The <span class="mathjax-formula">$p$</span>-adic local monodromy theorem for fake annuli

The p-adic Local Monodromy Theorem for Fake Annuli.

KIRANS. KEDLAYA(*)

ABSTRACT- We establish a generalization of thep-adic local monodromy theorem (of AndreÂ, Mebkhout, and the author) in which differential equations on rigid analytic annuli are replaced by differential equations on so-called ``fake annuli''.

The latter correspond loosely to completions of a Laurent polynomial ring with respect to a monomial valuation. The result represents a step towards a higher- dimensional version of thep-adic local monodromy theorem (the ``problem of semistable reduction''); it can also be viewed as a novel presentation of the ori- ginalp-adic local monodromy theorem.

1. Introduction.

This paper proves a generalization of the p-adic local monodromy theorem of Andre [1], Mebkhout [21], and the present author [11]. That theorem, originally conjectured by Crew [4] as an analogue in rigid (p-adic) cohomology of Grothendieck's local monodromy theorem in eÂtale (`-adic) cohomology, asserts the quasi-unipotence of differential modules with Frobenius structure on certain one-dimensional rigid analytic annuli.

Thep-adic local monodromy theorem has far-reaching consequences in the theory of rigid cohomology, particularly for curves [4]. However, al- though one can extend it to a relative form [15, Theorem 5.1.3] to obtain some higher-dimensional results, for some applications one needs a version of the monodromy theorem which is truly higher-dimensional.

This theorem takes an initial step towards producing such a higher- dimensional monodromy theorem, by proving a generalization in which the role of the annulus is replaced by a somewhat mysterious space, called a fake annulus, described by certain rings of multivariate power series.

When there is only one variable, the space is a true annulus, so the result

(*) Indirizzo dell'A.: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 USA.

E-mail: kedlaya@math.mit.edu

truly generalizes the original monodromy theorem. Indeed, this paper has the side effect of giving an exposition of the original theorem, albeit one somewhat encumbered by extra notation needed for the fake case.

In the remainder of this introduction, we explain further the context in which the p-adic local monodromy theorem arises, introduce and justify the fake analogue, and outline the structure of the paper.

1.1 ±Monodromy of p-adic differential equations.

LetK be a field of characteristic zero complete with respect to a non- archimedean absolute value, whose residue fieldkhas characteristicp>0.

Suppose we are given a rigid analytic annulus overK and a differential equation on the annulus, i.e., a module equipped with an integrable con- nection. We now wish to define the ``monodromy around the puncture'' of this connection, despite not having recourse to the analytic continuation we would use in the analogous classical setting. In particular, we would like to construct a representation of an appropriate eÂtale fundamental group, whose triviality or unipotence amounts to the existence of a full set of horizontal sections or log-sections; the latter relates closely to the ex- istence of an extension or logarithmic extension of the connection across the puncture (e.g., [16, Theorem 6.4.5]).

We can define a monodromy representation associated to a connection if we can find enough horizontal sections on some suitable covering space.

In particular, we are mainly interested in connections which become uni- potent, i.e., can be filtered by submodules whose successive quotients are trivial for the connection, on some cover of the annulus which is ``finite eÂtale near the boundary'' (in a sense that can be made precise). One can then construct a monodromy representation, using the Galois action on hor- izontal sections, giving an equivalence of categories between such quasi- unipotentmodules with connection and a certain representation category.

(Beware that if the fieldkis not algebraically closed, these representations are only semilinear over the relevant field, namely the maximal unramified extension ofK. See [13, Theorem 4.45] for a precise statement.)

In order for such an equivalence to be useful, we need to be able to establish conditions under which a module with connection is forced to be quasi-unipotent. As suggested in the introduction to [16], a natural, geo- metrically meaningful candidate restriction (analogous to the existence of a variation of Hodge structure for a complex analytic connection) is the existence of a Frobenius structure on the connection. For K discretely

valued, the fact that connections with a Frobenius structure are quasi- unipotent is the content of thep-adic local monodromy theorem (pLMT) of Andre [1], Mebkhout [21], and this author [11].

Note that in this paper, we will not go all the way to the construction of monodromy representations. These appear directly in AndreÂ's proof of the pLMT (at least for k algebraically closed); for a direct construction (of Fontaine type) assuming thepLMT, see [13]. See Remark 6.2.7 for more details.

1.2 ±Fake annuli.

The semistable reduction problem (or global quasi-unipotence pro- blem) for overconvergentF-isocrystals, as formulated by Shiho [23, Con- jecture 3.1.8] and reformulated in [16, Conjecture 7.1.2], is essentially to give a higher dimensional version of thepLMT. From the point of view of [16], this can be interpreted as proving a uniform version of the pLMT across all divisorial valuations on the function field of the original variety.

This interpretation immediately suggests that one needs to exploit the quasi-compactness of the Riemann-Zariski space associated to the function field of an irreducible variety; this observation is developed in more detail in [17].

The upshot is that one must prove the pLMT uniformly for the divi- sorial valuations in a neighborhood (in Riemann-Zariski space) of an ar- bitrary valuation, not just a divisorial one. Ideally, one could proceed by first verifying whether thepLMT itself makes sense and continues to hold true when one passes from a divisorial valuation to a more general one.

This entails replacing the annuli in the pLMT with some sort of ``fake annuli'' which cannot be described as rigid analytic spaces in the usual sense. Nonetheless, one can still sensibly define rings of analytic functions in a neighborhood of an irrational point, and thus set up a ring-theoretic framework in which an analogue of thep-adic local monodromy theorem can be formulated. (This allows us to get away with the linguistic swindle of speaking meaningfully about ``p-adic differential equations on fake annuli'' without giving the noun phrase ``fake annulus'' an independent meaning!) Indeed, this framework fits naturally into the context of the slope filtration theorem of [11]. That theorem, which gives a structural de- composition of a semilinear endomorphism on a finite free module over the Robba ring (of germs of rigid analytic functions on an open annulus with outer radius 1), does not make any essential use of the fact that the

Robba ring is described in terms of power series. Indeed, as presented in [14], the theorem applies directly to our fake annuli; thus to prove the analogue of the p-adic monodromy theorem, one need only analogize Tsuzuki's unit-root monodromy theorem from [24]. With a bit of effort, this can indeed be done, thus illustrating some of the power of the Frobenius-based approach to the monodromy theorem. Note that our definition of fake annuli will actually include true rigid analytic annuli, so the monodromy theorem given here will strictly generalize the p-adic local monodromy theorem.

