{o_{L} ,π}) we can consider the reduction \underline{M} ⊗_{A_{o_{L}}} A_{ℓ}, resp. \underline{M} ⊗_{A_{o_{L} ,π}} A_{ℓ}. Note, however, that this does not induce a functor from FMod(A_{o_{L}}), resp. FMod(A_{o_{L} ,π}) to FMod(A_{ℓ}), since the induced F -map need not be injective. This circumstance lies at the origin of our study of good models:

Definition 4.5. Let \underline{M} be a model of an F -module \underline{M}_{L} over A_{L}, resp. over A_{o_{L} ,π}[1/π]. Then \underline{M} is called a good model if \underline{M} /π\underline{M} is an F -module over A_{ℓ}, i.e. if the induced A_{ℓ}-linear map

σ̄^{∗}(M/πM) = (M/πM) ⊗_{A_{ℓ},σ̄} A_{ℓ} → M/πM

is injective.

If \underline{M}_{L} is an (analytic) Anderson motive there is an alternative notion of good reduction as follows.

Definition 4.6. Let \underline{M} be a model of an Anderson A-motive \underline{M}_{L}, resp. of an analytic Anderson A(1)- motive \underline{M}_{L}. Then \underline{M} is called a good model in the strong sense if coker(F_{M} ) is a finite free o_{L}-module and is annihilated by 𝔍^{d}, for some d ≥ 0. In this case we also say that \underline{M} has good reduction over o_{L}.

Theorem 4.7. Let \underline{M} be a model of an Anderson A-motive, resp. of an analytic Anderson A(1)-motive \underline{M}_{L}. Then \underline{M} is a good model in the weak sense of Definition 4.5 if and only if it is a good model in the strong sense of Definition 4.6.

Proof. Since σ^{∗}M is locally free over A_{o_{L}}, resp. over A_{o_{L} ,π}, the natural map σ^{∗}M → σ^{∗}M_{L} is injective and hence F_{M} is injective because F_{M_{L}} is. We obtain a short eqact sequence

0 −→ σ^{∗}M −⟶^{F_{M}} M −→ coker(F_{M} ) −→ 0 .  (4.2)

Let \underline{M} be a good model in the strong sense. Tensoring the short exact sequence (4.2) with ℓ over o_{L} and using that coker(F_{M} ) is supposed to be free over o_{L} shows that the induced A_{ℓ}-linear map σ̄^{∗}(M/πM) → M/πM remains injective. So \underline{M} is a good model in the weak sense.

Conversely suppose that \underline{M} is a good model in the weak sense. This time tensoring (4.2) with ℓ over o_{L} yields

0 −→ Tor_{1}^{o_{L}} cokerF_{M} , ℓ) −→ σ^{∗}M ⊗_{o_{L}} ℓ −−−−⟶^{F_{M} ⊗id_{ℓ}} M ⊗_{o_{L}} ℓ −→ coker(F_{M} ) ⊗_{o_{L}} ℓ −→ 0 .

By assumption F_{M} ⊗ id_{ℓ} is injective, and so 0 = Tor_{1}^{o_{L}} cokerF_{M} , ℓ) = {x ∈ coker(F_{M} ): πx = 0} and coker(F_{M} ) is flat over o_{L} by [Eis95, Corollary 6.3]. This implies coker(F_{M} ) ↪ coker(F_{M} ) ⊗_{o_{L}} L = coker(F_{M_{L}} ). Since coker(F_{M_{L}} ) is annihilated by 𝔍^{d} for some d, the same is true for coker(F_{M} ) which therefore is a finitely generated A_{o_{L}} /𝔍^{d}-module, resp. A_{o_{L} ,π}/𝔍^{d}-module, and a fortiori a finitely generated o_{L}-module. Being flat, coker(F_{M} ) is a finite free o_{L}-module. Thus M is a good model in the strong sense.

Remark 4.8. In [Gar03], Gardeyn develops a theory of semi-stable reduction of analytic Anderson A(1)- motives \underline{M}_{L}. He shows that after replacing L by a finite separable extension, \underline{M}_{L} has a model \underline{M} such that the reduction F_{M} ⊗ id_{ℓ} is not nilpotent [Gar03, Proposition 3.3]. If \overline{\underline{M}} ′ ⊆ \underline{M} /π\underline{M} is the maximal Frobenius A_{ℓ}-submodule with injective F\overline{M} ′, he further shows that the support of coker(F\overline{M} ′) is a finite set S ⊆ SpecA_{ℓ}. After removing S from 𝔄(1) := Sp(A_{o_{L} ,π}[1/π]) one can lift \overline{\underline{M}} ′ to an F -submodule \underline{M} ′⊆ \underline{M} |_{𝔄(1)∖S} which has good reduction in the weak sense of Definition 4.5; see [Gar03, Theorem 4.7]. As one sees from the following example, it is false in general that S is the zero locus of 𝔍 in SpecA_{ℓ} and so we cannot expect that \underline{M} ′ has good reduction in the strong sense of Definition 4.6.

Let A = 𝔽[z] and ζ = c^{∗}(z) ∈ 𝔪_{L}. Then 𝔍 = (z − ζ). Let M = o_{L}⟨z⟩^{⊕2} and F_{M} = \binom{0 π(z−ζ)}{π z−1}. Then \underline{M} = (M, F_{M} ) is a model of the analytic Anderson A(1)-motive \underline{M}_{L} := \underline{M} ⊗_{o_{L}} L. The reduction \underline{M} /π\underline{M} = ℓ[z]^{⊕2} , \binom{0 0}{0 z−1} contains the maximal Frobenius A_{ℓ}-submodule \overline{\underline{M}} ′= ℓ[z] · \binom{0}{1}, whose Frobenius is F\overline{M} ′ = z − 1. So S = V (z − 1) ≠ V (z) = V (𝔍).

