Generalized hypergeometric arithmetic D-modules under a p-adic non-Liouvilleness condition

TL;DR

This paper demonstrates that under a p-adic non-Liouvilleness condition, hypergeometric arithmetic D-modules can be decomposed into convolutions of rank-one modules, establishing their overholonomicity.

Contribution

It introduces a novel decomposition of hypergeometric arithmetic D-modules into rank-one convolutions under a p-adic non-Liouvilleness condition.

Findings

Hypergeometric arithmetic D-modules are described as convolutions of rank-one modules.

Overholonomicity of hypergeometric arithmetic D-modules is established.

The results depend on a p-adic non-Liouvilleness condition.

Abstract

We prove that the arithmetic $\mathscr{D}$-modules associated with the $p$-adic generalized hypergeometric differential operators, under a $p$-adic non-Liouvilleness condition on parameters, are described as an iterative multiplicative convolution of (hypergeometric arithmetic) $\mathscr{D}$-modules of rank one. As a corollary, we prove the overholonomicity of hypergeometric arithmetic $\mathscr{D}$-modules under a $p$-adic non-Liouvilleness condition.

Equations

$$\mathrm{Hyp}(\mathbf{\alpha };\mathbf{\beta })$$

$$H_{\pi }(\mathbf{\alpha };\mathbf{\beta })$$

$$f^{!}\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}\longrightarrow D^{\mathrm{b}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\big{)}.$$

$$f_{+}\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\big{)}\to D^{\mathrm{b}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}.$$

$$({\mathcal{Q}}^{\prime },D^{\prime })$$

$${\mathop{\otimes}\displaylimits^{\mathbb{L}}}^{\raisebox{-5.0pt}{\scriptsize\textdagger}}_{\mathscr{O}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})}\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}\times D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}\to D^{\mathrm{b}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}.$$

$$\widetilde{\otimes}_{(\mathscr{P},T)}\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}\times D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}\to D^{\mathrm{b}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}$$

$${\mathop{\boxtimes}\displaylimits^{\mathbb{L}}}^{\raisebox{-5.0pt}{\scriptsize\textdagger}}\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}_{1},\mathbb{Q}}({}^{{\dagger}}{T_{1}})\big{)}\times D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}_{2},\mathbb{Q}}({}^{{\dagger}}{T_{2}})\big{)}\to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}.$$

$$\mathcal{E}{⊠{\text{displaylimits}}^{\mathbb{L}}}^{†}\mathcal{F}\cong {\mathrm{pr}}_1^{!}\mathcal{E}{\tilde{\otimes }}_{(\mathcal{P},T)}{\mathrm{pr}}_2^{!}\mathcal{F}.$$

$$p_{1},p_{2}\colon(\mathscr{P},T)\mathrel{\vcentcolon=}\big{(}\widehat{\mathbb{P}^{1}_{V}},\{\infty\}\big{)}\times\big{(}\widehat{\mathbb{P}^{1}_{V}},\{\infty\}\big{)}\rightrightarrows\big{(}\widehat{\mathbb{P}^{1}_{V}},\{\infty\}\big{)}$$

$$\mathop{\mathrm{FT}}\nolimits_{\pi}\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)}\longrightarrow D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)}$$

$$\mathop{\mathrm{FT}}\nolimits_{\pi}(\mathscr{M})=p_{2,+}\big{(}p_{1}^{!}\mathscr{M}\operatorname{\widetilde{\otimes}}_{(\mathscr{P},T)}\mathscr{N}_{\pi}\big{)}.$$

$$A_{1}(K)^{{\dagger}}\mathrel{\vcentcolon=}\bigg{\{}\sum_{l,k\in\mathbb{N}}a_{l,k}x^{l}\partial^{[k]}\,\bigg{|}\,a_{l,k}\in K,\exists C>0,\exists\eta<1,|a_{l,k}|_{p}<C\eta^{l+k}\bigg{\}}.$$

$$\ast\colon D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)}\times D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)}\longrightarrow D^{\mathrm{b}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)}$$

$$\mathcal{E}\ast \mathcal{F}\cong {\mathrm{pr}}_{2,+}\left({\mathrm{pr}}_1^{!}{\mathrm{inv}}^{\ast }\mathcal{E}{\tilde{\otimes }}_{(\mathcal{P},T^{\prime })}{f}_{+}^{\prime }{\lambda }^{\prime !}\mathcal{F}\right)$$

$$\mathcal{E}\ast \mathcal{F}={\lambda }_{+}^{\prime }{f}^{\prime !}\left({\mathrm{pr}}_1^{!}\mathcal{E}{\tilde{\otimes }}_{(\mathcal{P},T^{\prime })}{\mathrm{pr}}_2^{!}\mathcal{F}\right)\cong {\mathrm{pr}}_{2,+}{\sigma }_{+}\left(f^{\prime !}{\mathrm{pr}}_1^{!}\mathcal{E}{\tilde{\otimes }}_{(\tilde{\mathcal{P}},{\overline{f}}^{-1}(T^{\prime }))}{f}^{\prime !}{\mathrm{pr}}_2^{!}\mathcal{F}\right).$$

$${\mathrm{pr}}_{2,+}{\sigma }_{+}\left(f^{\prime !}{\mathrm{pr}}_1^{!}\mathcal{E}\tilde{\otimes }{f}^{\prime !}{\mathrm{pr}}_2^{!}\mathcal{F}\right)$$

$$\cong {\mathrm{pr}}_{2,+}\left({\mathrm{pr}}_1^{!}{\mathrm{inv}}^{\ast }\mathcal{E}\tilde{\otimes }{\sigma }_{+}{f}^{\prime !}{\mathrm{pr}}_2^{!}\mathcal{F}\right).$$

$$j^{\ast}\big{(}\mathop{\mathrm{FT}}\nolimits_{\pi}(j_{+}\operatorname{inv}^{\ast}\mathscr{M})\big{)}\cong\mathscr{M}\ast(j^{\ast}\mathscr{L}_{\pi})[-1]$$

$$j^{\ast}\big{(}\mathop{\mathrm{FT}}\nolimits_{\pi}({j}_{+}\operatorname{inv}^{\ast}\mathscr{M})\big{)}=\mathop{\mathrm{pr}}\nolimits_{2,A,+}\big{(}{\mathop{\mathrm{pr}}\nolimits_{1,A}}^{!}{j}_{+}\operatorname{inv}^{\ast}\mathscr{M}\operatorname{\widetilde{\otimes}}j^{\prime\prime\ast}f_{+}\lambda^{!}\mathscr{L}_{\pi}\big{)}[-1],$$

$$\mathop{\mathrm{pr}}\nolimits_{1,A}^{!}j_{+}\operatorname{inv}^{\ast}\mathscr{M}\operatorname{\widetilde{\otimes}}j^{\prime\prime\ast}f_{+}\lambda^{!}\mathscr{L}_{\pi}\cong j^{\prime\prime\ast}\big{(}p_{1}^{!}j_{+}\operatorname{inv}^{\ast}\mathscr{M}\operatorname{\widetilde{\otimes}}f_{+}\lambda^{!}\mathscr{L}_{\pi}\big{)}$$

$${\mathrm{pr}}_{1,A}^{!}{j}_{+}{\mathrm{inv}}^{\ast }\mathcal{M}\cong j_{A,+}{\mathrm{pr}}_1^{!}{\mathrm{inv}}^{\ast }\mathcal{M}.$$

$${\mathrm{pr}}_{1,A}^{!}{j}_{+}{\mathrm{inv}}^{\ast }\mathcal{M}\tilde{\otimes }{j}^{\prime \prime \ast }{f}_{+}{\lambda }^{!}{\mathcal{L}}_{\pi }$$

$$\displaystyle\cong j_{A,+}\big{(}\mathop{\mathrm{pr}}\nolimits_{1}^{!}\,\operatorname{inv}^{\ast}\mathscr{M}\operatorname{\widetilde{\otimes}}j_{A,+}^{!}j^{\prime\prime!}f_{+}\lambda^{!}\mathscr{L}_{\pi}\big{)}$$

