Dwork's result

In this paragraph, we assume that $k$ is algebraically closed of characteristic $p\neq 2$, and $\mathcal {O}_K$ is the ring of Witt vectors over $k$. Let $X\to \mathbb {P}^1_K$ be Legendre family of elliptic curves defined by $E_t:w^2=z(z-1)(z-t)$. Then, the associated (first) Picard–Fuchs module corresponds to Gaussian hypergeometric differential operator $H:=t(1-t)\partial ^2+(1-t)\partial -1/4$ (see [Reference KedlayaKed10, Examples 22.2.1, 22.2.3]). Let $a\in \mathcal {O}_K$ be a lift of $\bar {a}\in k{\setminus} \{0,1\}$. We put $S_a:=\{x\in K[\![t-a]\!];t(1-t)d^2x/d(t-a)^2+(1-t){{d} x}/d(t-a)-x/4=0\}$, which is regarded as the set of local solutions of $Hx=0$ around $t=a$. We denote by $K\{t-a\}$ the subring of $K[\![t-a]\!]$ consisting of series converging on the open unit disc $|t-a|<1$. Then, we have $\dim _KS_a=2$, and $S_a\subset K\{t-a\}$. That is, the differential equation $Hx=0$ is solvable on the open unit disc $|t-a|<1$. However, one can prove that no nonzero $x\in S_a$ converges on the closed unit disc $|t-a|\leqslant 1$. To understand the asymptotic behavior of $x\in S_a$ as $|t-a|\to 1^{-}$, we introduce the log-growth filtration $\mathrm {Fil}_{\bullet }K\{t-a\}$ of $K\{t-a\}$. Let $\lambda \in \mathbb {R}_{\geqslant 0}$. A formal power series $x=\sum _{i\in \mathbb {N}}x_i(t-a)^i\in K[\![t-a]\!]$ with $x_i\in K$ has log-growth $\lambda$ if there exists $C\in \mathbb {R}$ such that $|x_i|\leqslant Ci^{\lambda }$ for all $i\in \mathbb {N}$. Note that if $x$ satisfies the condition, then $x\in K\{t-a\}$. We define $\mathrm {Fil}_{\lambda }K\{t-a\}$ as the $K$-subspace of $K[\![t-a]\!]$ consisting of series having log-growth $\lambda$. We put $\mathrm {Fil}_{\lambda }K\{t-a\}=0$ for $\lambda \in \mathbb {R}_{<0}$. Then, we put $S_{a,\lambda }:=S_a\cap \mathrm {Fil}_{\lambda }K\{t-a\}$ for $\lambda \in \mathbb {R}$. Dwork proves that the slope multiset of the filtration $S_{a,\bullet }$ is $\{0,1\}$ if $E_{\bar {a}}$ is ordinary, and $\{1/2,1/2\}$ if $E_{\bar {a}}$ is supersingular (see [Reference Chiarellotto and TsuzukiCT09, 7.4] for further explanation).