Unfortunately, it is not so clear how to prove a form of thepLMT for arbitrary valuations; in this paper, we restrict to a somewhat simpler class.

These are the monomial valuations, which correspond to monomial or- derings in a polynomial ring. For instance, these include valuations on k(x;y) in which the valuations ofxandyare linearly independent over the rational numbers. There are many valuations that do not take this form, namely theinfinitely singular valuations; the semistable reduction pro- blem for these must be treated in a more roundabout fashion, which we will not discuss further here.

1.3 ±Structure of the paper.

We conclude this introduction with a summary of the various sections of the paper.

In Section 2, we define the rings corresponding to fake annuli, and verify that they fit into the formalism within which slope filtrations are constructed in [14].

In Section 3, we defineF-modules,r-modules, and (F;r)-modules on fake annuli, and verify that the category of (F;r)-modules is invariant of the choice of a Frobenius lift (Proposition 3.4.7).

In Section 4, we give a fake annulus generalization of Tsuzuki's theo- rem on unit-root (F;r)-modules (Theorem 4.5.2).

In Section 5, we invoke the technology of slope filtrations from [11] (via [14]), and apply it to deduce from Theorem 4.5.2 a form of thep-adic local monodromy theorem for (F;r)-modules on fake annuli (Theorem 5.2.4).

In Section 6, we deduce some consequences of the p-adic local mono- dromy theorem. Namely, we calculate some extension groups in the ca- tegory of (F;r)-modules, establish a local duality theorem, and generalize some results from [5] and [12] on the full faithfulness of overconvergent-to- convergent restriction.

Acknowledgments. Thanks to Nobuo Tsuzuki for some helpful re- marks on the unit-root local monodromy theorem. Thanks also to Francesco Baldassarri and Pierre Berthelot for organizing useful workshops on F-isocrystals and rigid cohomology in December 2004 and June 2005. The author was supported by NSF grant number DMS-0400727.

2. Fake annuli.

In this section, we describe ring-theoretically the fake annuli to which we will be generalizing thep-adic local monodromy theorem, deferring to [14] for most of the heavy lifting.

First, we put in some notational conventions that will hold in force throughout the paper; for the most part, these hew to the notational reÂ- gime of [14] (which in turn mostly follows [11]), with a few modifications made for greater consistency with [16].

CONVENTION 2.0.1. Throughout this paper, letK be a complete dis- cretely valued field of characteristic 0, whose residue field k has char- acteristicp>0. Letoˆo_Kbe the ring of integers ofK, and letpdenote a uniformizer ofK. Letwbe the valuation ononormalized so thatw(p)ˆ1.

Letqbe a power ofp, and assume extant and fixed a ring endomorphism s_K:K!K, continuous with respect to thep-adic valuation, and lifting the q-power endomorphism onk. LetK_qbe the fixed field ofKunders_K, and let o_q be the fixed ring of o under s_K. Finally, let I_n denote the nn identity matrix over any ring.

REMARK2.0.2. As noted in the introduction, the restriction toKdis- cretely valued is endemic to the methods of this paper; see Remark 2.4.4 for further discussion.

2.1 ±Monomial fields

DEFINITION2.1.1. Letkbe a field. Anearly monomial field (of height1) over k is a field E equipped with a valuation v:E !R (also written v:E!R[ f‡1g) satisfying the following restrictions.

(a) The field E is a separable extension of k, i.e., kE and k\E^pˆk^p.

(b) The imagev(E) ofvis a finitely generatedZ-submodule ofR, andv(k)ˆ f0g.

(c) The fieldEis complete with respect tov.

(d) With the notations

oEˆ fx2E:v(x)0g m_Eˆ fx2E:v(x)>0g

k_Eˆo_E=m_E; the natural mapk!k_Eis finite.

Ifk_Eˆk, we sayEis amonomial field(orfake power series field) over k; in that case,kis integrally closed inE. We define therational rankofE to be the rank ofv(E) as aZ-module.

REMARK 2.1.2. One can also speak of monomial fields of height greater than 1, by allowing the valuation v to take values in a more general totally ordered abelian group. The techniques used in this paper do not apply to that case, so we will ignore it; in the semistable reduction context, one can eliminate the case of height greater than 1 by an in- ductive argument [17, Proposition 4.2.4].

EXAMPLE2.1.3. A monomial field of rational rank 1 is just a power series field, by the Cohen structure theorem. This characterization gen- eralizes to arbitrary monomial fields; see Definition 2.1.9. Note also that nearly monomial fields are examples ofAbhyankar valuations, i.e., valua- tions in which equality holds in Abhyankar's inequality [26, TheÂoreÁme 9.2].

REMARK2.1.4. If E is a nearly monomial field andE⁰=E is a finite separable extension, then E⁰ is also nearly monomial: E⁰ is separable over k, the valuation v extends uniquely to a valuation v⁰ on E⁰, E⁰ is complete with respect to v⁰, and the index [v⁰((E⁰)):v(E)] and degree [k_E⁰ :k_E] are both finite since their product is at most deg (E⁰=E) [26, Proposition 5.1]. Conversely, every nearly monomial field can be writ- ten as a finite separable extension of a monomial field; see Defini- tion 2.1.9.

REMARK 2.1.5. If E is a nearly monomial field over k and k_E=k is separable, then by Hensel's lemma, the integral closure k⁰ of k in E is isomorphic tokE; in other words, Eis a monomial field over k⁰. In parti-

cular, ifkis perfect, then any finite extension of a monomial field overkis a monomial field over some finite separable extension ofk. This fails ifkis not perfect, even for finite separable extensions of the monomial field: the fieldkis integrally closed in

k((t))[z]=(z^p z ct ^p) (c2knk^p);

but the latter has residue fieldk(c^1=p)6ˆk.

It will frequently be convenient to work with monomial fields in terms of coordinate systems.

DEFINITION2.1.6. Letmbe a nonnegative integer. LetLbe a lattice in R^m, i.e., aZ-submodule ofR^mwhich is free of rankm, and which spansR^m overR. LetL^_(R^m)^_denote the lattice dual toL:

L^_ˆ fm2(R^m)^_:m(z)2Z 8z2Lg:

Given a formal sumP

z2Lczfzg, with theczin some ring, define thesupportof the sum to be the set ofz2Lsuch thatcz6ˆ0; define the support of a matrix of formal sums to be the union of the supports of the entries. IfSLand a formal sum or matrix has support contained inS, we also say that the ele- ment or matrix is ``supported onS''. ForRa ring, letR[L] denote the group algebra ofLoverR, i.e., the set of formal sumsP

z2Lczfzgwith coefficients in Rand finite support.

