Index formulae for Stark units and their solutions

TL;DR

This paper proves index formulae for Stark units in abelian extensions of number fields and studies their solutions, providing partial evidence for the Stark conjecture in specific cyclic extensions.

Contribution

It establishes index formulae for Stark units assuming their existence and analyzes solutions in quadratic, quartic, and sextic cyclic extensions.

Findings

Index formulae for Stark units are proven under certain assumptions.

Solutions to the index formulae exist unconditionally in specific cyclic extensions.

Results include conditions when the Stark unit is a square or not.

Abstract

Let $K/k$ be an abelian extension of number fields with a distinguished place of $k$ that splits totally in $K$. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in $K$, called the Stark unit, constructed from the values of the $L$-functions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a result we deduce a weak version of the conjecture ("up to absolute values") in these cases and precise results on when the Stark unit, if it exists, is a square.