Estimates on Eisenstein distributions for reciprocals of p-adic L-functions: the case of irregular primes

TL;DR

This paper investigates the behavior of p-adic distributions derived from Eisenstein series, focusing on their measure properties and quantitative estimates in the case of irregular primes.

Contribution

It provides new quantitative estimates describing the behavior of these distributions when p is irregular, extending previous results on their measure properties.

Findings

Distributions are measures when p is regular.

Distributions fail to be measures when p is irregular.

Quantitative estimates describe their behavior for irregular primes.

Abstract

We consider the p-adic distributions derived from Eisenstein series in [GMPS], whose Mellin transforms are reciprocals of the Kubota-Leopoldt p-adic L-function. These distributions were shown there to be measures when p is regular. They fail to be measures when p is irregular; in this paper we give quantitative estimates that describe their behavior more precisely.