REMARK2.1.7. It is more typical to denote the class inR[L] of a lattice elementz2Lby [z], rather thanfzg. However, we need to use brackets to denote TeichmuÈller lifts, so we will stick to braces for internal consistency.

DEFINITION2.1.8. For any ringRand anyl2(Rⁿ)^_, letv_ldenote the valuation onR[L] given by

v_l X

z2L

c_zfzg

!

ˆminfl(z):z2L;c_z6ˆ0g:

LetPlLdenote the submonoid ofz2Lfor whichl(z)0, and letR[L]l

denote the monoid algebraR[Pl]. LetR[[L]]landR((L))_ldenote thevl-adic completions ofR[L]_landR[L], respectively.

DEFINITION 2.1.9. Given a lattice Land some l2(R^m)^_, we sayl is irrationalifL\ker (l)ˆ f0g. In this case,k((L))_lis a monomial field over

k. Conversely, given a nearly monomial fieldEover a fieldk, with valuation v, acoordinate systemforEis a sequencex1;. . .;xmof elements ofEsuch

that v(x₁);. . .;v(xm) freely generate v(E) as a Z-module. Given a co-

ordinate system, put LˆZ^m with generators z₁;. . .;z_m, and define l2(R^m)^_ by l(z_i)ˆv(x_i); then l is irrational, and the continuous map k((L))_l!Egiven by fzig 7!xiis injective. If we identifyk((L))_l with its image inE, thenEis finite separable overk((L))_l; ifEis monomial overk, then in fact Eˆk((L))_l by Proposition 2.1.10 below. This fact may be viewed as a monomial version of the Cohen structure theorem in equal characteristics.

PROPOSITION 2.1.10. Let L be a lattice in R^m, choose l2(R^m)^_ irrational, and letEbe a finite separable extension ofk((L))_l with value groupl(L)and residue fieldk. ThenEˆk((L))_l.

PROOF. The claim is equivalent to showing that k((L))_l is separably defectless in the sense of Ostrowski's lemma [22, TheÂoreÁme 2, p. 236]. In particular, since a tamely ramified extension ofEis necessarily without defect, it suffices to check that there is no Artin-Schreier defect extension.

IfEˆk((L))_l[z]=(z^p z x) were such an extension, we could rewrite it as k((L))_l[z]=(z^p z y) with the leading term ofybeing either an element of pLtimes a non-p-th powercink, or an element ofLnpLtimes a nonzero element ofk. But in the first case the residue field ofEwould bek(c^1=p), and in the second case we would have [v(E):l(L)]ˆp; in either case, we would contradict the assumption thatEis a defect extension. This contradiction

yields the claim. p

REMARK2.1.11. The term ``monomial field'' is modeled on the use of the term ``monomial valuation'', e.g., in [6], to refer to a valuation v of the sort considered in Definition 2.1.1. (Such valuations, each of which endows the lattice L with a total ordering, are more common in mathematics than one might initially realize: for example, they are used to define highest weights in the theory of Lie algebras, and they are sometimes used to construct term orders in the theory of GroÈbner bases.) In a previous version of this paper, the term ``fake power series field'' was used instead; we have decided that it would be better to save this term for describing the completion of a finitely generated field extension of k with respect to any valuation of height 1. (See Re- mark 2.3.7 for some reasons why we are not considering such valua- tions here.)

2.2 ±Witt rings and Cohen rings

We now enter the formalism of [14, § 2].

DEFINITION 2.2.1. Let K^perf be the completion of the direct limit K!^sK K!^sK for thep-adic topology; this is a complete discretely valued field of characteristic 0 with residue fieldk^perf, so it containsW(k^perf) by Witt vector functoriality. ForEa perfect field of characteristicpcontaining k, putG^EˆW(E)_W(k^perf₎O_K^perf; note that the valuationwextends natu- rally toG^E.

DEFINITION2.2.2. ForEa perfect field of characteristicpcontainingk, complete for a valuationvtrivial onk, define thepartial valuations v_non G^E[p ¹] as follows. Given x2G^E[p ¹], write xˆP

i [xi]pⁱ, where each xi2Eand the brackets denote TeichmuÈller lifts. Set

vn(x)ˆmin

infv(x_i)g:

As in [14, Definition 2.1.5], the partial valuations satisfy some useful iden- tities (here usingsto denote theq-power Frobenius):

v_n(x‡y)minfv_n(x);v_n(y)g (x;y2G^E[p ¹];n2Z) v_n(xy)min

m2Zfv_m(x)‡v_{n m}(y)g (x;y2G^E[p ¹];n2Z) vn(x^s)ˆqvn(x) (x2G^E[p ¹]; n2Z)

vn([x])ˆv(x) (x2E;n0):

In the first two cases, equality holds whenever the minimum is achieved exactly once. Define the levelwise topology(orweak topology) onG^E by declaring that a sequencefx_igconverges to zero if and only if for eachn, v_n(x_i)! 1asi! 1.

DEFINITION2.2.3. Forr>0, writevn;r(x)ˆrvn(x)‡n; forrˆ0, write conventionally

vn;0(x)ˆ n vn(x)<1 1 v_n(x)ˆ 1:

(

LetG^E_r be the subring ofG^Efor whichv_n;r(x)! 1asn! 1; thenssends G^E_r toG^E_r=q. Define the mapw_ronG^E_r by

wr(x)ˆmin

n fvn;r(x)g;

thenwris a valuation onGrby [14, Lemma 2.1.7], andwr(x)ˆw_r=q(x^s). Put G^E_con ˆ [_r>0G^E_r;

this is a henselian discrete valuation ring with maximal ideal pG^E_con and residue fieldE(see discussion in [14, Definition 2.2.13]).

CONVENTION2.2.4. ForEa not necessarily perfect field complete for a valuation v trivial on k, we write E^perf and E^alg for the completed (with respect tov) perfect and algebraic closures of E. WhenEis to be under- stood, we abbreviateG^Eperf andG^Ealg toG^perf andG^alg, respectively. Note that this is consistent with the conventions of [14] butnotwith those of [11], where the use of these superscripts is taken not to imply completion.

Since we are interested in constructing G^E for E a monomial field, which is not perfect, we must do a bit more work, as in [14, § 2.3].