$$\cong j_{A,+}\left({\mathrm{pr}}_1^{!}{\mathrm{inv}}^{\ast }\mathcal{M}\tilde{\otimes }{f}_{+}^{\prime }{\lambda }^{\prime !}{j}^{\ast }{\mathcal{L}}_{\pi }\right).$$

$${\mathrm{pr}}_{2,A,+}{j}_{A,+}\left({\mathrm{pr}}_1^{!}{\mathrm{inv}}^{\ast }\mathcal{M}\tilde{\otimes }{f}_{+}^{\prime }{\lambda }^{\prime !}{j}^{\ast }{\mathcal{L}}_{\psi }\right)\cong {\mathrm{pr}}_{2,+}\left({\mathrm{pr}}_1^{!}{\mathrm{inv}}^{\ast }\mathcal{M}\tilde{\otimes }{f}_{+}^{\prime }{\lambda }^{\prime !}{j}^{\ast }{\mathcal{L}}_{\psi }\right).$$

$$B_{1}(K)^{{\dagger}}\mathrel{\vcentcolon=}\bigg{\{}\sum_{l\in\mathbb{Z},k\in\mathbb{N}}a_{l,k}x^{l}\partial^{[k]}\,\bigg{|}\,a_{l,k}\in K,\exists C>0,\exists\eta<1,|a_{l,k}|<C\eta^{\max(l,-l)+k}\bigg{\}}.$$

$${\mathrm{Hyp}}_{\pi }(\mathbf{\alpha };\mathbf{\beta }):=\prod _{i=1}^m(x\partial -{\alpha }_i)-(-1{)}^{m+np}{\pi }^{m-n}x\prod _{j=1}^n(x\partial -{\beta }_j)$$

$$H_{\pi }(\mathbf{\alpha };\mathbf{\beta }):=B_1(K{)}^{†}/B_1(K{)}^{†}{\mathrm{Hyp}}_{\pi }(\mathbf{\alpha };\mathbf{\beta }).$$

$$\sum _{k\ge 0,k\ne \alpha }\frac{t^k}{\alpha -k}\text{or}\sum _{k\ge 0,k\ne -\alpha }\frac{t^k}{\alpha +k}$$

$$\sum _{k=0}^{\infty }\frac{x^k}{\alpha (1-\alpha )(2-\alpha )\dots (k-\alpha )}$$

Full text

0. Introduction.

N. M. Katz [Kat90] introduces the hypergeometric $\mathscr{D}$-modules $\mathop{\mathscr{H}\!\mathit{y}\mathit{p}}\nolimits(\boldsymbol{\alpha};\boldsymbol{\beta})$ over ${\mathbb{G}_{\mathrm{m}}}$ with complex parameters $\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})\in\mathbb{C}^{m}$ and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{n})\in\mathbb{C}^{n}$, and uses them to nicely describe an “inductive” structure of hypergeometric objects. To be precise, Katz proves that $\mathop{\mathscr{H}\!\mathit{y}\mathit{p}}\nolimits(\boldsymbol{\alpha};\boldsymbol{\beta})$ is described as iterative multiplicative convolution, denoted by $\ast$, of the hypergeometric $\mathscr{D}$-modules $\mathop{\mathscr{H}\!\mathit{y}\mathit{p}}\nolimits(\alpha_{i};\emptyset)$’s and $\mathop{\mathscr{H}\!\mathit{y}\mathit{p}}\nolimits(\emptyset;\beta_{j})$’s (i.e. those with only one parameter):

Theorem** ([Kat90, 5.3.1]).**

Let $\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})\in\mathbb{C}^{m}$ and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{n})\in\mathbb{C}^{n}$ be complex parameters, and assume that for any $i$ and $j$, $\alpha_{i}-\beta_{j}$ is not an integer.

Then, there exist isomorphisms

[TABLE]

In the previous article [Miy16], the author introduces the $p$-adic hypergeometric differential operators with $p$-adic parameters $\boldsymbol{\alpha}\in(\mathbb{Z}_{p})^{m}$ and $\boldsymbol{\beta}\in(\mathbb{Z}_{p})^{n}$ (under a choice of Dwork’s $\pi$) and also the arithmetic $\mathscr{D}$-modules $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ associated with such differential operators. The author then proves that, in the case where all components of $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ lie in $\frac{1}{q-1}\mathbb{Z}$, then $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ has an analogous description as the theorem above by using the multiplicative convolution of arithmetic $\mathscr{D}$-module on ${\mathbb{G}_{\mathrm{m}}}$ [Miy16, 3.2.5]. As an application of this theorem, the author proves that a $p$-adic hypergeometric differential operator defines an overconvergent $F$-isocrystal on $\mathbb{G}_{\operatorname{m}}$ if $m\neq n$, and on $\mathbb{G}_{\operatorname{m}}\setminus\{1\}$ if $m=n$ [Miy16, 4.1.3].

The goal of this article is to extend this decomposition of hypergeometric arithmetic $\mathscr{D}$-modules $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ to more general parameters which are not necessarily rational numbers (thus they do not necessarily come from a multiplicative character on the residue field). In fact, we prove this under a $p$-adic non-Liouvilleness condition on parameters:

Theorem** (Theorem 3.1.1).**

Let $\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})\in(\mathbb{Z}_{p})^{m}$ and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{n})\in(\mathbb{Z}_{p})^{n}$ be parameters in $p$-adic integers. Assume that, for any $i$ and $j$, $\alpha_{i}-\beta_{j}$ is not an integer nor a $p$-adic Liouville number.

Then, we have isomorphisms

[TABLE]

Since an algebraic number in $\mathbb{Z}_{p}$ is not a $p$-adic Liouville number, the theorem above is, in particular, applicable to any algebraic parameters with no integer differences.

As an application of this main theorem, we prove the quasi-$\Sigma$-unipotency in the sense of Caro [Car18], in particular the overholonomicity, of our $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ under a stronger condition of $p$-adic non-Liouvilleness:

Corollary** (Proposition 3.2.2).**

Let $\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})\in(\mathbb{Z}_{p})^{m}$ and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{n})\in(\mathbb{Z}_{p})^{n}$ be parameters in $p$-adic integers. Assume that $(m,n)\neq(0,0)$, that $\alpha_{i}-\beta_{j}\not\in\mathbb{Z}$ for any $(i,j)$, and that the subgroup $\Sigma$ of $\mathbb{Z}_{p}/\mathbb{Z}$ generated by $\alpha_{i}$’s and $\beta_{j}$’s does not have a $p$-adic Liouville number. Then, $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ is a quasi-$\Sigma$-unipotent $\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}$-module. In particular, it is an overholonomic $\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}$-module.

Contrary to the results in the previous article, our $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$’s do not necessarily have a Frobenius structure (in fact, for example, $\operatorname{\mathscr{H}}_{\pi}(\alpha;\emptyset)$ does not have a Frobenius structure if $\alpha$ is not rational). It is thus worth to remark that the corollary above gives examples of overholonomic $\mathscr{D}^{{\dagger}}$-modules without assuming the existence of a Frobenius structure.

We conclude this introduction by explaining the organization of this article.

In Section 1, after a quick review of the theory of cohomological operations on arithmetic $\mathscr{D}$-modules, we define the multiplicative convolution for arithmetic $\mathscr{D}$-modules and study the relationship with Fourier transform.

In Section 2, we firstly introduce the hypergeometric arithmetic $\mathscr{D}$-modules. Then, after recalling the notion of $p$-adic Liouvilleness, we give a crucial lemma on hypergeometric arithmetic $\mathscr{D}$-modules under a $p$-adic non-Liouvilleness condition, which we will need in proving the main theorem.

In Section 3, we establish the main theorem and give an application to the quasi-$\Sigma$-unipotence.

Acknowledgement

This work is supported by JSPS KAKENHI Grant Number 17K14170.

