2011/12/13 by Xavier-François Roblot, Roblot, Xavier-François
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1112.2820
openalex publication_date 2011/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.