An equivariant $p$-adic Artin conjecture

TL;DR

This paper formulates an equivariant version of Greenberg's $p$-adic Artin conjecture for $p$-adic $L$-functions in the context of $p$-adic Lie extensions of totally real fields, and proves its validity in specific cases.

Contribution

It introduces an equivariant formulation of the $p$-adic Artin conjecture and proves it in several cases using Wedderburn decomposition techniques.

Findings

Validity of the equivariant $p$-adic Artin conjecture in certain cases.

Application of Wedderburn decomposition to analyze the conjecture.

Extension of Greenberg's conjecture to equivariant settings.

Abstract

We formulate an equivariant version of Greenberg's $p$-adic Artin conjecture for smoothed equivariant $p$-adic Artin $L$-functions in the context of an arbitrary one-dimensional admissible $p$-adic Lie extension of a totally real number field. Using results of the author on the Wedderburn decomposition of the total ring of quotients of the Iwasawa algebra $\Lambda(\mathcal G)$, we deduce validity of the conjecture in several interesting cases.