Non-abelian pseudomeasures and congruences between abelian Iwasawa L-functions

TL;DR

This paper explores the extension of abelian pseudomeasures to non-abelian contexts in Iwasawa theory, focusing on necessary congruences and their relation to the equivariant main conjecture.

Contribution

It introduces the concept of congruences for abelian pseudomeasures as a step toward defining non-abelian $l$-adic L-functions in Iwasawa theory.

Findings

Proposes congruences for abelian pseudomeasures

Discusses the relation to the equivariant main conjecture

Highlights the need for these congruences to define non-abelian L-functions

Abstract

The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed.