p-adic heights and Vologodsky integration

TL;DR

This paper extends the Coleman-Gross p-adic height pairing to semi-stable curves using Vologodsky integration, demonstrating it coincides with Nekovar's pairing, thus broadening the applicability of p-adic height theories.

Contribution

It introduces a new extension of p-adic height pairings to semi-stable curves via Vologodsky integration, aligning with Nekovar's established pairing.

Findings

Vologodsky integration replaces Coleman integration for semi-stable curves.

The extended pairing matches Nekovar's pairing.

Broadens the scope of p-adic height pairings to more general curves.

Abstract

We extend the result of a previous work to the case of curves with semi-stable reduction. In this case, one can replace Coleman integration by Vologodsky integration to extend the Coleman-Gross definition of a $p$-adic height pairing. we show that this pairing still equals the one defined by Nekovar.