DEFINITION2.2.5. LetE be a nearly monomial field over kwith va- luationv. LetG^Ebe a complete discrete valuation ring of characteristic 0 containingoand having residue fieldE, such thatpgenerates the maximal ideal ofG^E. Suppose thatG^Eis equipped with aFrobenius lift, i.e., a ring endomorphismsextendings_Kono_Kand lifting theq-power Frobenius map onE. We may then embedG^EintoG^perfby mappingG^Einto the first term of the direct system G^E!^s G^E!^s , completing the direct system, and mapping the result intoG^perfvia Witt vector functoriality. In particular, we may use this embedding to induce partial valuations and a levelwise to- pology onG^E, taking care to remember that these depend on the choice ofs.

IfE⁰ is a finite separable extension ofE, and we start with a suitableG^E equipped with a Frobenius lifts, we may form the unramified extension of G^E with residue fieldE⁰; this will be a suitableG^E0, and carries a unique Frobenius lift extendings.

DEFINITION2.2.6. LetEbe a nearly monomial field overk, and fix a pair (G^E;s) as in Definition 2.2.5. Write

G^E_conˆG^E\G^perf_con;

with the intersection taking place withinG^perf. Forr>0, we say thatG^E has enough r-unitsifG^E\G^perf_r contains units lifting all nonzero elements ofE. We say thatG^Ehas enough units(or more properly, the pair (G^E;s) has enough units) ifG^Ehas enoughr-units for somer>0; this implies that G^E_con is a henselian discrete valuation ring with maximal idealpG^E_con and

residue fieldE. IfG^Ehas enough units, then so doesG^E0 for any finite se- parable extensionE⁰ofE[14, Lemma 2.2.12].

2.3 ±Toroidal interpretation.

The condition of having enough units is useful in the theory of slope filtrations, but is not convenient to check in practice. Fortunately, it has a more explicit interpretation in terms of certain ``naõÈve'' analogues of the functionsvnandwr, as in [11, § 2] or [14, § 2.3].

DEFINITION2.3.1. LetLbe a lattice inR^m and letl2(R^m)^_be an ir- rational linear functional. LetG^ldenote thep-adic completion ofo((L))_l; its elements may be viewed as formal sums P

z2Lczfzg with w(cz)! 1 as l(z)! 1. Define thenaõÈve partial valuationsonG^l[p ¹] by the formula

v^naive_n X c_zfzg

ˆminfl(z):z2L;w(c_z)ng;

where the minimum is infinite if the set of candidatez's is empty. These functions satisfy the identities

v^naive_n (x‡y)minfv^naive_n (x);v^naive_n (y)g (x;y2G^l[p ¹]) v^naive_n (xy)min

m2Zfv^naive_m (x)‡v^naive_{m n}(y)g (x;y2G^l[p ¹]);

with equality in each case if the minimum is achieved only once. Define the naõÈve levelwise topology(ornaõÈve weak topology) onG^lby declaring that a sequencefx_igconverges to zero if and only if for eachn,v^naive_n (x_i)! 1as i! 1.

DEFINITION2.3.2. Forr>0 andn2Z, write v^naive_n;r (x)ˆrv^naive_n (x)‡n;

extend the definition torˆ0 by setting

v^naive_n;0 (x)ˆ n v^naive_n (x)<1 1 v^naive_n (x)ˆ 1:

(

LetG^naive_r be the set ofx2G^lsuch thatv^naive_n;r (x)! 1asn! 1. Define the mapw^naive_r onG^naive_r by

w^naive_r (x)ˆmin

n2Zfv^naive_n;r (x)g;

thenw^naive_r is a valuation onG^naive_r [p ¹], as in [14, Lemma 2.1.7]. Put G^naive_con ˆ [_r>0G^naive_r :

REMARK 2.3.3. The ring G^naive_r is a principal ideal domain; this will follow from [14, Proposition 2.6.5] in conjunction with Definition 2.3.6 below.

We may view G^l as an instance of the definition of G^E in the case Eˆk((L))_l; this gives sense to the following result.

PROPOSITION 2.3.4. Let s be a Frobenius lift on G^EˆG^l for Eˆ

ˆk((L))_l. Then forr >0, the following are equivalent.

(a) s is continuous for the naõÈve levelwise topology (i.e., that to- pology coincides with the levelwise topology induced bys), and for each z2L nonzero,fzg^s=fzg^qis a unit in G^naive_r .

(b) For s2(0;qr], n0, and x2G^E, minjnfv_j;s(x)g ˆmin

jnfv^naive_j;s (x)g:

(2:3:4:1)

(c) G^E has enough qr-units, and for each z2L nonzero, fzg is a unit inG^E_qr.

In particular, in each of these cases, for s2(0;qr], G^naive_s ˆG^E_s and w_s(x)ˆw^naive_s (x)for all x2G_s.

PROOF. Given (a), fors2(0;qr], we have minjnfv^naive_j;s (x)g ˆmin

jnfv^naive_j;s=q(x^s)g (2:3:4:2)

for eachn0 and eachx2G, as in the proof of [14, Lemma 2.3.3]. We then obtain (b) as in the proof of [14, Lemma 2.3.5], from which (c) follows im- mediately.

Given (c), the equation (2.3.4.1) holds forxˆ fzgfor anyz2L, since the minima both occur forjˆ0. ForxˆP

czfzga finite sum, we have minjnfv^naive_j;s (x)g ˆmin

jnfmin

z2Lfv^naive_j;s (c_zfzg)gg (2:3:4:3)

and so the left side of (2.3.4.1) is greater than or equal to the right side. On the other hand, if jis taken to be the smallest value for which the outer minimum is achieved on the right side of (2.3.4.3), then the inner minimum is achieved by a unique value ofz. Thus we actually may deduce equality in

(2.3.4.1), again forxˆP

czfzga finite sum. For generalx, we may obtain the desired equality by replacing x by a finite sum x⁰ such that x x⁰ˆy‡zfor somey2G^Ewithw(y) greater thann, and somez2G^E_qr withw_qr(z) greater than either side of (2.3.4.1). Hence (c) implies (b).

Finally, note that (b) implies (a) straightforwardly. p REMARK 2.3.5. Note that in Proposition 2.3.4, conditions (a) and (c) may be checked for zrunning over a basis ofL. Note also that Proposi- tion 2.3.4 implies that forEˆk((L))_l, ifG^Ehas enough units, thenG^Eis isomorphic toG^l.

