Equicharacteristic Tate Conjecture for Drinfeld modules Oliver Percy Watson A Dissertation in Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

2003

Supervisor of Dissertation

Graduate Group Chairperson

ACKNOWLEGDEMENTS

I would like to thank my advisor, Professor Ching-Li Chai, for suggesting this problem and for his immense support both as advisor and mentor. The University of Pennylvania has been a wonderful place to work, and I would like to thank the Thouron foundation for supporting me during my first two years here. Steve Shatz, David Harbater, Jonathan Block and Tony Pantev have all helped me with mathematical guidance and friendship. Very special thanks are due to Clark Barwick for his great help as a friend and mathematician, and in particular for superhuman efforts to make this thesis readable. Thanks and good wishes are due to Dong Uk Lee for helpful conversations and mathematical fellowship. Lastly, I would like to thank my parents for their constant support throughout my time here, and Rachael and Xavier for making my stay here a very happy one.

ABSTRACT Equicharacteristic Tate Conjecture for Drinfeld modules

Oliver Percy Watson Advisor: Ching-Li Chai

In this paper we study an analogue of a theorem of de Jong on the homomorphisms of Barsotti–Tate groups, transposed in our case to the setting of Drinfeld modules and their generalizations. In particular we prove a result on the extension of homomorphisms of ℘-divisible groups over an equal characteristic discrete valuation ring, where ℘ is a prime ideal of the ring of regular functions on an affine curve. We then use this result to deduce the equal characteristic Tate Conjecture for Drinfeld modules, which was up till now the one remaining unproved case of the Tate Conjecture in this instance.

Contents

1 Introduction 1

2 Statement of the Problem 6

3 All sorts of τ–modules 9 3.1 τ–modules over Frobenius rings . . . . . . . . . . . . . . . . . . . . . 10 3.2 τ–modules over curves . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Formal and equicharacteristic τ–modules . . . . . . . . . . . . . . . . 11 3.4 Quasi–τ–modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Analytic τ–modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Results on formal τ–modules 15 4.1 Formal τ–modules and t–divisible k–schemes . . . . . . . . . . . . . . 16 4.2 Base change for τ–modules . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Tate uniformization and semistability . . . . . . . . . . . . . . . . . . 21

5 Convergent subrings of power series rings 23 5.1 Technical lemmas on convergent power series . . . . . . . . . . . . . . 24 5.2 Convergent power series over Laurent series . . . . . . . . . . . . . . 27

6 The slope filtration over L[[t]]^{c} 31 6.1 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 Existence of the slope filtration . . . . . . . . . . . . . . . . . . . . . 34

6.3 Descent for convergent τ–modules . . . . . . . . . . . . . . . . . . . . 37 7 Extension of endomorphisms 39 8 Proof of Theorem 7.1 41 9 Existence of Section 50 10 Reduction to transcendence degree 1 54 11 End of the proof 56

1 Introduction

Drinfeld modules were introduced in the landmark paper [4], as analogues of abelian varieties. Here we think of an abelian variety as the realization of ℤ as the endomorphisms of a geometric object, or in the case of a CM abelian variety, we take an order in the ring of integers of a number field. Let us then denote by A the ring of functions on some curve X over a finite field 𝔽_{q}, regular away from some 𝔽_{q}-rational point ∞. In the Drinfeld module case A will play the role of the ring of integers of a number field, and we will realize A as the ring of endomorphisms of an “additive” geometric object. To be precise, we take the underlying additive group of a line bundle ℒ over some base scheme S → Spec A, and equip this sheaf of additive groups in characteristic p with a non-trivial structure of A-module.

Let us look at the simplest example of this, which nonetheless serves to illustrate all the main ideas. As our geometric curve over Spec 𝔽_{q} we take ℙ^{1}, with the natural 𝔽_{q}–rational point ∞. The ring of functions on ℙ^{1} regular away from ∞ is simply the ring 𝔽_{q}[t] where t is some parameter of the affine patch. Let us now take a field k equipped with a map ι : 𝔽_{q}[t] → k. If the image of t under ι is non-constant, then clearly ι must be injective. This is the scenario corresponding to “characteristic 0” number theory. Alternatively if the image of t is constant, then there must exist a non-trivial prime ideal ℘ of the ring 𝔽_{q}[t] that is the kernel of ι, and the image of 𝔽_{q}[t] in k must be finite. This is the case corresponding to number theory over finite fields, and will be the case that is dealt with in this paper.

So anyway, once we have this setup, what is a Drinfeld module? Well, we have that the 𝔽_{q}–linear endomorphisms of the additive group of k are, since k is an 𝔽_{q}- algebra, just the ring k[τ] where τ is the map that sends c ∈ k to c^{q}. A Drinfeld module over k is an 𝔽_{q}-algebra map from ϕ : 𝔽_{q}[t] → k[τ] such that ϕ(t) = ι(t) + a_{1}τ + … + a_{r}τ^{r}, where the rank r of the Drinfeld module ϕ is > 0 - a non–triviality condition.

It turns out however that thinking of a Drinfeld module as above, namely as a commutative subring of the non–commutative ring k[τ] is not always the most useful way. Following work of Drinfeld [5] relating these objects to modifications of vector bundles on curves over finite fields, Anderson in [1] defined the notion of t-motive which at some point was further generalized to the notion of τ-sheaf.

To see how these objects (defined later) arise, let us return to the Drinfeld module case of a map ϕ : 𝔽_{q}[t] → k[τ]. In fact ϕ is just 𝔾_{a}/ Spec k equipped with a non-standard action of 𝔽_{q}[t] - i.e. elements in 𝔽_{q}[t] do not simply act by scalar multiplication. Let us take the set of maps in the category of algebraic groups from the underlying additive group of ϕ to 𝔾_{a}. This set has an obvious structure of (infinite dimensional) 𝔽_{q}–vector space. It has more however the structure of a right 𝔽_{q}[t]-module given by precomoposing a morphism by the image of an element of 𝔽_{q}[t] in the endomorphism ring of ϕ, and a left k-module structure given by postcomposing with the endomorphism of 𝔾_{a} given by scalar multiplication by an element of k (note that we need the fact that k is commutative here). These two operations obviously commute, since they are on different sides, thus giving k[τ] a structure of module over k ⊗_{𝔽_{q}} 𝔽_{q}[t] = k[t]. When we look at how the endomorphism τ interacts with this module structure, we see that it commutes with the 𝔽_{q}[t]- module part, and that τℓ = ℓ^{q}τ ℓ ∈ k. Anderson shows that in favorable cases (over a perfect field) we can recover the Drinfeld module from the module over k[t] together with its endomorphism τ which is skew-linear in the coefficients - namely it commutes with elements of k up to a q-power twist. Moreover the rank of the module over k[t] is just the rank of the Drinfeld module as originally defined.

This leads us to study modules of various types over R[t] for 𝔽_{q}–algebras R equipped with endomorphisms that are σ–linear, where σ acts by fixing t and as the q-th power on elements of R. These are τ–modules, and we should think of the study of these objects as somehow akin to the classical study of schemes over Spec R. If R is an “analytic” ring, for example a complete discretely valued field, or the ring of regular functions on some analytic space, then we can consider modules over R{{t}}, the functions with coefficients in R, which converge on some rigid subvariety of the rigid spectrum of R. For example, if R = k((x)) we could consider modules over the ring of power series in t with coefficients in R such that these power series actually converged for some open annulus contained in the non-archimedean open unit disk. These modules would be equipped with a σ-linear endomorphism, where σ acted as a q-th power on the coefficients in k((x)), and fixed t. Far–fetched though this example might sound, it is of crucial importance in this paper. The study of these objects should be though of as analogous to the study of rigid schemes in the classical setting. They are the natural setting for the analogues of results such as the Tate uniformization for degenerating elliptic curves. Finally of course we could just consider modules over R[[t]] together with a σ-linear endomorphism. This will be the setting for “formal” statements. For example, it is clear that the classical notion of a Tate module, or the Dieudonné crystal attached to a p-divisible group, are “formal” objects. The natural analogues of these objects, in this setting, will be of the above type. To get the general case, i.e. when the curve over 𝔽_{q} is not ℙ^{1}, we let A be the ring of functions on some affine curve over Spec 𝔽_{q}, regular away from some 𝔽_{q}–rational point that we again label ∞. Then all our previous discussion goes through, replacing R[t] by R ⊗_{𝔽_{q}} A. Notice that in the formal and analytic contexts there is no change, since we can always choose a local uniformizer t.

We should think of the Drinfeld module theory as coming in two flavors: the commutative subrings of non–commutative rings and the τ-module theory. The latter is the dual to the former in a precise sense. As we saw before, it comes from the construction \underline{Hom}(−, 𝔾_{a}). Many questions that arise from the theory in the setting of non–commutative rings are substantially easier to address in the τ-module setting, though this is not always true. The category of τ–modules can also be understood as the avatar, in this setting, of the as yet to be constructed category of motives in the classical setting.

In this thesis we investigate the analogue of the equal characteristic Tate Conjecture for abelian varieties proved by de Jong in [11]. Suppose we have a Drinfeld module ϕ; A → k[τ] over a field k. Let us denote by \overline{k} the algebraic closure of k. For any element α of A, we set ϕ[α] to be the Spec k–scheme in A–modules given by the kernel of the endomorphism of 𝔾_{a} given by ϕ(a). Suppose that ℘ is a prime ideal of A and that ℘ is not equal to the kernel of the structure map ι : A → k. The we can form the ℘-divisible group of ϕ as ϕ[℘^{∞}] := lim_{⟶} ϕ[℘^{n}], as well as the ℘–adic Tate module T_{℘}ϕ := Hom(ϕ[℘^{∞}].k_{℘}/A_{℘}), where here A_{℘} is the completion of the ring A at the prime ℘ and k_{℘} is the fraction field of A_{℘}. The Tate module T_{℘}ϕ is a free A_{℘}-module of rank r = deg ϕ, and is naturally equipped with a continuous A_{℘}–linear action of Gal(\overline{k}/k). There is an obvious map from End_{k}(ϕ)⊗_{A}A_{℘} → End_{A_{℘}}(T_{℘}(ϕ))^{Gal \overline{k}/k} from ℘–adic limits of endomorphisms of the Drinfeld module to the Galois equivariant equivariant endomorphisms of the Tate module. The Tate Conjecture asserts that this map is an isomorphism if k is finitely generated over its field of constants.This has been proved by Taguchi in [15] and considerably generalized in later work [16] and [17].

This is the étale version of of the Tate Conjecture for Drinfeld modules, i.e. we chose ℘ to be coprime to the characteristic ideal. Suppose now that ℘ is the characteristic ideal. Then we can still form the ℘–divisible group of ϕ. Now we have a natural map from End(ϕ) ⊗_{A} A_{℘} → End(ϕ[℘^{∞}]). The Tate Conjecture in this context asserts that this map is an isomorphism, and this is the case that we deal with in this paper.

The strategy is simple. We work in the dual setting to that of Drinfeld modules and their ℘–divisible groups, namely τ–modules and formal τ–modules. The Tate Conjecture is standardly proved as follows in the cases where it is known. We are trying to understand morphisms of ℘–divisible groups over fields of finite transcendence degree over their field of constants or number fields. The first thing one does is to reduce to the case of a number field or a field of transcendence degree one. Then one needs to understand the height of the endomorphisms at every place of that field. If one can obtain good lower bounds for every place, the result will follow easily by standard height-theoretic considerations. We prove an analogue of a theorem of de Jong [11] in the classical case on the extension of homomorphisms of formal τ–modules over discrete valuation rings. This gives a lower bound at almost every place - the places of good reduction of the Drinfeld module ϕ. At the other finitely many places, those of semistable reduction, we need to use the analytic structure of the τ-module associated to the Drinfeld module ϕ to get the bounds that we want. This theory is analogous to the Tate uniformization theory in the classical case. Finally putting this all together yields the required result.

2 Statement of the Problem

In this article we prove the remaining version of the Tate Conjecture for Drinfeld modules, namely, the equicharacteristic Tate Conjecture.

Fix a finite field 𝔽_{q}. All constructions will take place within the category of 𝔽_{q}– algebras. Hence in particular, any fields, rings, or schemes introduced in this paper will be over 𝔽_{q}.

Suppose A is the ring of functions of a curve X, L a field, finitely–generated over 𝔽_{q}, and ι : A → L a morphism with ℘ := ker ι nonzero. For these data we can form the ℘–divisible group ϕ[℘^{∞}] := lim_{⟶} ϕ[℘^{n}], and the Tate Conjecture in this context asserts that the canonical map

End ϕ ⊗_{A} A_{℘} → End(ϕ[℘^{∞}])

is an isomorphism.