Conventions and Notations.

In this article, $V$ denotes a complete discrete valuation ring of mixed characteristic $(0,p)$ whose residue field $k$ is a finite field with $q$ elements. The fraction field of $V$ is denoted by $K$. We denote by $|\cdot|$ the norm on $K$ normalized by $|p|=p^{-1}$. Throughout this article, we assume that there exists an element $\pi$ of $K$ that satisfies $\pi^{q-1}+(-p)^{(q-1)/(p-1)}=0$, and fix such a $\pi$.

1. Arithmetic $\mathscr{D}$-modules

1.1. Cohomological operations on D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}’s.

In this subsection, we recall some notation and fundamental properties concerning cohomological operations on D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}.

Definition 1.1.1**.**

(i) A d-couple is a pair $(\mathscr{P},T)$, where $\mathscr{P}$ is a smooth formal scheme over $\mathop{\mathrm{Spf}}\nolimits(V)$ and where $T$ is a divisor of the special fiber of $\mathscr{P}$ (an empty set is also a divisor). If a $k$-variety $X$ is the special fiber of $(\mathscr{P},T)$, we say that $(\mathscr{P},T)$ realizes $X$.

(ii) A morphism of d-couples $f\colon(\mathscr{P}^{\prime},T^{\prime})\to(\mathscr{P},T)$ is a morphism $\overline{f}\colon\mathscr{P}^{\prime}\to\mathscr{P}$ such that $\overline{f}(\mathscr{P}^{\prime}\setminus T^{\prime})\subset\mathscr{P}\setminus T$ and that $\overline{f}^{-1}(T)$ is a divisor (or empty). We say that $f$ realizes the morphism $f_{0}\colon X^{\prime}\to X$ of $k$-varieties if $(\mathscr{P}^{\prime},T^{\prime})$ (resp. $(\mathscr{P},T)$) realizes $X^{\prime}$ (resp. $X$) and if $f$ induces $f_{0}$.

Remark 1.1.2**.**

In the previous article [Miy16], we usually denote a morphism of d-couples by putting a tilde, like $\widetilde{f}$, and we use the notation $f$ for the morphism of $k$-varieties realized by $\widetilde{f}$. In this article, we do not put tildes on the name of a morphism of d-couples because we rarely need to write the name of the realized morphism of $k$-varieties.

1.1.3**.**

For each d-couple $(\mathscr{P},T)$, we denote by $\mathscr{O}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})$ the sheaf of functions on $\mathscr{P}$ with overconvergent singularities along $T$ [Ber96, 4.2.4], and denote by $\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})$ the sheaf of differential operators on $\mathscr{P}$ with overconvergent singularities along $T$ [Ber96, 4.2.5].

1.1.4**.**

Extraordinary pull-back functors [Car06, 1.1.6]. Let $f\colon(\mathscr{P}^{\prime},T^{\prime})\to(\mathscr{P},T)$ be a morphism of d-couples. Then, we have the extraordinary pull-back functor

[TABLE]

If $\overline{f}$ is smooth, or if $\overline{f}$ induces an open immersion $\overline{f}^{-1}(\mathscr{P}\setminus T)\hookrightarrow\mathscr{P}^{\prime}\setminus T^{\prime}$, then the essential image of $f^{!}$ lies in D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\big{)}. In the case where $\overline{f}$ is an isomorphism, we also denote $f^{!}$ by $f^{\ast}$. In this case, we have $f^{\ast}(\mathscr{M})=\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\otimes_{\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{f^{-1}(T)})}\overline{f}^{-1}(\mathscr{M})$.

Let $f^{\prime}\colon(\mathscr{P}^{\prime\prime},T^{\prime\prime})\to(\mathscr{P}^{\prime},T^{\prime})$ be another morphism of d-couples and let $\mathscr{M}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}. Then, as long as $f^{!}\mathscr{M}$ belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\big{)}, we have a natural isomorphism $f^{\prime!}f^{!}(\mathscr{M})=(f\circ f^{\prime})^{!}(\mathscr{M})$ of functors.

1.1.5**.**

Ordinary push-forward functors [Car06, 1.1.6]. Let $f\colon(\mathscr{P}^{\prime},T^{\prime})\to(\mathscr{P},T)$ be a morphism of d-couples. Then, we have a push-forward functor

[TABLE]

If $\overline{f}$ is proper and if $T^{\prime}=\overline{f}^{-1}(T)$, then the essential image of $f_{+}$ lies in D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}.

Let $f^{\prime}\colon(\mathscr{P}^{\prime\prime},T^{\prime\prime})\to(\mathscr{P}^{\prime},T^{\prime})$ be another morphism of d-couples, and let $\mathscr{M}^{\prime\prime}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime\prime}})\big{)}. Then, as long as $f^{\prime}_{+}\mathscr{M}^{\prime\prime}$ is an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\big{)}, we have a natural isomorphism $f_{+}f^{\prime}_{+}\mathscr{M}^{\prime\prime}\cong(f\circ f^{\prime})_{+}\mathscr{M}^{\prime\prime}$.

If $\mathscr{P}^{\prime}=\mathscr{P}$ and if $\overline{f}$ is the identity morphism on $\mathscr{P}$ (thus $f$ represents an open immersion), then $f_{+}$ is obtained by considering the complex of $\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})$-modules as a complex of $\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})$-module via the inclusion $\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\hookrightarrow\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})$.

The base change is also available. Suppose that we are given a cartesian diagram of d-couples

[TABLE]

and let $\mathscr{M}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{Q},\mathbb{Q}}({}^{{\dagger}}{D})\big{)}. If $f^{\prime!}\mathscr{M}$ belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{Q}^{\prime},\mathbb{Q}}({}^{{\dagger}}{D^{\prime}})\big{)} and if $g_{+}\mathscr{M}$ belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}, then we have a natural isomorphism $g^{\prime}_{+}f^{\prime!}\mathscr{M}\cong f^{!}g_{+}\mathscr{M}$ by [Abe14, Remark in 5.7].

1.1.6**.**

Interior tensor functor. Let $(\mathscr{P},T)$ be a d-couple. Then, we have an overconvergent tensor functor [Car15, 2.1.3]

[TABLE]

We define an interior tensor functor

[TABLE]

by $\mathscr{M}\widetilde{\otimes}_{(\mathscr{P},T)}\mathscr{N}\mathrel{\vcentcolon=}\mathscr{M}\displaystyle{\mathop{\otimes}\displaylimits^{\mathbb{L}}}^{\raisebox{-5.0pt}{\scriptsize\textdagger}}_{\mathscr{O}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})}\mathscr{N}[-\dim\mathscr{P}]$. If no confusion would occur, we omit the subscript $(\mathscr{P},T)$.

Let $f\colon(\mathscr{P}^{\prime},T^{\prime})\to(\mathscr{P},T)$ be a morphism of d-couples, and let $\mathscr{M}$ and $\mathscr{N}$ be objects of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}. Assume that $\mathscr{M}\operatorname{\widetilde{\otimes}}_{(\mathscr{P},T)}\mathscr{N}$ belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)} and that $f^{!}\mathscr{M}$ and $f^{!}\mathscr{N}$ belong to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}}\big{)}\big{)}. Then, we have an isomorphism $f^{!}(\mathscr{M}\widetilde{\otimes}_{(\mathscr{P},T)}\mathscr{N})\cong(f^{!}\mathscr{M})\widetilde{\otimes}_{(\mathscr{P}^{\prime},T^{\prime})}(f^{!}\mathscr{N})$. by [Car15, (2.1.9.1)].