Proposition 4.9. If \underline{M}_{L} is an Anderson A-Motive over L having a (good) model \underline{M} then its analytification \underline{M}_{L} ⊗_{A_{L}} A_{o_{L} ,π}[1/π] is an analytic Anderson A(1)-motive having the (good) model \underline{ ; M} := \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π} and the reduction \underline{ ; M} /π\underline{ ; M} of \underline{ ; M} is canonically isomorphic to the reduction \underline{M} /π\underline{M} of \underline{M} .

Proof. The statement without the properties of being a good model is obvious. From the isomorphism \underline{ ; M} /π\underline{ ; M} ^{∼} −→ \underline{M} /π\underline{M} it follows that \underline{M} is a good model in the sense of Definition 4.5 if and only if \underline{ ; M} is a good model in the sense of Definition 4.5.

Let us also mention the following result of Gardeyn on good reduction of Drinfeld A-modules.

Proposition 4.10. Let φ: A → L[τ] be a Drinfeld A-module over L; see [Dri76] or [Mat96]. Let \underline{M} = \underline{M}(φ) be the associated Anderson A-motive; see [And86, §4.1] or [Gar02, §8.1]. Then the following are equivalent:

(i) φ has good reduction over o_{L}, i.e. φ is isomorphic over L to a Drinfeld A-module ψ: A → L[τ] satisfying ψ(A) ⊆ o_{L}[τ] such that the reduction \overline{ψ} : A → o_{L}[τ] ↠ ℓ[τ] is a Drinfeld A-module over ℓ of the same rank as ψ and φ.

(ii) \underline{M} has good reduction over o_{L} in the weak and strong senses of Definitions 4.6 and 4.5.

Proof. Gardeyn [Gar02, Theorem 8.1] proved that φ has good reduction over o_{L} if and only if \underline{M} has a good model in the weak sense. So the proposition follows from Theorem 4.7.

5 Local shtukas and analytic Anderson motives

Anderson A-motives can be viewed as function-field analogs of abelian varieties. Barsotti-Tate groups, which can be associated with abelian varieties over ℤ_{p}-schemes, have effective local shtukas as function-field analogs.

Definition 5.1. An (effective) local shtuka at ε over o_{L} is an object \underline{M̂} = (M̂, FM̂) ∈ FMod(A_{o_{L} ,(ε,π)}) such that coker(FM̂) is a finite free o_{L}-module and is annihilated by a power of 𝔍.

Remark 5.2. If the residue field 𝔽_{ε} = A/ε of ε is larger than 𝔽, i.e. if the degree d_{ε} := [𝔽_{ε} : 𝔽] > 1, the ring A_{o_{L} ,(ε,π)} is not an integral domain but a product A_{o_{L} ,(ε,π)} = ∏_{i∈ℤ/d_{ε}ℤ} A_{o_{L} ,(ε,π)}/𝔞_{i} of integral domains. To

describe this product decomposition, note that A_{o_{L} ,(ε,π)} = \underleftarrow{lim}_{n} A_{o_{L}} /ε^{n} = \underleftarrow{lim}_{n} (A/ε^{n}) ⊗_{𝔽} o_{L} = A_{ε}⊗ _{𝔽}o_{L}. By Cohen’s structure theorem A_{ε} ≅ 𝔽_{ε}⟦z_{ε}⟧ for a uniformizer z_{ε} of A at ε. Then 𝔞_{i} = (α⊗1−1⊗c^{∗}(α)^{r^{i}} : α ∈ 𝔽_{ε} ⊆ A_{ε}), where we use that c^{∗}: A → o_{L} factors through c^{∗}: A_{ε} → o_{L}. The factors A_{o_{L} ,(ε,π)}/𝔞_{i} are isomorphic to o_{L}⟦z_{ε}⟧ and hence are integral domains. They are cyclically permuted by σ because σ(𝔞_{i}) = 𝔞_{i+1} . By [BH11, Proposition 8.8] the functor (M̂, FM̂) ↦ (M̂/𝔞_{0}M̂, (FM̂)^{d_{ε}}) is an equivalence between the category of effective local shtukas at ε over o_{L} as in Definition 5.1 and the category of pairs (M̂_{0}, ; FM̂) where M̂_{0} is a free module of finite rank over A_{o_{L} ,(ε,π)}/𝔞_{0} and ; FM̂ : (σ^{d_{ε}})^{∗}M̂_{0} → M̂_{0} is injective with coker( ; FM̂) being a finite free o_{L}-module. In [Har09, Har11] these pairs (M̂_{0}, ; FM̂) are called (effective) local shtukas.

The following criterion for good reduction of analytic Anderson A(1)-motives can be regarded as a good-reduction Local-Global Principle at the characteristic place.

Theorem 5.3. Let \underline{M}_{L} = (M_{L}, F_{M_{L}} ) be an analytic Anderson A(1)-motive over L such that coker(F_{M_{L}} ) is annihilated by 𝔍^{d} for some d. Then the following assertions are equivalent:

(i) \underline{M}_{L} admits a good model in the strong sense of Definition 4.6.

(ii) There is an effective local shtuka \underline{M̂} = (M̂, FM̂) at ε over o_{L} such that coker(FM̂) is annihilated by 𝔍^{d}, and an isomorphism \underline{M}_{L}⊗_{A_{o_{L} ,π} [1/π]}A_{o_{L} ,(ε,π)}[1/π] ≅ \underline{M̂} ⊗_{A_{o_{L} ,(ε,π)}} A_{o_{L} ,(ε,π)}[1/π] in FMod(A_{o_{L} ,(ε,π)}[1/π]).

Proof. 1. In order to show that (ii) implies (i), let f: M_{L} ⊗ A_{o_{L} ,(ε,π)}[1/π] ^{∼} −→ M̂ ⊗ A_{o_{L} ,(ε,π)}[1/π] =: M̂[1/π] be an F -equivariant isomorphism of A_{o_{L} ,(ε,π)}[1/π]-modules as in (ii). We have canonical F -equivariant A_{o_{L} ,π}-linear maps

i: M_{L} → M_{L} ⊗_{A_{o_{L} ,π} [1/π]} A_{o_{L} ,(ε,π)}[1/π], j: M̂ → M̂[1/π]

where i (resp., j) is injective since M_{L} (resp., M̂) is flat. Consider the A_{o_{L} ,π}-module M = im(i)∩f^{−1} (im(j)). We will show that M is a good model of \underline{M}_{L}. The inclusion M ↪ M_{L} gives rise to an A_{o_{L} ,π}[1/π]-linear embedding M[1/π] ↪ M_{L}[1/π] ≅ M_{L}, which is in fact an isomorphism, because if m ∈ M_{L} there is an s ≥ 0 such that π^{s}f(m ⊗ 1) ∈ im(j), i.e. π^{s}m ∈ M.

