Index formulae for stark units and their solutions (Q395035)

Let \(\mathbb{K}/k\) be an abelian extension of number fields with distinguished place, say \(v\), of \(k\) that splits totally in \(\mathbb{K}\) and \(G\) the Galois group of \(\mathbb{K}/k\). In that situation, the abelian rank-one Stark conjecture predicts the existence of unit in \(\mathbb{K}\), called the Stark unit, constructed from the values of the L-functions attached to the extension \(\mathbb{K}/k\).NEWLINENEWLINEIn this paper, the author, fixes a finite set S of places of \(k\) and makes additional assumptions as the following:NEWLINENEWLINE (A1) \(k\) is totally real and the infinite places of \(\mathbb{K}\) above \(v\) (assumed to be infinite in this paper) are real, the infinite places of \(\mathbb{K}\) not above \(v\) are complex.NEWLINENEWLINE (A2) The maximal totally real subfield \(\mathbb{K}^{+}\) of \(\mathbb{K}\) satisfies \([\mathbb{K}:\mathbb{K}^{+}]= 2\).NEWLINENEWLINE (A3) All the finite primes in S are either ramified or inert in \(\mathbb{K}/\mathbb{K}^{+}\).NEWLINENEWLINE Assuming the Stark unit exists, the author proves the index formulae that relates the index of the subgroup generated over \(\mathbb{Z}(G)\), by the Stark unit inside the minus-part of the group of units of \(\mathbb{K}\) to the cardinality of the minus-part of the class group of \(\mathbb{K}\), and an analogous of Rubin's index formula.NEWLINENEWLINE In a second part, the author supposes the existence of a unit which verifies two conditions more weaker than that of the existence of the Stark unit; and study the solutions of the index formulae and proves that they admit solutions unconditionally for quadratic, quartic and some sextic cyclic extensions. He then deduces a weak version of the conjecture ( up to absolute values) in these cases and specifies results on when the Stark unit, if it exists, is square.