DEFINITION2.3.6. By thestandard extensionofsKtoG^l, we will mean the Frobenius liftsdefined by

X

z2L

c_zfzg 7!X

z2L

c^s_z^Kfzg^q:

(We will also refer to such asas astandard Frobenius lift.) When equipped with a standard Frobenius lift,G^lhas enoughr-units for everyr>0; by Proposition 2.3.4, it follows that vn(x)ˆv^naive_n (x) for all n2Z and all x2G^l[p ¹]. Thus many of the results of [14, § 2], proved in terms of the Frobenius-based valuations, also apply verbatim to the naõÈve valuations.

REMARK 2.3.7. For applications to semistable reduction, one would also like to consider a similar situation in which the residue fieldk((L))_lis replaced by the completion of a finitely generated field extension ofkwith respect to an arbitrary valuation of height (real rank) 1, at least in the case where the transcendence degree overkis equal to 2. This would require a slightly more flexible set of foundations: one must work only with finitely generated k-subalgebras of the complete field, so that one has hope of having enough units. A more serious problem is how to perform Tsuzuki's method (a/k/a Theorem 4.5.2) in this context.

2.4 ±Analytic rings.

We now introduce ``analytic rings'', citing into [14] for their structural properties.

DEFINITION2.4.1. Let E be a nearly monomial field over k, or the completed perfect or algebraic closure thereof. In the first case, suppose

that G^E has enough r0-units for some r0>0 (otherwise take r0ˆ 1).

Let Ibe a subinterval of [0;r0) bounded away from r0 (i.e., I is a sub- interval of [0;r] for some r2(0;r₀)). Let G^E_I denote the FreÂchet com- pletion of G^E_r0[p ¹] under the valuations w_s for s2I; this ring is an in- tegral domain [14, Lemma 2.4.6]. If I is closed, then G^E_I is a principal ideal domain [14, Proposition 2.6.9]. Put

G^E_an;rˆG^E_(0;r];

this ring is a BeÂzout ring, i.e., a ring in which every finitely generated ideal is principal [14, Theorem 2.9.6]. PutG^E_an;conˆ [r>0G^E_an;r; thenG^E_an;conis also a BeÂzout ring. The group of units inG^E_an;conconsists of the nonzero elements of Gcon[p ¹] [14, Corollary 2.5.12]. For E⁰ finite separable over E, E⁰ˆE^perf, orE⁰ˆE^alg, by [14, Proposition 2.4.10] one has

G^E_an;con⁰ ˆG^E_an;conG^E_conG^E_con⁰ :

REMARK2.4.2. It is likely thatG^E_I is a BeÂzout ring for anyIas above.

However, this statement is not verified in [14], and we will not need it anyway, so we withhold further comment on it.

REMARK2.4.3. IfEˆk((t)) is a power series field, then the ringG^E_I is the ring of rigid analytic functions on the annulus w(t)2I in the t- plane. Thus our construction of fake annuli includes ``true'' one-dimen- sional rigid analytic annuli overK, and most of our results on fake an- nuli (like thep-adic local monodromy theorem) generalize extant theo- rems on true annuli. On the other hand, ifEˆk((L))<sub>l</sub> and rank(L)>1, then the ringG<sup>E</sup><sub>I</sub> is trying to be the ring of rigid analytic functions on a subspace of the rigid affine plane in the variables fz<sub>1</sub>g;. . .;fz<sub>m</sub>g for some basis z<sub>1</sub>;. . .;z<sub>m</sub> of L, consisting of points for which there exists r2Iwithw(fzig)ˆrl(zi) foriˆ1;. . .;m. IfIˆ[r;r], then this space is an affinoid space in the sense of Berkovich, but otherwise it is not (because one can only cut out an analytic subspace of the form w(x)ˆaw(y) for a rational). Indeed, as far as we can tell, this space is not a p-adic analytic space in either of the Tate or Berkovich senses, despite the fact that it has a sensible ring of analytic functions; hence the use of the adjective ``fake'' in the phrase ``fake annulus'', and the absence of an honest definition of that phrase.

REMARK2.4.4. Since one can sensibly define rigid analytic annuli over arbitrary complete nonarchimedean fields, Remark 2.4.3 suggests the

possibility of working with fake annuli over more general complete K.

However, the algebraic issues here get more complicated, and we have not straightened them out to our satisfaction. For example, the analogue of the ring G^naive_r fails to be a principal ideal ring if the valuation on K is not discrete; it probably still has the BeÂzout property (that finitely generated ideals are principal), but we have not checked this. In any case, the formalism of [14] completely breaks down whenKis not discretely valued, so an attempt here to avoid a discreteness hypothesis now would fail to improve upon our ultimate results; we have thus refrained from making such an attempt.

3. Frobenius and connection structures.

We now introduce a notion which should be thought of as ap-adic dif- ferential equation with Frobenius structure on a fake annulus. We start with some notational conventions.

CONVENTION 3.0.1. Throughout this section, assume that E is a monomial field and thatG^E is equipped with a Frobenius lift such that G^Ehas enoughr₀-units for somer₀>0; we viewG^Eas being equipped with a levelwise topology via the choice of a coordinate system. (This choice does not matter, as the topology can be characterized as the coarsest one under which the vn;r are continuous for all n2Z and all r2(0;r₀).) We suppressEfrom the notation, writingG forG^E,G_confor G^E_con, and so on.

CONVENTION3.0.2. When a valuation is applied to a matrix, it is de- fined to be the minimum value over entries of the matrix.

We also make a definition of convenience.

DEFINITION 3.0.3. Under Convention 3.0.1, we will mean by an ad- missible ringany one of the following topological rings.

The ringGorG[p ¹] with its levelwise topology.

The ringG_rorG_r[p ¹] with the FreÂchet topology induced byw_sfor alls2(0;r], forr2(0;r₀). Note that forG_r, this coincides with the topology induced byw_ralone.

The ringGconorGcon[p ¹] topologized as the direct limit of theGr

orGr[p ¹].

The ringGIwith the FreÂchet topology induced by thewsfors2I, for someI[0;r0) bounded away fromr0.

The ringGan;contopologized as the direct limit of theGan;r. By anearly admissible ring, we mean one of the above rings withE replaced by a finite separable extension.

3.1 ±Differentials.

DEFINITION 3.1.1. Let S=R be an extension of topological rings. A module of continuous differentialsis a topologicalS-moduleV¹_S=Requipped with a continuous R-linear derivationd:S!V¹_S=R, having the following universal property: for any topologicalS-moduleMequipped with a con- tinuous R-linear derivation D:S!M, there exists a unique morphism f:V¹_S=R!Mof topologicalS-modules such thatDˆfd. Since the de- finition is via a universal property, the module of continuous differentials is unique up to unique isomorphism if it exists at all.

