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Shun Ohkubo

Publications and source records attributed to Shun Ohkubo.

11 recordsLinked to original sources
arXiv2024

New cases of Dwork's conjecture on asymptotic behaviors of solutions of $p$-adic differential equations without solvability

One of the phenomena peculiar in the theory of $p$-adic differential equations is that solutions $f$ of $p$-adic differential equations defined on open discs may satisfy growth conditions at the boundaries. This phenomenon is first studied by Dwork, who proves the fundamental theorem asserting that if a $p$-adic differential equation defined on an open unit disc is solvable, then any solution $f$ has order of logarithmic growth at most $m-1$. In this paper, we study a conjecture proposed by Dwork on a generalization of this theorem to the case without solvability. We prove new cases of Dwork's conjecture by combining descending techniques of differential modules with the author's previous result on Dwork's conjecture in the rank $2$ case.

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arXiv2024

Duality for differential modules over complete non-archimedean valuation field of characteristic zero

Let $K$ be a complete non-archimedean valuation field of characteristic $0$, with non-trivial valuation, equipped with (possibly multiple) commuting bounded derivations. We prove a decomposition theorem for finite differential modules over $K$, where decompositions regarding the extrinsic subsidiary $\partial$-generic radii of convergence in the sense of Kedlaya-Xiao. Our result is a refinement of a previous decomposition theorem due to Kedlaya and Xiao. As a key step in the proof, we prove a decomposition theorem in a stronger form in the case where $K$ is equipped with a single derivation. To achieve this goal, we construct an object $f_{0*}L_0$ representing the usual dual functor and study some filtrations of $f_{0*}L_0$, which is used to construct the direct summands appearing in our decomposition theorem.

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arXiv2018

A note on logarithmic growth of solutions of $p$-adic differential equations without solvability

For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork's conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya's decomposition theorem of $p$-adic differential equations defined over annulus.

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arXiv2018

Logarithmic growth filtrations for $(φ,\nabla)$-modules over the bounded Robba ring

In this paper, we study the logarithmic growth (log-growth) filtration, a mysterious invariant found by B. Dwork, for $(φ,\nabla)$-modules over the bounded Robba ring. The main result is a proof of a conjecture proposed by B. Chiarellotto and N. Tsuzuki on a comparison between the log-growth filtration and Frobenius slope filtration. One of the ingredients of the proof is a new criterion for pure of bounded quotient, which is a notion introduced by Chiarellotto and Tsuzuki to formulate their conjecture. We also give several applications to log-growth Newton polygons, including a conjecture of Dwork on the semicontinuity, and an analogue of a theorem due to V. Drinfeld and K. Kedlaya on Frobenius Newton polygons for indecomposable convergent $F$-isocrystals.

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arXiv2015

On differential modules associated to de Rham representations in the imperfect residue field case

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$, whose residue field may not be perfect, and $G_K$ the absolute Galois group of $K$. In the first part of this paper, we prove that Scholl's generalization of fields of norms over $K$ is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor $\mathbb{N}_{\mathrm{dR}}(V)$ associating a de Rham representation $V$ with a $(φ,\nabla)$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of $\mathbb{N}_{\mathrm{dR}}(V)$ and Swan conductor of $V$, which generalizes Marmora's formula.

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arXiv2014

A note on logarithmic growth Newton polygons of $p$-adic differential equations

In this paper, we answer a question due to Y. André related to B. Dwork's conjecture on a specialization of the logarithmic growth of solutions of $p$-adic linear differential equations. Precisely speaking, we explicitly construct a $\nabla$-module $M$ over $\mathbb{Q}_p[[X]]_0$ of rank 2 such that the left endpoint of the special log-growth Newton polygon of $M$ is strictly above the left endpoint of the generic log-growth Newton polygon of $M$.

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arXiv2013

On Lie algebras arising from $p$-adic representations in the imperfect residue field case

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field $k_K$ such that $[k_K:k_K^p]=p^d<\infty$. Let $G_K$ be the absolute Galois group of $K$ and $ρ:G_K\to GL_h(\Q_p)$ a $p$-adic representation. When $k_K$ is perfect, Shankar Sen described the Lie algebra of $ρ(G_K)$ in terms of so-called Sen's operator $Θ$ for $ρ$. When $k_K$ may not be perfect, Olivier Brinon defined $d+1$ operators $Θ_0,...,Θ_d$ for $ρ$, which coincides with Sen's operator $Θ$ in the case of $d=0$. In this paper, we describe the Lie algebra of $ρ(G_K)$ in terms of Brinon's operators $Θ_0,...,Θ_d$, which is a generalization of Sen's result.

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arXiv2012

The p-adic monodromy theorem in the imperfect residue field case

Let K be a complete discrete valuation field of mixed characteristic (0,p) and G_K the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of G_K without any assumption on the residue field of K, for example the finiteness of a p-basis of the residue field of K. The main point of the proof is a construction of (phi,G_K)-module Nrig^+(V) for a de Rham representation V, which is a generalization of Pierre Colmez' Nrig^+(V). In particular, our proof is essentially different from Kazuma Morita's proof in the case when the residue field admits a finite p-basis. We also give a few applications of the p-adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the p-adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of G_K and the category of de Rham representations of an absolute Galois group of the canonical subfield of K. Finally, we compute H^1 of some p-adic representations of G_K, which is a generalization of Osamu Hyodo's results.

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arXiv2009

A note on Sen's theory in the imperfect residue field case

In Sen's theory in the imperfect residue field case, Brinon defined a functor from the category of C_p-representations to the category of linear representations of certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of C_p-representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo's calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.

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