# Pushforwards of $p$-adic differential equations

## TL;DR

This paper derives explicit formulas relating the convergence radii of $p$-adic differential equations under pushforward by étale morphisms, connecting topological invariants and ramification data, with applications to Newton polygons.

## Contribution

It provides a new explicit formula for the radii of convergence after pushforward, generalizing known cases and linking them to ramification and Newton polygon invariants.

## Key findings

- Explicit formula relating radii of convergence under pushforward.
- Connection between convergence radii and ramification jumps.
- Formula for the Laplacian of the Newton polygon height.

## Abstract

Given a differential equation on a smooth $p$-adic analytic curve, one may construct a new one by pushing forward by an \'etale morphism. The main result of the paper provides an explicit formula that relates the radii of convergence of the solutions of the two differential equations using invariants coming from the topological behavior of the morphism. We recover as particular cases the known formulas for Frobenius morphisms and tame morphisms. As an application, we show that the radii of convergence of the pushforward of the trivial differential equation at a point coincide with the upper ramification jumps of the extension of the residue field of the point given by the morphism. We also derive a general formula computing the Laplacian of the height of the Newton polygon of a $p$-adic differential equation.