Chiarellotto–Tsuzuki conjecture

In [Reference Chiarellotto and TsuzukiCT09], Chiarellotto and Tsuzuki formulated a conjecture called the Chiarellotto–Tsuzuki conjecture in this paper, which is regarded as a generalization of Dwork's result. Let us briefly recall the statement of the conjecture. Let $q$ be a positive power of $p$, and $\varphi _K$ an isometric ring endomorphism on $K$ which lifts the $q$-power map on $k$. We define $K[\![t]\!]_0:=\mathcal {O}_K[\![t]\!]\otimes _{\mathcal {O}_K}K$, that is, the subring of $K\{t\}$ consisting of series $x=\sum _{i\in \mathbb {N}}x_it^i$ with bounded coefficients, which is equipped with Gauss norm defined by $|x|_0=\sup _{i\in \mathbb {N}}|x_i|$. We endow $K\{t\}$ with the $K$-linear derivation $\partial =d/dt:K\{t\}\to K\{t\}$ and a ring endomorphism $\varphi$ of the form $\varphi (\sum _{i\in \mathbb {N}}x_it^i)=\sum _{i\in \mathbb {N}}\varphi _K(x_i)(S')^i$ for some $S'\in K[\![t]\!]_0$ with $|S'-t^q|_0<1$. Then, both $\varphi$ and $\partial$ restrict to $K[\![t]\!]_0$. Recall that for a ring $R$ with a ring endomorphism $\varphi$, a $\varphi$-module over $R$ is a finite free $R$-module $M$ equipped with a $\varphi$-semilinear endomorphism $\varphi _M$ such that the $R$-linear map $M\otimes _{R,\varphi }R\to M;m\otimes r\mapsto \varphi _M(m)r$ is an isomorphism. A $(\varphi ,\nabla )$-module over $K[\![t]\!]_0$ is a $\varphi$-module $M$ over $K[\![t]\!]_0$ equipped with a $K$-linear differential operator $\partial _M$ relative to $\partial$ satisfying the compatibility condition $\partial (\varphi (t))\cdot \varphi _M\circ \partial _M=\partial _M\circ \varphi _M$. We define the sets of analytic horizontal sections and analytic solutions of $M$, respectively, by $V(M):=\ker {(\partial _M\otimes \mathrm {id}_{K\{t\}}+\mathrm {id}_M\otimes \partial :M\otimes _{K[\![t]\!]_0}K\{t\}\to M\otimes _{K[\![t]\!]_0}K\{t\})}$ and $\mathrm {Sol}(M):=\{f\in \mathrm {Hom}_{K[\![t]\!]_0}(M,K\{t\});\partial \circ f=f\circ \partial _M\}$, which are equipped with a canonical perfect pairing $V(M)\otimes _K\mathrm {Sol}(M)\to K$. By Dwork's trick [Reference KedlayaKed10, Corollary 17.2.2], $V(M)$ and $\mathrm {Sol}(M)$ are regarded as $\varphi$-modules over $K$ of dimension equal to the rank of $M$, where $\varphi _{V(M)}$ and $\varphi _{\mathrm {Sol}(M)}$ are defined as the restriction of $\varphi _M\otimes \varphi$ and the unique $\varphi _K$-semilinear endomorphism on $\mathrm {Sol}(M)$ satisfying $\varphi _{\mathrm {Sol}(M)}(f)(\varphi _M(m))=\varphi (f(m))$ for $f\in \mathrm {Sol}(M),m\in M$, respectively. Consequently, $V(M)$ and $\mathrm {Sol}(M)$ are endowed with the Frobenius slope filtrations $S_{\bullet }(V(M))$ and $S_{\bullet }(\mathrm {Sol}(M))$, respectively (see Theorem 1.5(ii)). We also define the growth filtrations $\mathrm {Sol}_{\bullet }(M)$ and $V(M)^{\bullet }$ of $\mathrm {Sol}(M)$ and $V(M)$ respectively by $\mathrm {Sol}_{\lambda }(M)=\{f\in \mathrm {Sol}(M);f(M)\subset \mathrm {Fil}_{\lambda }K\{t\}\}$ and $V(M)^{\lambda }=\mathrm {Sol}_{\lambda }(M)^{\perp }$ for $\lambda \in \mathbb {R}$, where $\mathrm {Sol}_{\lambda }(M)^{\perp }$ denotes the orthogonal part of $\mathrm {Sol}_{\lambda }(M)$ with respect to the canonical pairing $V(M)\otimes _K \mathrm {Sol}(M)\to K$.