2. In order to show that M is a finitely generated A_{o_{L} ,π}-module we use the embedding ι: 𝔽[z] → A from Remark 2.1 and the induced maps L⟨z⟩ → A_{o_{L} ,π}[1/π] and o_{L}⟦z⟧ → A_{o_{L} ,(ε,π)} from (2.1). Let (e_{1}, ..., e_{m}) be a basis of M_{L} over the principal ideal domain L⟨z⟩. Furthermore, let (d_{1}, ..., d_{n}) be a basis for M̂ over the local ring o_{L}⟦z⟧. Note that the basis (e_{1}, ..., e_{m}) gives rise to an isomorphism M_{L}⊗_{L⟨z⟩}o_{L}⟦z⟧[1/π] ≅ o_{L}⟦z⟧[1/π]^{⊕m} . For every ν = 1, ..., n we consider f^{−1} (d_{ν}) and regard it as an element of the right-hand side of this isomorphism. We choose N ≥ 0 big enough, such that f^{−1} (π^{N}d_{ν}) ∈ o_{L}⟦z⟧^{⊕m} for all ν, say

f^{−1} (π^{N}d_{ν}) = (ρ_{ν,1}, ..., ρ_{ν,m})

where ρ_{ν,μ} ∈ o_{L}⟦z⟧. Now let x ∈ M. Via f we obtain f(x) = ∑ _{ν} λ_{ν}d_{ν} in M̂, with suitable λ_{ν} ∈ o_{L}⟦z⟧. Consequently f(π^{N}x) = ∑ _{ν} λ_{ν}(π^{N}d_{ν}), so that the image of π^{N}x in o_{L}⟦z⟧^{⊕m} satisfies π^{N}x = ∑ _{μ}(∑ _{ν} λ_{ν}ρ_{ν,μ})e_{μ}. The appearing scalars h_{μ} = ∑ _{ν} λ_{ν}ρ_{ν,μ} have, in fact, to be elements of L⟨z⟩ ∩ o_{L}⟦z⟧ = o_{L}⟨z⟩. Inside M_{L} we may write x = π^{−N} π^{N}x = ∑ _{μ} h_{μ}π^{−N} e_{μ}, so that we may conclude

M ⊆ ∑_{μ} o_{L}⟨z⟩π^{−N} e_{μ}.

Being a submodule of a finitely generated module over a noetherian ring, M has to be a finitely generated o_{L}⟨z⟩-module and hence a finitely generated A_{o_{L} ,π}-module.

3. We claim that M/πM is torsion-free and hence free over ℓ[z], because it is finitely generated. Let x ∈ M, and let λ ∈ o_{L}⟨z⟩ be such that λ ∉ πo_{L}⟨z⟩ and λx ∈ πM, say λx = πy for some y ∈ M. In order to prove that M/πM is torsion-free we must show that x ∈ πM. First suppose that λ ∈ o_{L}⟨z⟩ ∩ o_{L}⟦z⟧^{×} . We consider π^{−1} x ∈ M_{L}. In fact, this element lies in M, since we have f(π^{−1} x) = λ^{−1} f(y) ∈ M̂. Consequently x = π(π^{−1} x) ∈ πM.

Let us next assume that λ = z^{n} and show that z^{n}x ∈ πM implies x ∈ πM for any n ≥ 0. By induction, it suffices to consider the case n = 1. So suppose zx ∈ πM, say zx = πy. Let f(x) = ∑ _{ν} β_{ν}d_{ν}, where (d_{1}, ..., d_{n}) is the finite o_{L}⟦z⟧-basis of M̂ fixed before. The relation zx = πy implies that π | zβ_{ν} for every index ν, so that π | β_{ν} for every ν. Therefore π^{−1} x ∈ M_{L} necessarily maps via f to an element of M̂, i.e. x ∈ πM.

Finally we treat the case for general λ = ∑ _{s} λ_{s}z^{s} and suppose that λ ∉ o_{L}⟦z⟧^{×} , that is π | λ_{0}. This means we find λ′ ∈ o_{L}[z] and λ′′∈ o_{L}⟨z⟩ ∩ o_{L}⟦z⟧^{×} such that λ = πλ′+ z^{N}λ′′for some N ≥ 1. We have πy = λx = πλ′x + z^{N}λ′′x. In particular z^{N}λ′′x = π(y − λ′x) ∈ πM and by the above λ′′x ∈ πM and x ∈ πM.

Thus we have proved that M/πM is free over ℓ[z]. It follows that M/πM is locally free of finite rank over A_{ℓ}.

4. We claim that M is locally free of finite rank over A_{o_{L} ,π}. Since it is finitely generated it only remains to show that M is flat over A_{o_{L} ,π}. Since A_{o_{L} ,π} is π-adically complete and separated, πA_{o_{L} ,π} is contained in the Jacobson radical 𝔧(A_{o_{L} ,π}) by [Mat86, Theorem 8.2], and the A_{o_{L} ,π}-module M is finitely generated, so that M is π-adically ideally Hausdorff in the sense of [Bou67, III.5.1]. In the preceding step we have shown that M/πM is flat over A_{ℓ} ≅ A_{o_{L} ,π}/πA_{o_{L} ,π}, and we know that M has no π-torsion, so that the canonical map πA_{o_{L} ,π} ⊗_{A_{o_{L} ,π}} M → πM is an isomorphism. Therefore, by Bourbaki’s Flatness Criterion [Bou67, § III.5.2, Théorème 1(iii)], we may conclude that M is indeed flat over A_{o_{L} ,π}.