The projection formula is also available. Namely, let $f\colon(\mathscr{P}^{\prime},T^{\prime})\to(\mathscr{P},T)$ be a morphism of d-couples, let $\mathscr{M}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P}^{\prime},\mathbb{Q}}({}^{{\dagger}}{T^{\prime}})\big{)}, and let $\mathscr{N}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}. Assume that $f^{!}\mathscr{N}$, $\mathscr{M}\operatorname{\widetilde{\otimes}}_{(\mathscr{P}^{\prime},T^{\prime})}f^{!}\mathscr{N}$ and $f_{+}\mathscr{M}$ are all coherent objects. Then, we have an isomorphism $f_{+}(\mathscr{M}\operatorname{\widetilde{\otimes}}_{(\mathscr{P}^{\prime},T^{\prime})}f^{!}\mathscr{N})\cong(f_{+}\mathscr{M})\operatorname{\widetilde{\otimes}}_{(\mathscr{P},T)}\mathscr{N}$ by [Car15, 2.1.6].

1.1.7**.**

Exterior tensor functors [Car15, 2.3.3]. At last, we discuss the exterior tensor functor. Let $(\mathscr{P}_{1},T_{1})$ and $(\mathscr{P}_{2},T_{2})$ be two d-couples, and let $(\mathscr{P},T)\mathrel{\vcentcolon=}(\mathscr{P}_{1},T_{1})\times(\mathscr{P}_{2},T_{2})$ be the product of them, that is, $\mathscr{P}\mathrel{\vcentcolon=}\mathscr{P}_{1}\times\mathscr{P}_{2}$ and $T\mathrel{\vcentcolon=}(T_{1}\times P_{2})\cup(P_{1}\times T_{2})$. Then, we have an exterior tensor functor

[TABLE]

As usual, this functor can be described as follows. In the situation above, let $\mathop{\mathrm{pr}}\nolimits_{i}\colon(\mathscr{P},T)\to(\mathscr{P}_{i},T_{i})$ be projections for $i=1,2$. Then, we have an isomorphism [Car15, (2.3.5.2)]

[TABLE]

The Künneth formula is also available for this exterior tensor functor [Car15, (2.3.7.2)].

1.2. Fourier transform.

In this subsection, we recall the notion of Fourier transform for arithmetic $\mathscr{D}$-modules [NH04].

1.2.1**.**

Recall from Conventions and Notations that, in this article, we fix an element $\pi$ in $K$ that satisfies $\pi^{q-1}+(-p)^{(q-1)/(p-1)}=0$. Let $\mathscr{L}_{\pi}$ denote the Dwork isocrystal associated with $\pi$.

1.2.2**.**

Let us introduce notations which we need to define the Fourier transform.

[TABLE]

be the first and the second projection, respectively. There exists a smooth formal scheme $\widetilde{\mathscr{P}}$ and a projective morphism $\overline{f}\colon\widetilde{\mathscr{P}}\to\mathscr{P}=\operatorname{\widehat{\mathbb{P}^{1}_{V}}}\times\operatorname{\widehat{\mathbb{P}^{1}_{V}}}$ such that $\overline{f}$ induces an isomorphism $\overline{f}^{-1}\left(\operatorname{\widehat{\mathbb{A}^{1}_{V}}}\times\operatorname{\widehat{\mathbb{A}^{1}_{V}}}\right)\cong\operatorname{\widehat{\mathbb{A}^{1}_{V}}}\times\operatorname{\widehat{\mathbb{A}^{1}_{V}}}$ and that this isomorphism followed by the multiplication map $\operatorname{\widehat{\mathbb{A}^{1}_{V}}}\times\operatorname{\widehat{\mathbb{A}^{1}_{V}}}\to\operatorname{\widehat{\mathbb{A}^{1}_{V}}}$ extends to a morphism $\overline{\lambda}\colon\widetilde{\mathscr{P}}\to\operatorname{\widehat{\mathbb{P}^{1}_{V}}}$. Then, $\overline{f}$ (resp. $\overline{\lambda}$) defines the morphism of d-couples f\colon\big{(}\widetilde{\mathscr{P}},\overline{f}^{-1}(T)\big{)}\to(\mathscr{P},T) (resp. \lambda\colon\big{(}\widetilde{\mathscr{P}},\overline{f}^{-1}(T)\big{)}\to\big{(}\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{\infty\}\big{)}). Finally, we put $\mathscr{N}_{\pi}\mathrel{\vcentcolon=}f_{+}\lambda^{!}(\mathscr{L}_{\pi}[-1])$. Because $\mathscr{L}_{\pi}$ is an overconvergent isocrystal, $\mathscr{N}_{\pi}$ is an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)}.

Definition 1.2.3**.**

The functor

[TABLE]

is defined by sending $\mathscr{M}$ in D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)} to

[TABLE]

This object $\mathop{\mathrm{FT}}\nolimits_{\pi}(\mathscr{M})$ is called the geometric Fourier transform of $\mathscr{M}$.

Remark 1.2.4**.**

It is a central result of [NH04] that $\mathop{\mathrm{FT}}\nolimits_{\pi}$ sends D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)}.

The argument in loc. cit. also shows that, if $\mathscr{M}$ belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)}, then $p_{1}^{!}\mathscr{M}\operatorname{\widetilde{\otimes}}_{(\mathscr{P},T)}\mathscr{N}_{\pi}$ is also an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T})\big{)}. In fact, we may assume that $\mathscr{M}$ is a (single) coherent $\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})$-module placed at degree zero, and since such a coherent module has a free resolution [Huy98, 5.3.3, (ii)], we may assume that $\mathscr{M}=\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})$. The claim follows from the calculation in [NH04, 4.2.2].

1.2.5**.**

The geometric Fourier transform has another important description after passing to the global sections. Let $A_{1}(K)^{{\dagger}}$ be the ring defined by

[TABLE]

Then, by the $\mathscr{D}^{{\dagger}}$-affinity [Huy98, 5.3.3], the functor \Gamma\big{(}\widehat{\mathbb{P}^{1}_{V}},-\big{)} on the category of coherent $\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})$-modules is exact and gives an equivalence of this category with the category of coherent $A_{1}(K)^{{\dagger}}$-modules (cf. [Huy98, p.915]). Under this identification, the geometric Fourier transform is described as follows.

Proposition 1.2.6** ([NH04, 5.3.1]).**

Let $\varphi_{\pi}\colon A_{1}(K)^{{\dagger}}\to A_{1}(K)^{{\dagger}}$ be the ring automorphism defined by $\varphi_{\pi}(x)=-\partial/\pi$ and $\varphi_{\pi}(\partial)=\pi x$. Let $\mathscr{M}$ be a coherent $A_{1}(K)^{{\dagger}}$-module and denote by $\varphi_{\pi,\ast}\mathscr{M}$ the coherent $A_{1}(K)^{{\dagger}}$-module obtained by letting $A_{1}(K)^{{\dagger}}$ act on $\mathscr{M}$ via $\varphi_{\pi}$. Then, we have a natural isomorphism $\mathop{\mathrm{FT}}\nolimits_{\pi}(\mathscr{M})\cong\varphi_{\pi,\ast}\mathscr{M}[-1]$.

1.3. Multiplicative Convolutions

In this subsection, we define the notion of multiplicative convolution and study how it is related with Fourier transform.

1.3.1**.**

We follow the notation in the previous subsection. We put (\mathscr{P},T^{\prime})\mathrel{\vcentcolon=}\big{(}\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\}\big{)}\times\big{(}\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\}\big{)}, namely, $\mathscr{P}=\operatorname{\widehat{\mathbb{P}^{1}_{V}}}\times\operatorname{\widehat{\mathbb{P}^{1}_{V}}}$ (which is compatible with the notation in 1.2.2) and T^{\prime}\mathrel{\vcentcolon=}\big{(}\{0,\infty\}\times\mathbb{P}^{1}_{k}\big{)}\cup\big{(}\mathbb{P}^{1}_{k}\times\{0,\infty\}\big{)}. Let \mathop{\mathrm{pr}}\nolimits_{1},\mathop{\mathrm{pr}}\nolimits_{2}\colon(\mathscr{P},T^{\prime})\rightrightarrows\big{(}\widehat{\mathbb{P}^{1}_{V}},\{0,\infty\}\big{)} denote the first and the second projection, respectively. We denote by f^{\prime}\colon\big{(}\widetilde{\mathscr{P}},\overline{f}^{-1}(T^{\prime})\big{)}\to(\mathscr{P},T^{\prime}) (resp. \lambda^{\prime}\colon\big{(}\widetilde{\mathscr{P}},\overline{f}^{-1}(T^{\prime})\big{)}\to\big{(}\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\}\big{)}) the morphism of d-couples defined by $\overline{f}$ (resp. $\overline{\lambda}$).

