Log-Growth Filtration

and Frobenius Slope Filtration of F-Isocrystals

at the Generic and Special Points

Bruno Chiarellotto^{1} and Nobuo Tsuzuki^{2}

Received: October 9, 2009 Revised: February 25, 2010

Communicated by Peter Schneider

Abstract. We study, locally on a curve of characteristic p > 0, the relation between the log-growth filtration and the Frobenius slope filtration for F -isocrystals, which we will indicate as φ-∇-modules, both at the generic point and at the special point. We prove that a bounded φ-∇-module at the generic point is a direct sum of pure φ- ∇-modules. By this splitting of Frobenius slope filtration for bounded modules we will introduce a filtration for φ-∇-modules (PBQ filtration). We solve our conjectures of comparison of the log-growth filtration and the Frobenius slope filtration at the special point for particular φ-∇-modules (HPBQ modules). Moreover we prove the analogous comparison conjecture for PBQ modules at the generic point. These comparison conjectures were stated in our previous work [CT09]. Using PBQ filtrations for φ-∇-modules, we conclude that our conjecture of comparison of the log-growth filtration and the Frobenius slope filtration at the special point implies Dwork’s conjecture, that is, the special log-growth polygon is above the generic log-growth polygon including the coincidence of both end points.

2010 Mathematics Subject Classification: Primary 12H25; Secondary 11S80. Keywords and Phrases: p-adic differential equations, logarithmic growth, Frobenius slopes, Newton polygon.

1The author is supported by Progetto di Ateneo Padova “Rapresentations and Motives”

and Progetto Eccellenza “Differential methods in arithmetic geometry and algebra” (Cariparo Found.).

2The author is supported by Grant-in-Aid for Exploratory Research and Core-to-Core Program 18005 of JSPS.

Documenta Mathematica 16 (2011) 33–69

Contents

1 Introduction 34

2 Preliminaries 38 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Dwork’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Choices of Frobenius 44 3.1 Comparison morphism ϑ_{σ_{1},σ_{2}} . . . . . . . . . . . . . . . . . . . 44 3.2 Equivalence of categories . . . . . . . . . . . . . . . . . . . . . . 45

4 Boundedness and splitting of the Frobenius slope filtration 46 4.1 Splitting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Descent of splittings . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . 50

5 PBQ φ-∇-modules 55 5.1 Definition of PBQ φ-∇-modules . . . . . . . . . . . . . . . . . . 55 5.2 Existence of the maximally PBQ φ-∇-submodules over E … 56 5.3 Existence of the maximally PBQ φ-∇-submodules over K[[x]]_{0} . 56

6 Log-growth and Frobenius slope for HPBQ φ-∇-modules over K[[x]]_{0} 59 6.1 Log-growth for HPBQ φ-∇-modules . . . . . . . . . . . . . . . 59 6.2 Equislope φ-∇-modules over K[[x]]_{0} . . . . . . . . . . . . . . . . 63

7 Log-growth filtration and Frobenius filtration at the generic point 64 7.1 The log-growth of PBQ φ-∇-modules over E . . . . . . . . . . . 64 7.2 Rationality of breaks of log-growth filtrations . . . . . . . . . . 65

8 Toward Dwork’s conjecture LGF_{Dw} 65

8.1 The comparison at the special point and Dwork’s conjecture

LGF_{Dw} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.2 Dwork’s conjecture in the HBQ cases . . . . . . . . . . . . . . . 67 8.3 When do the generic and special log-growth polygons coincide? 67

1 Introduction

Log-growth Filtration and Frobenius Slope Filtration 35

The local behavior of p-adic linear differential equations is, in one sense, very easy. If the equation has a geometric origin (i.e., if it is furnished with a Frobenius structure), then the radius of convergence of solutions at any nonsigular point is at least 1. In general, the p-adic norm of the coefficients a_{n} in the Taylor series of a solution is an increasing function on n. However, one knows that some solutions are p-adically integral power series. B.Dwork discovered these mysterious phenomena and introduced a measure, called logarithmic growth (or log-growth, for simplicity), for power series in order to investigate this delicate difference systematically (see [Dw73] and [Ka73, Section 7]). He studied the log-growth of solutions of p-adic linear differential equations both at the generic point and at special points (see [Ro75], [Ch83]), and asked whether the behaviors are similar to those of Frobenius slopes or not. He conjectured that the Newton polygon of log-growth of solutions at a special point is above the Newton polygon of log-growth of solutions at the generic point [Dw73, Conjecture 2]. We refer to it as Conjecture LG_{Dw} when there are not Frobenius structures, and as Conjecture LGF_{Dw} where there are Frobenius structures (see Conjecture 2.7). He also proved that the Newton polygon of log-growth of solutions at the generic (resp. special) point coincides with the Newton polygon of Frobenius slopes in the case of hypergeometric Frobenius-differential systems if the systems are nonconstant, thus establishing the conjecture in these cases [Dw82, 9.6, 9.7, 16.9]. On the other hand P.Robba studied the generic log-growth of differential modules defined over the completion of ℚ(x) under the p-adic Gauss norm by introducing a filtration on them via p-adic functional analysis [Ro75] (see Theorem 2.2). His theory works on more general p-adically complete fields, for example our field E. Let k be a field of characteristic p > 0, let V be a discrete valuation ring with residue field k, and let K be the field of fractions of V such that the characteristic of K is 0. In [CT09] we studied Dwork’s problem on the log-growth for φ-∇-modules over E or K[[x]]_{0} which should be seen as localizations of F - isocrystals on a curve over k with coefficients in K. Here K[[x]]_{0} is the ring of bounded functions on the unit disk around x = 0, E is the p-adically complete field which is the field of fractions of K[[x]]_{0}, and φ (resp. ∇) indicates the Frobenius structure (resp. the connection) (See the notation and terminology introduced in Section 2). We gave careful attention to Dwork’s result on the comparison between the log-growth and the Frobenius slopes of hypergeometric Frobenius-differential equations. We compared the log-growth and the Frobenius slopes at the level of filtrations. Let M be a φ-∇-module over K[[x]]_{0}. Let M_{η} = M ⊗_{K[[x]]_{0}} E be a φ-∇-module over E which is the generic fiber of M and let V (M) be the φ-module over K consisting of horizontal sections on the open unit disk. Denote by M_{η}^{λ} the loggrowth filtration on M_{η} at the generic point indexed by λ ∈ ℝ, and by V (M)^{λ} be the log-growth filtration with real indices on the φ-module V (M). Furthermore, let S_{λ}( ·) be the Frobenius slope filtration such that S_{λ}( ·)/S_{λ−}( ·) is pure of slope λ. We proved that the log-growth filtration is included in the orthogonal part of the Frobenius slope filtration of the dual module under the natural perfect pairing M_{η} ⊗_{E} M_{η}^{∨} → E (resp. V (M) ⊗_{K} V (M^{∨}) → K) at the generic point (resp. the special point) [CT09, Theorem 6.17] (see the precise form in Theorem 2.3):

M_{η}^{λ} ⊂ (S_{λ−λ_{max}}(M_{η}^{∨}))^{⊥} (resp. V (M)^{λ} ⊂ (S_{λ−λ_{max}}(V (M^{∨})))^{⊥})

for any λ ∈ ℝ if λ_{max} is the highest Frobenius slope of M_{η}. We then conjectured: (a) the rationality of log-breaks λ (both at the generic and special fibers) and (b) if the bounded quotient M_{η}/M_{η}^{0} is pure as a φ-module then the inclusion relation becomes equality both at the generic and special points [CT09, Conjectures 6.8, 6.9]. The hypothesis of (b) will be called the condition of being “pure of bounded quotient” (PBQ) in Definition 5.1. Note that there are examples with irrational breaks, and that both M^{λ−} ⊋ M^{λ} and M^{λ} ⊋ M^{λ+} can indeed occur for log-growth filtrations in absence of Frobenius structures [CT09, Examples 5.3, 5.4]. We state the precise forms of our conjectures in Conjecture 2.4 on E and Conjecture 2.5 on K[[x]]_{0}, and denote the conjectures by LGF_{E} and LGF_{K[[x]]_{0}}, respectively. We have proved our conjectures LGF_{E} and LGF_{K[[x]]_{0}} if M is of rank ≤ 2 [CT09, Theorem 7.1, Corollary 7.2], and then we established Dwork’s conjecture LGF_{Dw} if M is of rank ≤ 2 [CT09, Corollary 7.3]. Let us now explain the results in the present paper. First we characterize bounded φ-∇-modules over E by using Frobenius structures (Theorem 4.1):

(1) A bounded φ-∇-module M over E (i.e., M^{0} = 0, which means that all the solutions on the generic disk are bounded) is isomorphic to a direct sum of several pure φ-∇-modules if the residue field k of V is perfect.

Note that the assertion corresponding to (1) is trivial for a φ-∇-module M over K[[x]]_{0} such that M_{η} is bounded by Christol’s transfer theorem (see [CT09, Proposition 4.3]). This characterization implies the existence of a unique increasing filtration {P_{i}(M)} of φ-∇-modules M over E such that P_{i}(M)/P_{i−1}(M) is the maximally PBQ submodule of M/P_{i−1}(M) (Corollary 5.5). This filtration is called the PBQ filtration. When we start with a φ-∇- module M over K[[x]]_{0}, we can introduce a similar PBQ filtration for M, i.e., a filtration consisting of φ-∇-submodules over K[[x]]_{0} whose generic fibers will induce the PBQ filtration of the generic fiber M_{η} (Corollary 5.10). To this end we use an argument of A.J. de Jong in [dJ98] establishing the full faithfulness of the forgetful functor from the category of φ-∇-modules over K[[x]]_{0} to the category of φ-∇-modules over E. The need to study horizontality behavior for the PBQ condition with respect to the special and generic points leads us to introduce a new condition for φ-∇-modules over K[[x]]_{0}, namely, the property of being “horizontally pure of bounded quotient ” (which, for simplicity, we abbreviate as HPBQ, cf. Definition 6.1). Then in Theorem 6.5 we prove that

(2) our conjecture LGF_{K[[x]]_{0}} (see 2.5) on the comparison between the loggrowth filtration and the Frobenius slope filtration at the special point holds for a HPBQ module.

A HPBQ module should be understood as a φ-∇-module for which the bounded quotient is horizontal and pure with respect to the Frobenius. Our method of proof will lead us to introduce the related definition of equislope φ-∇-modules over K[[x]]_{0} (Definition 6.7): they admit a filtration as φ-∇-modules over K[[x]]_{0} which induces the Frobenius slope filtration at the generic point. Note that a PBQ equislope object is HPBQ. Using this result, we prove in Theorem 7.1 that

(3) our conjecture LGF_{E} (see 2.4) on comparison between the log-growth filtration and the Frobenius slope filtration at the generic point holds for PBQ modules over E.

Indeed, for a φ-∇-module M over E, the induced φ-∇-module M_{τ} = M ⊗_{E} E_{t}[[X − t]]_{0} (where E_{t}[[X − t]]_{0} is the ring of bounded functions on the open unit disk at generic point t) is equislope. For the proof of comparison for HPBQ modules, we use an explicit calculation of log-growth for solutions of certain Frobenius equations (Lemma 4.8) and a technical induction argument. For a submodule L of a φ-∇-module M over E with N = M/L, the induced right exact sequence

L/L^{λ} → M/M^{λ} → N/N^{λ} → 0

is also left exact for any λ if L is a maximally PBQ submodule of M by Proposition 2.6. Since there do exist PBQ filtrations, the comparison between the log-growth filtrations and the Frobenius slope filtrations for PBQ modules both at the generic point and at the special point implies the rationality of breaks (Theorem 7.2 and Proposition 7.3) as well as Dwork’s conjecture (Theorem 8.1) that the special log-growth polygon lies above the generic log-growth polygon (including the coincidence of both end points):

(4) Our conjecture of comparison between the log-growth filtration and the Frobenius slope filtration at the special point (Conjecture LGF_{K[[x]]_{0}}, 2.5) implies Dwork’s conjecture (Conjecture LGF_{Dw}, 2.7).

As an application, we have the following theorem (Theorem 8.8) without any assumptions.

(5) The coincidence of both log-growth polygons at the generic and special points is equivalent to the coincidence of both Frobenius slope polygons at the generic and special points.

Let us also mention some recent work on log-growth. Y.André ([An08]) proved the conjecture LG_{Dw} of Dwork without Frobenius structures, that is, the loggrowth polygon at the special point is above the log-growth filtration at the generic point for ∇-modules, but without coincidence of both end points. (Note that his convention on the Newton polygon is different from ours, see Remark 2.8). He used semi-continuity of log-growth on Berkovich spaces. K.Kedlaya defined the log-growth at the special point for regular singular connections and studied the properties of log-growth [Ke09, Chapter 18]. This paper is organized in the following manner. In Section 2 we recall our notation and results from [CT09]. In Section 3 we establish the independence of the category of φ-∇-modules over E (resp. K[[x]]_{0}) of the choices of Frobenius on E (resp. K[[x]]_{0}). In Section 4 we study when the Frobenius slope filtration of φ-∇-modules over E is split and prove (1) above. In Section 5 we introduce the notion of PBQ and prove the existence of PBQ filtrations. In Section 6 we study the log-growth filtration for HPBQ φ-∇-modules over K[[x]]_{0} and prove the comparison (2) between the log-growth filtration and the Frobenius slope filtration. This comparison implies the comparison (3) for PBQ φ-∇-modules over E in Section 7. In Section 8 we show that (4) our conjecture of comparison at the special point implies Dwork’s conjecture.

2 Preliminaries

We fix notation and recall the terminology in [CT09]. We also review Dwork’s conjecture and our conjectures.