We can frame the equicharacteristic Tate Conjecture in the context of τ–modules on an 𝔽_{q}–algebra B equipped with an endomorphism τ. A τ-module on (B, τ) is a pair (M, σ) consisting of a free B–module M of finite rank together with an injective 𝔽_{q}–linear structure endomorphism σ which is τ–semi-linear; morphisms of τ–modules are ordinary module homomorphisms that commute with the structure endomorphisms. Endowing the tensor product of the modules of two τ–modules with the obvious structure morphism permits us to form tensor products in the category of τ–modules over B, and if (M, σ) is a τ–module on B, then one can form the submodule M^{σ} of elements of M fixed by the structure morphism σ.

Now suppose K an algebraic function field in one variable over 𝔽_{q}, ℘ a place of K, K_{℘} the completion of K at ℘, and L a field of finite type over 𝔽_{q}. We set K′ := K ⊗_{𝔽_{q}} L, and we denote by K′_{℘} the completion of K′ with respect to the ℘–adic topology. We extend the Frobenius morphism τ on L to the endomorphism τ ⊗ 𝟙 on K′; it extends naturally to an endomorphism (which by a slight abuse of notation we also call τ) of the completion K′_{℘}. The Tate conjecture in this context is then the following assertion:

2.1 Conjecture (Tate Conjecture). Suppose (M, σ) is a τ-module over K′. Then the natural map of K_{℘}–vector spaces

K_{℘} ⊗_{K} M^{τ} → (K′_{℘} ⊗_{K′} M)^{τ}

is an isomorphism.

Fixing an isomorphism M ≅ K^{′⊕r}, the Tate Conjecture can be restated in the following fashion (after [15]):

2.2 Conjecture (Tate Conjecture, Reformulation). Suppose A is a matrix in Mat_{r×r}(K′). Consider the linear Frobenius equation

(2.2.1)  AX^{τ} = X

where X is a variable taking values in K′_{℘}^{⊕r}, upon which τ acts component-wise. Let V (respectively, V_{℘}) be the space of solutions of (2.2.1) in K′^{⊕r} (resp., in K′_{℘}^{⊕r}). Then the natural homomorphism of K_{℘}-vector spaces K_{℘} ⊗_{K} V → V_{℘} is an isomorphism.

The injectivity of this map is easy, and one can reduce the problem without difficulty ([15]) to the case in which K = 𝔽_{q}(t), ℘ = (t), K′ is the ring L[t][S^{−1}], where S is the multiplicative system of all monic polynomials, and K′_{℘} = L((t)).

Now the Tate Conjecture falls naturally in two cases. In the first, M comes from a t–motive of characteristic different from ℘. This is the mixed characteristic case. The first important result in this direction was proved by Anderson, who showed that any endomorphism of the t–divisible group over the generic fiber of a discrete valuation ring in mixed characteristic extends to the closed fiber. Taguchi then proved ([15]) a special case of the Conjecture, and shortly thereafter Tamagawa verified the general mixed characteristic case (see [17] or [16] for an expository account).

In the second case, M comes from a t–motive of characteristic equal to ℘. This is the equicharacteristic case that we address in this paper. Our strategy will be as follows. We will use the main theorem of de Jong [11], and in particular the proof without the crystalline hypotheses given by Messing and Zink [13], translated into our context to prove the analogue of Anderson’s theorem. We then will use the height–theoretic methods of Taguchi to prove the equicharacteristic case of the Conjecture for the endomorphisms of the motive of a Drinfeld module.

3 All sorts of τ–modules

We give a short review of the basic definitions of τ–modules here. For further details, see [6].

3.1 τ–modules over Frobenius rings

3.1 Definition. A Frobenius ring is a pair (R, τ) consisting of a commutative ring R and an endomorphism τ of R, which we call the Frobenius of R. A morphism of Frobenius rings is a ring homomorphism that commutes with the Frobenius.

A τ–module over a Frobenius ring (R, τ) is a pair (M, σ) consisting of an R– module M and a τ–semilinear endomorphism σ of M (i.e., σ is additive, and for any a ∈ R and m ∈ M, σ(am) = a^{τ}σ(m)). The map σ is called the structure endomorphism or (by an abuse of terminology) the Frobenius of the τ–module (M, σ). A morphism of τ–modules over (R, σ) is a homomorphism of R–modules that commutes with the structure endomorphisms. 3.2 Remark. Observe that we have chosen to work systematically with τ–modules instead of more general objects such as τ–sheaves. As a result, many of the theorems we quote here will be stated only for τ-modules though they frequently hold more generally.

3.2 τ–modules over curves

A somewhat more restricted notion is that of a τ–module over a curve. 3.3 Definition. Suppose R a commutative 𝔽_{q}–algebra; set X := Spec R. Suppose C := Spec A an absolutely irreducible smooth affine curve with constant field 𝔽_{q}, and set F := Frac A. For the product C_{X} := C ×_{Spec 𝔽_{q}} X = Spec(A ⊗_{𝔽_{q}} R), Denote by f the Frobenius endomorphism of R given by the assignment s ↦ s^{q}. Then A ⊗_{𝔽_{q}} R is a Frobenius ring with Frobenius τ := 𝟙 ⊗ f. Then a τ–module of rank r over C_{X} is a pair (M, σ) consisting of a projective module M over A ⊗_{𝔽_{q}} R of rank r and an injective morphism σ : τ^{∗}M → M.

3.4 Example. Suppose K a field extension of 𝔽_{q}, and set C := 𝔸^{1} = Spec 𝔽_{q}[t] and X := Spec K. Then a τ–module over C_{X} is just a free finite module over K[t] together with a τ–semilinear injective endomorphism, where τ acts as Frobenius on K and fixes t.

3.3 Formal and equicharacteristic τ–modules

3.5 Definition. Suppose R an 𝔽_{q}–algebra, and let R[[t]] be the usual t–adic completion of R ⊗ 𝔽_{q}[[t]]. Then a formal τ–module over R is a τ–module over R[[t]], where τ acts as Frobenius on R and fixes t.

In this paper we will use a slightly more restrictive class of formal τ–modules that is more convenient for our purpose; in particular, the τ–modules we will consider will be defined to be equicharacteristic τ–modules.

3.6 Definition. A frame (A, τ) is a Frobenius ring (A, τ) such that A is an 𝔽_{q}[t]– algebra without t–torsion, and τ satisfies

τ(a) = a^{q} mod tA

and fixes t. A morphism of frames is a morphism of Frobenius rings.

3.7 Definition. Now let R be a reduced ring of characteristic p. Set A := R[[t]] and take for τ the endomorphism that acts as the q-th Frobenius on the coefficients of a power series in A and fixes t. Then (A, τ) is a frame, and an equicharacteristic τ–module of rank r over (A, τ) is a pair (M, σ) consisting of a finitely generated projective A–module M of rank r, together with a τ–linear endomorphism σ of M such that the linearized map

A ⊗_{(A,τ)} ∧^{r}M → ∧^{r}M

is of the form t^{s}u for some integer s and an isomorphism u : A ⊗_{(A,τ)} ∧^{r}M → ∧^{r}M.

Observe that if M is free, then the last requirement says that, in some basis of M, det σ = t^{s}ϵ for some ϵ ∈ A. We will write s = ord_{t} det F.

3.4 Quasi–τ–modules

As mentioned earlier, the category of τ–modules possesses a tensor product; likewise, the category of equicharacteristic τ–modules possesses a tensor product, and the equicharacteristic τ–module (A, τ) (where R is a reduced ring of characteristic p, A := R[[t]], and τ acts as the q-th Frobenius on R and fixes t) is a unit for − ⊗ −. In order to guarantee the existence of internal \underline{Hom}, it is necessary to work in a slightly bigger category, that of quasi–τ–modules:

3.8 Definition. A quasi–τ–module over A is a pair (M, σ) consisting of a projective A–module M and a τ–linear structure homomorphism

σ : M → M[1/t],

such that for some m ≥ 0, the pair (M, t^{m}σ) is a τ–module. A morphism of quasi–τ–modules is of course a homomorphism of modules that commutes with the τ–linear structure homomorphisms. 3.9 Definition. A homomorphism α : (M_{1}, σ_{1}) → (M_{2}, σ_{2}) of quasi–τ–modules of equal rank r is called an isogeny if ∧^{r}α is of the form t^{s}u for an integer s ≥ 0 and an isomorphism u : ∧^{r}M_{1} → ∧^{r}M_{2}.

In this article we will concentrate on formal τ–modules, which are essentially Tate modules in mixed characteristic and the duals of t–divisible groups in equal characteristic. A quasi–τ–module should be thought of as the generic fiber a formal τ–module. So in particular, suppose L a perfect field with a valuation ν, and let \overline{L} be the algebraic closure. Set R := \overline{L}, and let τ be the endomorphism of A such that τ(x) = x^{q} for any x ∈ R and τ(t) = t. The notion of a quasi–τ–module over \overline{L} is thus the function field analogue of an isocrystal in the classical case.

If (M, σ) is a quasi–τ–module over \overline{L}[[t]], then M⊗_{\overline{L}[[t]]}\overline{L}((t)) is a finite–dimensional vector space over \overline{L}((T)), and τ is bijective. Note that we lose no information when working in the category of quasi–τ–modules over k[[t]], where k is any field, by tensoring over k[[t]] with k((t)). We record here a theorem from [12], Theorem 2.4.5, page 32, special cases of which we will need. Notice that in [12], quasi–τ–modules are called Dieudonné modules.

3.10 Theorem. The category of quasi–τ–modules over \overline{L}[[t]] is \overline{L}((t))–linear and semisimple. The simple objects (up to isomorphism) are the objects (M_{r,s}, σ_{r,s}), where r and s are integers with r ≥ 1, (r, s) = 1, and

M_{r,s} = S^{⊕r}

and

σ_{r,s}(e_{i}) = \begin{cases}e_{i+1} & if i = 1, …, r − 1 \\ t^{s} & if i = r \end{cases}

where {e_{i}} is the standard basis of S^{⊕r}.

If the residue field L is not algebraically closed, then the category of quasi–τ– modules is L–linear, Artin, and noetherian. If L′ is a field extension of L, we have an obvious base–change functor L′ ⊗_{L} −. If (M, σ) is an indecomposable quasi–τ– module over a perfect field K, then there exists a unique triple of integers (r, s, t), with r, s ≥ 1 and (r, s) = 1 such that \overline{K} ⊗_{K} (M, σ) is isomorphic to (M_{r,s}, σ_{r,s})^{t}. We have the following definition:

3.11 Definition. The slope of (M, σ) is the rational number s/r.

3.5 Analytic τ–modules

We have discussed ordinary τ–modules, formal τ–modules, and what we have called equicharacteristic τ–modules. There are also analytic τ–modules, to which we now turn.

First we must recall some results and terminology from [6].

3.12 Definition. A τ–module over a discretely valued field K is said to have good reduction if there is a τ–module over R, the ring of integers of K, such that the generic fiber is the original τ–module. A τ–module over R is called non–degenerate if the fiber at the closed point contains a non–zero τ–module.

We can consider the formal spectrum of the ring R[[t]] and associate to it a “generic fiber” 𝒟_{K} in the sense of [3], which is an affinoid subspace of Spec K[t]. We will denote by K{{t}} its ring of functions.Given a τ–module M over K[t], we can base change to a τ–module M^{an} := M ⊗_{K[t]} K{{t}}. We call M^{an} the analytic τ–module associated to M.

Suppose M and M′ two τ–modules on C ×_{Spec 𝔽_{q}} Spec K. One can form the abelian group Hom^{an}(M, N) of analytic morphisms, namely the abelian group of K{{t}}–linear homomorphisms M^{an} → N^{an} commuting with the τ–actions.

4 Results on formal τ–modules

Fix a perfect extension field k of 𝔽_{q}. Our aim is now to prove the following analogue of a theorem of de Jong:

4.1 Theorem. Suppose M and N be equicharacteristic τ–modules over k[[t, x]], and suppose

ϕ : M ⊗_{k[[x,t]]} k((x))[[t]] → N ⊗_{k[[x,t]]} k((x))[[t]]

a morphism of τ–modules. Then ϕ(M) ⊂ N.

In order to proceed, we collect some of the key technical facts from the theory of formal τ–modules we will require in the sequel.

4.1 Formal τ–modules and t–divisible k–schemes

4.2 Definition. A t–divisible k–scheme in 𝔽_{q}[[t]]–modules is an inductive system of finite k–schemes in 𝔽_{q}[[t]]–modules

G = (G_{1} →^{i_{1}} G_{2} →^{i_{2}} G_{3} →^{i_{3}} …) = lim_{⟶} G_{n}

such that, for each integer n ≥ 1, the following axioms hold:

1. As a k–scheme in 𝔽_{q}–vector spaces, G_{n} can be embedded into 𝔾_{a,k}^{N} for some integer N ≥ 0. Note that since there is a canonical map from 𝔽_{q} to k, then 𝔾_{a,k} can be considered as a k–scheme in 𝔽_{q} vector spaces.

2. There exists an integer d ≥ 0—called the rank of G—such that as a k–scheme in 𝔽_{q}–vector spaces, the dimension of G_{n} is nd.

3. the sequence of k–schemes in 𝔽_{q}[[t]]–modules

0 → G_{n} →^{i_{n}} G_{n+1} →^{t^{n}} G_{n+1}

is exact. A morphism of t–divisible k–schemes in 𝔽_{q}[[t]]–modules is a morphism of the

corresponding inductive systems.

We will construct contravariant functors between this category and the category of τ–modules over k. More generally, let ℳ(k) be the category whose objects are pairs (M, f) consisting of a finite–dimensional k–vector–space M and a Frobenius– linear endomorphism f of M, and whose morphisms are linear maps commuting with these endomorphisms. Let 𝒮(k) be the full subcategory of the category of finite k–schemes in 𝔽_{q}–vector spaces spanned by those k–schemes in 𝔽_{q}–vector spaces that can be embedded in 𝔾_{a,k}^{×N} for some N ≥ 0.

Suppose (M, f) ∈ ℳ(k). We define a functor G_{k}(M, f) from the category of k–algebras to the category of 𝔽_{q}–vector spaces with

G_{k}(M, f)(R) := {g ∈ Hom_{k}(M, R) | ∀m ∈ M g(f(m)) = g(m)^{q}}

It is easy to see that G_{k}(M, f) is in fact an object of 𝒮(k) (after taking a basis of M over k). This defines a contravariant functor G_{k} from ℳ(k) to 𝒮(k).

If, conversely, G is an object of 𝒮(k), the group of homomorphisms of k–schemes in 𝔽_{q}–vector spaces

M_{k}(G) = Hom(G, 𝔾_{a,k})

is a finite dimensional k–vector space, and, moreover, the Frobenius endomorphism of 𝔾_{a,k} = Spec k[t] over 𝔽_{q} induces a Frobenius–linear morphism f_{k}(G) : M_{k}(G) → M_{k}(G). This defines a contravariant functor M_{k} from 𝒮(k) to ℳ(k).

We now state the following proposition from [12], Proposition 2.4.11, page 34. There the theorem is stated for k ⊂ \overline{𝔽_{p}} but one sees from the proof, in exact analogy to the case of F –crystals, that one needs only that the field in question be perfect.

4.3 Proposition. Let k be a perfect field containing 𝔽_{q}. The contravariant functors G_{k} and M_{k} between ℳ(k) and 𝒮(k) are exact and quasi–inverse to each other.

We now extend the definition of the functors G_{k} and M_{k} by setting:

G_{k}(M, f) := lim_{⟶} G_{k}(M/t^{n}M, f mod t^{n}M)

for a τ–module (M, f) over k and

(M_{k}(G), f_{k}(G)) = lim_{⟵}(M_{k}(G_{n}), f_{k}(G_{n}))

for any t–divisible k–scheme in 𝔽_{q}[[t]]–modules G = lim_{⟶} G_{n}. 4.4 Corollary. The functors G_{k} and M_{k} establish an anti–equivalence of categories between the category of formal τ–modules over k and the category of t–divisible k– schemes in 𝔽_{q}[[t]]–vector spaces.

This relates our main theorem to the Drinfeld module setting. More precisely, as an application of Theorem 4.1, it follows that the ring of endomorphisms of the t–divisible k–scheme associated to a Drinfeld module ϕ of positive characteristic over k is precisely the ring of endomorphisms of ϕ completed in the t–adic topology. This is the Tate Conjecture for Drinfeld modules of positive characteristic.

4.2 Base change for τ–modules

4.5 Proposition. Let ϕ : (A, τ_{A}) → (B, τ_{B}) be a homomorphism of frames. Assume that (t) is a prime ideal of A contained in the radical of A, and that ϕ(t) is not a unit of B.

Let α : (M_{1}, σ_{1}) → (M_{2}, σ_{2}) be a morphism of equicharacteristic τ–modules over A. If α ⊗ 𝟙_{B} is an isogeny (respectively, an isomorphism), then α is an isogeny (resp., an isomorphism).

Proof. The A–modules M_{1} and M_{2} have the same rank r. It suffices to show that ∧^{r}α is an isomorphism. Consider the commutative diagram:

\xymatrix{∧^{r}M_{1} \ar[d]_{∧^{r} det σ_{1}} \ar[r]^{∧^{r}α} & ∧^{r}M_{2} \ar[d]^{∧^{r} det σ_{2}} \\ ∧^{r}M_{1} \ar[r]_{∧^{r}α} & ∧^{r}M_{2}}

Now we write det F_{i} = t^{a_{i}}u_{i}, where a_{i} ∈ ℤ, and u_{i} is a τ–linear isomorphism. If we tensor the diagram with B we obtain a_{1} = a_{2}. Hence the diagram remains commutative if we replace det F_{i} by u_{i}. We divide ∧^{r}α by the maximal possible power of t, and call the resulting homomorphism β. This power is bounded because it is bounded over B. If we divide the morphisms of the above commutative diagram by the maximal power of t and consider the result modulo t, we obtain the commutative diagram:

\xymatrix{L_{1} \ar[r]^{\overline{β}} \ar[d]_{_{\overline{u}_{1}}} & L_{2} \ar[d]^{_{\overline{u}_{2}}} \\ L_{1} \ar[r]_{\overline{β}} & L_{2}}

We set \overline{A} := A/tA. The maps \overline{u}_{i} are τ–linear isomorphisms of \overline{A}–modules. It suffices to show that \overline{β} is an isomorphism. Take an open subset Spec \overline{A}_{f} ⊂ Spec \overline{A} such that there are isomorphisms L_{i} is isomorphic to \overline{A}_{f}. Then for some ρ ∈ \overline{A}_{f}, we may write: \overline{β}(x) = ρx for any x ∈ \overline{A}_{f}. We note that ρ ≠ 0; indeed, since \overline{A} is an integral domain, ρ = 0 would imply that \overline{β} = 0, but then contrary to our assumption, β would be divisible by t. Hence we write \overline{u}_{i}(x) = ϵ_{i}x^{q}, where ϵ_{i} ∈ \overline{A}_{f} are units for i = 1, 2. Because the last diagram is commutative we have ϵ_{2}ρ^{q} = ρϵ_{1}. Since \overline{A} is an integral domain, we may divide this equation by t. We obtain that ρ is a unit.

In the analogue 4.1 of de Jong’s theorem, we can replace equicharacteristic τ– modules by slightly more general objects. To see this we show:

4.6 Lemma. Suppose M a finitely–generated torsion–free k[[x, t]]–module. We set:

M_{1} := {m ∈ M_{(t)} | t^{s}m ∈ M, s ∈ ℕ}

Then M_{1} is a free k[[t, x]]–module.

Proof. Since k[[x, t]] is a UFD, we see that M_{1} = M if M is free. If M is not free, we find a free k[[x, t]]–module N, not of the same rank as M, such that M ⊂ N ⊂ M_{(t)}. The last k[[x, t]]–module is free, and since

N = N_{1} := {n ∈ N_{(t)} | t^{s}n ∈ N, s ∈ ℕ}

we find that M_{1} ⊂ N.

By definition, N/M_{1} has no t–torsion, and therefore depth_{k[[x,t]]} N/M_{1} ≥ 1. However, since k[[x, t]] is regular of dimension 2, this implies that the cohomological dimension of M_{1} is zero. This proves the lemma.

4.7 Remark. Let M be a torsion–free k[[x, t]]–module. Let σ be a τ–linear endomorphism of M such that σ ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) is an isomorphism. Since τ operates on k[[x, t]]_{(t)}, the map σ extends to a τ–linear endomorphism (which by an abuse we also call σ) of M_{1}, which induces an isomorphism when tensored with 𝔽_{q}(t). Hence there is an a ∈ k[[x, t]] and an integer m ≥ 0 such that a det τ = t^{m}. Since k[[x, t]] is a UFD, this implies that det σ = t^{m′}u, where u is a unit, and thus (M_{1}, σ) is a τ–module. In this sense the pair (M, σ) is very nearly a τ–module. One sees easily that the analogue of de Jong’s theorem holds if the pair (M, τ) is of this more general type, if it holds for τ–modules. We will not use this fact in the sequel.

4.3 Tate uniformization and semistability

We now recall Drinfeld’s result on Tate uniformization. Consider R, a complete discrete valuation 𝔽_{q}-algebra with finite residue field k and fraction field K. Suppose ϕ to be a Drinfeld module over K of rank r with bad reduction. There exists a finite totally ramified extension K′ of K and the Drinfeld module ϕ′ over K′ obtained by base extension such that at the closed point, ϕ′ is a Drinfeld module of rank s. Then the Tate uniformization theorem of Drinfeld states that there is an analytic morphism from a Drinfeld module ψ of rank s with good reduction over K to ϕ′. In [6], Gardeyn shows that there exists a way to think about this result in the τ– module setting. Here we state these results in a somewhat simpler setting, as that is all that is required.

4.8 Theorem. In the notation introduced earlier, suppose M and N to be the τ– modules associated to ϕ and ψ, then the analytic morphism e ∈ Hom(ψ, ϕ) of Drinfeld modules induces an analytic morphism

e^{∗} ∈ Hom^{an}(M, N)

Moreover there exists a τ–module P over K[t] , trivial over a finite extension of K, such that we have the following exact sequence:

0 → P̃ → M̃ → Ñ → 0

of τ–modules over K{{t}}.

We now recall some more results of Gardeyn from [7]. In line with our approach, we phrase them in the language of modules rather than of sheaves, though they hold more generally.

4.9 Definition. Let K be a discretely valued field with ring of integers R and residue field k. We will say that any property holds potentially if it holds after a finite extension of the base field K.

1. If (M, σ) is a τ–module over K[t], we say that a τ–module (ℳ, σ̃) over R is a model for (M, σ) if there is an isomorphism (ℳ⊗_{R}K, σ̃⊗_{R}K) → (M, σ). This model is called good if the induced map ℳ ⊗_{R} k → ℳ ⊗_{R} k is an injection.

2. If (M, σ) is a τ–module over K{{t}}, we say (ℳ, σ̃) is a model for (M, σ) if there is an isomorphism (ℳ ⊗_{R[[t]]} K{{t}}, σ̃ ⊗_{R[[t]]} K{{t}}) → (M, σ). The model is called good if the induced map M ⊗_{R} k → M ⊗_{R} k is an injection.

An analytic τ–module M^{an} associated to a τ-module M over K[t] is semistable if there exists a filtration

0 = N_{0}^{an} ⊂ N_{1}^{an} ⊂ … ⊂ N_{n}^{an} = M^{an}

of M^{an} by analytic τ–modules N_{i}^{an} such that the modules M^{an}/N_{i}^{an} are torsion free and such that the subquotient τ–modules M_{i}^{an} := N_{i}^{an}/N_{i−1}^{an} have good models over R[[t]].

We have the following special case of a theorem from [8]: 4.10 Theorem. Every analytic τ–module M^{an} coming from the motive M of a Drinfeld module is potentially semistable. 4.11 Remark. The above result is actually true for any motive built from the motives of Drinfeld modules by the operations of direct sum and tensor product. In fact, the methods of this paper will easily extend to prove the Tate conjecture in this case.

5 Convergent subrings of power series rings

To prove our theorem 4.1 we need to establish results on some special subrings of Laurent series rings in two variables. We start from a more general viewpoint, and then specialize to the case we need. Thus we suppose A a ring equipped with a discrete valuation ν. We wish to investigate particular subrings of the power series ring A[[t]].

5.1 Technical lemmas on convergent power series

5.1 Definition. A convergent power series with coefficients in A is an element f of A[[t]], where f = ∑ a_{i}t^{i} and there exist constants C_{1} ∈ ℝ and C_{2} ∈ ℝ_{>0} with the property that for every n ∈ ℕ, the inequality ν(a_{n}) ≥ C_{1} − C_{2}n holds.

Another (simpler) characterization of a convergent power series is that the Newton polygon of f as a function of t be bounded in the plane by some line of negative slope. Clearly this property is preserved under multiplication and addition, and therefore convergent power series form a ring, which we denote A[[t]]^{c}. A particular example of such rings will be given in the following easy proposition. We remind the reader again that all rings introduced are actually 𝔽_{q}–algebras.

5.2 Proposition. Let L be a field with ν : L → ℝ ∪ {∞} a nonarchimedean valuation. Then L[[t]]^{c} is a discrete valuation ring with residue field L and uniformizer t. If L is perfect, then the Frobenius map τ : ∑ a_{i}t^{i} → ∑ a_{i}^{q}t^{i} induces an automorphism of L[[t]]^{c}.

Proof. If f = ∑ a_{i}t^{i} ∈ L[[t]]^{c}, then we have ν(a_{i}) ≥ −C_{1} − C_{2}i for some C_{1}, C_{2} ∈ ℝ. Applying τ to f we get f^{(q)} := ∑ a_{i}^{q}t^{i}. Now ν(a_{i}^{q}) = qν(a_{i}) ≥ −qC_{1} − qC_{2}i; so f^{(q)} is convergent.

5.3 Proposition. Suppose C = (c_{ij}) be an h × h matrix with coefficients in A[[t]]^{c}. Let y ∈ A[[t]]^{⊕h} be a vector such that for some number r ≥ 0, Cy = τ^{r}y. Then we have y ∈ (A[[t]]^{c})^{⊕h}.

Proof. Let us first define for some f = ∑ a_{i}t^{i} ∈ A[[t]]^{c}:

ν(f, n) := − min{ν(a_{i}) | i = 0, …, n}

Note that this is an increasing function in n. We now write our equation in coordinates:

∑_{k=1}^{h} c_{lk}y_{k} = τ^{r}y_{l}, l = 1, …, h

We will show the following: Let M ≥ 0 be a constant such that for all l, k:

ν(c_{lk}, 0) ≤ M, ν(c_{lk}, n) ≤ Mn for n ≥ 1

Then for any k = 1, …, n we have:

ν(y_{k}, 0) ≤ M, ν(y_{k}, n) ≤ Mn for n ≥ 1.

We show this by induction beginning with the case n = 0. Choose i such that ν(y_{i}, 0) is maximal among the ν(y_{k}, 0) for k = 1, …, h. We obtain:

q^{r}ν(y_{i}, 0) = ν(τ^{r}y_{i}, 0) ≤ ν(c_{ij}y_{j}, 0) = ν(C_{ij}, 0) + ν(y_{j}, 0) ≤ ν(c_{ij}, 0) + ν(y_{i}, 0)

This implies

(q^{r} − 1)ν(y_{i}, 0) ≤ ν(c_{ij}, 0) ≤ M

Now we assume n ≥ 1 and ν(y_{k}, u) ≤ Mu for 1 ≤ u < n. Again we choose i such that ν(y_{i}, n) is maximal among ν(y_{k}, n), and j such that ν(c_{ij}y_{j}, n) is maximal among the ν(C_{ik}y_{k}, 0). Then we have:

q^{r}ν(y_{i}, n) ≤ ν(c_{ij}y_{j}, n) ≤ max{ν(y_{j}, u) + ν(c_{ij}, v) | u + v = n}

Note that the leftmost term is ν(τ^{r}y_{i}, n) and the rightmost term is ≤ Mu + Mv = Mn. We are done.

We set

Ã = {x ∈ A | ν(a) ≥ 0}

Thus Ã[[t]] is the set of power series with coefficients in Ã. 5.4 Corollary. (of the proof). Assume that the coefficients of C are in Ã[[t]]. Then y ∈ Ã[[t]]. 5.5 Proposition. Let ξ = ∑ x_{i}t^{i} ∈ A[[t]]^{c} such that x_{0} ∈ A is a unit. Then ξ is a unit in A[[t]]^{c}. Proof. First let us state two easy inequalities. Let α, β be two elements of A[[t]]. Then we have that for all n ∈ ℕ: ν(α + β, n) ≤ max{ν(α, n), ν(β, n)} (5.5.1) ν(αβ, n) ≤ max{ν(α, l) + ν(β, n − l) | 0 ≤ l ≤ n} (5.5.2) Moreover, if in either case there is a strict maximum on the right hand side, then these are equalities. Now let us take η ∈ A[[t]] such that ηξ = 1. We must prove that η is actually in A[[t]]^{c}. Now, since we have that x_{0} is a unit in A, we have that ν(ξ, 0) = ν(η, 0) = 0. Choose a constant C so that ν(ξ, n) ≤ Cn and ν(η, 1) ≤ C. Then we will prove by induction that ν(η, n) ≤ Cn.

We have by our second inequality (5.5.1) that: 0 = ν(1, n) ≤ max{ν(η, l) + ν(ξ, n − l)} and that the above is an equality if there is a strict maximum. Thus we must have that:

ν(η, n) + ν(ξ, 0) =≤ max{0, ν(η, l) + ν(ξ, n − l) | 0 ≤ l ≤ n − 1}

But now, using our assumptions together with the induction hypothesis we have that:

ν(η, n) ≤ max{0, Cl + (n − l)C | 0 ≤ l ≤ n − 1}

Thus we have proved that ν(η, n) ≤ nC and we are done.

5.2 Convergent power series over Laurent series

We now specialize to the case that we need. Let K := k((x)), the field of algebraic Laurent series for some perfect field k of finite characteristic. We will study τ– modules over K[[t]]^{c} in order to attack the Tate Conjecture. We have: 5.6 Corollary. Let L be a perfect discretely valued field. Then L[[t]]^{c} is a discrete valuation ring with residue field L and uniformizer t. Moreover τ induces an automorphism of L[[t]]^{c}.

Suppose k is an algebraically closed field, and let K^{perf} be the perfection of K := k((x)). The set {x^{α} | α ∈ ℤ[1/p], 0 < α < 1} is a basis of K^{perf} as a K–vector space. We therefore have that an arbitrary element ξ ∈ K^{perf}[[t]] can be written uniquely as a convergent sum:

ξ = ∑_{0<α<1} x^{α}f_{α}

where the f_{α} are in K[[t]]. Convergence means that for any m > 0, almost all the f_{α} are in t^{m}K[[t]]. It is easy to check that if ξ ∈ K^{perf}[[t]]^{c} then all the f_{α} are in K[[t]]^{c}, and conversely, if all the f_{α} have slopes bounded below by a uniform bound for α, then ξ is convergent.

5.7 Proposition. Set K := k((x)) and L := K^{perf}. Then the natural map

L[[t]]^{c} ⊗_{K[[t]]^{c}} K[[t]] → L[[t]]

is injective.

Proof. Both L[[t]]^{c} ⊗_{K[[t]]^{c}} K[[t]] and L[[t]] inherit the t–adic topology, and the completion of the left hand side in the t–adic topology is equal to the right hand side. Therefore it suffices to show that the left hand side is separated in the t–adic topology.

Assume first that k is algebraically closed. We mentioned that, using the decomposition of ξ, there is an injection of K[[t]]^{c}–modules:

L[[t]]^{c} → ∏_{α} K[[t]]^{c}

Both source and target of the above map possess a natural t–adic topology. Thus it suffices to show that (∏_{α} K[[t]]^{c}) ⊗_{K[[t]]^{c}} K[[t]] is t–adically separated. This in turn will follow from the injectivity of the map:

(∏_{α} K^{c}[[t]]) ⊗_{K[[t]]^{c}} K[[t]] → ∏_{α} K[[t]]

There is a standard proof for this: suppose M a finitely–generated K[[t]]^{c}–submodule of K[[t]]. Consider the commutative diagram

\xymatrix{(∏_{α} K[[t]]^{c}) ⊗_{K[[t]]^{c}} M \ar[d]_{π_{1}} \ar[r]|{ι} & ∏_{α} M \ar[d]^{π_{2}} \\ (∏_{α} K[[t]]^{c}) ⊗_{K[[t]]^{c}} K[[t]] \ar[r] & ∏_{α} K[[t]]}

Since any finitely generated module can be expressed as the cokernel of a map ψ of finite free modules, and ψ commutes with ⊗, then one sees immediately that ι is an isomorphism. Clearly π_{2} is injective. Now any element of (∏_{α} K[[t]]^{c})⊗_{K[[t]]^{c}} K[[t]] is in the image of π_{1} for some finitely generated module M. This proves our assertion, provided that k is algebraically closed.

Now suppose k is not algebraically closed, and \overline{k} is the algebraic closure of k. Set F := \overline{k}((x)). We have an injection

F [[t]]^{c} ⊗_{k[[t]]^{c}} K[[t]] → F [[t]]

Indeed, choose u_{i}, i ∈ I to be a basis of \overline{k} as a k vector space. Consider a series f ∈ F [[t]]:

f = ∑_{m∈ℤ} c_{m}x^{m}, c_{m} ∈ \overline{k}[[t]]

We write c_{m} = ∑_{i∈I} a_{m,i}u_{i} as a t–adic convergent sum. Then c_{m} ∈ t^{n}\overline{k}[[t]] if and only if, for any i ∈ I, a_{m,i} ∈ t^{n}k[[t]]. We set g_{i} := ∑_{m∈ℤ} a_{m,i}x^{m}. Clearly f ∈ F [[t]]^{c} only if for any i ∈ I, g_{i} ∈ K[[t]]^{c}. Thus we obtain an injection:

F [[t]]^{c} → ∏_{i∈I} K[[t]]^{c}

This proves the result.

5.8 Proposition. Let M and N be K[[t]]^{c}–modules, so that N is torsion–free. Suppose ϕ : M → N ⊗_{K[[t]]^{c}} K[[t]] is a homomorphism of K[[t]]^{c}–modules. Let ϕ̃ := ϕ ⊗_{K[[t]]^{c}} L[[t]]^{c}, where L = K^{perf}. Note that, by Proposition 5.7, the target of this morphism injects to N ⊗_{K[[t]]^{c}} L[[t]]. The the natural injection:

(Im ϕ ∩ N) ⊗_{K[[t]]^{c}} L[[t]]^{c} → Im ϕ̃ ∩ (N ⊗_{K[[t]]^{c}} L[[t]]^{c})

is bijective.

Proof. All tensor products will be taken over K[[t]]^{c}. We may assume that ϕ : M → N ⊗ K[[t]] is injective. Then by the previous proposition we have that the map

ϕ̃: M ⊗ L[[t]]^{c} → N ⊗ K[[t]] ⊗ L[[t]]^{c} → N ⊗ L[[t]]

is also injective. It is enough to show that the obvious inclusion:

ϕ^{−1}(N) ⊗ L[[t]]^{c} ⊂ ϕ̃^{−1}(N ⊗ L[[t]]^{c})

is actually an equality.

Consider the following injection induced by the map ϕ:

M/ϕ^{−1}(N) → (N ⊗ K[[t]])/N

If we tensor this map with L[[t]]^{c}, we obtain two injections, where the second comes from the previous proposition:

M ⊗ L[[t]]^{c}/ϕ^{−1}(N) ⊗ L[[t]]^{c} → (N ⊗ K[[t]]) ⊗ L[[t]]^{c}/N ⊗ L[[t]]^{c}
(N ⊗ K[[t]]) ⊗ L[[t]]^{c}/N ⊗ L[[t]]^{c} → N ⊗ L[[t]]/N ⊗ L[[t]]^{c}

Composing these two injections gives an injection induced by ϕ̃ (since we have tensored a map induced by ϕ with L[[t]]^{c}) from M ⊗ L[[t]]^{c}/ϕ^{−1}(N) ⊗ L[[t]]^{c} to N ⊗ L[[t]]/N ⊗ L[[t]]^{c}. Thus we have that:

ϕ̃^{−1}(N ⊗ L[[t]]^{c}) ⊂ ϕ^{−1}(N) ⊗ L[[t]]^{c}

since otherwise the composed map would have a non-trivial kernel. But this is what we needed to show.

6 The slope filtration over L[[t]]^{c}

In this section, L will be a perfect field with valuation ν. Then L[[t]] and L[[t]]^{c} will be discrete valuation rings whose maximal ideals are each generated by t. We will denote the Frobenius automorphism ∑ a_{i}t^{i} → ∑ a_{i}^{q}t^{i} by σ.

Let (M, τ) be a τ–module over L[[t]]. By [12] Appendix B, Theorem 2.4.5, we have that up to isogeny, (M, τ) is a direct sum of isoclinic τ–modules. This implies in particular that M has a filtration by τ–invariant submodules:

0 = M_{0} ⊂ M_{1} ⊂ … ⊂ M_{m},

such that M_{i}/M_{i−1} is a τ–module isoclinic of slope λ_{i}. Moreover we may arrange λ_{1} > … > λ_{m}, and then the filtration is unique. We will show that such a filtration exists for every τ–module over L[[t]]^{c}. Let (M, τ) and (N, τ) be quasi–τ–modules over L[[t]]. Let λ_{1}, …, λ_{h} be the slopes of M with multiplicities, where h = rank_{L[[t]]} M, and let μ_{1}, …, μ_{l} be the slopes of N with multiplicities, where l = rank_{L[[t]]} N. One sees easily that:

• The slopes of M ⊗ N with multiplicities are

λ_{i} + μ_{j}, i = 1, …, h, j = 1, …, l;

• the slopes of Hom_{L[[t]]}(M, N) with multiplicities are

μ_{j} − λ_{i}, i = 1, …, h, j = 1, …, l;

• the slopes of ∧^{k}M with multiplicities are

λ_{i_{1}} + … + λ_{i_{k}}

where the indices run through all tuples, such that 1 ≤ i_{1} < … < i_{k} ≤ h. Before we turn to τ–modules over L[[t]]^{c} we give some slightly more general results.

6.1 Technical lemmas

We now recall some technical results from [13] that we will need later on.

6.1 Lemma. Let R → R′ be an injective ring homomorphism. Let M be a finitely generated projective R–module, and let N′ ⊂ M ⊗_{R} R′ be a direct summand of constant rank r.

Then there is a direct summand N ⊂ M such that N′ = N ⊗_{R} R′ if and only if there is a direct summand L ⊂ ∧^{r}M such that L ⊗_{R} R′ = ∧^{r}N′.

Proof. Let X be the Grassmannian of rank r submodules of M and Y be ℙ(∧^{r}M^{∨}), where M^{∨} := Hom_{R}(M, R). The operation ∧^{r} induces the Plücker morphism X → Y which is a closed immersion. Corresponding to N′ and L we obtain a commutative diagram

\xymatrix{X \ar[r] & Y \\ Spec R′ \ar[u] \ar[r] & Spec R \ar[u]}

The lemma says that there is a lift Spec R → X. As X → Y is a monomorphism, this arrow is unique if it exists. Therefore the question of existence is local, and hence we may assume that Spec R → Y factors through an affine open Spec A ⊂ Y. Let us denote by Spec B ⊂ X the preimage of Spec A. Then we obtain a commutative diagram of rings:

(6.1.1)  \xymatrix{B \ar[d]_{v} & A \ar[d]^{u} \ar[l]_{π} \\ R′ & R \ar[l]}

We want to show that there is a w : B → R making the diagram commutative. Since R → R′ is injective, we have u(Ker π) = 0. But this suffices for the existence of w since π is surjective. This concludes the proof of the lemma.

Again following [13], Corollary 18, we show:

6.2 Corollary. Let S be a local ring, and G a set of endomorphisms of S. We denote by R = S^{G} the ring of invariants. Let M be a finitely generated free R– module. Assume that we are given a direct summand N′ of the S–module M ⊗_{R} S, such that gN′ ⊂ N′ for each g ∈ G. Then there is a unique direct summand N of the R–module M, such that N′ = N ⊗_{R} S

Proof. We note that R is also a local ring. The uniqueness follows as in lemma 6.1. The lemma also shows that we may assume that N′ is of rank 1. Let e_{1}, …, e_{r} be a basis of M and let n be a generator of N′. Then we write:

(6.2.1)  n = a_{1}e_{1} + … + a_{r}e_{r}

where a_{i} ∈ S. Since S is local we may assume without loss of generality that a_{1} = 1. By assumption there is for each g ∈ G an element λ(g) ∈ S, such that

g(n) = λ(g)n

Inserting for n the right hand side of (6.2.1) and comparing coefficients gives λ(g) = 1 and g(a_{i}) = a_{i}. But this implies a_{i} ∈ R. Then Rn ⊂ M is the desired direct summand.

6.2 Existence of the slope filtration

Let (M, τ) be a quasi τ–module over L[[t]]^{c}, where we recall that L is assumed to be a perfect field.

6.3 Proposition. Let

λ_{1} > λ_{2} > … > λ_{m}

be the slopes of the quasi τ–module (M, τ); then M has a unique filtration by quasi τ–submodules:

0 = M_{0} ⊂ M_{1} ⊂ … ⊂ M_{m}

such that M_{i}/M_{i−1} is a non–zero isoclinic quasi τ–module of slope λ_{i} for i = 1, …, m.

We need first the following lemma: 6.4 Lemma. Let (M, τ) be a quasi–τ–module over L[[t]]^{c}. Let λ = r/s where r and s are integers.

1. Assume that all slopes of (M, τ) are greater or equal to λ; then there is a quasi– τ–module (N, τ) which is isogenous to (M, τ), and such that τ^{s}N ⊂ t^{r}N.

2. Assume that all slopes of (M, τ) are less than or equal to λ; then there is a quasi–τ–module (N, τ) which is isogenous to (M, τ) and such that τ^{s}N ⊃ t^{r}N.

3. Assume that (M, τ) is isoclinic of slope λ. Then there is a quasi–τ–module

(N, τ) which is isogenous to (M, τ) and such that τ^{s}N = t^{r}N. Proof. The corresponding statement is true for M′ = L[[t]] ⊗_{L[[t]]^{c}} M. In the first case we find a submodule of finite index N′ ⊂ M′ such that τ^{s}N′ ⊂ t^{r}N′. We set N = N′ ∩ M ⊗ 𝔽_{q}(t). Then we find τ^{s}N ⊂ t^{r}N′ ∩ M ⊗ 𝔽_{q}(t) = t^{r}N. The other cases are similar. 6.5 Remark. We note that τ^{s}N ⊂ t^{r}N, if and only if t^{r}N^{∨} ⊂ τ^{s}N^{∨} holds for the dual module. This is because τ^{s}N^{∨} is the dual module to τ^{s}N with respect to the pairing:

(N ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)) × (N^{∨} ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)) → L[[t]]^{c} ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)

Proof of Proposition 6.3. To prove the proposition we may change M within its isogeny class. Let λ_{1} = λ = r/s be the highest slope. Then we find M ⊂ N ⊂ M ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) such that t^{r}N ⊂ τ^{s}N. Let N′ = L[[t]] ⊗_{L[[t]]^{c}} N. Let N′_{1} ⊂ N′ be the isoclinic part of slope λ. We have to show that there is a direct summand N_{1} ⊂ N, such that N′_{1} = L[[t]] ⊗_{L[[t]]^{c}} N_{1}. Applying lemma 6.1 we may assume that N′_{1} has rank 1. We set Φ = t^{r}τ^{−s} : N → N. Let n be a generator of N′_{1}. Then we find:

Φn = un

for some unit u ∈ L[[t]]. Let us first assume that L is algebraically closed. Then we may write u = a^{1−(q)}. Then we find that Φ(an) = an. Then proposition 5.3 shows that an ∈ N. This proves the proposition in the case that L is algebraically closed. In the general case it shows that there is a direct summand \overline{N}_{1} ⊂ \overline{L}[[t]]^{c} ⊗_{L[[t]]^{c}} N, where \overline{L} is the algebraic closure of L, such that \overline{L}[[t]] ⊗_{L[[t]]} N′_{1} = \overline{L}[[t]] ⊗_{\overline{L}[[t]]^{c}} \overline{N}_{1}. Since the left hand side of this equality is invariant under the Gal(\overline{L}/L), so is the direct summand \overline{N}_{1} ⊂ \overline{L}[[t]]^{c} ⊗_{L[[t]]^{c}} N. It is here that we use the fact that L is perfect. Hence this direct summand descends to a submodule \overline{N}_{1} ⊂ N, by corollary 6.2.

6.3 Descent for convergent τ–modules

6.6 Proposition. Let M, N be τ–modules over L[[t]]^{c}. Assume that M is isoclinic of slope λ, and that all the slopes of N are less than or equal to λ. Let

α : M ⊗_{L[[t]]^{c}} L[[t]] → N ⊗_{L[[t]]^{c}} L[[t]]

be a homomorphism of τ–modules. Then we have α(M) ⊂ N. Proof. We set U = (L[[t]]^{c}, σ), the unit τ–module. Tensoring α with the dual quasi τ–module M^{∨}, we obtain a morphism U ⊗_{L[[t]]^{c}} L[[t]] → N ⊗_{L[[t]]^{c}} M^{∨} ⊗_{L[[t]]^{c}} L[[t]] of quasi τ–modules. Twisting this morphism by U(ℓ)—the ℓ-th “Tate twist”, replacing τ by t^{m}τ—we obtain a morphism of τ–modules. Hence we assume without loss of generality that M = U(ℓ). Since the slopes of N are less than or equal to ℓ, we find an isogeny N → N′ such that t^{ℓ}τ^{−1}N′ ⊂ N′. We set n = α(1) ∈ N′ ⊗_{L[[t]]^{c}} L[[t]]. Then we have t^{ℓ}τ^{−1}n = n. By proposition 5.3 we find that n ∈ N′. This shows that α(M ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)) ⊂ N ⊗_{𝔽_{q}[t]} 𝔽_{q}(t). But this suffices since:

(N ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)) ∩ (L[[t]] ⊗_{L[[t]]^{c}} N) = N

Indeed, this follows because x ∈ L[[t]] is in L[[t]]^{c} if and only if tx ∈ L[[t]]^{c}

We recall that in our notation K = k((x)), the field of Laurent series over a perfect field k. 6.7 Proposition. Let M and N be τ–modules over K[[t]]^{c}. Assume that N is isoclinic of slope λ. Suppose we are given a morphism of τ–modules

ϕ : M ⊗_{K[[t]]^{c}} K[[t]] → N ⊗_{K[[t]]^{c}} K[[t]] such that ϕ ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) surjective. We set E = ϕ^{−1}(N) ∩ M and consider the map ψ : E → N induced by ϕ. Then the map ψ ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) is surjective too. Assume moreover that the map M → N ⊗_{K[[t]]^{c}} K[[t]] induced by ϕ is injective. Then all slopes of M are less than or equal to λ.

Proof. Let ϕ_{0} : M → N ⊗_{K[[t]]^{c}} K[[t]] be the restriction of ϕ. Clearly we may assume that this map is injective. If we tensor ϕ_{0} by ⊗_{K[[t]]^{c}}L[[t]]^{c}, where L is the perfection of K, we obtain by 5.7 an injection:

ϕ_{1} : M ⊗_{K[[t]]^{c}} L[[t]]^{c} → N ⊗_{K[[t]]^{c}} L[[t]]^{c} → N ⊗_{K[[t]]^{c}} L[[t]]

Let μ be the highest slope of M ⊗_{K[[t]]^{c}} L[[t]]^{c} and consider the first step of the slope filtration M(μ) ⊂ M ⊗_{K[[t]]^{c}} L[[t]]^{c}. Then M(μ) is a τ–module which is isoclinic of slope μ. Consider the injection

(6.7.1)  M(μ) → N ⊗_{K[[t]]^{c}} L[[t]]^{c}

induced by ϕ_{1}. If μ ≠ λ, the map M(μ) ⊗_{L[[t]]^{c}} L[[t]] → N ⊗_{K[[t]]^{c}} L[[t]] would be zero. This contradicts the injectivity of (6.7.1). Hence all slopes of M are less than or equal to λ.

It follows from proposition 6.6 that ϕ_{1} maps M(λ) to N ⊗_{K[[t]]^{c}} L[[t]]^{c}. Looking for the slope decomposition over L[[t]], we find that the map M(λ) ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) → N ⊗_{K[[t]]^{c}} L[[t]]^{c} ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) is surjective. This shows that Im(M ⊗_{K[[t]]^{c}} L[[t]]^{c}) ∩ (N ⊗_{K[[t]]^{c}} L[[t]]^{c}) has the same rank as N. It follows by corollary 5.8 that Im(M)∩N has the same rank as N.

7 Extension of endomorphisms

Our theorem depends on two further results, the first of which seems to require the theory of rigid analytic functions for its proof. After having stated them, we show how they imply the theorem, and then give the proofs in the following two sections. 7.1 Theorem. Let M be a τ–module over k[[t]][[x]]. Let N^{c} ⊂ M ⊗_{k[[x,t]]} k((x))[[t]]^{c} be an τ–invariant direct summand. Then there is a unique τ–submodule N ⊂ M, such that N^{c} = N ⊗_{k[[x,t]]} k((x))[[t]]^{c}. 7.2 Corollary. Given (M_{1}, τ_{M_{1}}) and (M_{2}, τ_{M_{2}}) τ–modules over k[[x, t]] and a morphism

ϕ^{c} : (M_{1} ⊗_{k[[t,x]]} K[[t]]^{c}, τ_{M_{1}}) → (M_{2} ⊗_{k[[x,t]]} K[[t]]^{c}, τ_{M_{2}})

there is a morphism ϕ : (M_{1}, τ_{M_{1}}) → (M_{2}, τ_{M_{2}}) such that ϕ^{c} = ϕ ⊗ K[[t]]^{c}. Proof. Apply the theorem to the graph of ϕ^{c}. 7.3 Proposition. Let M → N be a homomorphism of quasi–τ–modules over k[[x, t]]. Assume M ⊗_{k[[x,t]]} K[[t]] is isoclinic of slope λ and that all slopes of N ⊗_{k[[x,t]]} K[[t]] are less than or equal to λ. If the map

M ⊗_{𝔽_{q}[t]} 𝔽_{q}(t) → N ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)

is injective, then it admits a τ–equivariant retraction. Proof of Theorem 4.1. : We note first that it is enough to prove the “rational version” of the statement, namely that ϕ(M ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)) ⊂ N ⊗_{𝔽_{q}[t]} 𝔽_{q}(t). To prove this claim we need to check that

(7.3.1)  (N ⊗_{𝔽_{q}[t]} 𝔽_{q}(t)) ∩ (N ⊗_{k[[x,t]]} k((x))[[t]]) = N

We can assume for this that N = k[[x, t]]. Since k[[x, t]] is a UFD we see that k[[x, t]]_{(t)}/k[[x, t]] has no t-torsion. Since k((x))[[t]]/k[[x, t]]_{(t)} has no t Let N̂ be the quasi τ–module dual to N. Denote by U the unit τ–module. Then, in the category of quasi τ–modules over k[[x, t]] we have that (7.3.1) holds.

Hom(M, N) ≅ Hom(M ⊗ N̂, U)

Thus we may assume that N = U(ℓ), which is a rank one τ–module of slope ℓ. Consider the map ϕ̂: M → N ⊗_{k[[x,t]]} K[[t]]. Dividing by the kernel, we may assume that the map is injective. To show this consider M′, the factor of M by this kernel. If M′ is not free we consider the module M′_{1} given by Lemma 4.6. For a suitable integer a we obtain an injective map t^{a}ϕ̂: M′_{1} → N ⊗ K[[t]].

Assume then that ϕ̂ is injective, and denote by M^{c} the module M ⊗_{k[[x,t]]} K[[t]]^{c}. The map

M^{c} → N ⊗_{k[[x,t]]} K[[t]]

is also injective. Indeed by theorem (7.1), the kernel would be defined over k[[x, t]]. Thus it is zero since ϕ̂ is injective. Applying proposition (6.7) to the above morphism, we obtain that all slopes of M^{c} are less than or equal to λ. Moreover E^{c} = M^{c} ∩ N^{c} is a τ–submodule of rank 1 and slope ℓ of M^{c}.

By theorem (7.1) there is an E ⊂ M, such that E^{c} = E ⊗_{k[[x,t]]} k((x))[[t]]^{c}.

Applying the corollary to theorem (7.1) to the map E^{c} → N^{c} we obtain that ϕ̂(E) ⊂ N. Since the slopes of M^{c} are less than or equal to ℓ, the τ–submodule E ⊂ M has by Proposition 7.3 a complement E′ up to isogeny—i.e., the map E ⊕ E′ → M has cokernel supported on the closed point. Now we conclude ϕ̂(E′) ⊂ N by induction on the rank of M.

8 Proof of Theorem 7.1

In his original paper [11] de Jong proved his theorem for F –crystals rather than for Frobenius modules as in [13]. Thus he made use of the differential operator θ, given in the data of a crystal, in the proof of this result. The author suspected, given the analogy to the Drinfeld module case, that this was not needed, but it was only in discovering the paper of Zink and Messing that it became clear how to make this result apply.

To prove this theorem we must first recall some facts about non–archimedean rigid analysis. Let k be a perfect field, and let Ω be the algebraic closure of the field k((t)). We set |a| = p^{−ord_{t}a} for a ∈ Ω.

8.1 Definition. Let I be a nonempty interval of nonnegative real numbers. We denote by ℒ(I) the set of all formal Laurent series

∑_{n∈ℤ}a_{n}x^{n}, a_{n} ∈ k((t))

such that for all ρ ∈ I \ 0

lim_{|n|→∞} |a_{n}|ρ^{n} = 0

and if 0 ∈ I then a_{n} = 0 for n < 0. An element of ℒ(I) is called a Laurent series convergent in I.

Let f ∈ ℒ(I) for arbitrary I. An I–root of f will be an element z ∈ Ω such that |z| ∈ I, and ∑_{n} a_{n}z^{n} = 0. The following (see [10]) is a consequence of the theory of Newton polygons.

8.2 Theorem. Assume that a non–zero f ∈ ℒ(I) has only finitely many I–roots. Then there is a unique monic polynomial P ∈ k((t))[x] whose roots in Ω have absolute values in I, and a unit u in the ring ℒ(I) such that fu = P. In particular f ∈ ℒ(I)^{∗} iff f has no I–roots. If the interval I is compact, then f ∈ ℒ(I) has only finitely many I–roots.

8.3 Notation. We will have need of the following rings of Laurent series:

𝒟_{η} = ℒ([η, 1)), for 0 ≤ η < 1

We set 𝒟 = ∪_{η}𝒟_{η}. We have k((x))[[t]]^{c} ⊂ 𝒟. One can show that k((x))[[t]]^{c}[t^{−1}] consists of all the Laurent series f ∈ 𝒟 such that |a_{n}| is bounded, though we do not need this here. For any power series α with coefficients in t, α = ∑ α_{i}t^{i} we denote by α^{(q^{d})} the power series∑ α_{i}^{q^{d}}t^{i} for any integer d.

8.4 Lemma. Assume that a non–zero f ∈ ℒ satisfies an equation τ^{u}f = cf for some number u ≥ 1, and for some c ∈ K((t)). Then we have f ∈ K((t)) and c ∈ K[[t]]^{∗}. Proof. The assertion follows by comparing the coefficients of the equation:

∑_{n∈ℤ}a_{n}^{(q^{u})}x^{nq^{u}} = ∑_{n∈ℤ}ca_{n}x^{n}

The following proposition is an analogue of a result discovered by Dwork. 8.5 Proposition. Let k be algebraically closed. Let (M, τ_{M}) be a τ–module over k[[x, t]]. The 𝒟_{0}–module M ⊗_{k[[x,t]]} 𝒟_{0} possesses a basis {d_{1}, …, d_{r}} which satisfies τ_{M}^{n}d_{i} = t^{a_{i}}d_{i} for suitable integers n > 0, a_{1}, …, a_{r} ≥ 0. Proof. M/xM with the operator τ is a τ–module over k. Hence by the Dieudonné theory for Drinfeld modules [12], page 35, there exist elements e_{1}, … e_{r} ∈ M/xM which form a basis of M/xM ⊗_{k[[t]]} k((t)) and such that the following equations

(8.5.1)  τ_{M}^{n}e_{i} = t^{a_{i}}e_{i}, i = 1, …, r

hold for suitable integers n > 0 and a_{1}, …, a_{r} ≥ 0. Choose arbitrary lifts a′_{i} of these elements. We write

τ_{M}^{n}e′_{i} = t^{a_{i}}e′_{i} + xz_{i}

for suitable z_{i} ∈ M. Inductively we find for each number N:

τ^{Nn}e_{i} = t^{Nn}e_{i} ∑_{j=1}^{N} t^{(N−j)a_{i}}x^{q^{n(j−1)}}τ^{n(j−1)}(z_{i})

Consider for fixed i the sequence:

f_{i,n} = t^{−Na_{i}}τ^{Nn}e_{i}

By the last equation we obtain the congruence:

t^{−Na_{i}}τ^{Nn}e_{i} = t^{−(N−1)a_{i}}τ^{(N−1)n}e_{i} mod t^{−Na_{i}}x^{q^{n(N−1)}}M

We choose an isomorphism M ≅ k[[x, t]]^{r}. Clearly the sequence f_{i,n} converges in the x–adic topology to an element ẽ_{i} in M ⊗_{k[[x,t]]} k((t))[[x]].

Both sides of the above congruence differ by a vector with components t^{−Na_{i}}x^{q^{n(N−1)}}w_{i} where w_{i} ∈ k[[x, t]]. Since |t^{−Na_{i}}x^{q^{n(N−1)}}|_{ρ} = Na_{i}ρ^{q^{n(N−1)}} converges to zero for any ρ < 1 the components of the sequence f_{i,N} converge in the Banach algebra ℒ(ρ). Since this is true for any ρ < 1 we find ẽ_{i} ∈ M ⊗_{k[[x,t]]} 𝒟_{0}. From the equation t^{−a_{i}}τ^{n}f_{i,n} = f_{i,n+1} we obtain:

τ_{M}^{n} ẽ_{i} = t^{a_{i}} ẽ_{i}

It remains to be shown that the elements ẽ_{i} for i = 1, …, r form a basis of M ⊗ 𝒟_{0}, or equivalently that ẽ_{1} ∧ … ∧ ẽ_{r} is a basis of ⋀^{r} M ⊗_{k[[x,t]]} 𝒟_{0}. By the lemma below, there exists a generator y ∈ ⋀^{r} M such that τ_{M}(y) = t^{ℓ}y for some number ℓ. We write:

ẽ_{1} ∧ … ∧ ẽ_{r} = y ⊗ f, f ∈ 𝒟_{0}

Applying τ^{u} to this equation gives the equality:

t^{i}(ẽ_{1} ∧ … ∧ ẽ_{r}) = t^{j}y ⊗ f^{(q^{u})}

for some numbers i and j. We deduce that t^{i−j}f^{(q^{u})} = f. By lemma 8.4 we find i = j and f ∈ k((t)) It remains to verify that ẽ_{1} ∧ … ∧ ẽ_{r} ≠ 0. But by our construction we have that ẽ_{i} = e_{i} mod x and hence {ẽ_{1}, …, ẽ_{r}} gives a basis for the k–vector space

M ⊗_{k[[x,t]]} 𝒟_{0}/x(M ⊗_{k[[x,t]]} 𝒟_{0})

Therefore ẽ_{1} ∧ … ∧ ẽ_{r} is not zero. 8.6 Lemma. Let k be algebraically closed. Let (M, τ) be a τ–module over k[[x, t]] of rank 1. Then there is an element m ∈ M, such that τm = t^{l}m for some number l. Proof. We fix an isomorphism M ≅ k[[x, t]]. We set τ1 = λ. By assumption, λ = t^{l}η for some unit η ∈ k[[x, t]]. By the next lemma we find a unit u ∈ k[[x, t]] such that u^{(q)}η/u = 1. This implies τu = u^{(q)}t^{l}η = t^{l}u. 8.7 Lemma. Let η ∈ k[[x, t]] be a unit. Then there exists a unit ξ ∈ k[[x, t]] which solves the equation:

ξ^{(q)}/ξ = η

Proof. We write the equation modulo t:

ξ^{q−1} − η = 0 mod t

This has a solution in the algebraically closed field k which lifts by Hensel’s lemma to a solution in k[[x]]. Hence we obtain a solution modulo t. Let us assume by induction that we have a solution ξ modulo t^{n}:

ξ^{(q)} = ξη mod t^{n} It is enough to show that ξ lifts to a solution modulo t^{n+1}. We set ξ^{(q)} − ξη = t^{n}μ for μ ∈ k[[x, t]]. We have to find ρ ∈ k[[x, t]], such that

(ξ + t^{n}ρ)^{(q)} − (ξ + t^{n}ρ)η = 0 mod t^{n+1}

This amounts to finding a solution of the following congruence:

ρ^{q} − ρη + μ = 0 mod t

This congruence is an algebraic equation over k[[x]] which can be solved by Hensel’s lemma as above.

8.8 Lemma. Let k be algebraically closed. Let (M, τ_{M}) be a τ–module over k[[x, t]]. Let N′ ⊂ M ⊗_{k[[x,t]]}𝒟 be a τ–invariant direct summand, which is free of rank 1. Then there is a free direct summand N ⊂ M ⊗_{k[[x,t]]}𝒟_{0} of rank 1, such that N′ = N ⊗_{𝒟_{0}} 𝒟.

Proof. By the above proposition there exists a basis {d_{1}, …, d_{r}} of M ⊗_{k[[x,t]]} 𝒟_{0} such that for suitable numbers u and a_{i}:

τ^{u}d_{i} = t^{a_{i}}d_{i}

Let n be a generator of N′. We can write in M ⊗_{k[[x,t]]} 𝒟 = M ⊗_{k[[x,t]]} 𝒟_{0} ⊗_{𝒟_{0}} 𝒟:

(8.8.1)  n = ∑_{i=1}^{r} d_{i} ⊗ h_{i}, h_{i} ∈ 𝒟

Since N′ is invariant under τ we have

τ^{u}n = ∑_{i=1}^{r} d_{i} ⊗ t^{a_{i}}h_{i}^{(q^{u})} = hn

for some h ∈ 𝒟. comparing coefficients we find that t^{a_{i}}h_{i}^{(q^{u})} = hh_{i}. then we see that:

t^{a_{i}}h_{i}^{(q^{u})}h_{j} = t^{a_{j}}h_{j}^{(q^{u})}h_{i}

Assume we have chosen j such that h_{j} is not identically zero. we choose η such that all Laurent series h_{i} and h_{i}^{(q^{u})} are in 𝒟_{η}. Then we choose a transcendental ρ such that η < ρ < 1. We set s = q^{u}. One sees that x → x^{(q^{u})} induces a homomorphism of fields ℒ(ρ) → ℒ(ρ^{1/s}). Therefore we obtain in ℒ(ρ^{1/s}) the equation

h_{j}^{(q^{u})}(h_{i}h_{j})^{(q^{u})} = h_{i}^{(q^{u})}

Dividing this equation in the field ℒ(ρ^{1/s}) by h_{j}^{(q^{u})}h_{j} we obtain:

t^{a_{j}}h_{i}h_{j}^{−1} = t^{a_{j}}h_{i}^{(q^{u})}h_{j}^{−(q^{u})} = t^{a_{i}}(h_{i}h_{j})^{(q^{u})}

. By lemma 8.4 we obtain h_{i}h_{j}^{−1} = λ_{i} ∈ k((t)), wher λ_{i} = 0 if a_{i} ≠ a_{j}. We write f = h_{j} and rewrite (8.8.1) as:

n = ∑_{i=1}^{r} λ_{i}d_{i} ⊗ f

In the basis d_{1}, …, d_{r} we may replace the element d_{j} by ∑_{i=1}^{r} λ_{i}d_{i}. then we can write n = d_{j} ⊗ f. Since n generates a 𝒟 submodule of ⊕_{i=1}^{r}𝒟d_{i} that is in fact a direct summand, it follows that f is a unit in 𝒟. Hence N = 𝒟_{0}d_{j} is the direct summand that we wanted.

We have a Cartesian diagram of rings

\xymatrix{k[[x, t]] \ar[d] \ar[r] & K[[t]]^{c} \ar[d] \\ 𝒟_{0} \ar[r] & 𝒟}

which, when localized at the prime (t) of k[[x, t]] gives rise to a Cartesian diagram:

\xymatrix{B \ar[d] \ar[r] & K[[t]]^{c} \ar[d] \\ ℳ \ar[r] & 𝒟}

where here B = k[[x, t]]_{(t)} and ℳ = 𝒟_{0} ⊗_{k[[x,t]]} B.

8.9 Lemma. Let M be a finitely generated free B–module. Let N^{c} ⊂ M ⊗_{B} K[[t]]^{c} be a rank 1 direct summand, and let N′ ⊂ M ⊗_{B} ℳ be a direct summand, which is free of rank 1 as an ℳ–module. We assume that N^{c} ⊗_{K[[t]]^{c}} 𝒟 = N′ ⊗_{ℳ} 𝒟. Then there exists a unique direct summand N ⊂ M which induces N^{c} and N′.

Proof. Let f ∈ ℳ be an element which becomes a unit in 𝒟. We claim that f is a unit in ℳ. Assume f ∈ 𝒟_{0}. By assumption f is a unit in ℒ([ρ, 1)) for some ρ. By theorem 8.2 we may write in ℒ([0, ρ]) :

fu = P

In ℒ(ρ) we obtain u = P/f. The right hand side of this equation converges in [ρ, 1). We conclude that u converges in [ρ, 1) too, and finally u ∈ 𝒟_{0}. Therefore we find in ℳ the equation f(u/P) = 1. Let n^{c} respectively n′ be generators of N^{c} respectively N′. Let e_{1}, …, e_{r} be a basis for the B–module M. Then we write n^{c} = e_{1} ⊗ g_{1} + … + e_{r} ⊗ g_{r} (8.9.1) N′ = e_{1} ⊗ f_{1} + … + e_{r} ⊗ f_{r} (8.9.2) where the g_{i} ∈ K[[t]]^{c}, and f_{i} ∈ ℳ. Since K[[t]]^{c} is a local ring we may assume without loss of generality that g_{1} = 1. By assumption there is a unit h ∈ 𝒟 such that n′ = hn. We conclude that f_{1} = h, and by the assertion shown above that f_{1} is a unit in ℳ, on dividing by f_{1} we may assume f_{1} = h = 1. Then we see from our Cartesian diagram that f_{i} = g_{i} ∈ B. Proof. of theorem (7.1): Suppose given (M, τ_{M}) as before, and N^{c} ⊂ M ⊗_{k[[x,t]]} k((x))[[t]]^{c} a τ_{M}–invariant direct summand. We must show that there is a unique τ–submodule N of M so that N^{c} = N ⊗_{k[[x,t]]} K[[t]]^{c}. If the N in the statement of the theorem exists, then N_{1} = N^{c} ∩M is by lemma 4.6 another τ–submodule, which fulfills the theorem and contains N. Since N → N_{1} becomes an isomorphism over K[[t]]^{c}, by proposition 4.5, it is already an isomorphism. Thus N is unique.

Let B = k[[x, t]]_{(t)} be the localization in the prime (t). We set M′ = M ⊗ B. It is equivalent to show that there is a direct summand N′ of M′ as B–module such that N′ ⊗_{B} K[[t]]^{c} = N^{c}. Then we have that N′ = M′ ∩ N^{c} , and thus N′ is τ–invariant. By lemma 6.1 we may thus assume N^{c} has rank 1. Moreover we can assume that k is algebraically closed. Indeed, let \overline{k} be the algebraic closure of k. We denote by \overline{B} etc... the objects corresponding to \overline{k}. Then the Galois group G = Gal(\overline{k}/k) acts on \overline{B} and the invariants are B. Here again we use the fact that k is assumed perfect. If \overline{N}′ ⊂ M′ ⊗_{B} \overline{B} exists, it is stable by G since it is uniquely determined. Thus we may apply corollary 6.2 . Hence assume that k is algebraically closed and that N^{c} has rank 1. By lemma (8.8) we find a direct summand N′ ⊂ M ⊗_{k[[x,t]]} 𝒟_{0} of rank 1, such that N^{c} ⊗_{K[[t]]^{c}} 𝒟 = N′ ⊗_{𝒟_{0}} 𝒟. Thus by lemma (8.9) we are done.

9 Existence of Section

We now prove proposition 7.3 in its dual version:

9.1 Proposition. Let k be algebraically closed. Let M → N be a morphism of quasi–τ–modules over k[[x]], such that N is isoclinic of slope λ, and all slopes of M are greater or equal to λ. If the map M ⊗_{k[[t]]} k((t)) → N ⊗_{k[[t]]} k((t)) is surjective, then it admits a τ–equivariant section.

Proof. We set λ = r/s where r and s are integers, s > 0. We set U = t^{−r}τ^{s}. We may assume that UM ⊂ M, UN ⊂ N and that the cokernel of the map M → N is annihilated by a power of the maximal ideal of k[[x, t]]. Indeed, we set B = k[[x, t]]_{(t)}, M′ = M ⊗ B, and N′ = N ⊗ B. Let M′_{0} ⊂ M′ ⊗ k((t)) be a finitely generated free B–submodule which is mapped surjectively to N′_{0} = N′ and such that M′_{0} ⊗ k((t)) = M′ ⊗ k((t)). we set M′_{1} = ∑_{i=1}^{∞} U^{i}M′_{0} and N′_{1} = ∑_{i=1}^{∞} U^{i}N′_{0}. Then M′_{1}, N′_{1} are both U–invariant and finitely generated over B. The map M′_{1} → N′_{1} is surjective. Finally we consider M_{1} = M′_{1}∩(M ⊗k((t))) and N_{1} = N′_{1}∩(N ⊗k((t))).

These k[[x, t]]–modules are free by lemma 4.6. Then we have UM_{1} ⊂ M_{1} and UN_{1} ⊂ N_{1} because N_{1} is isoclinic of slope λ. Finally the map M_{1} → N_{1} becomes surjective if we tensor it with B. Hence the cokernel is annihilated by a power of the maximal ideal of k[[x, t]]. We define X and Y by the exact sequence:

0 → Y → M → N → X → 0

We deduce an exact sequence:

0 → (Y/xY) ⊗ k((t)) → (M/xM) ⊗ k((t)) → (N/xN) ⊗ k((t)) → 0

Then Y/xY is a U–invariant submodule of M/xM and \overline{Y} = (Y/xY) ⊗ k((t)) ∩ (M/xM) is U–invariant too. Let us consider the cokernel \overline{N}:

0 → \overline{Y} → M/xM → \overline{N} → 0

The operator U acts on \overline{N}. On the other hand, \overline{N} is isoclinic of slope λ, since \overline{N} ⊗ k((t)) = N/xN ⊗ k((t)) is. This shows that U induces an isomorphism \overline{N}^{(q^{s})} → \overline{N}. By the Dieudonné classification ([12], Theorem 2.4.5) we have a U–equivariant section \overline{N} ⊗ k((t)) → M/xM. Its image \overline{E} lifts by the Lemma 9.2 to a U–invariant direct summand E of M. Then the U–equivariant map:

Y ⊕ E → M

is an isogeny modulo x. Therefore it is an isogeny by proposition 4.5 and the remark after it. We find that E → N is an isogeny, and hence E ⊗ k((t)) → N ⊗ k((t)) is an isomorphism. This shows that M ⊗ k((t)) → N ⊗ k((t)) has a U–equivariant section β. But then

1/s ∑_{i=0}^{s−1} τ^{−i}βτ^{i}

is the desired τ–equivariant section.

Finally we must prove that \overline{E} lifts. We fix an integer s > 0 and we consider the endomorphism z → z^{(q^{s})} of k[[x, t]] 9.2 Lemma. Let P be a projective finitely generated k[[x, t]]–module and U : P → P a (q^{s})–linear endomorphism. We set P_{0} = P/xP. Then U induces a (q^{s})–linear endomorphism of the k[[t]]–module P_{0}.

Let E_{0} be a direct summand of P_{0}, such that U induces a (q^{s})–linear isomorphism

φ_{0} : E_{0} → E_{0}

Then there exists a direct summand E ⊂ P which is uniquely determined by the following properties:

1. U(E) ⊆ E.

2. E lifts E_{0}

3. U : E → E is a (q^{s})–linear isomorphism.

We prove this in a more general situation: Let f ∈ Γ^{c}. There exist constants C_{1}, C_{2} ∈ ℝ such that ν(f, n) ≤ C_{1} + nC_{2} for all nonnegative integers n. Let A be a commutative ring and 𝔞 ⊂ A an ideal consisting of nilpotent elements. We set A_{0} = A/𝔞 and more generally we denote for an A–module M the A_{0}–module M/𝔞M by M_{0}. Let σ : M → M be a ring homomorphism, such that σ(𝔞) ⊂ 𝔞 and such that there exists a natural number r with σ^{r}(𝔞) = 0. We denote by σ_{0} : A_{0} → A_{0} the induced morphism. We will be interested in the particular case where A = k[[x, t]]/(x^{m}) for an arbitrary natural number m, the endomorphism σ induced by σ on k[[x, t]] and the ideal 𝔞 = xA. Thus it is enough to prove the following proposition 9.3 Proposition. Let P be a finitely generated projective A–module and φ : P → P a σ–linear endomorphism. Let P_{0} = P ⊗_{A} A/𝔞. Then φ induces a σ_{0}–linear endomorphism φ_{0} : P_{0} → P_{0} of the A_{0}–module P_{0}. Let E_{0} be a direct summand of P_{0}, such that φ_{0} induces a σ_{0}–linear isomorphism :

φ_{0} : E_{0} → E_{0}

Then there exists a direct summand E ⊂ P which is uniquely determined by the following properties:

• φ(E) ⊂ E.

• E lifts E_{0}

• φ : E → E is a σ–linear isomorphism.

• Let C be an A–module, which is equipped with a σ–linear isomorphism ψ : C → C. Let α : (C, ψ) → (P, φ) be an A–module homomorphism such that α ∘ ψ = φ ∘ α. Let us assume that α_{0}(C_{0}) ⊂ E_{0}. Then we have α(C) ⊂ E.

Proof. This is just [13], Proposition 36.

10 Reduction to transcendence degree 1

We have so far only proved the analogue of de Jong’s theorem for the case of K = k[[x]] where k is a perfect field. We now show that it holds more generally for any discrete valuation. The proof is somewhat interesting for the following reason. In the analogous reduction of [11], de Jong makes critical use of the differential operator associated to an F -crystal to get around the fact that in the construction R → W (R), where R is a ring of characteristic p and W (R) is the ring of Witt vectors in R, there is no canonical lift of the Frobenius of R. However the noncanonicality of the lift is exactly made up for by the differential operator. This leads one to suspect that a classical statement about F -crystals that does not explicitly refer to the differential structure, will have an analogous statement for formal τmodules. Moreover, the same proof should carry over, with the modification that any step classically that uses the differential operator should be made up for by the use of the canonical Frobenius lift in the function field case.

Now let R be a discrete equal characteristic discrete valuation ring with K its fraction field. By standard commutative algebra, we can find a discrete valuation ring R′ which is complete with algebraically closed residue field k′ and so that e(R′/R) = 1, together with a local injective homomorphism R → R′. We have an obvious map R[[t]] → R′[[t]] together with the obvious lift of the absolute q-Frobenius on R, unlike the situation in [11]. Now if M_{1}, M_{2} are formal τ–modules over R then we claim that

Hom_{R}(M_{1}, M_{2}) = Hom_{K}(M_{1} ⊗_{R} K, M_{2} ⊗_{R} K) ∩ Hom_{R′}(M_{1} ⊗_{R} R′, M_{2} ⊗_{R} R′)

But this follows from looking at the cofficients of these maps in terms of a basis for M_{1} and M_{2} over R[[t]]. Thus our proof of 4.1 for a complete discrete valuation over an algebraically closed field (actually we only used perfect) suffices to prove the result for any discrete valuation ring.

Now we can reduce the proof of the equicharacteristic Tate Conjecture for Drinfeld modules to the case of a Drinfeld module over a field of transcendence degree one over a finite field. The case of a finite field is was proved by Drinfeld (see for instance: [12], Theorem 2.5.6).

Let us therefore take a Drinfeld module ϕ over a field L of transcendence degree greater than 1 over its (finite) field of constants. We know that ϕ has a model ϕ_{U} defined over an irreducible variety U with residue field L at the generic point. We can find an irreducible divisor Z ⊂ U so that the specialization map End(ϕ_{L}) → End(ϕ_{κ(Z)}) is an isomorphism, where κ(Z) is the residue field at the generic point of Z. For instance we could use the ℓ-adic representations for some prime ℓ away from the characteristic, where the Tate Conjecture is known from [15] and then use Bertini. On the other hand our extension of 4.1 to all discrete valuation rings gives us a specialization map

End(ϕ_{L}) → End(ϕ_{κ(Z)})

Thus if we prove the theorem for κ(Z) then the theorem for L follows. By induction we can assume that the transcendence degree of L is one.

11 End of the proof

We need one last proposition before we are ready to prove the promised version of the Tate Conjecture.

Let ϕ be a Drinfeld module for a ring A over the fraction field K of a discrete valuation ring R finite residue field, and let ℘ be a prime of A. Suppose moreover that ϕ has semistable reduction over R given by ϕ = ψ/Λ where ψ is a Drinfeld module over K with good reduction over R and Λ is a lattice in ψ (see [4] for further details). Consider the ℘–divisible group ϕ_{K}[℘^{∞}] of ϕ over K. The semistable structure of ϕ induces a semistable filtration

(11.0.1)  ϕ[℘^{∞}]^{c} ⊂ ϕ[℘^{∞}]^{st} ⊂ ϕ[℘^{∞}]

where ϕ[℘^{∞}] is the sub–℘–divisible group of ϕ[℘^{∞}] induced from the ℘–divisible group of ψ. We now claim: 11.1 Proposition. Every endomorphism of ϕ_{K}[℘^{∞}] respects the filtration (11.0.1)

The fact that any endomorphism of the ℘–divisible group takes the connected component ϕ[℘^{∞}] to itself follows immediately from the fact that there are no non– trivial morphisms from a purely inseparable scheme to an étale one. Thus in order to prove the proposition we need to find a characterization over K of the second step of the filtration. Since everything in question is now étale, we may view it as a question about the Galois representation ρ of G = Gal(\overline{K}/K) into the étale ℘–torsion of ϕ.

ρ : G → End(lim_{←} ϕ[℘^{n}]^{et})

Now there is a subrepresentation of ρ corresponding to the étale ℘ torsion of ψ. We know that after possibly passing to a finite extension of the base field K, the induced quotient representation is trivial, from the Tate uniformization theory. Since ψ has a good model over R, its ℘–torsion is étale, and the action of G is induced from the action of Frobenius on the ℘–torsion of the fibre of ψ over the residue field k of R. We assumed k to be finite, and in this situation it is well known that the ℘-adic valuations of Frobenius are all equal and greater than 0 (see for instance [9], Theorem 4.12.8, page 105). Thus we have characterized the semistable filtration purely in terms of the action of Gal(\overline{K}/K), and thus it must be preserved by endomorphisms over K.

11.2 Theorem. Let L be a field of finite type over 𝔽_{q}, and let M be the τ–module over 𝔽_{q}[t] ⊗ L corresponding to a Drinfeld module ϕ over L. Then the natural map of 𝔽_{q}[[t]]–modules:

𝔽_{q}[[t]] ⊗_{𝔽_{q}[t]} End(M, τ) → End(L[[t]] ⊗_{L[t]} M, τ)

is an isomorphism.

Proof. (Equicharacteristic Tate Conjecture)

We will proceed by the methods of [15]. We will first show the rational version of 11.2; namely that the map of 𝔽_{q}((t)) vector spaces:

𝔽_{q}((t)) ⊗_{𝔽_{q}[t]} End(M, τ) → End(M ⊗_{L[t]} L((t)), τ)

is an isomorphism. The integral statement then follows easily. For convenience we will denote by V the 𝔽_{q}(t)–vector space End(M, τ) ⊗_{L[t]} L(t) and by V_{t} the 𝔽_{q}((t))– vector space End(M ⊗_{L[t]} L((t)), τ).

We know that there are only finitely many places of L where ϕ has bad reduction and that this reduction becomes good or semistable after a finite extension of L. Thus we can pass to a finite extension K of L such that ϕ has semistable reduction at a finite number of places of K, and good reduction everywhere else. Now let v be a place of K where ϕ has semistable reduction, and set M_{v} = M ⊗_{K[t]} K_{v}[t]. By the results of Gardeyn, we know that M̃_{v} := M_{v} ⊗_{K[t]} K{{t}} decomposes into an exact sequence:

(11.2.1)  0 → M̃_{v}^{triv} → M̃_{v} → M̃_{v}^{good} → 0

where the map M_{v} → M_{v}^{good} is the map induced by the exponential map from the good reduction of ψ to ϕ, where ψ is the good model of ϕ. This decomposition of analytic motives corresponds to a decomposition of the t–divisible group associated to ϕ. Moreover by Proposition 11.1 any endomorphism of the t–divisible group respects this decomposition. By the equivalence of Corollary 4.4 this means that any endomorphism of the analytic motive respects the decomposition. Thus if we have that τ_{M} is given by

τ_{M} = ( \begin{matrix}τ^{triv} & τ^{ext} \\ 0 & τ^{good} \end{matrix} )

Then any endomorphism of the t–divisible group corresponds to a matrix A with coefficients in K[[t]] such that

(11.2.2) ( \begin{matrix}τ^{triv} & τ^{ext} \\ 0 & τ^{good} \end{matrix} ) ( \begin{matrix}A_{11}^{(q)} & A_{21}^{(q)} \\ A_{12}^{(q)} & A_{22}^{(q)} \end{matrix} ) = ( \begin{matrix}A_{11} & A_{21} \\ A_{12} & A_{22} \end{matrix} ) ( \begin{matrix}τ^{triv} & τ^{ext} \\ 0 & τ^{good} \end{matrix} )

The fact that a morphism of the t–divisible groups respects the Tate data, gives us that A_{12} = 0. We now conclude the following:

1. A_{11} has bounded coefficients. That is A_{11} = (a_{ij}) where the a_{ij} ∈ K[[t]], a_{ij} = ∑ b_{ijk}t^{k} and v(b_{ijk}) ≥ c for some fixed c.

2. A_{22} has integral coefficients.

3. A_{21} has bounded coefficients.

Proof. 1. We will introduce the following notation for convenience. Given a discretely valued field K with a valuation ν, we will say that an element α = ∑ α_{i}t^{i} ∈ K[[t]] lies in K[[t]]^{bdd} if α has bounded coefficients. Now by a result of Gardeyn we know that M_{n}^{triv} becomes trivial after base change to a finite extension of K, and a change of basis of bounded valuation. That is, there exists a matrix L = (l_{ij}) with coefficients in K′[[t]]^{bdd}, such that

Lτ^{triv}L^{−(q)} = Id

In fact we can choose L to be a matrix of constant functions, though we do not need this fact. Now let N be a square matrix with coefficients in 𝔽_{q}[[t]] corresponding to an endomorphism of M_{v}^{triv}, written in this new basis. Then in our old basis, this morphism is given by: L^{−1}NL = A_{11}. The fact that the coefficients have bounded valuation follows immediately from the corresponding fact for L.

2. This follows immediately from the analogue of de Jong’s theorem, since the determinant of the τ–module corresponding to the good reduction of ϕ is equal to tu where u is a unit in K.

3. Let L be the change of basis matrix for M_{V}^{triv} such that Lτ^{triv}L^{−(q)} = Id. We change basis for M_{v} by the matrix:

( \begin{matrix}L & 0 \\ 0 & 1 \end{matrix} )

Then by (11.2.2), we have the following identity:

A_{21}^{(q)} + Lτ^{ext}A_{22}^{(q)} = A_{11}Lτ^{ext} + AA_{21}τ^{good}

The result will follow from the following more general lemma.

11.3 Lemma. Let A_{i}, B_{i}, X be matrices with coefficients in K[[t]] for K, ρ a valued field and suppose that the coefficients of A_{i} and B_{i} actually lie in M_{n×n}(K[[t]]^{bdd}).

Suppose that X satisfies a “τ–equation” of the form:

X^{(q^{n})} + A_{n}X^{(q^{n−1})}B_{n} + … + A_{1}XB_{1} + A_{0}

Then X ∈ M_{n×n}(K[[t]]^{bdd})

Proof. We proceed by induction on k, where we assume that the hypothesis is true for the coefficients of t^{k} in X. More precisely, let N be min{ρ(coefficients of A_{i}, B_{i})}. We claim that the valuation all coefficients of X are bounded below by 2N. Suppose this is false for the constant coefficients in X. Denote by x_{ij}^{0} the constant coefficient of X having minimum valuation, and let v = ρ(x_{ij}^{0}) be this valuation. We have that

(11.3.1)  q^{n}v = min{α_{n}, …, α_{0}}

Where here α_{i} is the valuation of the constant coefficient of the ijth entry of the matrix A_{i}X^{(q^{i−1})}B_{i}. Clearly α_{i} ≥ q^{i−1}v + 2N. Thus we have that:

(11.3.2)  q^{n}v ≥ 2N + q^{n−1}v

We conclude that v is bounded below by 2N/q^{n−1}(q − 1) and a fortiori by 2N.

Now the general case follows easily. For the induction step we suppose that ρ(x_{ij}^{l}) ≥ 2N ∀l < k where here x_{ij}^{l} is the coefficient of t^{l} in the ij position of the matrix X. Then if now v is taken to be the minimal valuation of the coefficient of t^{k} in X, we have the same equation as (11.3.1) for the new v. The result follows immediately.

Now for the next step. The decomposition of (11.2.1) takes place over K{{t}}, the ring of analytic functions with coefficients in K. These are power series in K[[t]] that converge at any point of K. In particular they lie K[[t]]^{bdd}. Thus, Gardeyn shows that if (M, τ_{M}) is the t–motive associated to ϕ, then there exists a matrix P ∈ GL(K{{t}}), such that:

P τ_{M}P^{−(q)} = ( \begin{matrix}τ^{triv} & τ^{ext} \\ 0 & τ^{good} \end{matrix} )

Thus if A is the matrix in (11.2.2), then P^{−1}AP is the matrix representing an endomorphism of (M, τ_{M}). Since we have proved that A and P have coefficients in K[[t]]^{bdd}, so does B := P^{−1}AP. Thus we have that B satisfies

(11.3.3)  τ_{M}B^{(q)} = Bτ_{M}

. Writing τ_{M} as ∑_{i=0}^{N} C_{i}t^{i} and B = ∑_{i=0}^{∞} where C_{i}, B_{i} ∈ M_{n×n} and setting equal the coefficients of t^{k} we get

(11.3.4)  ∑_{i=0}^{N} C_{i}B_{k−i}^{q} = ∑_{i=0}^{N} B_{k−i}C_{i}

Now suppose that this equation has a solution \overline{B} in M_{n×n}(L((t))). We will show that it has a solution B in L(t) sufficiently close in the t–adic topology to \overline{B}, so that, if (\overline{B}_{i}) is a basis for V_{t}, then B_{i} is a basis for V.

For every valuation ν on L, let us denote by ν(B) the minimum of the valuations of the entries of B_{i}, ∀i ≥ 0. We have shown that ν(B) is bounded below for all valuations ν, and bounded by 0 for all but finitely many places ν. Thus by the bounded height theorem, there appear only finitely many matrices in the sequence B_{0}, B_{1}, … We call a solution B = ∑ B_{i}t^{i} to (11.3.4) periodic if, apart from a finite number of initial terms we have for some integer I the relation B_{i} = B_{i+I}. However, since there appear only finitely many equations in (11.3.4), then one can choose a periodic solution B = ∑ B_{i}t^{i} in M_{n×n}(L((t))) close enough to the original \overline{B}. Periodicity implies that B is rational with denominator in 𝔽_{q}[t], and hence B ∈ M_{n×n}(L(t)).

11.4 Remark. The above methods should extend without too much difficulty to prove the Tate conjecture for τ-modules of equal characteristic which are analytically semistable. These will include for instance t-motives formed out of natural operations such as tensor and direct sum from the t-motives of Drinfeld modules. Unfortunately very little is known about the analytic structure of general τ-modules at the characteristic. In particular, the methods of Gardeyn do not apply there.

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