Regulators and derivatives of Vologodsky functions with respect to log(p)

TL;DR

This paper investigates the behavior of p-adic regulators and Vologodsky integrals in bad reduction scenarios, revealing how derivatives with respect to log(p) relate continuous and discrete components.

Contribution

It introduces a method to differentiate Vologodsky integrals with respect to log(p), connecting continuous and discrete parts of p-adic regulators in bad reduction cases.

Findings

Differentiation of Vologodsky integrals relates to the discrete component of the regulator.

The continuous component depends on the choice of the p-adic logarithm branch.

The study clarifies the structure of p-adic regulators in bad reduction situations.

Abstract

We describe several instances of the following phenomenon: In bad reduction situations the \( p \)-adic regulator has a continuous and a discrete component. The continuous component is computed using Vologodsky integrals. These depend on a choice of the branch of the \( p \)-adic logarithm, determined by \( \log (p) \). They can be differentiated with respect to this parameter and the result is related to the discrete component.