Definition 1.3.2**.**

We define a multiplicative convolution functor

[TABLE]

by \displaystyle\mathscr{E}\ast\mathscr{F}\mathrel{\vcentcolon=}\lambda^{\prime}_{+}f^{\prime!}\big{(}\mathscr{E}{\mathop{\boxtimes}\displaylimits^{\mathbb{L}}}^{\raisebox{-5.0pt}{\scriptsize\textdagger}}\mathscr{F}\big{)}=\lambda^{\prime}_{+}f^{\prime!}\big{(}\mathop{\mathrm{pr}}\nolimits_{1}^{!}\mathscr{E}\operatorname{\widetilde{\otimes}}_{(\mathscr{P},T^{\prime})}\mathop{\mathrm{pr}}\nolimits_{2}^{!}\mathscr{F}\big{)}.

In the following, we let \operatorname{inv}\colon\big{(}\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\}\big{)}\to\big{(}\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\}\big{)} denote the morphism of d-couples defined by $\overline{\operatorname{inv}}\colon\operatorname{\widehat{\mathbb{P}^{1}_{V}}}\to\operatorname{\widehat{\mathbb{P}^{1}_{V}}};x\mapsto x^{-1}$.

Lemma 1.3.3**.**

Let $\mathscr{E}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)} and let $\mathscr{F}$ be an overconvergent isocrystal on $\mathbb{G}_{\operatorname{m},k}$ considered as an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)}. Then, we have a natural isomorphism

[TABLE]

in D^{\mathrm{b}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)}.

Proof.

Let \sigma\colon\big{(}\widetilde{\mathscr{P}},\overline{f}^{-1}(T^{\prime})\big{)}\to(\mathscr{P},T^{\prime}) denote the morphism defined by $\overline{\sigma}=(\overline{\operatorname{inv}}\circ\overline{\mathop{\mathrm{pr}}\nolimits_{1}}\circ\overline{f},\overline{\lambda})$. Note that $\sigma$ represents the isomorphism $\mathbb{G}_{\operatorname{m},k}\times\mathbb{G}_{\operatorname{m},k}\to\mathbb{G}_{\operatorname{m},k}\times\mathbb{G}_{\operatorname{m},k};(x,y)\mapsto(x^{-1},xy)$. Since $\lambda^{\prime}=\mathop{\mathrm{pr}}\nolimits_{2}\circ\sigma$ and since $\sigma_{+}$ preserves coherence, we have an identification $\lambda^{\prime}_{+}=\mathop{\mathrm{pr}}\nolimits_{2,+}\circ\sigma_{+}$. By using this fact, we have

[TABLE]

Moreover, since $\mathop{\mathrm{pr}}\nolimits_{1}\circ f^{\prime}=\operatorname{inv}\circ\mathop{\mathrm{pr}}\nolimits_{1}\circ\sigma$, and since each of $f^{\prime!}$, $\operatorname{inv}^{!}$ and $\mathop{\mathrm{pr}}\nolimits_{1}^{!}$ preserves coherence, we have an identification $f^{\prime!}\mathop{\mathrm{pr}}\nolimits_{1}^{!}=\sigma^{!}\mathop{\mathrm{pr}}\nolimits_{1}^{!}\operatorname{inv}^{\ast}$ and therefore

[TABLE]

Since $\sigma$ represents an involution on $\mathbb{G}_{\operatorname{m},k}\times\mathbb{G}_{\operatorname{m},k}$ and since $\mathscr{F}$ is an overconvergent isocrystal, we have $\sigma_{+}f^{\prime!}\mathop{\mathrm{pr}}\nolimits_{2}^{!}\mathscr{F}=f^{\prime}_{+}\sigma^{!}\mathop{\mathrm{pr}}\nolimits_{2}^{!}\mathscr{F}=f^{\prime}_{+}\lambda^{\prime!}\mathscr{F}$, which completes the proof. ∎

Proposition 1.3.4**.**

We denote by j\colon\big{(}\widehat{\mathbb{P}^{1}_{V}},\{0,\infty\}\big{)}\to\big{(}\widehat{\mathbb{P}^{1}_{V}},\{\infty\}\big{)} the morphism of d-couples such that $\overline{j}=\mathop{\mathrm{id}}\nolimits_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}}}$. (Thus, $j$ realizes the inclusion $\mathbb{G}_{\operatorname{m},k}\hookrightarrow\mathbb{A}^{1}_{k}$.) Let $\mathscr{M}$ be an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)}, and assume that $j_{+}\operatorname{inv}^{\ast}\mathscr{M}$ belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\widehat{\mathbb{P}^{1}_{V}},\mathbb{Q}}({}^{{\dagger}}{\{\infty\}})\big{)}. Then, we have a natural isomorphism

[TABLE]

Proof.

Put (\mathscr{P},T_{A})\mathrel{\vcentcolon=}\big{(}\widehat{\mathbb{P}^{1}_{V}},\{\infty\}\big{)}\times\big{(}\widehat{\mathbb{P}^{1}_{V}},\{0,\infty\}\big{)}. Let \mathop{\mathrm{pr}}\nolimits_{1,A}\colon(\mathscr{P},T_{A})\to\big{(}\widehat{\mathbb{P}^{1}_{V}},\{\infty\}\big{)} (resp. \mathop{\mathrm{pr}}\nolimits_{2,A}\colon(\mathscr{P},T_{A})\to\big{(}\widehat{\mathbb{P}^{1}_{V}},\{0,\infty\}), $j_{A}\colon(\mathscr{P},T^{\prime})\to(\mathscr{P},T_{A})$) be the morphisms of d-couples defined by the first projection (resp. the second projection, the identity morphism) on $\mathscr{P}=\widehat{\mathbb{P}^{1}_{V}}\times\widehat{\mathbb{P}^{1}_{V}}$. This morphism represents the first projection $\mathbb{A}^{1}_{k}\times\mathbb{G}_{\operatorname{m},k}\to\mathbb{A}^{1}_{k}$ (resp. the second projection $\mathbb{A}^{1}_{k}\times\mathbb{G}_{\operatorname{m},k}\to\mathbb{G}_{\operatorname{m},k}$, and the inclusion $\mathbb{G}_{\operatorname{m},k}\times\mathbb{G}_{\operatorname{m},k}\hookrightarrow\mathbb{A}^{1}_{k}\times\mathbb{G}_{\operatorname{m},k}$).

Then, the definition of Fourier transform, we obtain a natural identification

[TABLE]

where $j^{\prime\prime}\colon(\mathscr{P},T_{A})\to(\mathscr{P},T)$ is the morphism of d-couples defined by $\overline{j^{\prime\prime}}=\mathop{\mathrm{id}}\nolimits_{\mathscr{P}}$, thus represents the inclusion $\mathbb{A}^{1}_{k}\times\mathbb{G}_{\operatorname{m},k}\hookrightarrow\mathbb{A}^{1}_{k}\times\mathbb{A}^{1}_{k}$. Here, in the right-hand side, by the coherence assumption and Remark 1.2.4,

[TABLE]

belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T_{A}})\big{)}.

