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

TL;DR

This paper advances the understanding of $p$-adic differential equations by proving new cases of Dwork's conjecture, which relates to the growth of solutions without the solvability condition, using innovative techniques.

Contribution

It generalizes Dwork's theorem to non-solvable cases by combining differential module techniques with previous rank 2 results, expanding the conjecture's validity.

Findings

Proved new cases of Dwork's conjecture for non-solvable $p$-adic differential equations.

Established growth bounds for solutions without solvability.

Enhanced methods for analyzing $p$-adic differential equations.

Abstract

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.