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

TL;DR

This paper verifies Dwork's conjecture on the logarithmic growth of solutions for rank two $p$-adic differential equations without solvability, using Kedlaya's decomposition theorem.

Contribution

It provides the first non-trivial proof of Dwork's conjecture in the rank two case, extending understanding of solution growth without solvability assumptions.

Findings

Confirmed Dwork's conjecture for rank two cases

Applied Kedlaya's decomposition theorem to non-solvable equations

Established logarithmic growth behavior in broader $p$-adic differential equations

Abstract

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.