Vologodsky integration on curves with semi-stable reduction

TL;DR

This paper establishes a detailed relationship between Vologodsky integrals and Coleman integrals on curves with semi-stable reduction over p-adic fields, providing a complete characterization of the Vologodsky integral.

Contribution

It proves that Vologodsky integrals can be fully described using Coleman integrals on components and harmonic cochains on connecting annuli for curves with semi-stable reduction.

Findings

Vologodsky integrals restrict to Coleman integrals on smooth components.

Differences of Coleman integrals on connecting annuli form harmonic cochains.

Results are exemplified in the case of Tate elliptic curves.

Abstract

We prove that the Vologodsky integral of a mermorphic one-form on a curve over a $p$-adic field with semi-stable reduction restrict to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. We analyze the behavior of the integral on the connecting annuli and we explain the results in the case of a Tate elliptic curve.