On the rationality and continuity of logarithmic growth filtration of solutions of $p$-adic differential equations

TL;DR

This paper investigates the asymptotic behavior of solutions to Frobenius equations over overconvergent series and proves a conjecture on the rationality and continuity of Dwork's logarithmic growth filtrations for certain $p$-adic differential equations.

Contribution

It establishes the rationality and right continuity of Dwork's logarithmic growth filtrations for ordinary linear $p$-adic differential equations with Frobenius structures, confirming a conjecture.

Findings

Proves the rationality of Dwork's logarithmic growth filtrations.

Shows the right continuity of these filtrations.

Provides insights into the asymptotic behavior of solutions to Frobenius equations.

Abstract

We study the asymptotic behavior of solutions of Frobenius equations defined over the ring of overconvergent series. As an application, we prove Chiarellotto-Tsuzuki's conjecture on the rationality and right continuity of Dwork's logarithmic growth filtrations associated to ordinary linear $p$-adic differential equations with Frobenius structures.