5. We note that σ^{∗}M = σ^{∗}im(i) ∩ (σ^{∗}f)^{−1} (σ^{∗}im(j)) because the functor σ^{∗} is exact by Lemma 3.1. By the F -equivariance of f we obtain a Frobenius F_{M} : σ^{∗}M → M. It is injective because F_{M_{L}} is. We set \underline{M} := (M, F_{M} ).

6. Next we claim that 𝔍^{d}coker(F_{M} ) = 0. Let x = ∑ _{ν} h_{ν}m_{ν} ∈ 𝔍^{d}M where h_{ν} ∈ 𝔍^{d} and m_{ν} ∈ M. Since coker(F_{M_{L}} ) is annihilated by 𝔍^{d}, there is a (unique) y ∈ σ^{∗}M_{L} such that x = ∑ _{ν} h_{ν}m_{ν} = F_{M_{L}} (y). We have to show that y ∈ σ^{∗}M = σ^{∗}im(i) ∩ (σ^{∗}f)^{−1} (σ^{∗}im(j)). So it remains to see that (σ^{∗}f)(y) ∈ im(σ^{∗}j). Indeed, inside M̂[1/π] we have f(x) = f(F_{M_{L}} (y)) = FM̂((σ^{∗}f)(y)). On the other hand, the linearity of f and j gives that f(x) = ∑ _{ν} h_{ν}f(m_{ν} ⊗ 1) = j(y′) for some y′ ∈ 𝔍^{d}M̂ ⊆ im(FM̂), say y′ = FM̂(y′′) for a y′′∈ σ^{∗}M̂. Thus f(x) = FM̂((σ^{∗}j)(y′′)). So finally, since FM̂ : σ^{∗}M̂[1/π] → M̂[1/π] is injective, we obtain that (σ^{∗}f)(y) = (σ^{∗}j)(y′′), as desired.

7. Finally we show that the kernel V of \overline{F}: σ^{∗}(M/πM) → M/πM is trivial. This implies that \underline{M} is a good model of \underline{M}_{L} in the weak sense of Definition 4.5, which is enough by Theorem 4.7.

We have already shown that 𝔍^{d}M ⊆ im(F_{M} ). Since (z − ζ) ∈ 𝔍 for ζ := c^{∗}(z) ∈ o_{L} we have a chain of o_{L}⟨z⟩-modules (z − ζ)^{d}M ⊆ im(F_{M} ) ⊆ M. The element ζ ∈ o_{L} is zero mod π, and we obtain

z^{d}(M/πM) ⊆ im(\overline{F}) ⊆ M/πM.  (5.3)

We know that M/πM is finite free over ℓ[z]. Therefore the middle term W := im(\overline{F}) in the latter chain has full rank inside M/πM. Finally, taking ranks in the (split) short exact sequence of finite free ℓ[z]-modules

0 → V → σ^{∗}(M/πM) −→^{\overline{F}} W → 0

accomplishes the proof that V indeed is trivial.

8. Conversely, in order to show that (i) implies (ii), suppose that (\underline{M} , α) is a good model of \underline{M}_{L}. We define

\underline{M̂} = \underline{M} ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)},

i.e. \underline{M̂} equals the completion of \underline{M} for the (ε, π)A_{o_{L} ,π}-adic topology. It is clear that the F -equivariant isomorphism α: M_{L} ^{∼} −→ M[1/π] of A_{o_{L} ,π}[1/π]-modules gives rise to a natural F -equivariant A_{o_{L} ,(ε,π)}[1/π]- linear isomorphism M_{L} ⊗_{A_{o_{L} ,π} [1/π]} A_{o_{L} ,(ε,π)}[1/π] ≅ M̂[1/π].

We claim that \underline{M̂} is a local shtuka. Indeed, by base change, M̂ is again locally free of finite rank. Furthermore, since the completion map A_{o_{L} ,π} → A_{o_{L} ,(ε,π)} is Frobenius-equivariant and flat, we obtain an injective map M̂ ⊗_{(A_{o_{L} ,(ε,π)} ),σ} A_{o_{L} ,(ε,π)} → M̂. Let C′ be its cokernel, and let C = coker(F_{M} ), i.e. C′ ≅ C ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)}. Since C is annihilated by 𝔍^{d} the module C′ equals C and it is finite free over o_{L}. Thus \underline{M̂} is an effective local shtuka over o_{L}.

Remark 5.4. Steps 1-4 in the previous proof suggest that there is an equivalence of categories

F: \{ \begin{matrix}finite locally free \\ A_{o_{L} ,π}-modules M \end{matrix} \} ⟷^{∼}  \{ \begin{matrix}triples (M_{L}, M̂, f) consisting of \\ • a finite locally free A_{o_{L} ,π}[1/π]-module M_{L}, \\ • a finite locally free A_{o_{L} ,(ε,π)}-module M̂, and \\ • an isomorphism of A_{o_{L} ,(ε,π)}[1/π]-modules \\ f: M_{L} ⊗_{A_{o_{L} ,π} [1/π]} A_{o_{L} ,(ε,π)}[1/π] ^{∼} −→ M̂ ⊗_{A_{o_{L} ,(ε,π)}} A_{o_{L} ,(ε,π)}[1/π] \end{matrix} \}

M ⟼→ M ⊗_{A_{o_{L} ,π}} A_{o_{L} ,π}[1/π], M ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)}, id_{M⊗A_{o_{L} ,(ε,π)} [1/π]} ,

where on the right a morphism \underline{h} = (h_{L}, ĥ): (M_{L}, M̂, f) → (M′_{L}, M̂′, f′) consists of a morphism h_{L}: M_{L} → M′_{L} and a morphism ĥ: M̂ → M̂′ such that f′∘ (h_{L} ⊗ id_{A_{o_{L} ,(ε,π)} [1/π]}) = (ĥ ⊗ id_{A_{o_{L} ,(ε,π)} [1/π]}) ∘ f.

