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.
Keywords
Stark conjectures, index formula, sextic extensions