The Chiarellotto–Tsuzuki conjecture concerns a comparison between $V(M)^{\bullet }$ (respectively, $\mathrm {Sol}_{\bullet }(M)$) and $S_{\bullet }(V(M))$ (respectively, $S_{\bullet }(\mathrm {Sol}(M))$). To state the conjecture precisely, we briefly recall the notion of being pure of bounded quotient (PBQ), which is a condition on the generic fiber of $M$ (see [Reference Chiarellotto and TsuzukiCT11, 5.1] for details). Let $\mathcal {E}$ be the Amice ring, that is, the completion of the fraction field of $K[\![t]\!]_0$ for the norm $|\cdot |_0$, endowed with natural extensions of $\varphi$ and $\partial$. We define $\mathcal {E}[\![X-t]\!]_0$ as before with $X-t$ a variable. We define the ring homomorphism $\tau :\mathcal {E}\to \mathcal {E}[\![X-t]\!]_0;f\mapsto \sum _{i\in \mathbb {N}}\partial ^i(f)\cdot (X-t)^i/i!$. We endow $\mathcal {E}[\![X-t]\!]_0$ with the $\mathcal {E}$-linear derivation $\partial =d/d(X-t):\mathcal {E}[\![X-t]\!]_0\to \mathcal {E}[\![X-t]\!]_0$ and a ring endomorphism $\varphi$ defined by $\varphi (\sum _{i\in \mathbb {N}}x_i(X-t)^i)=\sum _{i\in \mathbb {N}}\varphi (x_i)(\tau (\varphi (t))-S')^i$, which commute with $\tau$. Let $M'$ be a $(\varphi ,\nabla )$-module over $\mathcal {E}$. Let $\lambda _{\rm max}(M')$ denote the maximum Frobenius slope of $M'$ (see Definition 1.3). Regarding $\tau ^*(M'):=M'\otimes _{\mathcal {E},\tau }\mathcal {E}[\![X-t]\!]_0$ as a $(\varphi ,\nabla )$-module over $\mathcal {E}[\![X-t]\!]_0$ (see § 4.1), we define a $\varphi$-module $V(\tau ^*(M'))$ over $\mathcal {E}$ equipped with a growth filtration $V(\tau ^*(M'))^{\bullet }$ as before. For $\lambda \in \mathbb {R}$, by a theorem of Robba, we have a (unique) subobject $(M')^{\lambda }$ of $M'$ in the category of $(\varphi ,\nabla )$-modules over $\mathcal {E}$ such that there exists a canonical isomorphism $V(\tau ^*(M'/(M')^{\lambda }))\cong V(\tau ^*(M'))^{\lambda }$, which forms a decreasing filtration of $M'$ called the log-growth filtration of $M'$ (see [Reference Chiarellotto and TsuzukiCT09, 3.2 and 3.3]). We say that $M'$ is PBQ if $M'/(M')^0$ is pure as a $\varphi$-module over $\mathcal {E}$, that is, the Frobenius Newton polygon of $M'/(M')^0$ is a straight line.

We quickly review known results on the Chiarellotto–Tsuzuki conjecture. If $M$ is of rank $1$, then the conjecture obviously holds as $M$ is trivial as a $\nabla$-module over $K[\![t]\!]_0$ (see [Reference KedlayaKed10, Proposition 18.4.3]). If $M$ arises from the Picard–Fuchs module associated with the Legendre family of elliptic curves, then Dwork's result implies part (ii) of the conjecture (see [Reference Chiarellotto and TsuzukiCT09, 7.4] for details). Chiarellotto and Tsuzuki proved the conjecture in the rank $2$ case [Reference Chiarellotto and TsuzukiCT09, Theorem 7.1(1)]. The present author proved part (i) of the conjecture without assumptions [Reference OhkuboOhk17, Theorem 3.7(i)].

A generalization of Chiarellotto–Tsuzuki conjecture

In this paper, we prove Theorem 0.1, which is regarded as a generalization of the Chiarellotto–Tsuzuki conjecture to $(\varphi ,\nabla )$-modules over the bounded Robba ring $\mathcal {R}^{\mathrm {bd}}$. For $r\in \mathbb {R}_{\geqslant 0}$ and a formal Laurent series over $K$ denoted by $x=\sum _{i\in \mathbb {Z}}x_it^i$ with $x_i\in K$, we put $|x|_r=\sup _{i\in \mathbb {Z}}|x_i|e^{-ri}\in [0,\infty ]$. We define the (respectively, bounded) Robba ring $\mathcal {R}$ (respectively, $\mathcal {R}^{\mathrm {bd}}$) as the ring consisting of series $x$ such that there exists $r>0$ such that $|x|_s<+\infty$ for all $s\in (0,r]$ (respectively, $s\in [0,r]$). We fix an element $X\in \mathcal {O}_K[\![t]\!]$ of the form $t^iv$ with $i\in \mathbb {N}_{\geqslant 1}$ and $v\in \mathcal {O}_K[\![t]\!]^{\times }$, and define $\mathcal {R}_{\rm log}$ as the polynomial ring $\mathcal {R}[\ell _X]$ with a variable $\ell _X$. Then, the ring $\mathcal {R}^{\mathrm {bd}}$ is a field, and we have the following commutative diagram of rings, where the hooked arrows denote the inclusions.