2.1 Notation

Let us fix the basic notation which follows from [CT09].

p : a prime number.

K : a complete discrete valuation field of mixed characteristic (0, p).

V : the ring of integers of K.

k : the residue field of V.

m : the maximal ideal of V.

| | : a p-adically absolute value on K and its extension as a valuation field, which is normalized by |p| = p^{−1}.

q : a positive power of p.

σ : (q-)Frobenius on K, i.e., a continuous lift of q-Frobenius endomorphism (a ↦ a^{q} on k). We suppose the existence of Frobenius on K. We also denote by σ a K-algebra endomorphism on A_{K}(0, 1^{−}), which is an extension of Frobenius on K, such that σ(x) is bounded and |σ(x) − x^{q}|_{0} < 1. Then K[[x]]_{λ} is stable under σ. We also denote by σ the unique extension of σ on E, which is a Frobenius on E. In the case we only discuss φ-∇- modules over E, one can take a Frobenius σ on K such that σ(x) ∈ E with |σ(x) − x^{q}|_{0} < 1.

{ \atop K^{perf}} : the p-adic completion of the inductive limit K^{perf} of K→^{σ}K→^{σ} · · ·. Then { \atop K^{perf}} is a complete discrete valuation field such that the residue field of the ring of integers of { \atop K^{perf}} is the perfection of k and that the value group of { \atop K^{perf}} coincides with the value group of K. The Frobenius σ uniquely extend to { \atop K^{perf}}. Moreover, taking the p-adic completion K̂^{al} of the maximally unramified extension K^{al} of K^{perf}, we have a canonical extension of K as a discrete valuation field with the same value group such that the residue field of the ring of integers is algebraically closed and the Frobenius extends on it. We use the same symbol σ for Frobenius on the extension.

q^{λ} : an element of K with log_{q}|q^{λ}| = −λ for a rational number λ such that σ(q^{λ}) = q^{λ}. Such a q^{λ} always exists if the residue field k is algebraically closed and λ ∈ log_{q}|K^{×}|. In particular, if k is algebraically closed, then there exists an extension L of K as a discrete valuation field with an extension of Frobenius such that q^{λ} is contained in L for a fix λ. In this paper we freely extend K as above.

A_{K}(c, r^{−}) : the K-algebra of analytic functions on the open disk of radius r at the center c, i.e.,

A_{K}(c, r^{−}) = \{ ∑_{n=0}^{∞} a_{n}(x − c)^{n} ∈ K[[x − c]] | \begin{matrix}|a_{n}|γ^{n} → 0 as n → ∞ \\ for any 0 < γ < r \end{matrix} \} .

K[[x]]_{0} : the ring of bounded power series over K, i.e.,

K[[x]]_{0} = { ∑_{n=0}^{∞} a_{n}x^{n} ∈ A_{K}(0, 1^{−}) | sup_{n} |a_{n}| < ∞} .

An element of K[[x]]_{0} is said to be a bounded function.

K[[x]]_{λ} : the Banach K-module of power series of log-growth λ in A_{K}(0, 1^{−}) for a nonnegative real number λ ∈ ℝ_{≥0}, i.e.,

K[[x]]_{λ} = { ∑_{n=0}^{∞} a_{n}x^{n} ∈ A_{K}(0, 1^{−}) | sup_{n} |a_{n}|/(n + 1)^{λ} < ∞} ,

with a norm | ∑_{n=0}^{∞} a_{n}x^{n}|_{λ} = sup_{n} |a_{n}|/(n + 1)^{λ}. K[[x]]_{λ} is a K[[x]]_{0}- modules. K[[x]]_{λ} = 0 for λ < 0 for the convenient. An element f ∈ K[[x]]_{λ} which is not contained in K[[x]]_{γ} for γ < λ is said to be exactly of loggrowth λ.

E : the p-adic completion of the field of fractions of K[[x]]_{0} under the Gauss norm | |_{0}, i.e.,

E = { ∑_{n=−∞}^{∞} a_{n}x^{n} | a_{n} ∈ K, sup_{n} |a_{n}| < ∞, |a_{n}| → 0 (as n → −∞)} .

E is a complete discrete valuation field under the Gauss norm | |_{0} in fact K is discrete valuated. The residue field of the ring O_{E} of integers of E is k((x)).

t : a generic point of radius 1.

E_{t} : the valuation field corresponding to the generic point t, i.e., the same field as E in which x is replaced by t: we emphasize t in the notation with the respect to [CT09]. We regard the Frobenius σ as a Frobenius on E_{t}.

E_{t}[[X − t]]_{0} : the ring of bounded functions in A_{E_{t}}(t, 1^{−}). Then

τ : E → E_{t}[[X − t]]_{0} τ(f) = ∑_{n=0}^{∞} \frac{1}{n!}(\frac{d^{n}}{dx^{n}} f)|_{x=t}(X − t)^{n}

is a K-algebra homomorphism which is equivariant under the derivations \frac{d}{dx} and \frac{d}{dX}. The Frobenius σ on E_{t}[[X − t]]_{0} is defined by σ on E_{t} and σ(X − t) = τ(σ(x)) − σ(x)|_{x=t}. τ is again σ-equivariant.

For a function f on R and for a matrix A = (a_{ij}) with entries in R, we define f(A) = (f(a_{ij})). In case where f is a norm | |, then |A| = sup_{i,j} |a_{ij}|. We use 1 (resp. 1_{r}) to denote the unit matrix of suitable degree (resp. of degree r). For a decreasing filtration {V^{λ}} indexed by the set ℝ of real numbers, we put

V^{λ−} = ∩_{μ<λ} V^{μ}, V^{λ+} = ∪_{μ>λ} V^{μ}.

We denote by W_{λ−} = ∪_{μ<λ} W_{μ} and W_{λ+} = ∩_{μ>λ} W_{μ} the analogous objects for an increasing filtration {W_{λ}}_{λ}, respectively.

2.2 Terminology

We recall some terminology and results from [CT09]. Let R be either K (K might be E) or K[[x]]_{0}. A φ-module over R consists of a free R-module M of finite rank and an R-linear isomorphism φ : σ^{∗}M → M. For a φ-module over K, there is an increasing filtration {S_{λ}(M)}_{λ∈ℝ} which is called the Frobenius slope filtration. Then there is a sequence λ_{1} < ⋯ < λ_{r} of real numbers, called the Frobenius slopes of M, such that S_{λ_{i}}(M)/S_{λ_{i}−}(M) is pure of slope λ_{i} and M ⊗ K̂^{al} ≅ ⊕_{i} S_{λ_{i}}(M) ⊗_{K} K̂^{al}/S_{λ_{i}−}(M) ⊗_{K} K̂^{al} is the Dieudonné-Manin decomposition as φ-modules over K̂^{al}. We call λ_{1} the first Frobenius slope and λ_{r} the highest Frobenius slope, respectively. Let R be either E or K[[x]]_{0}. A φ-∇-module over R consists of a φ-module (M, φ) over R and a K-connection ∇ : M → M ⊗_{R} Ω_{R}, where Ω_{R} = Rdx, such that φ ∘ σ^{∗}(∇) = ∇ ∘ φ. For a basis (e_{1}, ⋯, e_{r}), the matrices A and G with entries R,

φ(1 ⊗ e_{1}, ⋯, 1 ⊗ e_{r}) = (e_{1}, ⋯, e_{r})A, ∇(e_{1}, ⋯, e_{r}) = (e_{1}, ⋯, e_{r})Gdx are called the Frobenius matrix and the connection matrix of R, respectively. Then one has

\frac{d}{dx}A + GA = (\frac{d}{dx}σ(x))Aσ(G)  (FC)

by the horizontality of φ. We denote the dual of M by M^{∨} . Let M be a φ-∇-module over K[[x]]_{0}. We define the K-space

V (M) = {s ∈ M ⊗_{K[[x]]_{0}} A_{K}(0, 1^{−}) | ∇(s) = 0}

of horizontal sections and the K-space of solutions,

Sol(M) = Hom_{K[[x]]_{0}[∇]}(M, A_{K}(0, 1^{−})),

on the unit disk. Both dim_{K} V (M) and dim_{K} Sol(M) equal to rank_{K[[x]]_{0}} M by the solvability. If one fixes a basis of M, the solution Y of the equations

\{ \begin{matrix}A(0)σ(Y) = Y A \\ \frac{d}{dx}Y = Y G \\ Y (0) = 1 \end{matrix}

in A_{K}(0, 1^{−}) is a solution matrix of M, where A(0) and Y (0) are the constant terms of A and Y, respectively. The log-growth filtration {V (M)^{λ}}_{λ∈ℝ} is defined by the orthogonal space of the K-space Sol_{λ}(M) = Hom_{K[[x]]_{0}[∇]}(M, K[[x]]_{λ}) under the natural bilinear perfect pairing

V (M) × Sol(M) → K.

Then V (M)^{λ} = 0 for λ >> 0 by the solvability of M and the log-growth filtration is a decreasing filtration of V (M) as φ-modules over K. The following proposition allows one to change the coefficient field K to a suitable extension K′.

Proposition 2.1 ([CT09, Proposition 1.10]) Let M be a φ-module over K[[x]]_{0}. For any extension K′ over K as a complete discrete valuation field with an extension of Frobenius, there is a canonical isomorphism V (M ⊗_{K[[x]]_{0}} K′[[x]]_{0}) ≅ V (M) ⊗_{K} K′ as log-growth filtered φ-modules.

The induced φ-∇-module M_{η} = M ⊗_{K[[x]]_{0}} E over E is said to be the generic fiber of M, and the K-module V (M) is called the special fiber of M.

Let M be a φ-∇-module over E. We denote by M_{τ} the induced φ-∇-module M ⊗_{E} E_{t}[[X − t]]_{0} over E_{t}[[X − t]]_{0}. Applying the theory of Robba [Ro75], we have a decreasing filtration {M^{λ}}_{λ∈ℝ} of M as φ-∇-modules over E which is characterized by the following universal property.

Theorem 2.2 [Ro75, 2.6, 3.5] (See [CT09, Theorem 3.2].) For any real number λ, M/M^{λ} is the maximum quotient of M such that all solutions of loggrowth λ of M_{τ} on the generic unit disk come from the solutions of (M/M^{λ})_{τ}.

The filtration {M^{λ}} is called the log-growth filtration of M. Note that M^{λ} = M for λ < 0 by definition and M^{λ} = 0 for λ >> 0 by the solvability. The quotient module M/M^{0} is called the bounded quotient, and, in particular, if M^{0} = 0, then M is called bounded.

Our main theorem in [CT09] is the following:

Theorem 2.3 ([CT09, Theorem 6.17])

(1) Let M be a φ-∇-module over E. If λ_{max} is the highest Frobenius slope of M, then M^{λ} ⊂ (S_{λ−λ_{max}}(M^{∨}))^{⊥}.

(2) Let M be a φ-∇-module over K[[x]]_{0}. If λ_{max} is the highest Frobenius slope of M_{η}, then V (M)^{λ} ⊂ (S_{λ−λ_{max}}(V (M^{∨})))^{⊥}.

Here S^{⊥} denotes the orthogonal space of S under the natural bilinear perfect pairing

M ⊗_{E} M^{∨} → E or V (M) ⊗_{K} V (M^{∨}) → K.

We conjectured that equalities hold in Theorem 2.3 if M is PBQ (Definition 5.1) in [CT09], and proved them if M is of rank ≤ 2 [CT09, Theorem 7.1, Corollary 7.2].

Conjecture 2.4 ([CT09, Conjectures 6.8]) Let M be a φ-∇-module over E.

(1) All breaks of log-growth filtration of M are rational and M^{λ} = M^{λ+} for any λ.

(2) Let λ_{max} be the highest Frobenius slope of M. If M/M^{0} is pure as φmodule (PBQ in Definition 5.1 (1)), then M^{λ} = (S_{λ−λ_{max}}(M^{∨}))^{⊥}.

We denote Conjecture 2.4 above by LGF_{E}.

Conjecture 2.5 ([CT09, Conjectures 6.9]) Let M be a φ-∇-module over K[[x]]_{0}.

(1) All breaks of log-growth filtration of V (M) are rational and V (M)^{λ} = V (M)^{λ+} for any λ.

(2) Let λ_{max} be the highest Frobenius slope of M_{η}. If M_{η}/M_{η}^{0} is pure as φ-module (PBQ in Definition 5.1 (2)), then V (M)^{λ} = (S_{λ−λ_{max}}(V (M)^{∨}))^{⊥}.

We denote Conjecture 2.5 above by LGF_{K[[x]]_{0}}.

Note that we formulate the theorem and the conjecture in the case where λ_{max} = 0 in [CT09]. However, the theorem holds for an arbitrary λ_{max} by Proposition 2.1 (and the conjecture should also hold). Moreover, it suffices to establish the conjecture when the residue field k of V is algebraically closed.

In section 7 we will reduce the conjecture LGF_{E} (1) (resp. LGF_{K[[x]]_{0}} (1)) to the conjecture LGF_{E} (2) (resp. LGF_{K[[x]]_{0}} (2)) by applying the proposition below to the PBQ filtration which is introduced in section 5. The following proposition is useful for attacking log-growth questions by induction.

Proposition 2.6 Let 0 → L → M → N → 0 be an exact sequence of φ-∇- modules over E (resp. K[[x]]_{0}) and let λ_{max} be the highest Frobenius slope of M and L (resp. M_{η} and L_{η}).

(1) Suppose that L^{λ} = (S_{λ−λ_{max}}(L^{∨}))^{⊥} for λ. Then the induced sequence

0 → L/L^{λ} → M/M^{λ} → N/N^{λ} → 0

is exact.

