A higher order p-adic class number formula

TL;DR

This paper generalizes Leopoldt's p-adic class number formula to relate p-adic regulators and zeta function values, providing conditions for non-vanishing modulo p at s=1 in real abelian extensions.

Contribution

It extends Leopoldt's formula to higher order p-adic regulators and establishes new non-vanishing results for relative zeta functions at s=1.

Findings

Generalized Leopoldt's formula for p-adic regulators

Linked p-adic regulator modulo p to zeta function values

Proved non-vanishing conditions for zeta function at s=1

Abstract

We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of Q with the value of the relative Dedekind zeta function at s=2-p. We use this generalization to give a statement on the non-vanishing modulo p of this relative zeta function at the point s=1 under a mild condition.