We endow $\mathcal {R}$ with the $K$-linear derivation $\partial =d/dt:\mathcal {R}\to \mathcal {R}$ and a ring endomorphism $\varphi$ of the form $\varphi (\sum _{i\in \mathbb {N}}x_it^i)=\sum _{i\in \mathbb {N}}\varphi _K(x_i)S^i$ for some $S\in \mathcal {R}^{\mathrm {bd}}$ with $|S-t^q|_0<1$. Then, $\varphi$ and $\partial$ restrict to $\mathcal {R}^{\mathrm {bd}}$, and extend to $\mathcal {E}$, $\mathcal {R}_{\rm log}$ by setting $\varphi (\ell _X)=\log {(\varphi (X)/X^q)}+q\ell _X$ and $\partial (\ell _X)=\partial (X)/X$, respectively (see Definitions 2.1(ii), (iii), and 2.2). We define the log-growth filtrations of $\mathcal {R}$ and $\mathcal {R}_{\rm log}$ by $\mathrm {Fil}_{\lambda }\mathcal {R}=\{x\in \mathcal {R};|x|_r=O(r^{-\lambda })\text { as }r\to +0\}$ if $\lambda \in \mathbb {R}_{\geqslant 0}$, $\mathrm {Fil}_{\lambda }\mathcal {R}=0$ if $\lambda \in \mathbb {R}_{<0}$, and $\mathrm {Fil}_{\lambda }\mathcal {R}_{\rm log}:=\bigoplus _{i\in \mathbb {N}}\mathrm {Fil}_{\lambda -i}\mathcal {R}\cdot \ell _X^i$ for $\lambda \in \mathbb {R}$. We can define the notion of $(\varphi ,\nabla )$-modules over $\mathcal {R}^{\mathrm {bd}}$ as before. For a literal generalization of the Chiarellotto–Tsuzuki conjecture, we should study the whole category of $(\varphi ,\nabla )$-modules $M$ over $\mathcal {R}^{\mathrm {bd}}$, where we encounter problems caused by the absence of a naïve analogue of Dwork's trick. To avoid complications, in this paper, we assume that $M_{\mathcal {R}}:=M\otimes _{\mathcal {R}^{\mathrm {bd}}}\mathcal {R}$ is unipotent as a $\nabla$-module over $\mathcal {R}$. Thanks to the assumption, we can define $V(M),\mathrm {Sol}(M)$, etc., as before, where $K[\![t]\!]_0$ and $K\{t\}$ are replaced by $\mathcal {R}^{\mathrm {bd}}$ and $\mathcal {R}_{\rm log}$, respectively. The main result of this paper is as follows.

We prove part (ii) of Theorem 0.1 in § 8. We give two proofs of part (i) of Theorem 0.1; the first given in § 5 is done by a reduction to the case $X=t$, in which case the assertion is proved in [Reference OhkuboOhk17, Theorem 4.19], and the second given in § 10 is simple and self-contained.

In the rest of this subsection, we put $S=S'\in K[\![t]\!]_0$ so that the inclusion $K[\![t]\!]_0\to \mathcal {R}^{\mathrm {bd}}$ is $\varphi$-equivariant. Moreover, we admit part (ii) of Theorem 0.1 and the property of the log-growth filtrations of $\mathcal {R}$ and $\mathcal {R}_{\rm log}$ (see § 3.2). Under this setup, we will prove the following result.

Note that even a log analogue of the Chiarellotto–Tsuzuki conjecture holds (Theorem 10.6).