(2) Suppose that V (L)^{λ} = (S_{λ−λ_{max}}(V (L)^{∨}))^{⊥} for λ. Then the induced sequence

0 → V (L)/V (L)^{λ} → V (M)/V (M)^{λ} → V (N)/V (N)^{λ} → 0

is exact.

Proof. (1) Since

L/L^{λ} → M/M^{λ} → N/N^{λ} → 0

is right exact by [CT09, Proposition 3.6], we have only to prove the injectivity of the first morphism. There is an inclusion relation

M^{λ} ⊂ (S_{λ−λ_{max}}(M^{∨}))^{⊥} = S_{(λ_{max}−λ)−}(M)

by Theorem 2.3 and the equality

L^{λ} = (S_{λ−λ_{max}}(L^{∨}))^{⊥} = S_{(λ_{max}−λ)−}(L).

holds by our hypothesis on L. Since the Frobenius slope filtrations are strict for any morphism, the bottom horizontal morphism in the natural commutative diagram

L/L^{λ} ⟶ M/M^{λ}
=↓  ↓
L/S_{(λ_{max}−λ)−}(L) ⟶ M/S_{(λ_{max}−λ)−}(M)

is injective. Hence we have the desired injectivity.

(2) The proof here is similar to that of (1) on replacing [CT09, Proposition 3.6] by [CT09, Proposition 1.8]. □

2.3 Dwork’s conjecture

We recall Dwork’s conjecture. We have proved it in the case where M is of rank ≤ 2 [CT09, Corollary 7.3].

Conjecture 2.7 ([Dw73, Conjecture 2], [CT09, Conjecture 4.9]) Let M be a φ-∇-module over K[[x]]_{0}. Then the special log-growth is above the generic loggrowth polygon (with coincidence at both endpoints).

We denote Conjecture 2.7 above by LGF_{Dw}. We will prove that the conjecture LGF_{Dw} follows from the conjectures LGF_{E} and LGF_{K[[x]]_{0}} in section 8. There is also a version of Dwork’s conjecture without Frobenius structures, we denote it by LG_{Dw}. Let us recall the definition of the log-growth polygon: the generic log-growth polygon is the piecewise linear curve defined by the vertices

(0, 0), (dim_{E} \frac{M_{η}}{M_{η}^{λ_{1}+}} , λ_{1} dim_{E} \frac{M_{η}^{λ_{1}−}}{M_{η}^{λ_{1}+}} ), ⋯, (dim_{E} \frac{M_{η}}{M_{η}^{λ_{i}+}} , ∑_{j=1}^{i} λ_{j} dim_{E} \frac{M_{η}^{λ_{j} −}}{M_{η}^{λ_{j} +}} ),

⋯, (dim_{E} M_{η}, ∑_{j=1}^{r} λ_{j} dim_{E} \frac{M_{η}^{λ_{j} −}}{M_{η}^{λ_{j} +}} ),

where 0 = λ_{1} < ⋯ < λ_{r} are breaks (i.e., M^{λ−} ≠ M^{λ+}) of the log-growth filtration of M_{η}. The special log-growth polygon is defined in the same way using the log-growth filtration of V (M).

Remark 2.8 (1) The convention of André’s polygon of log-growth [An08] is different from ours. His polygon at the generic fiber is ∑_{j=1}^{r} λ_{j} dim_{E} \frac{M^{λ_{j} −}}{M^{λ_{j} +}} below our polygon in the direction of the vertical axis and the starting point of the polygon is (dim_{E} M, 0), and the same at the special fiber. André proved the conjecture LG_{Dw} except the coincidence of both endpoints in [An08].

(2) If the special log-growth polygon lies above the generic log-growth polygon in both conventions of André’s and ours, then both endpoints coincide with each other. However even if this is the case, we cannot prove M_{η}^{λ} = M_{η}^{λ+} (resp. V (M)^{λ} = V (M)^{λ+}) for a break λ.

3 Choices of Frobenius

Let us recall the precise form of equivalence between categories of φ-∇-modules with respect to different choices of Frobenius on E (resp. K[[x]]_{0}) (see [Ts98a, Section 3.4] for example). We will use it in the next section.

3.1 Comparison morphism ϑ_{σ_{1},σ_{2}}

Let σ_{1} and σ_{2} be Frobenius maps on E (resp. K[[x]]_{0}) such that the restriction of each σ_{i} to K is the given Frobenius on K. Let M be a φ-∇-module. We define an E-linear (resp. K[[x]]_{0}-linear) morphism

ϑ_{σ_{1},σ_{2}} : σ_{1}^{∗}M → σ_{2}^{∗}M

by

ϑ_{σ_{1},σ_{2}}(a ⊗ m) = a ∑_{n=0}^{∞} (σ_{2}(x) − σ_{1}(x))^{n} ⊗ \frac{1}{n!}∇(\frac{d^{n}}{dx^{n}})(m).

Since M is solvable and |σ_{2}(x) − σ_{1}(x)| < 1, the right hand side converges in σ_{2}^{∗}M. As a matrix representation, the transformation matrix is

H = ∑_{n=0}^{∞} σ_{2}(G_{n})\frac{(σ_{2}(x) − σ_{1}(x))^{n}}{n!}

for the induced basis 1 ⊗ e_{1}, ⋯, 1 ⊗ e_{r}, where G is the matrix of connection, G_{0} = 1 and G_{n+1} = GG_{n} + \frac{d}{dx}G_{n} for n ≥ 0.

Proposition 3.1 Let σ_{1}, σ_{2}, σ_{3}, σ be Frobenius maps of E (resp. K[[x]]_{0}) as above. Then we have the cocycle conditions:

(1) ϑ_{σ_{2},σ_{3}} ∘ ϑ_{σ_{1},σ_{2}} = ϑ_{σ_{1},σ_{3}}.

(2) ϑ_{σ,σ} = id_{σ^{∗}M}.

Proposition 3.2 Let M be a φ-∇-module pure of slope λ over E and let A be the Frobenius matrix of M with respect to a basis. Suppose that |A−q^{λ}1|_{0} ≤ q^{−μ} for μ ≥ λ. Then the representation matrix H of the comparison morphism ϑ_{σ_{1},σ_{2}} with respect to the bases which are the pull-backs by σ_{1} and σ_{2} respectively, satisfies |H − 1|_{0} < q^{λ−μ}.

Proof. By replacing the Frobenius φ by q^{−λ}φ, we may assume that λ = 0. The assertion then follows from the fact that under these assumptions the solution matrix Y at the generic point satisfies Y ≡ 1 (mod (X −t)m^{n}O_{E_{t}}[[X − t]]). Here n is the least integer such that |m^{n}| ≤ q^{−μ}. □

3.2 Equivalence of categories

Let R be either E or K[[x]]_{0} and let σ_{1} and σ_{2} be Frobenius maps on R as in the previous subsection. We define a functor

ϑ_{σ_{1},σ_{2}}^{∗} : (φ-∇-modules over (R, σ_{2})) → (φ-∇-modules over (R, σ_{1}))

by (M, ∇, φ) ↦ (M, ∇, φ ∘ ϑ_{σ_{1},σ_{2}}). Here ϑ_{σ_{1},σ_{2}} is defined as in the previous section. The propositions of the previous subsection then give

Theorem 3.3 ϑ_{σ_{1},σ_{2}}^{∗} is an equivalence of categories which preserves tensor products and duals. Moreover, ϑ_{σ_{1},σ_{2}}^{∗} preserves the Frobenius slope filtration and the log-growth filtration of M (resp. V (M)) for a φ-∇-module M over E (resp. K[[x]]_{0}).

4 Boundedness and splitting of the Frobenius slope filtration

4.1 Splitting theorem

Theorem 4.1 Suppose that the residue field k of V is perfect. A φ-∇-module M over E is bounded if and only if M is a direct sum of pure φ-∇-modules, that is,

M ≅ ⊕_{i=1}^{r} S_{λ_{i}}(M)/S_{λ_{i}−}(M)

as φ-∇-modules, where λ_{1} < λ_{2} < ⋯ < λ_{r} are Frobenius slopes of M.

Since any pure φ-∇-module over E is bounded by [CT09, Corollary 6.5]. Hence, Theorem 4.1 above follows from the next proposition.

Proposition 4.2 Suppose that the residue field k of V is perfect. Let 0 → L → M → N → 0 be an exact sequence of φ-∇-modules over E such that both L and N are pure of Frobenius slope λ and ν, respectively. If one of the conditions

(1) ν − λ < 0;

(2) ν − λ > 1;

(3) M is bounded and 0 < ν − λ ≤ 1,

holds, then the exact sequence is split, that is, M ≅ L ⊕ N as φ-∇-modules.

In the case (1) the assertion easily follows from the fact that, for a ∈ E with |a|_{0} < 1, aσ is a contractive operator on the p-adic complete field E. The rest of this section will be dedicated to proving the assertion in cases (2) and (3).

4.2 Descent of splittings

Proposition 4.3 Let 0 → L → M → N → 0 be an exact sequence of φmodules over E such that L and N are pure and the two slopes are different. Let E′ be one of the following:

(i) E′ is a p-adic completion of an unramified extension of E;

(ii) E′ is the p-adic completion of E ⊗_{K} K′ for some extension K′ of K as a complete discrete valuation field with an extension σ′ of σ such that, if G is the group of continuous automorphisms of K′ over K, then the invariant subfield of K′ by the action of G is K.

If the exact sequence is split over E′, then it is split over E. The same holds for φ-∇-modules over E.

Proof. In each case we may assume that E is the invariant subfield of E′ by the action of continuous automorphism group G. Let e_{1}, ⋯, e_{r}, e_{r+1}, ⋯, e_{r+s} be a basis of M over E such that e_{1}, ⋯, e_{r} is a basis of L. Put

φ(e_{1}, ⋯, e_{r}, e_{r+1}, ⋯, e_{r+s}) = (e_{1}, ⋯, e_{r}, e_{r+1}, ⋯, e_{r+s}) ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) ,

where A_{11} is of degree r and A_{22} is of degree s, respectively, and all entries of A_{11}, A_{12} and A_{22} are contained in E. By the hypothesis of splitting over E′ there exists a matrix Y with entries in E′ such that

A_{11}σ(Y) − Y A_{22} + A_{12} = 0.

For any ρ ∈ G, ρ(Y) also gives a splitting. Hence A_{11}σ(Y − ρ(Y)) = (Y − ρ(Y))A_{22}. By the assumption on slopes, ρ(Y) = Y. Therefore, all entries of Y are contained in E and the exact sequence is split over E. □

Definition 4.4 An extension E′ (resp. K′) of E (resp. K) is allowable if E′ is a finitely successive extension of E (resp. K) of type in (i) or (ii) (resp. (ii)) of Proposition 4.3.

4.3 Preparations

In this subsection we assume that the residue field k of V is algebraically closed. Moreover we assume that the Frobenius on E (resp. K[[x]]_{0}) is defined by σ(x) = x^{q}. For an element a = ∑_{n} a_{n}x^{n} in E (resp. K[[x]]) we define the subseries a^{(q)} by ∑_{n} a_{qn}x^{qn}.

Lemma 4.5 Let ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) be an invertible matrix of degree r + s over E

(resp. K[[x]]_{0}) with A_{11} of degree r and A_{22} of degree s such that the matrix satisfies the conditions:

(i) A_{11} = A_{11}^{(q)} and A_{11} = P^{−1} for a matrix P over E (resp. K[[x]]_{0}) with |P |_{0} < 1,

(ii) A_{22} = A_{22}^{(q)} and |A_{22} − 1_{s}|_{0} < 1.

Suppose that A_{12}^{(q)} ≠ 0. Then there exists an r × s matrix Y over E (resp. K[[x]]_{0}) with |Y |_{0} < |A_{12}^{(q)}|_{0} such that, if one puts B = A_{11}σ(Y) − Y A_{22} + A_{12}, then |B^{(q)}|_{0} < |A_{12}^{(q)}|_{0}. Moreover, there exists an r × s matrix Y over E (resp. K[[x]]_{0}) such that if one defines B_{12} by

( \begin{matrix}A_{11} & B_{12} \\ 0 & A_{22} \end{matrix} ) = ( \begin{matrix}1_{r} & −Y \\ 0 & 1_{s} \end{matrix} ) ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) ( \begin{matrix}1_{r} & σ(Y) \\ 0 & 1_{s} \end{matrix} ) ,

then B_{12}^{(q)} = 0.

Proof. Take a matrix Y such that σ(Y) = −P A_{12}^{(q)}. Such a Y exists since the residue field k of V is perfect. Then |Y |_{0} < |A_{12}^{(q)}|_{0} and B = A_{11}σ(Y) − Y A_{22} + A_{12} = A_{11}P A_{12}^{(q)} − Y A_{22} + A_{12} = A_{12} − A_{12}^{(q)} − Y A_{22}. Hence |B^{(q)}|_{0} = |Y A_{22}^{(q)}|_{0} < |A_{12}^{(q)}|_{0} and we have the first assertion. Applying the first assertion inductively on the value |A_{12}^{(q)}|_{0}, we have a desired matrix Y of the second assertion since E (resp. K[[x]]_{0}) is complete under the norm | |_{0}. □

We give a corollary of the preceding lemma for φ-∇-modules over E.

Proposition 4.6 Let 0 → L → M → N → 0 be an exact sequence of φ-∇- modules over E. Suppose that N is pure of Frobenius slope ν and all Frobenius slopes of L are less than ν. Then there exist an allowable extension E′ of E and a basis e_{1}, ⋯, e_{r}, e_{r+1}, ⋯, e_{r+s} of M ⊗_{E} E′ with respect to the exact sequence such that, if one fixes an element x′ in the ring O_{E′} of integers of E′ whose image gives a uniformizer of the residue field of O_{E′} and a Frobenius σ′ on E′

with σ′(x′) = x′^{q}, then the Frobenius matrix ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) of M ⊗_{E} E′ with

respect to σ′ (here we use Theorem 3.3) has the following form:

(i) A_{11} = A_{11}^{(q)} and A_{11} = P^{−1} for a matrix |P |_{0} < q^{ν},

(ii) A_{22} = A_{22}^{(q)} and |A_{22} − q^{ν}1_{s}|_{0} < q^{−ν},

(iii) A_{12}^{(q)} = 0,

where a^{(q)} is defined by using the parameter x′. Moreover, one can replaces the inequality |A_{22} − q^{ν}1_{s}|_{0} < q^{−ν} in (ii) by the inequality |A_{22} − q^{ν}1_{s}|_{0} < q^{−ν}η for a given 0 < η ≤ 1 (the extension E′ depends on η).

Proof. Since k is algebraically closed, there is a uniformizer π of K such that σ(π) = π. Let K_{m} be a Galois extension K(π^{1/m}, ζ_{m}) of K for a positive integer m, where ζ_{m} denotes a primitive m-th root of unity. Then σ on K extends on K_{m}. If we choose a positive integer m such that m/log_{q}|π| is a common multiple of denominators of ν and the highest Frobenius slope of L, then ν and the highest Frobenius slope of L are contained in log_{q}|K_{m}^{×}|. Hence we may assume that ν = 0 and all Frobenius slopes of the twist πφ_{L} of the Frobenius φ_{L} of L are less than or equal to 0.

Let A = ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) be a Frobenius matrix of M with respect to the

given exact sequence. Since any φ-module over E has a cyclic vector [Ts96, Proposition 3.2.1], we may assume that A_{22} ∈ GL_{s}(O_{E}) by ν = 0. Then there is a matrix X ∈ GL_{s}(O_{E′}) such that X^{−1}A_{22}σ(X) ≡ 1_{s} (mod mO_{E′}) for some finite unramified extension E′ over E by [Ts98b, Lemma 5.2.2]. By applying the existence of a cyclic vector again, we may assume that the all entries of Frobenius matrix of L^{∨} are contained in mO_{E} by the hypothesis on Frobenius slopes of L.

Now we fix a parameter x′ of E′ and change a Frobenius σ′ on E′ such that σ′(x′) = x′^{q}. The the hypothesis of the matrices A_{11} and A_{12} are stable by Theorem 3.3. If one replaces the basis (e_{1}, ⋯, e_{r+s}) by (e_{1}, ⋯, e_{r+s})A, then the Frobenius matrix becomes σ′(A). Since the hypothesis in Lemma 4.5 hold in our Frobenius matrix A, we have the assertion. □

Now a variant of Proposition 4.6 for φ-∇-modules over K[[x]]_{0}, which we use it in section 6, is given.

Proposition 4.7 Let 0 → L → M → N → 0 be an exact sequence of φ- ∇-modules over K[[x]]_{0}. Suppose that N_{η} is pure of Frobenius slope ν and all Frobenius slopes of L_{η} are less than ν. Then there exist an allowable extension K′ of K with an extension of Frobenius σ′ and a basis e_{1}, ⋯, e_{r}, e_{r+1}, ⋯, e_{r+s} of M ⊗_{K[[x]]_{0}} K′[[x]]_{0} with respect to the exact sequence such that the Frobenius

matrix ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) of M ⊗_{K[[x]]_{0}} K′[[x]]_{0} with respect to σ′ has the following

form:

(i) A_{11} = A_{11}^{(q)} and A_{11} = P^{−1} for a matrix |P |_{0} < q^{ν},

(ii) A_{22} = q^{ν}1_{s},

(iii) A_{12}^{(q)} = 0

Proof. We may assume μ = 0 and the highest Frobenius slope of L_{η} is contained in log_{q}|K_{m}^{×}| as in the proof of Proposition 4.6. Then N is a direct sum of copies of the unit object (K[[x]]_{0}, d, σ)’s since k is algebraically closed. In order to find the matrix P, we apply the isogeny theorem [Ka79, Theorem 2.6.1] and the existence of a free lattice over V[[x]] in [dJ98, Lemma 6.1] for L^{∨}. The rest is again same as the proof of Proposition 4.6. □

Lemma 4.8 Let ν be a nonnegative rational number. Suppose that y ∈ xK[[x]] satisfies a Frobenius equation

y − q^{−ν}aσ(y) = f.

for a ∈ K with |a| = 1 and for f = ∑_{n} f_{n}x^{n} ∈ xK[[x]].

(1) Suppose that f^{(q)} = 0. If f ∈ K[[x]]_{ν} \ {0}, then y ∈ K[[x]]_{ν} \ K[[x]]_{ν−}, and if f ∈ K[[x]]_{λ} \ K[[x]]_{λ−} for λ > ν, then y ∈ K[[x]]_{λ} \ K[[x]]_{λ−}.

(2) Let l be a nonnegative integer with q |̸ l. If f ∈ K[[x]]_{0} and |f_{l}| > |q^{ν}f|_{0} = q^{−ν}|f|_{0} ≠ 0, then y ∈ K[[x]]_{ν} \ K[[x]]_{ν−}.

Proof. Since the residue field k of V is algebraically closed, we may assume that a = 1. Formally in K[[x]],

y = ∑_{n} ∑_{m=0}^{∞} (q^{−ν})^{m}σ^{m}(f_{n})x^{q^{m}n}

is a solution of the equation.

(1) If q^{m}n = q^{m′}n′, then m = m′ and n = n′ because q |̸ n, n′. Hence, y ≠ 0. By considering a subseries ∑_{m=0}^{∞} (q^{−ν})^{m}σ^{m}(f_{n})x^{q^{m}n} for f_{n} ≠ 0, y is of log-growth equal to or greater than ν. Moreover, we have

|(q^{−ν})^{m}σ^{m}(f_{n})|/(q^{m}n + 1)^{ν} = |f_{n}|/(n + 1/q^{m})^{ν}

Hence, if f ∈ K[[x]]_{ν}, then y is exactly of log-growth ν. Suppose f ∈ K[[x]]_{λ} \ K[[x]]_{λ−}. Since for each m, n

|(q^{−ν})^{m}σ^{m}(f_{n})|/(q^{m}n + 1)^{λ} = |f_{n}|/(q^{m(1−ν/λ)}n + 1/q^{mν/λ})^{λ},

the log-growth of y is exactly λ.

(2) There exists z ∈ xK[[x]]_{0} with |z|_{0} ≤ |q^{ν}f|_{0} = q^{−ν}|f|_{0} such that, if g = f −z +q^{−ν}σ(z) = ∑_{n} g_{n}x^{n}, then g^{(q)} = 0 and g_{l} ≠ 0 by the same construction of the proof of Lemma 4.5. The assertion now follows from (1). □

4.4 Proof of Proposition 4.2

Replacing K by an extension, we may assume that k is algebraically closed and that λ = 0, ν > 0 and ν ∈ log_{q}|K^{×}| by Proposition 4.3 (see the beginning of proof of Proposition 4.6). We may also assume σ(x) = x^{q} by Theorem 3.3.

Let A = ( \begin{matrix}A_{11} & A_{12} \\ 0 & A_{22} \end{matrix} ) be a Frobenius matrix of M with respect to the basis

which is compatible with the given extension (i.e., the (1, 1)-part (resp. (2, 2)-

part) corresponds to L (resp. N)) and let G = ( \begin{matrix}G_{11} & G_{12} \\ 0 & G_{22} \end{matrix} ) be the matrix of

the connection, respectively. The commutativity of Frobenius and connection (the relation (FC) in section 2.2) gives the relation

1^{∘}  \frac{d}{dx}A_{12} + G_{11}A_{12} + G_{12}A_{22} = qx^{q−1}(A_{11}σ(G_{12}) + A_{12}σ(G_{22}))

of the (1, 2)-part of the matrix. We may assume that

2^{∘} A_{11} = A_{11}^{(q)}, |A_{11} − 1_{r}|_{0} ≤ q^{−1} and hence |G_{11}|_{0} < q^{−1} (r is rank of L);

3^{∘} A_{22} = A_{22}^{(q)}, |A_{22} − q^{ν}1_{s}|_{0} ≤ q^{−ν−1} and |G_{22}|_{0} < q^{−1} (s is rank of N);

4^{∘} A_{12}^{(q)} = 0.

by Proposition 4.6 Note that both inequalities |G_{11}|_{0} < q^{−1} and |G_{11}|_{0} < q^{−1} above follow from the relation (FC) in section 2.2 for L and N, respectively. When ν ≠ 1, we will first prove A_{12} = 0 and then prove G_{12} = 0. When ν = 1, we will first prove G_{12} = 0 and then prove A_{12} = 0. Hence, we will have a splitting in all cases.

4.4.1 The case where ν > 1

Suppose ν > 1 (and λ = 0). Assume that A_{12} ≠ 0. By 4^{∘} we have |\frac{d}{dx}A_{12}|_{0} > |qA_{12}|_{0} = q^{−1}|A_{12}|_{0}. Then |G_{11}A_{12}|_{0} < q^{−1}|A_{12}|_{0} < |\frac{d}{dx}A_{12}|_{0} and |qx^{q−1}A_{12}σ(G_{22})|_{0} < q^{−1}|A_{12}|_{0} < |\frac{d}{dx}A_{12}|_{0}. On the other hand, |G_{12}A_{22}|_{0} < |qx^{q−1}A_{11}σ(G_{12})|_{0} by ν > 1 since A_{11} (resp. A_{22}) is a unit matrix (resp. a unit matrix times q^{ν}) modulo mO_{E} (resp. q^{ν}mO_{E}) by 2^{∘} (resp. 3^{∘}). So we have

\frac{d}{dx}A_{12} ≡ qx^{q−1}A_{11}σ(G_{12}) (mod q^{−log_{q}| \frac{d}{dx} A_{12}|_{0}}mO_{E})

But, on comparing the x-adic order of both sides, this is seen to be impossible by 2^{∘}, 3^{∘} and 4^{∘}. Hence A_{12} = 0. Now the commutativity of Frobenius and connection (the relation 1^{∘}) is just

G_{12}A_{22} = qx^{q−1}A_{11}σ(G_{12}).

Since any morphism between pure φ-modules with different Frobenius slopes are 0, we have G_{12} = 0 by ν > 1.

4.4.2 The case where 0 < ν < 1

Suppose 0 < ν < 1 (and λ = 0). Assuming that A_{12} ≠ 0, we will show the existence of unbounded solutions on the generic disk by applying Lemma 4.8 (2). This is a contradiction to our hypothesis of boundedness of M, and thus we must have A_{12} = 0. Since ν ≠ 1, we again have G_{12} = 0 by the slope reason. Therefore, the extension is split. Assume that A_{12} = ∑_{n} A_{12,n}x^{n} ≠ 0. Since |G_{12}A_{22}|_{0} = q^{−ν}|G_{12}|_{0}, |qx^{q−1}A_{11}σ(G_{12})|_{0} = q^{−1}|G_{12}|_{0}, and |\frac{d}{dx}A_{12}|_{0} > q^{−1}|A_{12}|_{0} by 3^{∘}, 2^{∘} and our hypothesis, respectively, the formula 4^{∘} gives estimates

5^{∘} q^{−1}|A_{12}|_{0} < q^{−ν}|G_{12}|_{0} = |G_{12}A_{22}|_{0} = |\frac{d}{dx}A_{12}|_{0} ≤ |A_{12}|_{0}.

We also claim that

6^{∘} there is a positive integer m with q |̸ m such that |\frac{1}{m!} \frac{d^{m}}{dx^{m}} A_{12}|_{0} = |A_{12}|_{0}

by 1^{∘}. Indeed, let l be an integer such that |A_{12,l}| = |A_{12}|_{0}. When l > 0, we put m = l. Then the coefficient of \frac{1}{m!} \frac{d^{l}}{dx^{l}} A_{12} in the 0-th term x^{0} is A_{12,l} and we have |\frac{1}{l!} \frac{d^{l}}{dx^{l}} A_{12}|_{0} = |A_{12,l}| = |A_{12}|_{0}. When l < 0, we put m = q^{−l} + l (remark that any sufficient large power of q can be replaced by q^{−l}). Then the coefficient

of \frac{1}{m!} \frac{d^{m}}{dx^{m}} A_{12} in the l −m(= −q^{−l})-th term x^{l−m} is (−1)^{m} ( \begin{matrix}m − l − 1 \\ m \end{matrix} ) A_{12,l}

and we have |\frac{1}{m!} \frac{d^{m}}{dx^{m}} A_{12}|_{0} = |A_{12,l}| = |A_{12}|_{0} since (−1)^{m} ( \begin{matrix}m − l − 1 \\ m \end{matrix} ) is a

p-adic unit. In proving the assertion, we will consider the following two cases for A_{12}:

(i) |\frac{d}{dx}A_{12}|_{0} > q^{−ν}|A_{12}|_{0}.