Constructing modules of continuous differentials is tricky in general (imitating the usual construction of the module of KaÈhler differentials requires a topological tensor product, which is a rather delicate matter);

however, for fake annuli, the construction is straightforward.

DEFINITION3.1.2. By acoordinate systemforG, we will mean a latticeL in some R^m, an irrational linear functional l2(R^m)^_, an isomorphism G^lGcarryingz2Lto a unit inG_r0 for each nonzeroz2L, and a basis z₁;. . .;z_mofL. Such data always exist thanks to Proposition 2.3.4.

DEFINITION3.1.3. For the remainder of this subsection, choose a co- ordinate system for G, and let m₁;. . .;m_m2L^_ denote the basis dual to z1;. . .;zm. Form2L^_ andSan admissible ring, let@mbe the continuous derivation onSdefined by the formula

@_m X

z

c_zfzg

!

ˆX

z

m(z)c_zfzg;

note thatm(z)2Z, so it may sensibly be viewed as an element ofo. (The continuity of @m is clear in terms of naõÈve partial valuations, so Proposi- tion 2.3.4 implies continuity in terms of the Frobenius-based valuations.) For mˆm_i, write @_i for @m_i. Define V¹_S=o to be the free S-module S dfz₁g S dfz_mg, equipped with the natural induced topology and

with the continuouso-linear derivationd:S!V¹_S=ogiven by dxˆX^m

iˆ1

@_i(x)dlogfz_ig (wheredlog (f)ˆdf=f).

PROPOSITION 3.1.4. The module V¹_S=o is a module of continuous derivations forSovero. In particular, the construction does not depend on the choice of the coordinate system.

PROOF. This is a straightforward consequence of the fact that one of o[fz_ig¹] oro[p ¹;fz_ig¹] is dense inS. p REMARK3.1.5. Note that Proposition 3.1.4 also allows us to construct the module of continuous differentials V¹_S=o when S is only nearly ad- missible.

REMARK3.1.6. For rank(L)ˆ1 andm2Lnonzero, the image of@mis closed; however, this fails for rank(L)>1, because boundingl(z) does not in any way limit thep-adic divisibility ofzwithinL. This creates a striking difference between the milieux of true and fake annuli, from the point of view of the study of differential equations. On true annuli, one has the rich theory of p-adic differential equations due to Dwork-Robba, Christol- Mebkhout, et al. On fake annuli, much of that theory falls apart; the parts that survive are those that rest upon Frobenius structures, whose behavior differs little in the two settings.

3.2 ±r-modules.

DEFINITION3.2.1. LetSbe a nearly admissible ring. Define ar-module overSto be a finite freeS-moduleMequipped with an integrableo-linear connection r:M!MV¹_S=o; here integrability means that, lettingr1

denote the induced map

M_SV¹_S=o^r1! M_SV¹_S=oSV¹_S=o^{1^}! M_S^²_SV¹_S=o;

the composite mapr₁ ris zero. We sayv2Mishorizontalifr(v)ˆ0.

DEFINITION3.2.2. SupposeSis admissible, and fix a coordinate system forG. Given ar-moduleMoverS, form2L^_, define the mapDm:M!M

by writingr(v)ˆP^m

iˆ1w_idlogfz_igwithw_i2M, and setting D_m(v)ˆX^m

iˆ1

m(z_i)w_i: Also, writeD_iforD_mi.

REMARK3.2.3. The mapsD_msatisfy the following properties.

The mapL^_M !M given by (m;v)7!D_m(v) is additive in each factor.

For allm2L^_,s2S, andv2M, we have the Leibniz rule D_m(sv)ˆsD_m(v)‡@_m(s)v:

Form₁;m₂2L^_, the mapsD_m1;D_m2commute.

Conversely, given a finite free S-module M equipped with maps Dm:M!Mfor eachm2L^_satisfying these conditions, one can uniquely reconstruct ar-module structure onMthat gives rise to theDm.

REMARK 3.2.4. Note that for true annuli (i.e., rank(L)ˆ1), the in- tegrability restriction is empty becauseV¹_S=ohas rank 1 overS. However, for fake annuli, integrability is a real restriction: even though the ring theory looks one-dimensional, the underlying ``fake space'' is reallym-di- mensional, inasmuch asV¹_S=ohas rankmoverS.

DEFINITION3.2.5. LetMbe ar-module overG^E_an;con⁰ , forE⁰a finite se- parable extension ofE. W e sayMis:

constantifM admits a horizontal basis (a basis of elements of the kernel ofr);

quasi-constantif there exists a finite separable extensionE⁰⁰ofE⁰ such thatMG^E_an;con⁰⁰ is constant;

unipotent if M admits an exhaustive filtration by saturated r- submodules, whose successive quotients are constant;

quasi-unipotentifMadmits an exhaustive filtration by saturated r-submodules, whose successive quotients are quasi-constant.

We extend these definitions to (F;r)-modules by applying them to the underlyingr-module.

REMARK 3.2.6. IfM is quasi-unipotent, then there exists a finite se- parable extensionE⁰⁰ofE⁰such thatMG^E_an;con⁰⁰ is unipotent. The converse is also true: if MG^E_an;con⁰⁰ is unipotent, then the shortest unipotent fil- tration ofMG^E_an;con⁰⁰ is unique, so descends toG^E_an;con⁰ .

3.3 ±Frobenius structures

DEFINITION3.3.1. LetSbe a nearly admissible ring stable unders; for instance,G;Gcon;Gan;conare permitted, butGris not. Define anF-module overS(with respect tos) to be a finite freeS-moduleMequipped with aS- module homomorphismF:sM!M which is an isogeny, i.e., which be- comes an isomorphism upon tensoring withS[p ¹]. We typically viewFas a s-linear map fromMto itself; we occasionally viewMas a left module for the twisted polynomial ring Sfsg. Given an F-module M over S and an integerc, which must be nonnegative ifp ¹2S, define the= twist M(c) ofMto be a copy ofMwith the action ofFmultiplied byp^c.

DEFINITION 3.3.2. Let S be a nearly admissible ring stable under s.