Lemma 0.3 Let $M$ be a $(\varphi ,\nabla )$-module over $K[\![t]\!]_0$. We put $M_{\mathcal {R}^{\mathrm {bd}}}:=M\otimes _{K[\![t]\!]_0}\mathcal {R}^{\mathrm {bd}}$, which is regarded as a $(\varphi ,\nabla )$-module over $\mathcal {R}^{\mathrm {bd}}$. Then, $(M_{\mathcal {R}^{\mathrm {bd}}})_{\mathcal {R}}$ is unipotent as a $\nabla$-module over $\mathcal {R}$. Moreover, there exist canonical isomorphisms of $\varphi$-modules over $K$

\begin{gather*} \alpha:V(M)\to V(M_{\mathcal{R}^{\mathrm{bd}}}),\\ \beta:\mathrm{Sol}(M)\to \mathrm{Sol}(M_{\mathcal{R}^{\mathrm{bd}}}), \end{gather*}

which are compatible with the canonical pairings. Furthermore, the maps $\alpha$ and $\beta$ induce isomorphisms $V(M)^{\bullet }\to V(M_{\mathcal {R}^{\mathrm {bd}}})^{\bullet }$ and $\mathrm {Sol}_{\bullet }(M)\to \mathrm {Sol}_{\bullet }(M_{\mathcal {R}^{\mathrm {bd}}})$ of filtrations, respectively. In particular, the log-growth Newton polygon $\mathrm {NP}(M)$ of $M$ (see [Reference Chiarellotto and TsuzukiCT11, 2.3] or the next subsection) coincides with the log-growth Newton polygon $\mathrm {NP}(M_{\mathcal {R}^{\mathrm {bd}}})$ of $M_{\mathcal {R}^{\mathrm {bd}}}$ (Definition 10.1(i)).

Proof. As $M\otimes _{K[\![t]\!]_0}K\{t\}$ is trivial as a $\nabla$-module over $K\{t\}$ by Dwork's trick [Reference KedlayaKed10, Corollary 17.2.2], $(M_{\mathcal {R}^{\mathrm {bd}}})_{\mathcal {R}}\cong (M\otimes _{K[\![t]\!]_0}K\{t\})\otimes _{K\{t\}}\mathcal {R}$ is unipotent (even trivial) as a $\nabla$-module over $\mathcal {R}$. We define the maps

\begin{gather*} \alpha':M\otimes_{K[\![t]\!]_0}K\{t\}\to (M\otimes_{K[\![t]\!]_0}\mathcal{R}^{\mathrm{bd}})\otimes_{\mathcal{R}^{\mathrm{bd}}}\mathcal{R}_{{\rm log}};m\otimes r\mapsto m\otimes 1\otimes r,\\ \beta':\mathrm{Hom}_{K[\![t]\!]_0}(M,K\{t\})\to\mathrm{Hom}_{\mathcal{R}^{\mathrm{bd}}}(M\otimes_{K[\![t]\!]_0}\mathcal{R}^{\mathrm{bd}},\mathcal{R}_{{\rm log}});f\mapsto (m\otimes r\mapsto f(m)r), \end{gather*}

which are compatible with Frobenius actions and differential operators. Let $\alpha :V(M)\to V(M_{\mathcal {R}^{\mathrm {bd}}})$ and $\beta :\mathrm {Sol}(M)\to \mathrm {Sol}(M_{\mathcal {R}^{\mathrm {bd}}})$ be the morphisms of $\varphi$-modules over $K$ induced by $\alpha '$ and $\beta '$, respectively. As $\alpha '$ and $\beta '$ are injective, the maps $\alpha$ and $\beta$ are isomorphisms by comparing dimensions. By definition, $\alpha$ and $\beta$ are compatible with the canonical pairings. To complete the proof, we have only to prove, by duality, that for $f\in \mathrm {Sol}(M)$, we have $f\in \mathrm {Sol}_{\lambda }(M)$ if and only if $\beta (f)\in \mathrm {Sol}_{\lambda }(M_{\mathcal {R}^{\mathrm {bd}}})$. If $f\in \mathrm {Sol}_{\lambda }(M)$, then $\beta (f)\in \mathrm {Sol}_{\lambda }(M_{\mathcal {R}^{\mathrm {bd}}})$ by

