Divisibility of partial zeta function values at zero for degree 2p extensions

TL;DR

This paper investigates the divisibility properties of partial zeta function values at zero for specific degree 2p abelian extensions, providing new criteria and conditions related to number theory conjectures.

Contribution

It extends previous work by establishing divisibility criteria for non-cohomologically trivial roots of unity in degree 2p extensions, and offers new sufficient conditions for key conjectures.

Findings

Criteria for p-valuation of zeta function values in specific extensions

New conditions supporting the p-local Brumer-Stark conjecture

Results relating to Leopoldt's conjecture and Zp-extensions

Abstract

Let K/k be an Abelian extension of number fields, S be a set of places of k, and p be an odd prime number. We continue an earlier investigation of the author into the values at zero of the S-imprimitive partial zeta functions of K/k. An earlier result provides, under the assumption that the p-power roots of unity in K are cohomologically trivial, a criterion for the values to have larger than expected p-valuation. The present paper provides such a criterion for a special class of degree 2p extensions for which the p-power roots of unity are not cohomologically trivial. For such extensions, new sufficient conditions are also given for the p-local Brumer-Stark conjecture for K/k and for Leopoldt's conjecture on the number of independent Zp-extensions of k.