INDEX FORMULAE FOR STARK UNITS AND THEIR SOLUTIONS

Description

Niveau: Supérieur, Doctorat, Bac+8
INDEX FORMULAE FOR STARK UNITS AND THEIR SOLUTIONS XAVIER-FRANC¸OIS ROBLOT Abstract. 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. 1. Introduction Let K/k be an abelian extension of number fields. Denote by G its Galois group. Let S∞ and Sram denote respectively the set of infinite places of k and the set of finite places of k ramified in K/k. Let S(K/k) := S∞ ? Sram. Fix a finite set S of places of k containing S(K/k) and of cardinality at least 2. Assume that there exists at least one place in S, say v, that splits totally in K/k and fix a place w of K dividing v.

• jsps global


• group generated


• g?g


• cannot exist


• clk ?


• takes also


• index formulae


• galois group


• dm ?

Sujets

• Galois group

Informations

• Publié par: mijec
• Nombre de lectures: 166

Extrait

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