Conjectures, consequences, and numerical experiments for p-adic Artin L-functions

TL;DR

This paper formulates and tests conjectures about the non-vanishing and properties of p-adic Artin L-functions for various characters over number fields, supported by theoretical insights and extensive computational evidence.

Contribution

It introduces new conjectures on p-adic L-functions, provides computational verification for many cases, and explores the behavior of related Iwasawa series and regulators.

Findings

Conjecture that p-adic L-functions are non-zero at all non-zero integers for certain characters.

Computational evidence supporting the conjectures over rationals and real quadratic fields.

Verification of Gross' conjecture and analysis of Iwasawa series behavior.

Abstract

We conjecture that the p-adic L-function of a non-trivial irreducible even Artin character over a totally real field is non-zero at all non-zero integers. This implies that a conjecture formulated by Coates and Lichtenbaum at negative integers extends in a suitable way to all positive integers. We also state a conjecture that for certain characters the Iwasawa series underlying the p-adic L-series have no multiple roots except for those corresponding to the zero at s=0 of the p-adic L-function. We provide some theoretical evidence for our first conjecture, and prove both conjectures by means of computer calculations for a large set of characters (and integers where appropriate) over the rationals and over real quadratic fields, thus proving many instances of conjectures by Coates and Lichtenbaum and by Schneider. The calculations and the theoretical evidence also prove that certain p-adic regulators corresponding to 1-dimensional characters for the rational numbers are units in many cases. We also verify Gross' conjecture for the order of the zero of the p-adic L-function at s=0 in many cases. We gather substantial statistical data on the constant term of the underlying Iwasawa series, and propose a model for its behaviour for certain characters.