Vol. 266, No. 2, 2013

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Index formulae for Stark units and their solutions

Xavier-François Roblot

Vol. 266 (2013), No. 2, 391–422

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.

Stark conjectures, index formula, sextic extensions
Mathematical Subject Classification 2010
Primary: 11R27, 11R29, 11R42
Received: 14 March 2012
Revised: 11 June 2012
Accepted: 16 June 2012
Published: 12 November 2013
Xavier-François Roblot
Institut Camille Jordan
Université Claude Bernard Lyon 1
43 boulevard du 11 novembre 1918
69622 Villeurbanne cedex