Define an (F;r)-moduleover Sto be a finite freeS-moduleMequipped with the structures of both anF-module and ar-module, which are com- patible in the sense of making the following diagram commute:

Mƒƒ^r!MV¹_S=o

#^F

#^Fds Mƒƒ^r!MV¹_S=o

REMARK 3.3.3. We may regardV¹_S=o itself as anF-module viads, in which case the compatibility condition asserts that r:M!MV¹_S=ois an F-equivariant map. The fact that @m(f^s)0 modp for any f 2Gcon

means that V¹_Gcon=o is isomorphic as an F-module to N(1), for some F- module NoverG_con. In the language of [14], this means that the generic HN slopes ofV¹_Gcon=oare positive [14, Proposition 5.1.3].

DEFINITION3.3.4. For a a positive integer, define an F^a-module or (F^a;r)-moduleas anF-module or (F;r)-module relative tos^a. Given an F-module M, viewed as a leftSfsg-module, define theF^a-module [a]M to be the left Sfs^ag-module given by restriction along the inclusion Sfs^ag,!Sfsg; in other words, replace the Frobenius action by its a-th

power. Given anF^a-moduleN, viewed as a leftSfs^ag-module, define theF- module [a]Mto be the leftSfsg-module

[a]MˆSfsg _Sfs^a_gM;

then the functors [a]and [a]are left and right adjoints of each other. See [14, § 3.2] for more on these operations.

3.4 ±Change of Frobenius.

The category of (F;r)-modules overGan;conrelative tosturns out to be canonically independent of the choice ofs, by a Taylor series argument (as in [25, § 3.4]).

CONVENTION3.4.1. Throughout this subsection, fix a coordinate sys- tem onG. Given anm-tupleJˆ(j1;. . .;jm) of nonnegative integers, write J!ˆj1! jm!; ifUˆ(u1;. . .;um), writeU^Jˆu^j₁¹ u^j_m^m. Also, define the

``falling factorials''

@^JˆY^m

iˆ1

Y

ji 1

lˆ0

(@_i l)

D^JˆY^m

iˆ1

Y

ji 1

lˆ0

(D_i l);

with the convention that@⁰andD⁰are the respective identity maps. (The use of falling factorial notation is modeled on [8].)

LEMMA 3.4.2. Let M be a r-module over G_an;con. Then for any r2Gan;con, anyv2M, and any m-tuple J of nonnegative integers,

1

J!D^J(rv)ˆ X

J1‡J2ˆJ

1 J1!@^J1(r)

1

J2!D^J2(v)

:

PROOF. Since

@i(@i 1) (@i j‡1)ˆ fzig^j(fzig ¹@i)^j

and similarly forDi, this amounts to a straightforward application of the

Leibniz rule. p

LEMMA 3.4.3. For any u1;. . .;um2Gcon with w(ui)>0 for iˆ1;. . .;m, and any x2Gan;con, the series

f(x)ˆ X¹

j1;...;jmˆ0

1

J!U^J@^J(x)

converges in G_an;con, and the map x7!f(x) is a continuous ring homo- morphism sendingfz_igto u_i.

PROOF. Pick r>0 such that u1;. . .;um;x2Gan;r and wr(ui)>0 for

iˆ1;. . .;m. WritexˆP

z2Lczfzg; note that for eachJ, 1

J!@^J(x)ˆX

z2L

Y^m

iˆ1

m_i(z) ji

! czfzg;

so that w_s(@^J(x)=J!)w_s(x) for s2(0;r]. This yields the desired con- vergence, as well as continuity of the mapx7!f(x). Moreover,f is a ring homomorphism on o[fz1g;. . .;fzmg] by Lemma 3.4.2, so must be a ring homomorphism onGan;con by continuity; the fact that it sendsfz_igtou_iis

apparent from the formula. p

LEMMA3.4.4. Let M be ar-module overG_an;rfor some r>0. Suppose that for some positive integer h, M admits a basise1;. . .;en such that the nn matrices N1;. . .;Nmdefined byD_i(e_l)ˆP

j (N_i)_jle_jsatisfy wr(N_i)>

>w((p^h)!)for iˆ1;. . .;m. For J an m-tuple of nonnegative integers, define the nn matrix N_Jby

D^J(e_l)ˆX

j

(N_J)_jle_j: Then

wr(N_J)w(J!) w(p)(j1‡ ‡jm)=(p^h(p 1)):

PROOF. The condition thatwr(N_i)>w((p^h)!) means that for anya2Z and anyb2 f0;. . .;p^h 1g, if we write

vˆX^m

jˆ1

x_je_j (x_j2Gan;r)

(D_i ap^h)(D_i ap^h 1) (D_i ap^h b)vˆX^m

jˆ1

y_ijabe_j (y_ijab2G_an;r);

then minjfwr(yijab)g minjfwr(xj)g ‡w(b!) (i.e., the same bound as for the trivial connection withe1;. . .;enhorizontal). This gives the bound

wr(NJ)w(J!)‡X^m

iˆ1

w(ji!)‡ bji=p^hcw((p^h)!)‡w((ji p^hbji=p^hc)!) w(J!) X^m

iˆ1

w(p)ji=(p^h(p 1)) using the fact thatw(ji!)ˆ P¹

gˆ1w(p)bji=p^gc. This yields the claim. p LEMMA3.4.5. Let M be an(F;r)-module overG_an;conor overG_con[p ¹], and lete₁;. . .;e_nbe a basis of M. For each nonnegative integer g, define the nn matrices Ng;1;. . .;Ng;m by Di(F^gel)ˆP

j (Ng;i)jl(F^gej). Then there exist r₁2(0;r₀)and c>0such that for each nonnegative integer g and for each of iˆ1;. . .;m, Ng;ihas entries inGan;r1q ^gand

wrq ^g(N_g;i)g c (r2[r1=q;r1]):

Moreover, if M is defined over G_con[p ¹], we can also ensure that w(N_g;i)g c.

PROOF. Defineahi2Gconby the formula

@_i(x^s)ˆX^m

hˆ1

a_hi(@_hx)^s (x2G_con);

thenw(a_hi)1 as in Remark 3.3.3. In particular, we can chooser₁2(0;r₀) as in Proposition 2.3.4 such that foriˆ1;. . .;m,ai2Gr1,wr1(ahi)1, and N_0;ihas entries inGan;r1. Then the formula

Ng‡1;iˆX^m

hˆ1

ahiN^s_g;h

yields the claim for anyc with minifmin_r2[r1=q;r1]fwr(N0;i)gg c and (in caseMis defined overGcon[p ¹]) min_ifw(N_0;i)g c.