However, this is false as can be seen from the following example, where we take A = 𝔽[z]. We choose an

element a ∈ ℓ⟦z⟧ ⊆ ℓ((z)) such that a ∉ ℓ(z), and we let Δ = \begin{matrix}1 & π^{−1} a \\ 0 & π^{−1} \end{matrix} . Set M_{L} = L⟨z⟩^{⊕2} , M̂ = Δ · o_{L}⟦z⟧^{⊕2}

and f = id_{o_{L} ⟦z⟧[1/π]^{2}}. Then Δ^{−1} = ( \begin{matrix}1 & −a \\ 0 & π \end{matrix} ) ∈ o_{L}⟦z⟧^{2×2} and

o_{L}⟦z⟧^{⊕2} = Δ · Δ^{−1} o_{L}⟦z⟧^{⊕2} ⊆ M̂ ⊆ π^{−1} o_{L}⟦z⟧^{⊕2} .

If there was a finite free A_{o_{L} ,π}-module M with (h_{L}, ĥ): F(M) ^{∼} −→ (M_{L}, M̂, f), then it had to satisfy M ≅ M_{L} ∩ M̂ with h_{L} and ĥ induced from the inclusions M_{L} ∩ M̂ ⊆ M_{L} and M_{L} ∩ M̂ ⊆ M̂. So we may take directly M := M_{L} ∩ M̂. It satisfies o_{L}⟨z⟩^{⊕2} ⊆ M ⊆ π^{−1} o_{L}⟨z⟩^{⊕2} . We claim that, in fact, the first inclusion is an equality. Namely let ( \begin{matrix}v \\ w \end{matrix}) = ( \begin{matrix}π^{−1} v_{0}+v′ \\ π^{−1} w_{0}+w′ \end{matrix} ) ∈ M with v_{0}, w_{0} ∈ ℓ[z] and v′, w′ ∈ o_{L}⟨z⟩.

Then Δ^{−1} ( \begin{matrix}v \\ w \end{matrix}) = \begin{matrix}π^{−1} v_{0}+v′−π^{−1} aw_{0}−aw′ \\ w_{0}+πw′ \end{matrix} ∈ o_{L}⟦z⟧^{⊕2} . This implies v_{0} = aw_{0} in ℓ⟦z⟧. If w_{0} ≠ 0 we get

a = v_{0}/w_{0} ∈ ℓ(z) in contradiction to our assumption. So w_{0} = v_{0} = 0 and ( \begin{matrix}v \\ w \end{matrix}) ∈ o_{L}⟨z⟩^{⊕2} . This proves our claim that M = o_{L}⟨z⟩^{⊕2} . We conclude that F(M) ≅ (M_{L}, M̂, f) and F is not an equivalence of categories.

After this example the following result is even more surprising.

Corollary 5.5. Let \underline{M}_{L} be an analytic Anderson A(1)-motive over L. Then there is an equivalence of categories

\{ \begin{matrix}good models (\underline{M} , α) of & \underline{M} & L & in the \\ sense of Definitions 4.6 and 4.5 \end{matrix} \} ⟷^{∼}  \{ \begin{matrix}pairs (\underline{M̂}, f) consisting of \\ • a local shtuka \underline{M̂} at ε over o_{L}, and \\ • an isomorphism in FMod(A_{o_{L} ,(ε,π)}[1/π]) \\ f: \underline{M}_{L} ⊗ A_{o_{L} ,(ε,π)}[1/π] ^{∼} −→ \underline{M̂}[1/π] \end{matrix} \}

(\underline{M} , α) ⟼→ (\underline{M} , α) ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)} ,

where on the right-hand side a morphism of pairs β̂: (\underline{M̂}, f) ^{∼} −→ (\underline{M̂}′, f′) is defined to be an isomorphism of local shtukas β̂: \underline{M̂} ^{∼} −→ \underline{M̂}′ satisfying f′= β̂ ∘ f.

Proof. Suppose that (\underline{M} , α) is a good model of \underline{M}_{L}. In the proof of 5.3 we have seen that its completion \underline{M̂} := \underline{M} ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)} is a local shtuka at ε. The F -equivariant isomorphism α: M_{L} ^{∼} −→ M[1/π] of A_{o_{L} ,π}[1/π]-modules induces the isomorphism

f := α ⊗ id_{A_{o_{L} ,(ε,π)} [1/π]}: M_{L} ⊗_{A_{o_{L} ,π} [1/π]} A_{o_{L} ,(ε,π)}[1/π] ^{∼} −→ M̂ ⊗_{A_{o_{L} ,(ε,π)}} A_{o_{L} ,(ε,π)}[1/π]

which is F -equivariant, and satisfies M = f(M_{L}) ∩ M̂ , because A_{o_{L} ,π} = A_{o_{L} ,π}[1/π] ∩ A_{o_{L} ,(ε,π)}.

To see that this functor is fully faithful let (\underline{M} , α) and (\underline{M} ′, α′) be good models of \underline{M}_{L} and let β̂: (\underline{M̂} , f) := (\underline{M} , α) ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)} ^{∼} −→ (\underline{M̂} ′, f′) := (\underline{M} ′, α′) ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)} be an isomorphism. This means f′= β̂ ∘ f. Applying M = f(M_{L}) ∩ M̂ and M′= f′(M_{L}) ∩ M̂′ we see that β̂(M) = M′. Therefore β := β̂|_{M} : M ^{∼} −→ M′ is the desired isomorphism satisfying β ⊗ id_{A_{o_{L} ,(ε,π)}} = β̂. This implies α′= β ∘ α and the F -equivariance of β, and hence β: (\underline{M} , α) ^{∼} −→ (\underline{M} ′, α′).