Now, again by the coherence assumption, we have a base change isomorphism

[TABLE]

Moreover, since $j^{!}_{A,+}j^{\prime\prime!}f_{+}\lambda^{!}\mathscr{L}_{\pi}\cong f^{\prime}_{+}\lambda^{\prime!}j^{\ast}\mathscr{L}_{\pi}$, we see that

[TABLE]

Since this object belongs to D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}({}^{{\dagger}}{T_{A}})\big{)} and $\mathop{\mathrm{pr}}\nolimits_{2}=\mathop{\mathrm{pr}}\nolimits_{2,A}\circ j_{A}$, we see that

[TABLE]

By Lemma 1.3.3, this is isomorphic to the right-hand side of (1) as desired. ∎

2. Hypergeometric arithmetic $\mathscr{D}$-modules.

2.1. Definitions and fundamental properties.

2.1.1**.**

Firstly, let us define a hypergeometric arithmetic $\mathscr{D}$-module on $\mathbb{G}_{\operatorname{m},k}$ as a coherent $\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})$-module. Note that the category of coherent $\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})$-modules is identified with the category of coherent $B_{1}(K)^{{\dagger}}$-modules [Huy98, 5.3.3 and p.915], where

[TABLE]

Definition 2.1.2**.**

Let $\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{n}$ be elements of $K$. We write the sequence $\alpha_{1},\ldots,\alpha_{m}$ by $\boldsymbol{\alpha}$ and $\beta_{1},\ldots,\beta_{n}$ by $\boldsymbol{\beta}$.

• (i)

We define the hypergeometric operator $\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})=\operatorname{Hyp}_{\pi}(\alpha_{1},\ldots,\alpha_{m};\beta_{1},\ldots,\beta_{n})$ to be

[TABLE]

• (ii)

We define a $B_{1}(K)^{{\dagger}}$-module $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})=\operatorname{\mathscr{H}}_{\pi}(\alpha_{1},\ldots,\alpha_{m};\beta_{1},\ldots,\beta_{n})$ by

[TABLE]

This is also considered as an object of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})\big{)} by putting it on degree zero.

Remark 2.1.3**.**

By definition, $\operatorname{\mathscr{H}}_{\pi}(\emptyset;\emptyset)$ is the delta module at $1$.

If $(m,n)=(1,0)$, we may immediately check the isomorphism $\operatorname{\mathscr{H}}_{\pi}(\alpha;\emptyset)\cong j^{\ast}\mathscr{L}_{\pi}\otimes^{{\dagger}}\mathscr{K}_{\alpha}$, where $\mathscr{K}_{\alpha}$ is the Kummer isocrystal associated with $\alpha$. Similarly, if $(m,n)=(1,0)$, we get \operatorname{\mathscr{H}}_{\pi}(\emptyset;\beta)\cong\operatorname{inv}^{\ast}\big{(}j^{\ast}\mathscr{L}_{(-1)^{p}\pi}\otimes^{{\dagger}}\mathscr{K}_{-\beta}). (Recall that $\operatorname{inv}\colon(\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\})\to(\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\})$ denotes the morphism of d-couples defined by $\overline{\operatorname{inv}}\colon x\mapsto x^{-1}$.)

2.1.4**.**

The goal of this article is to prove, under a $p$-adic non-Liouville condition, that $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ can be obtained inductively in terms of multiplicative convolution.

2.1.5**.**

The following lemma is obtained by a straight-forward calculation as in [Miy16, Lemma 3.1.3]. (In loc. cit., (ii) is stated in the case where $\gamma\in\frac{1}{q-1}\mathbb{Z}$, but this condition is not necessary.)

Lemma 2.1.6** ([Miy16, Lemma 3.1.3]).**

Under the notation in Definition 2.1.2, $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ has the following properties.

• (i)

$\operatorname{inv}^{\ast}\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$* is isomorphic to $\operatorname{\mathscr{H}}_{(-1)^{p}\pi}(-\boldsymbol{\beta},-\boldsymbol{\alpha})$, where $-\boldsymbol{\alpha}$ (resp. $-\boldsymbol{\beta}$) denotes the sequence $-\alpha_{1},\ldots,-\alpha_{m}$ (resp. $-\beta_{1},\ldots,-\beta_{n}$).*

• (ii)

Let $\gamma$ be an element of $\mathbb{Z}_{p}$. Then, $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})\otimes^{{\dagger}}_{\mathscr{O}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}({}^{{\dagger}}{\{0,\infty\}})}\mathscr{K}_{\gamma}$ is isomorphic to $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha}+\gamma;\boldsymbol{\beta}+\gamma)$, where $\boldsymbol{\alpha}+\gamma$ (resp. $\boldsymbol{\beta}+\gamma$) denotes the sequence $\alpha_{1}+\gamma,\ldots,\alpha_{m}+\gamma$ (resp. $\beta_{1}+\gamma,\ldots,\beta_{n}+\gamma$).

2.2. $p$-adic Liouville numbers.

In this subsection, we recall the notion of $p$-adic Liouville numbers and give a lemma which we need later.

Definition 2.2.1**.**

Let $\alpha$ be an element of $\mathbb{Z}_{p}$. We say that $\alpha$ is a $p$-adic Liouville number if one of the two power series,

[TABLE]

has radius of convergence strictly less than $1$.

Proposition 2.2.2** ([Ked10, 13.1.7]).**

Let $\alpha$ be an element of $\mathbb{Z}_{p}\setminus\mathbb{Z}$ which is not a $p$-adic Liouville number. Then, the power series

[TABLE]

has radius of convergence greater than or equal to $p^{-1/(p-1)}$.

Lemma 2.2.3**.**

Let $l$ be a non-negative integer and let $\alpha$ be an element of $\mathbb{Z}_{p}$.

(i)* For any non-negative integer $N\geq l$, the following inequality holds:*

[TABLE]

(ii)* Assume that $\alpha$ is neither an integer nor a $p$-adic Liouville number. Then, for all positive real number $r$ with $r<p^{-\frac{1}{p-1}}$, we have*

[TABLE]

Proof.

(i) The proof is the same as that of the first inequality of [Miy16, 3.1.5]. We include a proof here for the convenience for the reader.

Since the inequality is trivial if $\alpha\in\{l,\dots,N\}$, we assume that this is not the case. For each positive integer $m$, let $t_{m}$ denote the number of $(s-\alpha)$’s for $s=l,...,N$ that belongs to $p^{m}\mathbb{Z}_{p}$:

[TABLE]

Then, we have $v_{p}\left(\prod_{s=l}^{N}(s-\alpha)\right)=\sum_{m=1}^{\infty}t_{m}$ (note that the right-hand side is essentially a finite sum). Now, since there is exactly one multiple of $p^{m}$ in every $p^{m}$ successive $(s-\alpha)$’s, we have t_{m}\geq\big{\lfloor}\frac{N-l+1}{p^{m}}\big{\rfloor}. This shows that

[TABLE]

The right-hand side equals v_{p}\big{(}(N-l+1)!\big{)} and it is well-known that, for any positive integer $M$ we have $v_{p}(M!)\geq\frac{M}{p-1}-\log_{p}M-1$. Therefore, we have $v_{p}\left(\prod_{s=l}^{N}(s-\alpha)\right)\geq\frac{N-l+1}{p-1}-1-\log_{p}(N-l+1)$, from which the assertion follows.

(ii) Since $l-\alpha$ is neither an integer nor a $p$-adic Liouville number, Proposition 2.2.2 shows that the power series

[TABLE]

has radius of convergence greater than or equal to $p^{-\frac{1}{p-1}}$. This means that for all $r\in(0,p^{-\frac{1}{p-1}})$, we have

[TABLE]

which shows the claim. ∎

2.3. A lemma on hypergeometric arithmetic $\mathscr{D}$-modules under a $p$-adic non-Liouvilleness condition.

In this subsection, we establish the following lemma that generalizes [Miy16, Proposition 3.1.4]. This lemma plays a central role in proving the main theorem in this article.

Lemma 2.3.1**.**

Let $\alpha_{1},\dots,\alpha_{m}$ and $\beta_{1},\dots,\beta_{n}$ be elements of $\mathbb{Z}_{p}$, and assume that $\alpha_{i}$’s does not have an integer nor have a $p$-adic Liouville numbers. Let $j\colon(\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{0,\infty\})\to(\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\{\infty\})$ be the morphism of d-couples defined by $\overline{j}=\mathop{\mathrm{id}}\nolimits_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}}}$. Then, the following assertions hold.

