Explicit Vologodsky Integration for Hyperelliptic Curves

TL;DR

This paper presents an algorithm for computing Vologodsky integrals on hyperelliptic curves with bad reduction, extending previous methods and enabling practical calculations in arithmetic geometry.

Contribution

It introduces a new algorithm for Vologodsky integrals on bad reduction hyperelliptic curves, broadening computational capabilities in arithmetic geometry.

Findings

Algorithm successfully computes Vologodsky integrals numerically.

Extends previous work to all meromorphic differential forms.

Implemented and tested with Sage examples.

Abstract

Vologodsky's theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry. In contrast with the theory developed by Coleman, it has the advantage of being insensitive to the reduction type at $p$. Building on recent work of Besser and Zerbes, we describe an algorithm for computing Vologodsky integrals on bad reduction hyperelliptic curves. This extends previous joint work with Katz to all meromorphic differential forms. We illustrate our algorithm with numerical examples computed in Sage.