To prove essential surjectivity, let a local shtuka \underline{M̂} together with an isomorphism f: \underline{M}_{L} ⊗_{A_{o_{L} ,π} [1/π]} A_{o_{L} ,(ε,π)}[1/π] ^{∼} −→ \underline{M̂}[1/π] be given. It remains to show that the (ε, π)A_{o_{L} ,π}-adic completion \underline{M̂} := \underline{M} ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)} of the good model M = M_{L} ∩ f^{−1} (M̂) gained in the proof of 5.3 gives back \underline{M̂}. Then we take α as the canonical isomorphism id: M ⊗_{A_{o_{L} ,π}} A_{o_{L} ,π}[1/π] ^{∼} −→ M_{L}. By construction of \underline{M} , the map f restricts to an embedding M ↪ M̂, which in turn induces an F -equivariant and A_{o_{L} ,(ε,π)}-linear map ψ := f|M̂ : M̂ → M̂, which becomes an isomorphism after inverting π. Our aim is to show that already the map ψ is an isomorphism (\underline{M} , id) ⊗_{A_{o_{L} ,π}} A_{o_{L} ,(ε,π)} ^{∼} −→ (\underline{M̂}, f). According to Remark 5.4 we have to use the Frobenius morphisms FM̂ and FM̂ in an essential way.

We know that M is finite free over o_{L}⟨z⟩ and that rk_{o_{L} ⟦z⟧}(M̂ ) = rk_{o_{L} ⟦z⟧}(M̂) =: s. We fix an o_{L}⟦z⟧-basis 𝔅 (resp., ℭ) of M̂ (resp., of M̂) and let A = _{ℭ}[ψ]_{𝔅} ∈ o_{L}⟦z⟧^{s×s} be the matrix which describes ψ with respect to 𝔅 and ℭ. Likewise, we let

T = _{𝔅}[FM̂ ]_{σ^{∗}𝔅}, T′= _{ℭ}[FM̂]_{σ^{∗}ℭ}

be the matrices corresponding to FM̂ and FM̂, so that AT = T′σ(A) by virtue of the F -equivariance of ψ. In order to see that ψ is an isomorphism, we need to show that det(A) is a unit in o_{L}⟦z⟧. To begin with, an elementary application of the Weierstraß Division Theorem for o_{L}⟦z⟧ ([Bou67, VII.3.8.5]) shows that the kernel of the epimorphism o_{L}⟦z⟧ → o_{L}, z ↦ ζ, is generated by z − ζ, so that the latter is a prime element of o_{L}⟦z⟧. Furthermore, recall that o_{L}⟦z⟧, being a regular local ring, is factorial ([Mat86], 20.3). We know that \underline{M̂} is a local shtuka, so that FM̂ becomes an isomorphism after inverting z − ζ which means that det(T)^{−1} lies in o_{L}⟦z⟧[\frac{1}{z−ζ} ]. Say we have a relation (z − ζ)^{e} = det(T)u in o_{L}⟦z⟧, for some e ≥ 0 and some u ∈ o_{L}⟦z⟧. By a comparison of powers of z − ζ, we may assume that u is not divisible by z − ζ. In this equation there is only one prime element of o_{L}⟦z⟧ occurring on both sides, which, by factoriality, implies that u has to be a unit in o_{L}⟦z⟧. Let (z − ζ)^{e′} = det(T′)u′ be the corresponding relation for the local shtuka M̂, with a unit u′ ∈ o_{L}⟦z⟧^{×} and some suitable e′ ≥ 0. Since M̂ → M̂ becomes an isomorphism after inverting π, we see that det(A) ∈ o_{L}⟦z⟧[1/π]^{×} . Note that the natural reduction-mod-z map o_{L}⟦z⟧ → o_{L}, h ↦ h(0), induces an epimorphism of abelian groups o_{L}⟦z⟧[\frac{1}{π}]^{×} → L^{×} , so that the absolute term δ := det(A)(0) of det(A) lies in L^{×} . By virtue of the relations derived above, the equation det(A) det(T) = det(T′)σ(det(A)) yields

det(A)u^{−1} (z − ζ)^{e} = u^{′−1}(z − ζ)^{e′}σ(det(A))

which modulo z gives δ^{q−1} = \frac{u′(0)}{u(0)} (−ζ)^{e−e′} in L^{×} . Suppose for a moment that e = e′. In this case it follows at once that δ is a unit in o_{L}, so that det(A) is a unit in o_{L}⟦z⟧. Therefore it remains to verify that our assumption e = e′ is justified. This can be seen as follows: The reduction-mod-π map o_{L}⟦z⟧ → ℓ⟦z⟧ is an epimorphism with kernel πo_{L}⟦z⟧, and via applying the functor · ⊗_{o_{L} ⟦z⟧} ℓ⟦z⟧ to FM̂ : σ^{∗}M̂ → M̂ we obtain a commutative diagram

\xymatrix{σ^{∗}M̂ = M̂ ⊗_{o_{L} ⟦z⟧,σ} o_{L}⟦z⟧ \ar[r] \ar[d] & M̂ \ar[d] \\ σ̄^{∗} M̂/πM̂ = M̂/πM̂ ⊗_{ℓ⟦z⟧,σ̄} ℓ⟦z⟧ \ar[r] & M̂/πM̂}

where in the upper row (resp., the bottom row) both modules are finite free of the same rank over o_{L}⟦z⟧ (resp., over ℓ⟦z⟧) and the arrow is given by FM̂ (resp., by F̄ = FM̂ ⊗id_{ℓ⟦z⟧}). The reduced matrix \overline{T′} ∈ ℓ⟦z⟧^{s×s} describes the map F̄ with respect to the ℓ⟦z⟧-bases \overline{σ^{∗}ℭ} = σ̄^{∗} ℭ̄ of σ̄^{∗} M̂/πM̂ and ℭ̄ of M̂/πM̂ respectively, and from what we have seen before, we derive the relation det(\overline{T′})\overline{u′} = z^{e′}, i.e. e′ = ord_{z}(det(\overline{T′})), the latter being true since \overline{u′} ∈ ℓ⟦z⟧^{×} . In particular we have det(\overline{T′}) ∈ ℓ⟦z⟧ − \{0\}. A similar observation for the local shtuka M̂ instead of M̂ shows that e = ord_{z}(det(\overline{T})). Let C = coker(FM̂ ) and C′ = coker(FM̂). Multiplication with the matrix \overline{T′} gives rise to a finite presentation ℓ⟦z⟧^{s} → ℓ⟦z⟧^{s} → C′/πC′ → 0. Taking determinants in an equation of the form S_{1}\overline{T′}S_{2} = Diag(a_{1}, ..., a_{d}, 0, 0, ..., 0), where S_{1}, S_{2} ∈ Gl_{s}(ℓ⟦z⟧) are suitable matrices such that a_{1}, ..., a_{d} ∈ ℓ⟦z⟧ − \{0\} are the elementary divisors of \overline{T′} (see [Bou81], VII.4.5.1), yields that necessarily d = s, so that C′/πC′ is a torsion ℓ⟦z⟧-module and