• (i)

j^{\ast}\big{(}A_{1}(K)^{{\dagger}}/A_{1}(K)^{{\dagger}}\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})\big{)}* is isomorphic to $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$.*

• (ii)

The natural morphism

[TABLE]

is an isomorphism.

Proof.

(i) follows from the exactness of $j^{\ast}$ on the category of coherent $A_{1}(K)^{{\dagger}}$-modules. The proof of (ii) is, as in the proof of [Miy16, Proposition 3.1.4], reduced to the following Lemma. ∎

Lemma 2.3.2**.**

Let $\alpha_{1},\dots,\alpha_{m}$ and $\beta_{1},\dots,\beta_{n}$ be elements of $\mathbb{Z}_{p}$. Assume that $\alpha_{i}$’s does not have an integer nor have a $p$-adic Liouville number. Then, on $A_{1}(K)^{{\dagger}}/A_{1}(K)^{{\dagger}}\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$, the multiplication by $x$ from the left is bijective.

Proof.

Firstly, we prove the injectivity.

To prove this, it suffices to show that if $P,Q\in A_{1}(K)^{{\dagger}}$ satisfy $xP=Q\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ then $Q\in xA_{1}(K)^{{\dagger}}$. In fact, then since $x$ is not a zero-divisor in $A_{1}(K)^{{\dagger}}$, we get that $P\in A_{1}(K)^{{\dagger}}\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ and the injectivity follows.

In order to show that $Q\in xA_{1}(K)^{{\dagger}}$, we may assume that $Q$ is of the form $Q=\sum_{l=0}^{\infty}c_{l}\partial^{[l]}$, where $c_{l}$’s are elements of $K$ satisfying $\exists C>0,\exists\eta<1,\forall l,|c_{l}|<C\eta^{l}$. Then, by using the congruence $\partial^{[l]}x\equiv\partial^{[l-1]}\pmod{xA_{1}(K)^{{\dagger}}}$, we have

[TABLE]

modulo $xA_{1}(K)^{{\dagger}}$. By assumption , the left-hand side belongs to $xA_{1}(K)^{{\dagger}}$, which shows the recurrence relation

[TABLE]

Now, fix a non-negative integer $l$ that exceeds all $\beta_{j}$’s which are integers. Then, by the recurrence relation, we have

[TABLE]

Let us choose $C>0$ and $\eta<1$ such that $\forall l,|c_{l}|<C\eta^{l}$. The series $\left\{\eta^{-k}|c_{l+k}|\right\}_{k=0}^{\infty}$ is then bounded.

Now, put $r\mathrel{\vcentcolon=}\eta^{1/2m}p^{-1/(p-1)}$; if $m=0$, we interpret $\eta^{1/2m}=1$. Lemma 2.2.3 (i) shows that $\left|\dfrac{1}{(l+k-1-\beta_{j})\dots(l-\beta_{j})}\right|\geq p^{k/(p-1)-1}k^{-1}$ for each $j=1,\dots,n$. Moreover, Lemma 2.2.3 (ii) shows that \big{|}(l+k-1-\alpha_{i})\dots(l-\alpha_{i})\big{|}r^{-k}\to\infty as $k\to\infty$ for each $i=1,\dots,m$. We therefore have, since $|\pi|=p^{-1/(p-1)}$,

[TABLE]

If $|c_{l}|\neq 0$, then the right-hand side tends to $\infty$ as $k\to\infty$, which contradicts the fact that $\left\{\eta^{-k}|c_{l+k}|\right\}_{k=0}^{\infty}$ is bounded. Therefore we have $c_{l}=0$, and consequently $c_{l+k}=0$ for all $k\geq 0$. Now, by the recurrence relation (2) and the assumption that $\alpha_{i}$’s are not integers, we get that $Q=0$.

Nextly, we prove the surjectivity.

Given $P\in A_{1}(K)^{{\dagger}}$, we have to show that there exists $Q,R\in A_{1}(K)^{{\dagger}}$ such that $xQ=P+R\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$. To prove this, we may and do assume that $P$ is of the form $P=\sum_{l=0}^{\infty}c_{l}\partial^{[l]}$, where $c_{l}$’s are elements of $K$ satisfying $\exists C>0,\exists\eta<1,\forall l,|c_{l}|<C\eta^{l}$; under this assumption, we show that there exists $R\in A_{1}(K)^{{\dagger}}$ of the form $R=\sum_{d=0}^{\infty}d_{l}\partial^{[l]}$ that satisfies $P+R\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})\in xA_{1}(K)^{{\dagger}}$. We define a number $l_{0}$ as follows: $l_{0}$ is the greatest number in $\Set{\beta_{j}+1}{j\in\{1,\ldots,n\}}\cap\mathbb{Z}_{\geq 0}$ if this set is not empty; we set $l_{0}=0$ if it is empty.

To prove the existence of $R\in A_{1}(K)^{{\dagger}}$ as above, we may assume that $c_{l}=0$ if $l<l_{0}$ by the following reason. If $A_{1}(K)$ denotes the usual Weyl algebra with coefficients in $K$, then since $\alpha_{i}$’s are not integers, the right multiplication by $\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ is bijective on $A_{1}(K)/xA_{1}(K)$ [Kat90, 2.9.4, (3)$\Rightarrow$(2)]. This shows that there exists $R^{\prime}\in A_{1}(K)$ such that $\sum_{l=0}^{l_{0}-1}c_{l}\partial^{[l]}+R^{\prime}\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})\in xA_{1}(K)$ (The proof in the reference [Kat90] is given over $\operatorname{\mathbb{C}}$, but it remains valid for all field of characteristic [math]). Now, we assume that $c_{l}=0$ if $l<l_{0}$.

We put $d_{l}=0$ if $l<l_{0}$, and for each $s\geq 0$ we put

[TABLE]

let us firstly check that this infinite series actually converges. Lemma 2.2.3 (i) shows that \big{|}(l_{0}+t-1-\beta_{j})\dots(l_{0}+s-\beta_{j})\big{|}\leq p^{-(t-s)/(p-1)+1}(t-s). Let $C>0$ and $\eta<1$ be numbers such that $\forall l,|c_{l}|<C\eta^{l}$, and put $r\mathrel{\vcentcolon=}\eta^{1/2m}p^{-1/(p-1)}$ (as before, if $m=0$, then we interpret $\eta^{1/2m}=1$). Then, Lemma 2.2.3 (ii) shows that \dfrac{1}{\big{|}(l_{0}+t-\alpha_{i})\dots(l_{0}+s-\alpha_{i})\big{|}}r^{t-s}\to 0 as $t\to\infty$. Therefore, the norm of each summand in the right-hand side of (3) is bounded from above by

[TABLE]

and the right-hand side converges to [math] as $t\to\infty$. We have now checked that the right-hand side of (3) converges and that thus $d_{l_{0}+s}$ is well-defined.

Nextly, we put $R\mathrel{\vcentcolon=}\sum_{l=0}^{\infty}d_{l}\partial^{[l]}$ and prove that $R\in A_{1}(K)^{{\dagger}}$. By the bound of the each summand of (3) given above, we have

[TABLE]

If $m=0$, then it is easy to check that there exists a constant $C^{\prime}>0$ such that $|d_{l_{0}+s}|\leq C^{\prime}\eta^{s/2}$. We thus assume that $m>0$.

For each $i=1,\dots,m$, Lemma 2.2.3 (i) shows the inequality

[TABLE]

By Lemma 2.2.3 (ii) , the fraction \dfrac{r^{t}}{\big{|}(l_{0}+t-\alpha_{i})\dots(l_{0}-\alpha_{i})\big{|}} is bounded by a constant independent of $t$. Therefore, by looking at (4), there exists a constant $C_{1}>0$ such that

[TABLE]

Now, $C_{1}\max_{t\geq 0}\left\{t^{n}\eta^{t/2}\right\}\left(s^{n}\eta^{s/4}\right)$ is bounded by a constant $C_{2}$ independent of $s$ and we have $|d_{l_{0}+s}|\leq C_{2}\eta^{s/4}$ for all $s\geq 0$. This proves that $R=\sum_{l=0}^{\infty}d_{l}\partial^{[l]}$ belongs to $A_{1}(K)^{{\dagger}}$.