\[ \beta(f)(M_{\mathcal{R}^{\mathrm{bd}}})=f(M)\cdot\mathcal{R}^{\mathrm{bd}}\subset (\mathrm{Fil}_{\lambda}K\{t\})\cdot\mathcal{R}^{\mathrm{bd}}\subset\mathrm{Fil}_{\lambda}\mathcal{R}_{{\rm log}}\cdot \mathrm{Fil}_{0}\mathcal{R}_{{\rm log}}\subset\mathrm{Fil}_{\lambda}\mathcal{R}_{{\rm log}}, \]

where we use Corollary 3.7(i) and (iii). Conversely, if $\beta (f)\in \mathrm {Sol}_{\lambda }(M_{\mathcal {R}^{\mathrm {bd}}})$, then $f\in \mathrm {Sol}_{\lambda }(M)$ by

\[ f(M)\subset K\{t\}\cap \beta(f)(M_{\mathcal{R}^{\mathrm{bd}}})\subset K\{t\}\cap \mathrm{Fil}_{\lambda}\mathcal{R}_{{\rm log}}=K\{t\}\cap \mathrm{Fil}_{\lambda}\mathcal{R}=\mathrm{Fil}_{\lambda}K\{t\}, \]

where we use Corollary 3.6.

Application to Dwork's conjecture

Recall that the Grothendieck–Katz specialization theorem for $\varphi$-modules $M$ over $K[\![t]\!]_0$ asserts that the Frobenius Newton polygon of $M/tM$ lies on or above the Frobenius Newton polygon of $M_{\mathcal {E}}$ with the same endpoints [Reference KedlayaKed10, Theorem 15.3.2]. Chiarellotto and Tsuzuki formulated Dwork's conjecture as an analogue of the Grothendieck–Katz specialization theorem for $(\varphi ,\nabla )$-modules $M$ over $K[\![t]\!]_0$ (see also Remark 10.5). Recall that a slope of $V(M)^{\bullet }$ is a real number $\lambda$ such that ${\bigcap} _{\mu <\lambda }V(M)^{\mu }\neq \bigcup _{\mu >\lambda }V(M)^{\mu }$. Let $\{s_1(M)\leqslant \cdots \leqslant s_n(M)\}$ denote the multiset of slopes $\lambda$ of $V(M)^{\bullet }$ with multiplicity $\dim _K{\bigcap} _{\mu <\lambda }V(M)^{\mu }-\dim _K\bigcup _{\mu >\lambda }V(M)^{\mu }$, where $n$ denotes the rank of $M$. Then, we define the log-growth Newton polygon $\mathrm {NP}(M)$ of $M$ as the boundary of the lower convex hull of the points $(0,0)$ and $(i,s_1(M)+\dots +s_i(M))$ for $i\in \{1,\dots ,n\}$. Similarly, we define the Newton polygon $\mathrm {NP}(M)$ of a $(\varphi ,\nabla )$-module $M$ over $\mathcal {E}$ (Definition 10.1(ii)).

Dwork's conjecture is previously known in the case where the rank is less than or equal to two [Reference Chiarellotto and TsuzukiCT09, Corollary 7.3]. In § 10, we prove the following result.

If we admit Theorem 0.4, then we have the following.

Structure of the paper

We prove part (ii) of Theorem 0.1 in § 8. If the reader admits the key ingredients, that is, Propositions 3.11 and 7.2, then the (short) proof can be easily understood. We prove Propositions 3.11 and 7.2 in §§ 3 and 7 respectively, after giving preparations in §§ 1, 2 and §§ 4–6, respectively. We prove part (i) of Theorem 0.1 and Theorem 0.4 in § 10 by using the PBQ filtrations of $(\varphi ,\nabla )$-modules over $\mathcal {R}^{\mathrm {bd}}$, which are studied in § 9.

In § 6, we prove the invariance of the growth filtration under the base change of the coefficient field $K$, which can be skipped by assuming that $k$ is algebraically closed. Parts of §§ 2 and 3 (specifically, §§ 2.2, 2.3, 3.1, and 3.3) are rather technical because the extended Robba ring is involved. Hence, the reader may postpone these parts by admitting results proved there.

Finally, we note that to make the paper concise, some results of an earlier version of this paper are not available in the current version, and will be included in a forthcoming paper(s).

