Effective convergence bounds for Frobenius structures on connections

TL;DR

This paper establishes effective bounds on the convergence of Frobenius structures for p-adic meromorphic connections, with applications to Picard-Fuchs and Gauss-Manin equations, demonstrating near-optimality of these bounds.

Contribution

It provides explicit convergence bounds for Frobenius structures on p-adic connections, improving understanding of their analytic behavior and stability under Frobenius lift changes.

Findings

Derived effective convergence bounds for Frobenius structures

Demonstrated bounds are essentially optimal through examples

Analyzed the impact of changing Frobenius lifts on convergence

Abstract

Consider a meromorphic connection on P^1 over a p-adic field. In many cases, such as those arising from Picard-Fuchs equations or Gauss-Manin connections, this connection admits a Frobenius structure defined over a suitable rigid analytic subspace. We give an effective convergence bound for this Frobenius structure by studying the effect of changing the Frobenius lift. We also give some examples indicating that our bound is essentially optimal.