It remains to prove that $R$ satisfies $P+R\operatorname{Hyp}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})\in xA_{1}(K)^{{\dagger}}$, and this is just a formal calculation. In fact, it is equivalent to showing that

[TABLE]

for all $l\geq 0$. It trivially holds if $l<l_{0}-1$ because $d_{l}=d_{l+1}=c_{l}=0$ in this case; it also holds if $l=l_{0}-1$ because $d_{l}=c_{l}=0$ and $l-\beta_{j}=0$ for some $j$; otherwise, we may check it directly by using (3). ∎

3. Hypergeometric Arithmetic $\mathscr{D}$-modules and Multiplicative Convolution.

3.1. Main Theorem.

Now, we are ready to state and prove the main theorem of this article.

Theorem 3.1.1**.**

Let $\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})$ and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{n})$ be sequences of elements of $\mathbb{Z}_{p}$. Assume that, for any $i$ and $j$, $\alpha_{i}-\beta_{j}$ is not an integer nor a $p$-adic Liouville number.

(i)* Assume that $m\geq 1$ and put $\boldsymbol{\alpha}^{\prime}=(\alpha_{2},\dots,\alpha_{m})$. Then, we have an isomorphism*

[TABLE]

(ii)* Assume that $n\geq 1$ and put $\boldsymbol{\beta}^{\prime}=(\beta_{1},\dots,\beta_{n})$. Then, we have an isomorphism*

[TABLE]

Proof.

We prove (i) and (ii) by induction on $m+n$. If $(m,n)=(1,0)$ (resp. $(m,n)=(0,1)$), then (i) (resp. (ii)) follows from the fact that $\operatorname{\mathscr{H}}_{\pi}(\emptyset;\emptyset)$ is a unit object for the multiplicative convolution. The latter fact can be checked as in the proof of [Miy16, 2.1.2].

Now, assume that $m+n\geq 2$ and let us prove the assertions (i) and (ii). In fact, Lemma 2.1.6 (i) and the isomorphism $\operatorname{inv}^{\ast}(\mathscr{M}\ast\mathscr{N})\cong(\operatorname{inv}^{\ast}\mathscr{M})\ast(\operatorname{inv}^{\ast}\mathscr{N})$, whose proof is straightforward and left to the reader, show that (ii) is deduced from (i).

The proof of (i) is reduced to the case where $\alpha_{1}=0$ as follows. Because of the isomorphism $\lambda^{\prime!}\mathscr{K}_{\alpha_{1}}\cong f^{\prime!}\left(\mathop{\mathrm{pr}}\nolimits_{1}^{!}\mathscr{K}_{\alpha_{1}}\operatorname{\widetilde{\otimes}}\mathop{\mathrm{pr}}\nolimits_{2}^{!}\mathscr{K}_{\alpha_{1}}\right)[1]$, we have

[TABLE]

Therefore, if the assertion (i) is proved for $\alpha_{1}=0$, then we get the desired theorem for general $\alpha_{1}$ by tensoring $\mathscr{K}_{\alpha_{1}}$, with the aid of Lemma 2.1.6 (ii).

In the case where $\alpha_{1}=0$, we may prove the assertion in the same way as [Miy16, Theorem 3.2.5]. We include here a sketch of the proof.

By the induction hypothesis, $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha}^{\prime};\boldsymbol{\beta})\cong\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha}^{\prime}+1;\boldsymbol{\beta}+1)$ because for the Kummer isocrystals $\mathscr{K}_{\gamma}$ we have an isomorphism $\mathscr{K}_{\gamma}\cong\mathscr{K}_{\gamma+1}$. Therefore, since $(-\beta_{j}-1)$’s do not have a $p$-adic Liouville number, we see by Lemma 2.1.6 (i) and Lemma 2.3.1 that

[TABLE]

Because this is a coherent $A_{1}(K)^{{\dagger}}$-module, Proposition 1.3.4 shows that

[TABLE]

Finally, by a direct calculation using Proposition 1.2.6, we may prove the isomorphism

[TABLE]

(cf. the proof of [Miy16, 3.2.5]). Now the assertion follows by Lemma 2.3.1 (i). ∎

3.2. Quasi-$\Sigma$-unipotence.

In this last subsection, we discuss the quasi-$\Sigma$-unipotence of arithmetic hypergeometric $\mathscr{D}$-modules.

3.2.1**.**

Let $\Sigma$ be the subgroup of $\mathbb{Z}_{p}/\mathbb{Z}$ that does not contain a $p$-adic Liouville number. Caro [Car18, 3.3.5] defines, for each smooth formal scheme $\mathscr{P}$ over $V$, the subcategory D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}\big{)} of D^{\mathrm{b}}_{\mathrm{coh}}\big{(}\mathscr{D}^{{\dagger}}_{\mathscr{P},\mathbb{Q}}\big{)} consisting of “quasi-$\Sigma$-unipotent” objects. These categories are stable under Grothendieck’s six operations.

Proposition 3.2.2**.**

Let $\boldsymbol{\alpha}=(\alpha_{1},\dots,\alpha_{m})$ and $\boldsymbol{\beta}=(\beta_{1},\dots,\beta_{n})$ be sequences of elements of $\mathbb{Z}_{p}$, and assume that $(m,n)\neq(0,0)$. Let $\Sigma$ be the subgroup of $\mathbb{Z}_{p}/\mathbb{Z}$ generated by the canonical images of $\alpha_{i}$’s and $\beta_{j}$’s. Assume that $\alpha_{i}-\beta_{j}\not\in\mathbb{Z}$ for any $i,j$, and that $\Sigma$ does not contain the canonical image of a $p$-adic Liouville number.

Then, $\operatorname{\mathscr{H}}_{\pi}(\boldsymbol{\alpha};\boldsymbol{\beta})$ is an object of D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}\big{)}. In particular, it is an overholonomic $\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}$-module.

Proof.

If the canonical image of a $p$-adic number $\gamma\in\mathbb{Z}_{p}$ in $\mathbb{Z}_{p}/\mathbb{Z}$ belongs to $\Sigma$, then $\mathscr{K}_{\gamma}$ is an object of D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}\big{)} because it is (the realization on $(\mathbb{G}_{\operatorname{m},k},\operatorname{\widehat{\mathbb{P}^{1}_{V}}})$ of) an overconvergent isocrystal on $\mathbb{G}_{\operatorname{m},k}$ whose exponent is $\gamma\in\Sigma$ (resp. $-\gamma\in\Sigma$) at [math] (resp. at $\infty$), and because D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}\big{)} contains all such objects by construction.

We may also show that $\mathscr{L}_{\pi}$ is also an object of D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}\big{)}. In fact, it is a direct factor of the push-forward of the trivial isocrystal on $\mathbb{A}^{1}_{k}$ along the Artin–Schreier morphism. Now, the trivial isocrystal on $\mathbb{A}^{1}_{k}$ is an object of D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}\big{(}\mathscr{D}^{{\dagger}}_{\operatorname{\widehat{\mathbb{P}^{1}_{V}}},\mathbb{Q}}\big{)} (the exponent at $\infty$ is $0\in\Sigma$). Since $D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}$ is stable under push-forward and direct factor, the claim follows.

Now, by Remark 2.1.3, the corollary holds for $(m,n)=(1,0),(0,1)$. For general $(m,n)$, Theorem 3.1.1 and the stability of $D^{\mathrm{b}}_{\operatorname{\mathrm{q}}\mathchar 45\relax\Sigma}$ under Grothendieck’s six functors show the assertion. ∎