Notation and terminology

(1) In this paper, a ring $R$ is a commutative ring with $1$ unless otherwise mentioned. For a ring homomorphism $\psi :R\to S$, we define the base change functor $\psi ^*(-)$ from the category of $R$-modules to the category of $S$-modules by $M\mapsto \psi ^*M=M_S=M\otimes _RS$. When $\psi$ is injective, we identify $R$ with $\psi (R)$, and regard $\psi$ as the inclusion.

(2) We recall some terminology of difference modules [Reference KedlayaKed10, § 14]. A difference ring (respectively, field) is a ring (respectively, field) $R$ equipped with a ring endomorphism $\phi$. Let $R$ be a difference ring. A difference module over $R$ is an $R$-module $M$ equipped with a $\phi$-semilinear endomorphism $\phi _M$. A $\phi$-module over $R$ is a difference module $M$ over $R$ such that $M$ is finite free as an $R$-module, and the $R$-linear map $\phi ^*M=M\otimes _{R,\phi }R\to M;m\otimes r\mapsto r\phi _M(m)$ is an isomorphism. The category of $\phi$-modules over $R$ is equipped with the following operations: the internal Hom $\mathrm {Hom}_R(-,-)$, the tensor product $(-)\otimes _R(-)$, the dual $(-)^\vee =\mathrm {Hom}_R(-,R)$, and the natural pairing $(-)\otimes _R(-)^\vee \to R$. For $c\in R^{\times }$, let $R(c)=Re_c$ be the $\phi$-module over $R$ defined by $\phi _{R(c)}(e_c)=ce_c$. We define the twist of a $\phi$-module $M$ over $R$ by $c$ to be $M(c)=M\otimes _R R(c)$. For $d\in \mathbb {N}_{>0}$, we define the $d$-pushforward functor $[d]_*(-)$ from the category of $\phi$-modules over $R$ to the category of $\phi ^d$-modules over $R$ by $(M,\phi _M)\mapsto (M,\phi _M^d)$. For a morphism $\psi :R\to S$ of difference rings, we define a base change functor $\psi ^*(-)$, as in part (1), from the category of $\phi$-modules over $R$ to the category of $\phi$-modules over $S$. Then, the $d$-pushforward functor and the base change functor are compatible with internal Homs, tensor products, duals, and natural pairings in an obvious sense.

Let $F$ be a difference field with respect to a ring endomorphism $\phi$. We say that $F$ is inversive if $\phi$ is bijective. We say that $F$ is weakly difference-closed if any equation of the form $\phi (x)=cx$ with $c\in F^{\times }$ always has a solution $x\in F^{\times }$. We say that $F$ is strongly difference-closed if $F$ is weakly difference-closed and inversive.

(3) Let $V$ be a finite-dimensional vector space over a field $F$ with $n=\dim _FV$, equipped with a filtration of subspaces which is exhaustive and separated. We recall the definition of the Newton polygon associated with the filtration. For simplicity, we assume that our filtration is decreasing, which is denoted by $\{V^{\lambda };\lambda \in \mathbb {R}\}$. We put $m({\lambda })=\dim _F({\bigcap} _{\varepsilon >0}V^{\lambda -\varepsilon })/(\bigcup _{\varepsilon >0}V^{\lambda +\varepsilon })=\lim _{\varepsilon \to 0^+}\dim _FV^{\lambda -\varepsilon }-\lim _{\varepsilon \to 0^+}\dim _FV^{\lambda +\varepsilon }$. Note that $m({\lambda })=0$ for all but finitely many $\lambda$, and $\sum _{\lambda \in \mathbb {R}}m(\lambda )=n$. If $m({\lambda })\neq 0$, then we call $\lambda$ a slope of $V^{\bullet }$. The slope multiset of $V^{\bullet }$ is the multiset consisting of slopes $\lambda$ of $V^{\bullet }$ with multiplicity $m(\lambda )$, which is denoted by $\{s_1\leqslant \cdots \leqslant s_n\}$. We define the Newton polygon $\mathrm {NP}(V^{\bullet })$ of $V^{\bullet }$ by the boundary of the lower convex hull in the $xy$-plane of the set of points $(0,0)$ and $(i,s_1+\dots +s_i)$ for $i\in \{1,\dots ,n\}$.