LEMMA 3.4.6. Let M be an (F;r)-module over G_an;con (resp. over G_con[p ¹]). Then for any u₁;. . .;u_m2G_conwith w(u_i)>0for iˆ1;. . .;m, and anyv2M, the series

f(v)ˆ X¹

j1;...;jmˆ0

1

J!U^JD^J(v)

converges for the natural topology of M, and the mapv7!f(v)is semilinear for the map defined by Lemma3.4.3.

PROOF. Pick a basise1;. . .;en ofM; for each nonnegative integerg, define thennmatricesN_g;1;. . .;N_g;mbyD_i(F^ge_l)ˆP

j (N_g;i)_jl(F^ge_j). By Lemma 3.4.5, we can choose r₁2(0;r₀) such that for some c>0, wrq^g(Ng;i)g c for all nonnegative integers g and all r2[r1=q;r1];

moreover, if we are working overGcon[p ¹], we can ensure thatw(Ng;i) g c.

Now choose a positive integerhwithw(p)=(p^h(p 1))<1=2. Then by the previous paragraph, for each sufficiently small r>0, there exists a basisv1;. . .;vnofM(depending onr) on which eachDiacts via a matrixNi

withwr(Ni)>w((p^h)!). By Lemma 3.4.4, the matrixNJ defined by D^J(v_l)ˆX

j

(N_J)_jlv_j satisfieswr(N_J)w(J!) (j1‡ ‡jm)=2.

On the other hand, since w(u_i)1 for iˆ1;. . .;m, we have that w_r(u_i)>1=2 forrsufficiently small. We conclude that for each sufficiently small r>0, there exists a basis v1;. . .;vn such that the series defining each of f(v1);. . .;f(vn) converges under wr. By Lemma 3.4.2 and Lem- ma 3.4.3, for each Gan;r-linear combinationv of v1;. . .;vn, the series de- finingf(v) converges underw_r. By the same token, in caseM is defined over G_con[p ¹], for each G_r[p ¹]-linear combination v of v₁;. . .;v_n, the series defining f(v) converges under w. This yields the desired con- vergence of f; again, the semilinearity follows from Lemma 3.4.2 and

Lemma 3.4.3. p

PROPOSITION3.4.7. Lets₁ ands₂be Frobenius lifts onGsuch thatG has enough units with respect to each of s1 ands2, and for each z2L nonzero,fzgis a unit inGconunder both definitions. (By Proposition2.3.4, it is equivalent to require that the definitions ofGconwith respect tos1and tos₂coincide.) Then there is a canonical equivalence of categories between (F;r)-modules overG_an;con(resp. overG_con[p ¹]) relative tos₁and relative tos₂, acting as the identity on the underlyingr-modules.

PROOF. Putuiˆ fzig^s2=fzig^s1 1. LetMbe ar-module admitting a compatible Frobenius structureF1relative tos1. Forv2M, define

F2(v)ˆ X¹

j1;...;jmˆ0

1

J!U^JF1(D^J(v));

this series converges thanks to Lemma 3.4.6. Moreover, the result iss2-

linear thanks to Lemma 3.4.2. p

REMARK 3.4.8. By tweaking the proof of Proposition 3.4.7, one can also obtain the analogous independence from the choice ofsfor the cate- gory of (F;r)-modules over Gcon. We will not use this result explicitly, though a related construction will occur in Subsection 4.2.

4. Unit-root (F,r)-modules (after Tsuzuki).

In this section, we give the generalization to fake annuli of Tsuzuki's unit-root local monodromy theorem [24], variant proofs of which are given by Christol [2] and in the author's unpublished dissertation [10]. Our ar- gument here draws on elements of all of these; its specialization to the case of true annuli constitutes a novel (if only slightly so) exposition of Tsuzuki's original result.

CONVENTION 4.0.1. Throughout this section, let E denote a nearly monomial field over k, viewed in a fixed fashion as a finite separable ex- tension of a monomial field overk. We assume that any Frobenius lift s considered onGˆG^Eis chosen so thatGhas enough units. In particular, GˆG^E and Gcon ˆG^E_con are nearly admissible in the sense of Defini- tion 3.0.3.

4.1 ±Unit-root F-modules

DEFINITION4.1.1. We say anF-moduleMoverG^EorG^E_con, with respect to some Frobenius lifts, isunit-root(oreÂtale) if the mapF:sM!Mis an isomorphism (not just an isogeny). We say an (F;r)-module overG^EorG_con is unit-root if its underlyingF-module is unit-root.

We will frequently calculate on such modules in terms of bases, so it is worth making the relevant equations explicit.

REMARK4.1.2. Assume thatEis a monomial field, and fix a coordinate system forG. LetMbe ar-module overGorGconwith basise1;. . .;en. Given m2L^_, define the nn matrixNm byDm(e_l)ˆP

j (Nm)_jle_j; if we identify vˆc1e1‡ ‡cnen 2M with the column vector with entries c1;. . .;cn,

then we have

Dm(v)ˆNmv‡@m(v):

Given anF-module with the same basise1;. . .;en, define thennmatrixA by F(e_l)ˆP

j A_jle_j; then with the same identification of vwith a column vector, we have

F(v)ˆAv^s:

In case the Frobenius liftsis standard, the compatibility of Frobenius and connection structures is equivalent to the equations

N_mA‡@_m(A)ˆqAN^s_m (m2L^_);

of course it is only necessary to check this on a basis ofL^_.

REMARK4.1.3. It is also worth writing out how the equations in Re- mark 4.1.2 transform under change of basis. First, if U is an invertible nnmatrix, then

N_mA‡@_m(A)ˆ0 () (U ¹N_mU‡U ¹@_m(U))(U ¹A)‡@_m(U ¹A)ˆ0:

Second, in casesis standard, the equations

N_mA‡@_m(A)ˆqAN_m^s and N_m⁰A⁰‡@_m(A⁰)ˆqA⁰(N⁰_m)^s are equivalent for

N_m⁰ ˆU ¹NmU‡U ¹@m(U) A⁰ˆU ¹AU^s:

4.2 ±Unit-root F-modules and Galois representations

We now consider unit-root F-modules over G, obtaining the usual Fontaine-style setup.

LEMMA4.2.1. Let`be a separably closed field of characteristic p>0, and lettdenote the q-power Frobenius on`. Let A be an invertible nn matrix over`.

(a) There exists an invertible nn matrix U over ` such that U ¹AU^tis the identity matrix.

(b) For any 1n column vector v over `, there are exactly q<sup>n</sup> distinct1n column vectorswover`for which Aw^t wˆv.