C′/πC′ ≅ ℓ⟦z⟧/a_{1}ℓ⟦z⟧ ⊕ ... ⊕ ℓ⟦z⟧/a_{s}ℓ⟦z⟧ ≅ ℓ^{n_{1}} ⊕ ... ⊕ ℓ^{n_{s}}

where n_{j} = ord_{z}(a_{j}) and ∑ _{j} n_{j} = e′, i.e. e′ = ord_{z}(det(\overline{T′})) = rk_{ℓ}(C′/πC′) = rk_{o_{L}} (C′), the latter equation being valid since C′/πC′ ≅ C′⊗_{o_{L} ⟦z⟧} ℓ⟦z⟧. Finally, imitating this argument for the local shtuka M̂ yields that e = ord_{z}(det(\overline{T})) = rk_{ℓ}(C/πC) = rk_{o_{L}} (C). So it remains to show that rk_{o_{L}} (C) = rk_{o_{L}} (C′). Indeed, we know that ψ: M̂ → M̂ gives back f in the generic fiber, which means that ψ is an isomorphism after inverting π. Therefore, inverting π in the commutative diagram with exact rows

\xymatrix{0 \ar[r] & σ^{∗}(M̂ ) \ar[r] \ar[d]_{σ^{∗}ψ} & M̂ \ar[r] \ar[d]_{ψ} & C \ar[r] \ar[d] & 0 \\ 0 \ar[r] & σ^{∗}M̂ \ar[r] & M̂ \ar[r] & C′ \ar[r] & 0}

exhibits (σ^{∗}ψ)[1/π] = σ^{∗}(ψ[1/π]) and ψ[1/π] as o_{L}⟦z⟧[1/π]-linear isomorphisms, so that the Snake Lemma yields C′[1/π] ≅ C[1/π], and we obtain rk_{o_{L}} (C′) = dim_{L}(C′[1/π]) = dim_{L}(C[1/π]) = rk_{o_{L}} (C), as desired.

6 The reduction criterion for Anderson motives

Definition 6.1. (a) Let \underline{M} ∈ FMod(A_{o_{L}}). Following Gardeyn [Gar03], \underline{M} is called A_{o_{L}} -maximal if for every \underline{N} ∈ FMod(A_{o_{L}}) the canonical map

Hom_{FMod(A_{o_{L}} )}(\underline{N} , \underline{M} ) → Hom_{FMod(A_{L})}(\underline{N} [1/π], \underline{M} [1/π])

is surjective (and hence bijective).

(b) An object \underline{M} ′∈ FMod(A_{o_{L} ,π}) is called A_{o_{L} ,π}-maximal if for every \underline{N}′∈ FMod(A_{o_{L} ,π}) the canonical map

Hom_{FMod(A_{o_{L} ,π} )}(\underline{N}′, \underline{M} ′) → Hom_{FMod(A_{o_{L} ,π} [1/π])}(\underline{N}′[1/π], \underline{M} ′[1/π])

is surjective (and hence bijective).

(c) Let \underline{M} ∈ FMod(A_{L}). An object \underline{M} ∈ FMod(A_{o_{L}}) is called an A_{o_{L}} -maximal model for \underline{M} if \underline{M} [1/π] ≅ \underline{M} inside FMod(A_{L}) (i.e. \underline{M} is a model for \underline{M}) and if \underline{M} is A_{o_{L}} -maximal. Correspondingly, given \underline{M}′ ∈ FMod(A_{o_{L} ,π}[1/π]), an object \underline{M} ′∈ FMod(A_{o_{L} ,π}) is called an A_{o_{L} ,π}-maximal model for \underline{M}′if \underline{M} ′[1/π] ≅ \underline{M}′ inside FMod(A_{o_{L} ,π}[1/π]) and if \underline{M} ′ is A_{o_{L} ,π}-maximal.

The existence of (A_{o_{L}} - and A_{o_{L} ,π}-)maximal models has been established in [Gar03].

Proposition 6.2 ([Gar03, Proposition 2.13]). Let \underline{M} ∈ FMod(A_{L}). Then the following assertions hold:

(i) \underline{M} admits an A_{o_{L}} -maximal model, which is unique up to unique isomorphism.

(ii) If a model \underline{M} ∈ FMod(A_{o_{L}}) of \underline{M} is good in the weak sense of Definition 4.5, then it is A_{o_{L}} -maximal.

The next proposition is a variant of Gardeyn’s theory of maximal models.

Proposition 6.3. The following assertions hold:

(i) Every \underline{M} ∈ FMod(A_{o_{L} ,π}[1/π]) admits a maximal model, which is unique up to unique isomorphism.

(ii) If \underline{M} ∈ FMod(A_{L}) is given and if \underline{M} ∈ FMod(A_{o_{L}}) is an A_{o_{L}} -maximal model of \underline{M} then \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π} ∈ FMod(A_{o_{L} ,π}) is an A_{o_{L} ,π}-maximal model of \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π] ∈ FMod(A_{o_{L} ,π}[1/π]).

(iii) Let \underline{M} ∈ FMod(A_{o_{L} ,π}[1/π]) and let \underline{M} ∈ FMod(A_{o_{L} ,π}) be a model of \underline{M}. If \underline{M} is a good model in the weak sense of Definition 4.5, then it is A_{o_{L} ,π}-maximal.

