Index formulae for Stark units and their solutions

Xavier-François Roblot (1)

1 ICJ - Institut Camille Jordan

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.

Domaines: Théorie des nombres [math.NT]

Xavier-François Roblot. Index formulae for Stark units and their solutions. Pacific Journal of Mathematics, 2013, 266 (2), pp.391-422. ⟨hal-00863158⟩