(ii) |\frac{d}{dx}A_{12}|_{0} ≤ q^{−ν}|A_{12}|_{0}. (Hence we have |G_{12}|_{0} ≤ |A_{12}|_{0} by 5^{∘})

In order to prove the existence of unbounded solutions above, let us reorganize the matrix representation by using changes of basis of M, a change of Frobenius and an extension of scalar field. Let us consider the induced φ-∇-module M_{τ} = M ⊗_{E} E_{t}[[X − t]]_{0} over the bounded functions E_{t}[[X − t]]_{0} at the generic disk. Since L_{τ} and N_{τ} are pure, we have bounded solution matrices Y_{11} of L and Y_{22} of N, that is,

7^{∘} \begin{matrix}L : \{ \begin{matrix}A_{11}(t)σ(Y_{11}) = Y_{11}τ(A_{11}) \\ \frac{d}{dX} Y_{11} = Y_{11}τ(G_{11}) \\ Y_{11} ∈ 1_{r} + q(X − t)Mat_{r}(O_{E_{t}}[[X − t]]) \end{matrix} \\ N : \{ \begin{matrix}A_{22}(t)σ(Y_{22}) = Y_{22}τ(A_{22}) \\ \frac{d}{dX} Y_{22} = Y_{22}τ(G_{22}) \\ Y_{22} ∈ 1_{s} + q(X − t)Mat_{s}(O_{E_{t}}[[X − t]]) \end{matrix} \end{matrix}

by 2^{∘} and 3^{∘}. Note that τ(f) = ∑_{n} \frac{1}{n!}(\frac{d^{n}}{dx^{n}} f)(t)(X − t)^{n} for f ∈ E and it is an isometry. Consider a change of basis of M_{τ} by the matrix Y^{−1} =

( \begin{matrix}Y_{11}^{−1} & 0 \\ 0 & Y_{22}^{−1} \end{matrix} ). Then the new Frobenius matrix and the new connection

matrix are as follows:

\begin{matrix}A^{τ} = Y Aσ(Y)^{−1} = ( \begin{matrix}A_{11}(t) & Y_{11}τ(A_{12})σ(Y_{22})^{−1} \\ 0 & A_{22}(t) \end{matrix} ) \\ G^{τ} = Y \frac{d}{dX} Y^{−1} + Y GY^{−1} = ( \begin{matrix}0 & Y_{11}τ(G_{12})Y_{22}^{−1} \\ 0 & 0 \end{matrix} ) . \end{matrix}

Let us put A_{12}^{τ} = ∑_{n} A_{12,n}^{τ}(X − t)^{n} (resp. G_{12}^{τ}) to be the (1, 2)-part of the Frobenius matrix A^{τ} (resp. G^{τ}), and define B_{12}^{τ} = ∑_{n>0} A_{12,n}^{τ}(X − t)^{n} by the subseries of positive powers. Then we have

8^{∘} |B_{12}^{τ}|_{0} = |A_{12}|_{0}

9^{∘} |G_{12}^{τ}|_{0} = |τ(G_{12})|_{0} = |G_{12}|_{0}.

by 6^{∘} and Y ≡ 1_{r+s} (mod q(X − t)O_{E_{t}}[[X − t]]). Now we consider a change of Frobenius. At first our Frobenius on E is given by σ(x) = x^{q}. Hence the induced Frobenius on the generic disk is given by σ(X − t) = ((X − t) + t)^{q} − t^{q}. Let us replace σ by the Frobenius σ̃ defined by σ̃(X − t) = (X − t)^{q}. Note that

10^{∘} σ(X − t) − σ̃(X − t) ≡ qt^{q−1}(X − t) (mod p(X − t)^{2}O_{E_{t}}[[X − t]]).

Since |\frac{1}{n!} \frac{d^{n−1}}{dX^{n−1}} G_{12}^{τ}|_{0} ≤ |n|^{−1}|G_{12}|_{0} and |p^{n}/n| ≤ |p| for all n ≥ 1, the matrix H of comparison transform ϑ_{σ̃,σ}^{∗}(M_{τ}) in section 3.1 satisfies the congruence relation

11^{∘} \begin{aligned}H & = & 1_{r+s} + ∑_{n=1}^{∞} \frac{1}{n!} ( \begin{matrix}0 & σ(\frac{d^{n−1}}{dX^{n−1}} G_{12}^{τ}) \\ 0 & 0 \end{matrix} ) (σ(X − t) − σ̃(X − t))^{n} \\ & ≡ & 1_{r+s} + qt^{q−1} ( \begin{matrix}0 & σ(G_{12}(t)) \\ 0 & 0 \end{matrix} ) (X − t) \\ & & (mod pq^{−log_{q}|G_{12}|_{0}}(X − t)^{2}O_{E_{t}}[[X − t]]) \end{aligned}

by 9^{∘} and 10^{∘}. Our Frobenius matrix of M_{τ} with respect to the Frobenius σ̃ is

Ã = A^{τ} H = ( \begin{matrix}A_{11}(t) & A_{12}^{τ} + A_{11}(t)H_{12} \\ 0 & A_{22}(t) \end{matrix} )

by the definition of the equivalence (Theorem 3.3), where H_{12} = ∑_{n} H_{12,n}(X− t)^{n} is the (1, 2)-part of H. If we put Ã_{12} = ∑_{n} Ã_{12,n}(X − t)^{n} to be the (1, 2)- part of Ã and put B̃_{12} = ∑_{n>0} Ã_{12,n}(X − t)^{n}, then

12^{∘} there is a positive integer m with q |̸ m such that |Ã_{12,m}|_{0} > q^{−ν}|B̃_{12}|_{0}.

Indeed, in the case (i) for A_{12}, since Ã_{12,1} = A_{12,1}^{τ} +A_{11}(t)H_{12,1} and |H_{12,1}|_{0} ≤ q^{−1}|G_{12}|_{0}, we have |Ã_{12,1}|_{0} = |A_{12,1}^{τ}|_{0} = |\frac{d}{dx}A_{12}|_{0} by 5^{∘} and 11^{∘}. On the other hand, |B̃_{12}|_{0} ≤ max\{|B_{12}^{τ}|_{0}, |H_{12}|_{0}\} ≤ max\{|A_{12}|_{0}, |p||G_{12}|_{0}\} < q^{ν}|\frac{d}{dx}A_{12}|_{0} by 5^{∘}, 8^{∘} and 11^{∘} because of our hypothesis (i), |\frac{d}{dx}A_{12}|_{0} > q^{−ν}|A_{12}|_{0}. Hence we can take m = 1. In the case (ii), we take a positive integer m such as 6^{∘}. Since |G_{12}|_{0} ≤ |A_{12}|_{0} by the hypothesis (ii), we have |B̃_{12}|_{0} ≤ max{|B_{12}^{τ}|_{0}, |H_{12}|_{0}} = |A_{12}|_{0} by 8^{∘} and 11^{∘}. By Proposition 2.1 we may replace E_{t} by the p-adic completion Ê_{t}^{ur} of the maximally unramified extension of E_{t}. Then we may assume that Ã_{11} = 1_{r} and Ã_{22} = q^{ν}1_{s} since the solutions of both (1, 1)-part and (2, 2)-part is 1 modulo q

by 2^{∘} and 3^{∘}. The solution matrix of M_{τ} ⊗_{E_{t}} Ê_{t}^{ur} has a form Z = ( \begin{matrix}1_{r} & Z_{12} \\ 0 & 1_{s} \end{matrix} )

satisfying Ã|_{X=t}σ̃(Z) = ZÃ and Z_{12}|_{X=t} = 0. In particular, Z_{12} satisfies the relation

σ̃(Z_{12}) = q^{ν}Z_{12} + B̃_{12}.

On applying Lemma 4.8 (2) to Z_{12}, one sees that one of entries of Z_{12} must be exactly of log-growth ν by 12^{∘}. Hence the non-vanishing of A_{12} implies that M is unbounded. This completes the proof for the case 0 < ν < 1.

4.4.3 The case where ν = 1

Suppose that ν = 1. Suppose that G_{12} ≠ 0. Let us develop G_{12} = ∑_{n} G_{12,n}x^{n} and let m be the least integer such that |G_{12,m}| = |G_{12}|_{0}. If A_{12} ≠ 0, we have |\frac{d}{dx}A_{12}|_{0} > q^{−1}|A_{12}|_{0} by 4^{∘}. So the relation 1^{∘} induces a congruence

13^{∘} \frac{d}{dx}A_{12} + qG_{12} ≡ qx^{q−1}σ(G_{12}) (mod q^{1+log_{q}|G_{12}|_{0}}mO_{E})

by 2^{∘} and 3^{∘}. This congruence 13^{∘} also holds when A_{12} = 0. Suppose that m < −1. The least power of x which should appear in the right hand side of the congruence 13^{∘} above is qm + q − 1. Since qm + q − 1 < m, this is precluded by 4^{∘}. Suppose that m = −1. Then

14^{∘} \begin{matrix}τ(G_{12}) = ∑_{n=0}^{∞} \frac{1}{n!}(\frac{d^{n}}{dx^{n}} G_{12})(t)(X − t)^{n} \\ = ∑_{n=0}^{∞} (G_{12,−1} \frac{(−1)^{n}}{t^{n}} + q^{−log_{q}|G_{12}|_{0}}Mat(t^{1−n}V[[t]]_{0} + mO_{E}))(X − t)^{n}. \end{matrix}

Let us calculate the solution matrix of M_{τ} by using 7^{∘} as in the previous case.

By changing a basis of M_{τ} by the invertible matrix Y = ( \begin{matrix}Y_{11}^{−1} & 0 \\ 0 & Y_{22}^{−1} \end{matrix} ) as

before, our differential equation becomes

\frac{d}{dX} ( \begin{matrix}1 & Z \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}1 & Z \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & Y_{11}τ(G_{12})Y_{22}^{−1} \\ 0 & 0 \end{matrix} ) .

in Z, that is, \frac{d}{dX} Z = Y_{11}τ(G_{12})Y_{22}^{−1}. Since all the coefficients of all the power series which appear on the entries of Y_{11}τ(G_{12})Y_{22}^{−1} do not vanish modulo q^{−log_{q}|G_{12}|_{0}}mO_{E} by 7^{∘} and 14^{∘}, at least one of entries of Z is exactly of loggrowth 1. This contradicts to our hypothesis of boundness of M. Hence, m ≠ −1. Suppose that m > 0. Then we have

\begin{matrix}G_{12} ≡ −q^{−1}x^{−1} (x\frac{d}{dx}A_{12} + σ(x\frac{d}{dx}A_{12}) + σ^{2}(x\frac{d}{dx}A_{12}) + ⋯) \\ (mod q^{−log_{q}|G_{12}|_{0}}mO_{E}) \end{matrix}

by 4^{∘} and 13^{∘}. The case where A_{12} = 0 is impossible since G_{12} ≠ 0. If A_{12} ≠ 0, then we have a solution exactly of log-growth 1 on the generic disk by the similar construction in the case m = −1. This contradicts our hypothesis. Therefore, we have G_{12} = 0 in any case. Now we prove A_{12} = 0. Suppose that A_{12} ≠ 0. Then the relation 1^{∘} is

\frac{d}{dx}A_{12} + G_{11}A_{12} = qx^{q−1}A_{12}σ(G_{22}).

This is impossible by 2^{∘}, 3^{∘} and 4^{∘}. Hence, A_{12} = 0. This completes the proof of Proposition 4.2. □

Remark 4.9 There is another proof of Proposition 4.2: one can reduce Proposition 4.2 to the case where q = p, that is, the Frobenius σ is a p-Frobenius. Then, in the proof of the case 0 < ν < 1, it is enough to discuss only in the case |\frac{d}{dx}A_{12}|_{0} = |A_{12}|_{0}.

5 PBQ φ-∇-modules

5.1 Definition of PBQ φ-∇-modules

Definition 5.1 (Definition of “PBQ” φ-∇-modules)

(1) A φ-∇-module M over E is said to be pure of bounded quotient (called PBQ for simplicity) if M/M^{0} is pure as a φ-module.

(2) A φ-∇-module M over K[[x]]_{0} is said to be pure of bounded quotient (called PBQ for simplicity) if the generic fiber M_{η} of M is PBQ as a φ-∇-module over E.

The notion “PBQ” depends only on the Frobenius slopes of the bounded quotient of the generic fiber of φ-∇-modules. As we saw in Theorem 4.1, the bounded quotient of the generic fiber always admits a splitting by Frobenius slopes when it has different slopes.

Example 5.2 (1) A bounded φ-∇-module M over E is PBQ if and only if M is pure as a φ-module. In particular, any φ-∇-module M over E of rank 1 is PBQ.

(2) Any φ-∇-module M over E of rank 2 which is not bounded is PBQ [CT09, Theorem 7.1].

(3) Let us fix a Frobenius on σ with σ(x) = x^{q}. Let M be a φ-∇-module over K[[x]]_{0} with a basis (e_{1}, e_{2}, e_{3}) such that the Frobenius matrix A and the connection matrix G are defined by

A = ( \begin{matrix}1 & −q^{1/2}x & −qx \\ 0 & q^{1/2} & 0 \\ 0 & 0 & q \end{matrix} ) , G = ( \begin{matrix}0 & ∑_{n=0}^{∞} q^{n/2}x^{q^{n}−1} & ∑_{n=0}^{∞} x^{q^{n}−1} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

Then M_{η} is not bounded and M is not PBQ. Indeed, the K[[x]]_{0}-submodule L generated by e_{1} is a φ-∇-submodule of M such that the quotient (M/L)_{η} is bounded and (M/L)_{η} is not pure. On the other hand the dual M^{∨} of M is PBQ.

Proposition 5.3 Any quotient of PBQ φ-∇-modules over E (resp. K[[x]]_{0}) is PBQ.

Proof. Let M be a PBQ φ-∇-module over E and let M′ be a quotient of M. The assertion follows from that the natural morphism M/M^{0} → M′/(M′)^{0} is surjective by [CT09, Corollary 3.5]. □

5.2 Existence of the maximally PBQ φ-∇-submodules over E

Proposition 5.4 Suppose that the residue field k of V is perfect. Let M be a φ-∇-module over E with highest Frobenius slope λ_{max} and let N′ be a φ-∇- submodule of M/S_{λ_{max}−}(M). Then there is a unique φ-∇-submodule N of M such that N is PBQ and the natural morphism N/N^{0} → M/S_{λ_{max}−}(M) gives an isomorphism between N/N^{0} and N′. When N′ = M/S_{λ_{max}−}(M), we call the corresponding N the maximally PBQ submodule of M.

Proof. First we prove the uniqueness of N. Let N_{1} and N_{2} be a PBQ submodule of M such that both natural morphisms N_{1}/N_{1}^{0} → M/S_{λ_{max}−}(M) ← N_{2}/N_{2}^{0} give isomorphisms with N′. Let N be the image of N_{1} ⊕ N_{2} → M (a, b) ↦ a+b. Then N is PBQ by Proposition 5.3. Since N_{1}/N_{1}^{0} ⊕N_{2}/N_{2}^{0} → N/N^{0} is surjective by [CT09, Proposition 3.6], the natural morphism N/N^{0} → M/S_{λ_{max}−}(M) gives an isomorphism with N′. If N_{1} (resp. N_{2}) is not N, then the quotient N/N_{1} (resp. N/N_{2}) has a bounded solution at the generic disk whose Frobenius slope is different from λ_{max}. But this is impossible because N is PBQ. Hence N = N_{1} = N_{2}. Now we prove the existence of N. We use the induction on the dimension of M. Let f : M → M/M^{0} be a natural surjection. Since M/M^{0} is bounded, M/S_{λ_{max}−}(M) is a direct summand of M/M^{0} by the maximality of slopes by Theorem 4.1. Put L = f^{−1}(N′). If L is PBQ, then one can put N = L. If L is not PBQ, then L is a proper submodule of M and there is a PBQ submodule L′ of L such that L′/(L′)^{0} ≅ L/S_{λ_{max}−}(L) = N′ by the induction hypothesis.

□

Corollary 5.5 Suppose that the residue field k of V is perfect. Let M be a φ-∇-module over E. Then there is a unique filtration 0 = P_{0}(M) ⊊ P_{1}(M) ⊊ ⋯ ⊊ P_{r}(M) = M of φ-∇-modules over E such that P_{i}(M)/P_{i−1}(M) is the maximally PBQ submodule of M/P_{i−1}(M) for any i = 1, ⋯, r. We call {P_{i}(M)} the PBQ filtration of M.

5.3 Existence of the maximally PBQ φ-∇-submodules over K[[x]]_{0}

Theorem 5.6 Suppose that the residue field k of V is perfect. Let M be a φ-∇-module over K[[x]]_{0}. Then there is a unique φ-∇-submodule N of M over K[[x]]_{0} such that the generic fiber N_{η} of N is the maximally PBQ submodule of the generic fiber M_{η} of M. We call N the maximally PBQ submodule of M.

Proof. The proof of uniqueness of the maximally PBQ submodules is same to the proof of Proposition 5.4. We prove the existence of the maximally PBQ submodules by induction on the rank of M. If M is of rank 1, then the assertion is trivial. For general M, if M is PBQ, then there is nothing to prove. Suppose that M is not PBQ. Then there is a direct summand L_{η} of M_{η}/M_{η}^{0} such that L_{η} is pure with the Frobenius slope which is less than the highest slope λ_{max} of M by Theorem 4.1. Consider the composite of natural morphisms M → M_{η}/M_{η}^{0} → L_{η}. It is not injective by Lemma 5.7 below. Put M′ to be the kernel. Then M′ is a φ-∇-submodule of M such that M′_{η}/S_{λ_{max}}(M′_{η}) ≅ M_{η}/S_{λ_{max}}(M_{η}). By the induction hypothesis there is a maximally PBQ submodule N of M′ which becomes the maximally PBQ submodule N of M. □

Lemma 5.7 Suppose that the residue field k of V is perfect. Let M be a φmodule over K[[x]]_{0} such that the highest Frobenius slope of the generic fiber M_{η} of M is λ_{max}. Suppose that there exists an injective K[[x]]_{0}-homomorphism f : M → L_{η} which is φ-equivariant, i.e., φ ∘ f = f ∘ φ for a pure φ-module L_{η} over E. Then the Frobenius slope of L_{η} is λ_{max}.

Proof. In [dJ98, Corollary 8.2] A.J. de Jong proved this assertion when L_{η} is a generic fiber of a rank 1 pure φ-∇ module L over K[[x]]_{0}. (Indeed, he proved a stronger assertion.) We give a sketch of the proof which is due to [dJ98, Propositions 5.5, 6.4 and 8.1]. Our E (resp. E^{†}, resp. Ẽ, resp. Ẽ^{†} introduced below) corresponds to Γ (resp. Γ_{c}, resp. Γ_{2}, resp. Γ_{2,c}) in [dJ98]. We also remark that Ẽ^{†} is the extended bounded Robba ring R̃^{bd} in [Ke08, 2.2]. We may assume that the residue field k of V is algebraically closed and all slopes of M are contained in the value group of log_{q}|K^{×}|. We may also assume that σ(x) = x^{q} by Theorem 3.3. Let us define K-algebras

Ẽ = \{∑_{n∈ℚ} a_{n}x^{n} | \begin{matrix}a_{n} ∈ K, sup_{n}|a_{n}| < ∞, |a_{n}| → −∞ (n → −∞), \\ \{n | |a_{n}| ≥ α\} is a well-ordered set with respect to \\ the order ≤ for any α ∈ ℝ. \end{matrix} \}

Ẽ^{†} = { ∑_{n∈ℚ} a_{n}x^{n} ∈ Ẽ| |a_{n}|η^{n} → 0 (n → −∞) for some 0 < η < 1.} .

Both Ẽ and Ẽ^{†} are discrete valuation fields such that both ring of integers have a same residue field

k((x^{ℚ})) = \{ ∑_{n∈ℚ} a_{n}x^{n} | \begin{matrix}a_{n} ∈ k, \{n | a_{n} ≠ 0\} is a well-ordered set \\ with respect to the order ≤ . \end{matrix} \} ,

which includes an algebraic closure of k((x)) [Ke01], and that the p-adic completion of Ẽ^{†} is Ẽ. Ẽ is naturally an E-algebra and σ naturally extends to Ẽ by σ(∑_{n} a_{n}x^{n}) = ∑_{n} σ(a_{n})x^{qn}. Put

E^{†} = Ẽ^{†} ∩ E.

Then E^{†} is stable under σ and the K-derivation d/dx. We also denote by OẼ^{†} the ring of integer of Ẽ^{†}. By explicit calculations we have the following sublemmas.

Sublemma 5.8 For 0 < η < 1 and for ∑_{n∈ℚ} a_{n}x^{n} ∈ Ẽ^{†}, let us consider a

condition:

(∗)_{η} : sup_{n}|a_{n}| max{η^{n}, 1} ≤ 1.

If f and g in Ẽ^{†} satisfy the condition (∗)_{η}, then so are f +g and fg. Moreover, if f = ∑_{n} a_{n}x^{n} satisfies the condition (∗)_{η} and |a_{0}| = 1, then so is f^{−1}.

Note that, if η < μ, then the condition (∗)_{η} implies the condition (∗)_{μ}.

Sublemma 5.9 (1) Let A = 1 + B be a square matrix such that 1 is the unit matrix and all entries of B contained in m^{n}OẼ^{†} for a positive integer n. Suppose that all entries of A satisfy the condition (∗)_{η} in Sublemma 5.8. Then there is a matrix Y = 1 + Z with Aσ(Y) = Y such that all entries of Z are contained in m^{n}OẼ^{†} and satisfy the condition (∗)_{η^{q}}.

(2) Let C be a matrix such that all entries are contained in m^{n}OẼ^{†} for a nonnegative integer n and satisfy the condition (∗)_{η}. Then there is a matrix Z satisfying σ(Z)−Z = C such that all entries of Z are contained in m^{n}OẼ^{†} and satisfy the condition (∗)_{η^{q}}.

Proof. (1) follows from (2) by considering a congruence equation Aσ(Y) ≡ Y (mod m^{l}OẼ^{†}) inductively on l.

(2) Since the residue field k of V is perfect, σ is bijective. Put C = ∑_{n} C_{n}x^{n} = C_{−} + C_{0} + C_{+}, where they are subseries of negative powers, a constant term, and subseries of positive powers, respectively. The series Z_{−} = ∑_{n<0} ∑_{i=1}^{∞} σ^{−i}(C_{n})x^{n/q^{i}} converges and all entries of Z_{−} satisfies the condition (∗)_{η^{q}}, and the equation σ(Z_{−}) − Z_{−} = C_{−} holds. Since k is algebraically closed, there is a matrix Z_{0} over V with |Z_{0}| ≤ |C_{0}| such that σ(Z_{0}) − Z_{0} = C_{0}. The series Z_{+} = − ∑_{i=0}^{∞} σ(C_{+}) converges and satisfies σ(Z_{+}) − Z_{+} = C_{+}. Hence, Z = Z_{−} + Z_{0} + Z_{+} is the desired solution. □

If N^{†} is a φ-∇-submodule of M ⊗_{K[[x]]_{0}} E^{†} over E^{†}, then there is a φ-∇- submodule N of M over K[[x]]_{0} with N ⊗_{K[[x]]_{0}} E^{†} ≅ N by [dJ98, Proposition 6.4]. Hence, the induced morphism M ⊗_{K[[x]]_{0}} E^{†} → L_{η} is also injective. Moreover, since Ẽ^{†} ⊗_{E^{†}} E → Ẽ is injective (the similar proof of [dJ98, Proposition 8.1] works), the induced morphism M ⊗_{K[[x]]_{0}} Ẽ^{†} → L_{η} ⊗_{E} Ẽ is again injective. Let λ_{1} < ⋯ < λ_{r}(= λ_{max}) be Frobenius slopes of M_{η}. One can prove that there exists an increasing filtration 0 = M̃_{0} ⊊ M̃_{1} ⊊ ⋯ ⊊ M̃_{r} = M ⊗_{K[[x]]_{0}} Ẽ^{†} of φ-modules over Ẽ^{†} such that (M̃_{i}/M̃_{i−1}) ⊗Ẽ^{†} Ẽ is pure of slope λ_{r−i+1}. This existence of filtration of opposite direction corresponds to Proposition 5.5 in [dJ98]. Indeed, since the residue field k((x^{ℚ})) includes an algebraic closure of k((x)), there is a basis of M ⊗_{K[[x]]_{0}} Ẽ^{†} such that the Frobenius matrix of M ⊗_{K[[x]]_{0}} Ẽ^{†} has a form

( \begin{matrix}q^{λ_{1}}1 \\ & . & . & . \\ & & & & q^{λ_{r}}1 \end{matrix} ) + (a square matrix with entries in m^{n}OẼ)

by Dieudonné-Manin classification of φ-modules and the density of Ẽ^{†} in Ẽ. Here q^{λ} is a element of K with log_{q}|q^{λ}| = −λ, 1 is the unit matrix with a certain size (the first matrix is a diagonal matrix), and n is sufficiently large. One can find a basis of M ⊗_{K[[x]]_{0}} Ẽ^{†} such that the Frobenius matrix of M ⊗_{K[[x]]_{0}} Ẽ^{†} is a lower triangle matrix

( \begin{matrix}q^{λ_{1}}1 & & & & 0 \\ & . & . & . \\ ∗ & & & & q^{λ_{r}}1 \end{matrix} )

by Sublemmas 5.8 and 5.9. Hence, one has a filtration of opposite direction. Since M̃_{1} is pure of slope λ_{r} = λ_{max} and the inclusion M̃_{1} ⊂ L_{η} ⊗_{E} Ẽ is φequivariant, the slope of L_{η} must be λ_{max}. □

Corollary 5.10 Suppose that the residue field k of V is perfect. Let M be a φ- ∇-module over K[[x]]_{0}. Then there is a unique filtration 0 = P_{0}(M) ⊊ P_{1}(M) ⊊ ⋯ ⊊ P_{r}(M) = M as φ-∇-modules over K[[x]]_{0} such that P_{i}(M)/P_{i−1}(M) is the maximally PBQ submodule of M/P_{i−1}(M) for any i = 1, ⋯, r. We call {P_{i}(M)} the PBQ filtration of M.

Example 5.11 Let M be a φ-∇-module over K[[x]]_{0} which is introduced in Example 5.2 (3). If P_{1}(M) is a φ-∇-submodule of M over K[[x]]_{0} generated by e_{1} and e_{3}, the sequence 0 = P_{0}(M) ⊊ P_{1}(M) ⊊ P_{2}(M) = M is the PBQ filtration of M.

6 Log-growth and Frobenius slope for HPBQ φ-∇-modules over K[[x]]_{0}

6.1 Log-growth for HPBQ φ-∇-modules

Definition 6.1 (1) A φ-∇-module M over K[[x]]_{0} is horizontal of bounded quotient (HBQ for simplicity) if there is a quotient N of M as a φ-∇- module over K[[x]]_{0} such that the canonical surjection induces an isomorphism M_{η}/M_{η}^{0} ≅ N_{η} at the generic fiber.

(2) A φ-∇-module M over K[[x]]_{0} is horizontally pure of bounded quotient (HPBQ for simplicity) if M is PBQ and HBQ.

Example 6.2 (1) A bounded φ-∇-module M over K[[x]]_{0} is HBQ. A bounded φ-∇-module M over K[[x]]_{0} is HPBQ if and only if M_{η} is pure as a φmodule.

(2) Let M be a φ-∇-module M over K[[x]]_{0} of rank 2 which arises from the first crystalline cohomology of a projective smooth family E of elliptic curves over Spec k[[x]]. Then M is HBQ if and only if either (i) E is a non-isotrivial family over Spec k[[x]] and the special fiber E_{s} of E is ordinary or (ii) E is an isotrivial family over Spec k[[x]]. In the case (i) M is HPBQ, but in the case (ii) M is HPBQ if and only if E is an isotrivial family of supersingular elliptic curves.

(3) Let M be a φ-∇-module over K[[x]]_{0} which is introduced in Example 5.2 (3). Then M is HBQ but is not HPBQ. The dual M^{∨} of M is HPBQ.

Proposition 6.3 Let M be a φ-∇-module over K[[x]]_{0}. Then M is HBQ if and only if

dim_{K} V (M)/V (M)^{0} = dim_{E}M_{η}/M_{η}^{0}.

Moreover, when M is HBQ, the natural pairing M ⊗_{K} Sol_{0}(M) → K[[x]]_{0} induces an isomorphism

M_{η}/M_{η}^{0} ≅ V (M)/V (M)^{0} ⊗_{K} E

as φ-∇-modules.

Proof. Suppose that M is HBQ. Let N be the quotient as in Definition 6.1 (1). Since N_{η} is bounded, we have V (N)^{0} = 0 by Christol’s transfer theorem (see [CT09, Proposition 4.3]) and dim_{K} V (M)/V (M)^{0} ≥ dim_{K} V (N)/V (N)^{0} = rank_{K[[x]]_{0}} N = dim_{E}M_{η}/M_{η}^{0}. On the other hand, one knows an inequality dim_{K} V (M)/V (M)^{0} ≤ dim_{E}M_{η}/M_{η}^{0} by [CT09, Proposition 4.10]. Hence, we have an equality dim_{K} V (M)/V (M)^{0} = dim_{E}M_{η}/M_{η}^{0}. Now we prove the inverse. The natural pairing M ⊗_{K} Sol_{0}(M) → K[[x]]_{0} induces the surjection M → V (M)/V (M)^{0} ⊗_{K} K[[x]]_{0}. If dim_{K} V (M)/V (M)^{0} = dim_{E}M_{η}/M_{η}^{0}, we have an isomorphism M_{η}/M_{η}^{0} ≅ V (M)/V (M)^{0} ⊗_{K} E since V (M)/V (M)^{0} ⊗_{K} E is bounded. □

Since any quotient of bounded φ-∇-modules over E is again bounded, the proposition below follows from the chase of commutative diagrams.

Proposition 6.4 Any quotient of HBQ φ-∇-modules over K[[x]]_{0} is HBQ. In particular, any quotient of HPBQ modules is HPBQ.

Proof. We may assume that the residue field of V is algebraically closed and q^{λ_{max}} ∈ K. Since M is HBQ, there is a surjection M → V (M)/V (M)^{0} ⊗_{K} K[[x]]_{0} by Propoition 6.3 whose kernel is denoted by L. Then M_{η}^{0} = L_{η}. If f : M → N be the given surjection, N/f(L) is a quotient of V (M)/V (M)^{0} ⊗_{K} K[[x]]_{0} and hence a direct sum of copies of (K[[x]]_{0}, q^{λ}σ, d) for some λ. Since f gives a surjection from M_{η}^{0} to N_{η}^{0} by [CT09, Proposition 3.6], we have

dim_{K} V (N)/V (N)^{0} ≥ rank_{K[[x]]_{0}} N/f(L) = N_{η}/N_{η}^{0}.

On the other hand, dim_{K} V (N)/V (N)^{0} ≤ dim_{E} N_{η}/N_{η}^{0} by [CT09, Proposition 4.10]. Hence dim_{K} V (N)/V (N)^{0} = dim_{E} N_{η}/N_{η}^{0}. The rest follows from Proposition 5.3. □

Note that the notion PBQ is determined only by the generic fiber. On the other hand, for ”HPBQ”, the bounded quotient is horizontal.

Theorem 6.5 Let M be a φ-∇-module M over K[[x]]_{0} which is HPBQ. Then the conjecture LGF_{K[[x]]_{0}} (see 2.5) holds for M.

Proof. We have only to prove the conjecture LGF_{K[[x]]_{0}} (2) for M. Then the property of Frobenius slopes implies the conjecture the conjecture LGF_{K[[x]]_{0}} (1) for M. We may assume that the residue field of V is algebraically closed and all Frobenius slopes of V (M) are contained in the valued group log_{q}|K^{×}| by Proposition 2.1. We may also assume that our Frobenius σ is defined by σ(x) = x^{q} by Theorem 3.3. Let us denote by λ_{max} the highest Frobenius slope of M_{η} (= the highest Frobenius slope of V (M)). Let 0 = M_{0} ⊊ M_{1} ⊊ ⋯ ⊊ M_{r} = M be a filtration of M as φ-∇-modules over K[[x]]_{0} such that M_{i}/M_{i−1} (i = 1, ⋯, r) is irreducible (i.e., it has no nontrivial φ-∇-submodule over K[[x]]_{0}). We will prove the induction on r. If r = 1, then M ≅ (K[[x]]_{0}, q^{λ_{max}}σ, d) and the assertion is trivial. Now suppose r > 1. We may also assume dim_{K} V (M)/V (M)^{0} = 1, hence M_{r}/M_{r−1} ≅ (K[[x]]_{0}, q^{λ_{max}}σ, d). Indeed, suppose that s = dim_{K} V (M)/V (M)^{0} > 1. By our assumption, there is a φ-∇-submodule L′ over K[[x]]_{0} such that the highest Frobenius slope of L′ is λ_{max} with multiplicity 1 (note that L′ is M_{r−s+1}). Take the maximally PBQ submodule L of L′. Then L is HPBQ such that the highest Frobenius slope is λ_{max} with multiplicity 1. Since both highest Frobenius slopes of L and M/L are λ_{max}, the assertion follows from the induction hypothesis by Propositions 2.6 and 6.4. Since all Frobenius slopes of (M_{r−1})_{η} are less than λ_{max}, one can take a basis

e_{1}, ⋯, e_{s} of M such that the Frobenius matrix A = ( \begin{matrix}A_{1} & B \\ 0 & q^{λ_{max}} \end{matrix} ) (A_{1} is

the Frobenius matrix of M_{r−1}) satisfies (i) all entries of A_{1} are contained in K[[x]]_{0}∩x^{q}K[[x^{q}]] and (ii) all entries of B are contained in xK[[x]]_{0}\x^{q}K[[x^{q}]]∪{0} by Proposition 4.7. Moreover B ≠ 0 by Lemma 6.6 below since M is PBQ. Let G be the matrix of connection of M. Then the identification

Sol(M) = \{y ∈ A_{K}(0, 1^{−}) | \frac{d}{dx}y = yG\}

is given by f ↦ (f(e_{1}), ⋯, f(e_{s})). The inclusion relation in Theorem 2.3 for the solution space is

Sol_{λ}(M) ⊃ S_{λ−λ_{max}}(Sol(M)).

Then it is sufficient to prove the inclusion is equal for all λ. The φ-module is a direct sum of 1-dimensional φ-spaces, on which φ acts by q^{δ}σ for some rational number δ such that λ_{max} − δ is a Frobenius slope of M, by our assumption of K. Let f ∈ Sol_{λ}(M) with φ(f) = q^{δ}f. Then the restriction of f on M_{r−1} gives a (φ, \frac{d}{dx})-equivariant morphism

M_{r−1} → (A_{K}(0, 1^{−}), q^{−δ}σ, d).

The kernel L of f is a φ-∇-module over K[[x]]_{0} and f is a solution of M/L of log-growth λ. Suppose that L ≠ 0. Then the length of M/L is smaller than M and M/L is HPBQ by Proposition 6.4. Considering f as a solution of M/L, we have δ ≤ λ − λ_{max} by the hypothesis of induction. Suppose that L = 0. The Frobenius relation φ(f) = q^{δ}f is equivalent to

q^{−δ}σ(f(e_{1}), ⋯, f(e_{s})) = (f(e_{1}), ⋯, f(e_{s}))A.

By the assumption of A_{1} we have f(e_{i}) ∈ A_{K}(0, 1^{−}) ∩ x^{q}K[[x^{q}]]. Let us focus on the s-th entry, then it is

q^{−δ}σ(f(e_{s})) = q^{λ_{max}}f(e_{s}) + (f(e_{1}), ⋯, f(e_{s−1}))B.

Since the highest Frobenius slope of M_{r−1} is less than λ_{max}, the log-growth of the restriction of f on M_{r−1} is of log-growth less than λ_{max} + δ, and so is (f(e_{1}), ⋯, f(e_{s−1}))B. Since f is injective, (f(e_{1}), ⋯, f(e_{s−1}))B ∈ A_{K}(0, 1^{−}) \ x^{q}K[[x^{q}]] is not 0. Hence, f(e_{s}) is exactly of log-growth λ_{max} + δ by Lemma 4.8 (1). This provides an inequality λ_{max} + δ ≤ λ, and we have δ ≤ λ − λ_{max}. Therefore, f ∈ S_{λ−λ_{max}}(Sol(M)). This completes the proof of Theorem 6.5.

□

Lemma 6.6 Let 0 → L → M → N → 0 be an exact sequence of φ-∇-modules over K[[x]]_{0}. If the exact sequence is split as φ-modules, then it is split as φ-∇-modules.

Proof. Let A = ( \begin{matrix}A_{1} & B \\ 0 & A_{2} \end{matrix} ) and G = ( \begin{matrix}G_{1} & H \\ 0 & G_{2} \end{matrix} ) be the matrices of

Frobenius and connection, respectively. We should prove that B = 0 implies H = 0. It is sufficient to prove the assertion above as A_{K}(0, 1^{−})-modules with Frobenius and connection. Solving the differential modules L and N, we may assume that A_{1} and A_{2} are constant regular matrices and G_{1} = G_{2} = 0. Then the horizontality of Frobenius structure means he relation

HA_{2} = qx^{q−1}A_{1}σ(H).

Then we have H = 0 by comparing the x-adic order of both sides. □

6.2 Equislope φ-∇-modules over K[[x]]_{0}

Definition 6.7 A φ-∇-module M over K[[x]]_{0} is equislope if there is an increasing filtration {S_{λ}(M)}_{λ∈ℝ} of φ-∇-module over K[[x]]_{0} such that S_{λ}(M)⊗E gives the Frobenius slope filtration of the generic fiber M_{η} of M. We also call {S_{λ}(M)}_{λ∈ℝ} the Frobenius slope filtration of M.

By [Ka79, 2.6.2] (see [CT09, Theorem 6.21]) we have

Proposition 6.8 A φ-∇-module M over K[[x]]_{0} is equislope if and only if both the special polygon and generic polygon of Frobenius slopes of M coincides with each other.

Corollary 6.9 Any subquotients, direct sums, extensions, tensor products, duals of equislope φ-∇-modules over K[[x]]_{0} are equislope.

Proposition 6.10 Let M be an equislope φ-∇-module over K[[x]]_{0}.

(1) M is HBQ. In particular, if M is PBQ, then M is HPBQ.

(2) If V (M)/V (M)^{0} is pure as a φ-module, then M is HPBQ.

Proof. (1) We may assume that the residue field of V is algebraically closed and all slopes of M_{η} is contained in the value group log_{q}|K^{×}| of K^{×} by Proposition 2.1. Let us take a φ-∇-submodule L such that its generic fiber L_{η} is M_{η}^{0}. Such an L exists by Lemma 6.11 below. Since (M/L)_{η} ≅ M_{η}/L_{η} is bounded, M is HBQ by definition.

(2) The assertion follows from (1) and Proposition 6.3. □

Lemma 6.11 Let M be an equislope φ-∇-module over K[[x]]_{0}. Suppose that the residue field of V is algebraically closed and all slopes of M_{η} are contained in the valued group log_{q}|K^{×}|. The map taking generic fibers gives a bijection from the set of φ-∇-submodules of M to the set of φ-∇-submodules of M_{η}.

Proof. Since the functor from the category φ-∇-module over K[[x]]_{0} to the category φ-∇-module over E is fully faithful, it is sufficient to prove the surjectivity [dJ98, Theorem 1.1]. We may assume that σ(x) = x^{q} by Theorem 3.3. We use the induction on the number of Frobenius slopes of M in order to prove the existence of a submodule N over K[[x]]_{0} for a given submodule N_{η} over E. Suppose that M is pure of slope λ. There are a basis e_{1}, ⋯, e_{r} of M such that the Frobenius matrix is q^{λ}1_{r} since M is bounded. Let N_{η} be a φ-∇-submodule of M_{η} over E which is generated by (e_{1}, ⋯, e_{r})P for P ∈ Mat_{rs}(E) with s = dim_{C} N_{η}. Since N_{η} is a φ-submodule, there is a B ∈ GL_{s}(E) such that q^{λ}σ(P) = P B. Since rank(P) = s, there is a regular minor Q of P of degree s such that q^{λ}σ(Q) = QB. If one puts R = P Q^{−1} ∈ Mat_{rs}(E), then σ(R) = R. Hence, R ∈ Mat_{rs}(K). Since (e_{1}, ⋯, e_{r})R is a basis of N_{η} such that (e_{1}, ⋯, e_{r})R are included in M, the submodule N is given by the K[[x]]_{0}-submodule of M generated by (e_{1}, ⋯, e_{r})R. Let λ_{1} be the first slopes of M_{η}. By the induction hypothesis there are a φ- ∇-submodule N_{1} of S_{λ_{1}}(M) such that the generic fiber (N_{1})_{η} of N_{1} is N_{η} ∩ S_{λ_{1}}(M_{η}) and a φ-∇-submodule N_{2} of M/S_{λ_{1}}(M) such that the generic fiber of N_{2} is N_{η}/(S_{λ_{1}}(M_{η}) ∩ N_{η}) = N_{η}/(N_{1})_{η}. Let N_{3} be the inverse image of N_{2} by the surjection M/N_{1} → M/S_{λ_{1}}(M). Since the intersection of N_{η}/(N_{1})_{η} and S_{λ_{1}}(M_{η})/(N_{1})_{η} is 0 in M_{η}/(N_{1})_{η}, (N_{3})_{η} is a direct sum of N_{η}/(N_{1})_{η} and S_{λ}(M_{η})/(N_{1})_{η}. By applying the fully faithfulness of the functor from the category of φ-∇-modules over K[[x]]_{0} to the category of φ-∇-modules over E [dJ98, Theorem 1.1], there is a direct summand N_{4} of N_{3} as φ-∇-module over K[[x]]_{0} such that the generic fiber of N_{4} is N_{η}/(N_{1})_{η}. Then the inverse image N of N_{4} by the surjection M → M/N_{1} is our desired one. □

Theorem 6.12 The conjecture LGF_{K[[x]]_{0}} (see 2.5) holds for any equislope and PBQ φ-∇-module over K[[x]]_{0}.

Proof. The assertion follows from Theorem 6.5 and Proposition 6.10 (1). □

7 Log-growth filtration and Frobenius filtration at the generic point

7.1 The log-growth of PBQ φ-∇-modules over E

Theorem 7.1 The conjecture LGF_{E} (see 2.4) holds for any PBQ φ-∇-module over E.

Proof. Let M be a PBQ φ-∇-module over E such that λ_{max} is the highest Frobenius slope of M, and let us consider a φ-∇-module M_{τ} = M ⊗_{E} E_{t}[[X −t]]_{0} over the E_{t}-algebra E_{t}[[X − t]]_{0} of bounded functions on the generic disk. Then M_{τ} is equislope since {(S_{λ}(M))_{τ} } gives a Frobenius slope filtration of M_{τ}. Moreover, since M is PBQ, Sol_{0}(M, A_{E_{t}}(t, 1^{−})) is a pure φ-module. Hence V (M_{τ})/V (M_{τ})^{0} is pure, and M_{τ} is HPBQ by Proposition 6.10 (2). Applying Theorem 6.5 to M_{τ}, we have

dim_{E} M/M^{λ} = dim_{E_{t}} Sol_{λ}(M, A_{E_{t}}(t, 1^{−})) = dim_{E_{t}} V (M_{τ})/V (M_{τ})^{λ}
= dim_{E_{t}} V (M_{τ}^{∨}) − dim_{E_{t}} (S_{λ−λ_{max}}(V (M_{τ}^{∨})))^{⊥}
= dim_{E} M^{∨} − dim_{E} (S_{λ−λ_{max}}(M^{∨}))^{⊥}
= dim_{E} M^{∨}/(S_{λ−λ_{max}}(M^{∨}))^{⊥}

for any λ. Hence, M^{λ} = (S_{λ−λ_{max}}(M^{∨}))^{⊥} by Theorem 2.3. Therefore, the conjecture LGF_{E} holds for M. □

7.2 Rationality of breaks of log-growth filtrations

Theorem 7.2 Let M be a φ-∇-module over E and let λ be a break of loggrowth filtration of M, i.e., M^{λ−} ⊋ M^{λ+}. Then λ is rational and M^{λ} = M^{λ+}. In other words, the conjecture LGF_{E} (1) (see 2.4) holds for any φ-∇-modules over E.

Proof. We may assume that the residue field k of V is perfect by Proposition 2.1. Suppose that λ_{max} be the maximal Frobenius slope of M. If M is PBQ, then M^{λ} = (S_{λ−λ_{max}}(M^{∨}))^{⊥} = S_{(λ_{max}−λ)−}(M) for any λ by Theorem 7.1. Then we have

M^{λ+} = ∪_{μ>λ} S_{(λ_{max}−μ)−}(M) = ∪_{μ>λ} S_{(λ_{max}−μ)}(M) = S_{(λ_{max}−λ)−}(M) = M^{λ}.

If λ is a break of log-growth filtration, then

S_{λ_{max}−λ}(M) = S_{(λ_{max}−λ)+}(M) = M^{λ−} ⊋ M^{λ} = S_{(λ_{max}−λ)−}(M)

and λ is also a Frobenius slope filtration. Hence λ is rational. For a general M, we use the induction on the length of the PBQ filtration of M. Let L be the maximally PBQ submodule of M and suppose N = M/L. Then we have the assertion by Proposition 2.6 (1), the PBQ case and the induction hypothesis on L and N. □

Proposition 7.3 Suppose that the residue field of V is perfect. Let M be a φ- ∇-module over K[[x]]_{0} and let λ be a break of log-growth filtration of V (M), i.e., V (M)^{λ−} ⊋ V (M)^{λ+}, and let {P_{i}(M)} be the PBQ filtration of M. Suppose that the conjecture LGF_{K[[x]]_{0}} (2) (see 2.5) holds for all P_{i}(M)/P_{i−1}(M). Then λ is rational and V (M)^{λ} = V (M)^{λ+}. In particular, the conjecture LGF_{K[[x]]_{0}} (2) implies the conjecture LGF_{K[[x]]_{0}} (1) for any φ-∇-modules over K[[x]]_{0}.

Proof. The proof is similar to that of Theorem 7.2 by replacing Proposition 2.6 (1) by Proposition 2.6 (2). □

8 Toward Dwork’s conjecture LGF_{Dw}

8.1 The comparison at the special point and Dwork’s conjecture

LGF_{Dw}

Theorem 8.1 The conjecture LGF_{K[[x]]_{0}} (2) (see 2.5) implies the conjecture LGF_{Dw} (see 2.7), that is, the special log-growth polygon lies above the generic log-growth polygon (and they have the same endpoints).

The theorem above follows from the proposition below by Proposition 2.1.

Proposition 8.2 Suppose that the residue field k of V is perfect. Let M be a φ-∇-module over K[[x]]_{0} and let {P_{i}(M)} be the PBQ filtration of M. Suppose that the conjecture LGF_{K[[x]]_{0}} (2) (see 2.5) holds for all P_{i}(M)/P_{i−1}(M). Then the special log-growth polygon of M lies above the generic log-growth polygon of M (and they have the same endpoints).

Proof. For the PBQ φ-∇-modules arising from the PBQ filtration of M, the log-growth polygons at the generic (resp. special) fiber coincides with the Newton polygon of Frobenius slopes of the dual at the generic (resp. special) fiber under the suitable shifts of Frobenius actions by Theorem 7.1 (resp. our hypothesis). The assertion follows from Proposition 2.6, Lemma 8.3 below and the fact that the special Newton polygon of Frobenius slopes is above the generic Newton polygon of Frobenius slopes and they have the same endpoints.

□

Lemma 8.3 Let 0 → L → M → N → 0 be an exact sequence of φ-∇-modules over K[[x]]_{0} such that the induced sequences

\begin{aligned}0 & → & L_{η}/L_{η}^{λ} & → & M_{η}/M_{η}^{λ} & → & N_{η}/N_{η}^{λ} & → & 0 \\ 0 & → & V (L)/V (L)^{λ} & → & V (M)/V (M)^{λ} & → & V (N)/V (N)^{λ} & → & 0 \end{aligned}

on both the generic fiber and the special fiber are exact for any λ.

(1) If the special log-growth polygon lies above the generic log-growth polygon (the endpoints might be different) for both L and N, then the same holds for M.

(2) If the special log-growth polygon and the generic log-growth polygon have the same endpoints for both L and N, then the same holds for M.

(3) Suppose that the special log-growth polygon lies above the generic loggrowth polygon for both L and N. Then both the special and the generic log-growth polygons coincide with each other for M if and only if the same hold for L and N.

Proof. Let r be the rank of M. Let λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{r} be breaks of log-growth filtration of M_{η} with multiplicities, and put b_{0}(M_{η}) = 0 and

b_{j}(M_{η}) = λ_{1} + ⋯ + λ_{j}

for 1 ≤ j ≤ r. Then the generic log-growth polygon of M is a polygon which connects points (0, b_{0}(M_{η})), (1, b_{1}(M_{η})), ⋯, (r, b_{r}(M_{η})) by lines. We also define b_{j}(V (M)) for the special log-growth of M. Then the exactness for any λ implies the equality

b_{j}(M_{η}) = min \{b_{i}(L_{η}) + b_{k}(N_{η}) | \begin{matrix}0 ≤ i ≤ rank L, 0 ≤ k ≤ rank N, \\ i + k = j \end{matrix} \}

for all 0 ≤ j ≤ r, and the same holds for the special log-growth. The special log-growth polygon lies above the generic log-growth polygon for M if and only if b_{j}(M_{η}) ≤ b_{j}(V (M)) for all j, the special log-growth polygon and the generic log-growth polygon have the same endpoints for M if and only if b_{r}(M_{η}) = b_{r}(V (M)), and both the special and the generic log-growth polygons coincide with each other for M if and only if b_{j}(M_{η}) = b_{j}(V (M)) for all j. Hence we have the assertions. □

Remark 8.4 If L is supposed to be HPBQ in the short exact sequence of the previous lemma, then the induced sequences are automatically exact for all λ: in fact one has Theorems 7.1 and 6.5 and can apply Proposition 2.6.

Remark 8.5 If one assumes that the conjecture LGF_{K[[x]]_{0}} (2) (see 2.5) for any PBQ φ-∇-module over K[[x]]_{0} of rank ≤ r, then the proofs of Proposition 7.3 and Theorem 8.1 works for any φ-∇-module over K[[x]]_{0} of rank ≤ r.

8.2 Dwork’s conjecture in the HBQ cases

Lemma 8.6 Let M be a HBQ φ-∇-module over K[[x]]_{0} and let N be a φ-∇- submodule of M over K[[x]]_{0} which is PBQ. Then N is HPBQ. In particular, suppose that the residue field of V is perfect and let {P_{i}(M)} be the PBQ filtration of M, then P_{i}(M)/P_{i−1}(M) is HPBQ for all i.

Proof. We have dim_{K} V (M)/V (M)^{0} = dim_{E} M_{η}/M_{η}^{0} and dim_{K} V (M/N)/V (M/N)^{0} = dim_{E} (M/N)_{η}/(M/N)_{η}^{0} by Proposition 6.3 since the quotient M/N is HBQ by Proposition 6.4. Comparing the induced exact sequence 0 → N_{η}/N_{η}^{0} → M_{η}/M_{η}^{0} → (M/N)_{η}/(M/N)_{η}^{0} → 0 at the generic point by Theorem 7.1 and Proposition 2.6 (1) to the corresponding right exact sequence at the special point, we have an inequality dim_{K} V (N)/V (N)^{0} ≥ dim_{E} N_{η}/N_{η}^{0}. On the contrary, we know the inequality dim_{K} V (N)/V (N)^{0} ≤ dim_{E} N_{η}/N_{η}^{0} by [CT09, Proposition 4.10]. Hence, dim_{K} V (N)/V (N)^{0} = dim_{E} N_{η}/N_{η}^{0} and N is HPBQ. The rest follows from the first part and Proposition 6.4. □

Theorem 8.7 Let M be a HBQ φ-∇-module over K[[x]]_{0}. Then the conjecture LGF_{K[[x]]_{0}} (1) (see 2.5) and the conjecture LGF_{Dw} (see 2.7) hold for M.

Proof. The assertions follows from the similar arguments of Theorems 7.2 and 8.1, respectively, by using Theorem 6.5 and Lemma 8.6. □

8.3 When do the generic and special log-growth polygons coincide?

Theorem 8.8 Let M be a φ-∇-module over K[[x]]_{0}. The special log-growth polygon and the generic log-growth polygon coincide with each other if and only if M is equislope.

Proof. We may assume that the residue field of V is algebraically closed by Proposition 2.1. Let {P_{i}(M)} be the PBQ filtration of M (Theorem 5.6). Each condition (i) the coincidence of special and generic log-growth polygons or (ii) equislope implies that P_{i}(M)/P_{i−1}(M) is HPBQ and M/P_{i}(M) is HBQ for all i by Propositions 6.3, 6.4, and Lemma 8.6 for (i) and by Corollary 6.9 and Proposition 6.10 (1) for (ii). Then we can apply Lemma 8.3 (3) inductively on i by Remark 8.4 and Theorem 8.7. Hence it is sufficient to prove the assertion when M is HPBQ by Corollary 6.9. Then the coincidence of the log-growth filtration and the Frobenius slope filtration both at the special point (Theorem 8.7) and at the generic point (Theorem 7.1) implies our desired equivalence.

□

Example 8.9 (1) Let M be a φ-∇-module over K[[x]]_{0} such that M_{η} is bounded. Then there is a φ-module L over K such that M ≅ L⊗_{K} K[[x]]_{0} by Christol’s transfer theorem (see [CT09, Proposition 4.3]). Hence, M is equislope.

(2) Let M be a φ-∇-module over K[[x]]_{0} of rank 2 such that M_{η} is not bounded. Then we have identities M^{λ} = (S_{λ−λ_{max}}(M^{∨}))^{⊥} and V (M)^{λ} = (S_{λ−λ_{max}}(V (M^{∨})))^{⊥} for any λ [CT09, Theorem 7.1], where λ_{max} is the highest Frobenius slope of M_{η}. Hence the special log-growth polygon and the generic log-growth polygon coincide with each other if and only if M is equislope.

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Bruno Chiarellotto

Dipartimento Matematica Università di Padova Via Trieste 63 35121 Padova Italy chiarbru@math.unipd.it Nobuo Tsuzuki Mathematical Institute Tohoku University Aramaki Aza-Aoba 6-3 Aobaku Sendai 980-8578 Japan tsuzuki@math.tohoku.ac.jp 70

Documenta Mathematica 16 (2011)