Proof. For (i) (resp. (ii); resp. (iii)), see [Gar03], 3.3(i) (resp. 3.4(i); resp. 2.13(ii)). Note that strictly speaking Gardeyn proves these statements for the rings Γ(𝔄(∞), O_{𝔄(∞)} ) instead of A_{o_{L} ,π}[1/π] and Γ(𝔄(∞), O_{𝔄(∞)} )∩ A_{o_{L} ,π} instead of A_{o_{L} ,π}. His arguments carry over literally to our rings.

We may conclude:

Proposition 6.4. In the weak sense of Definition 4.5 a Frobenius A_{L}-module \underline{M} admits a good model over A_{o_{L}} if and only if \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π] ∈ FMod(A_{o_{L} ,π}[1/π]) admits a good model over A_{o_{L} ,π}. If this is the case, the functor (\underline{M} , α) ↦ (\underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π}, α ⊗ id_{A_{o_{L} ,π} [1/π]}) is an equivalence of categories between the good models of \underline{M} and the good models of \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π].

Proof. First suppose that \underline{M} admits a good model \underline{M} ∈ FMod(A_{o_{L}}). It follows that \underline{M} is an A_{o_{L}} -maximal model of \underline{M}. Furthermore, its image \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π} inside FMod(A_{o_{L} ,π}) is an A_{o_{L} ,π}-maximal model of \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π]. Since the reduction of \underline{M} is canonically isomorphic to the reduction of \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π} by Proposition 4.9, it follows that the latter is a good model.

Conversely, suppose that \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π] admits a good model \underline{ ; M} ∈ FMod(A_{o_{L} ,π}). Necessarily \underline{ ; M} is a maximal model by Proposition 6.3(iii). We know that there is an A_{o_{L}} -maximal model \underline{M} ∈ FMod(A_{o_{L}}) of \underline{M} such that \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π} ≅ \underline{ ; M} , and that the reduction of \underline{ ; M} is canonically isomorphic to the reduction of \underline{M} by Propositions 6.2, 6.3(ii) and 4.9. Since \underline{ ; M} is a good model, so is \underline{M} . This proves the first statement and it also proves essential surjectivity of the functor.

To prove full faithfulness let (\underline{M} , α) and (\underline{M} ′, α′) be good models of \underline{M} and let β̂ : \underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π} ^{∼} −→ \underline{M} ′⊗_{A_{o_{L}}} A_{o_{L} ,π} be an isomorphism in FMod(A_{o_{L} ,π}) satisfying α′⊗ id = β̂∘ (α ⊗ id). Since A_{o_{L}} = A_{L} ∩ A_{o_{L} ,π} inside A_{o_{L} ,π}[1/π], we can recover M as M = α(M)∩M⊗_{A_{o_{L}}} A_{o_{L} ,π}. This implies β̂(M) = M′and β := β̂|_{M} is the desired isomorphism β: \underline{M} ^{∼} −→ \underline{M} with α′= β ∘ α. This proves full faithfulness.

For Anderson A-motives Proposition 6.4 and Theorem 4.7 imply the following

Corollary 6.5. Let \underline{M} be an Anderson A-motive over L. Then in the strong sense of Definition 4.6, \underline{M} admits a good model \underline{M} if and only if the associated analytic Anderson A(1)-motive \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π] admits a good model \underline{M} ′. If this is the case, the functor (\underline{M} , α) ↦ (\underline{M} ⊗_{A_{o_{L}}} A_{o_{L} ,π}, α ⊗ id_{A_{o_{L} ,π} [1/π]}) is an equivalence of categories between the good models of \underline{M} and the good models of \underline{M} ⊗_{A_{L}} A_{o_{L} ,π}[1/π].

This corollary together with Theorem 5.3 and Corollary 5.5 implies the following criterion for good reduction of Anderson A-motives, which can be regarded as an analog of the reduction criteria for abelian varieties of Grothendieck [SGA 7, Proposition IX.5.13] and de Jong [dJ98, 2.5].

Corollary 6.6. Let \underline{M} be an Anderson A-motive over L such that coker(F_{\underline{M}}) is annihilated by 𝔍^{d} for some d. Then the following assertions are equivalent:

(i) \underline{M} admits a good model (\underline{M} , α) in the strong sense of Definition 4.6, i.e. there is an object \underline{M} ∈ FMod(A_{o_{L}}) such that coker(F\underline{M} ) is a finite free o_{L}-module and is annihilated by 𝔍^{d}, together with an isomorphism α: \underline{M} ^{∼} −→ \underline{M} [1/π] inside FMod(A_{L}); (ii) There is an effective local shtuka \underline{M̂} at ε over o_{L} such that coker(FM̂) is annihilated by 𝔍^{d}, and an

isomorphism \underline{M} ⊗_{A_{L}} A_{o_{L} ,(ε,π)}[1/π] ≅ \underline{M̂}[1/π] inside FMod(A_{o_{L} ,(ε,π)}[1/π]). Moreover, there is an equivalence of categories

\{ \begin{matrix}good models (\underline{M} , α) of & \underline{M} & in the \\ sense of Definitions 4.6 and 4.5 \end{matrix} \} ⟷^{∼}  \{ \begin{matrix}pairs (\underline{M̂}, f) consisting of \\ • a local shtuka \underline{M̂} at ε over o_{L}, and \\ • an isomorphism in FMod(A_{o_{L} ,(ε,π)}[1/π]) \\ f: \underline{M} ⊗_{A_{L}} A_{o_{L} ,(ε,π)}[1/π] ^{∼} −→ \underline{M̂}[1/π] \end{matrix} \}
(\underline{M} , α) ⟼→ (\underline{M} , α) ⊗_{A_{o_{L}}} A_{o_{L} ,(ε,π)} ,

where on the right-hand side a morphism of pairs β̂: (\underline{M̂}, f) ^{∼} −→ (\underline{M̂}′, f′) is defined to be an isomorphism of local shtukas β̂: \underline{M̂} ^{∼} −→ \underline{M̂}′ satisfying f′= β̂ ∘ f.

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Urs Hartl

Universität Münster

Mathematisches Institut

Einsteinstr. 62

D – 48149 Münster

Germany

www.math.uni-muenster.de/u/urs.hartl/ Simon Hüsken Universität Münster Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany