Effective Convergence Bounds for Frobenius Structures on Connections

Abstract

Consider a meromorphic connection on P1 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 an example indicating that our bound is optimal.