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

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.