On a conjecture concerning minus parts in the style of Gross

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Authors

GREITHER Cornelius KUČERA Radan

Year of publication 2008
Type Article in Periodical
Magazine / Source Acta Arithmetica
MU Faculty or unit

Faculty of Science

Citation
Field General mathematics
Keywords Stark units; regulators; Gross conjecture on tori
Description This paper is devoted to Gross's conjecture on tori over the base field Q. We call it the Minus Conjecture, since it involves a regulator built from units in the minus part. We recall and develop its relation to a conjecture of Burns, which is now known to hold generally in the absolutely abelian setting; however in many situations it is not clear at all how one should deduce the Minus Conjecture from it. We prove a somewhat weaker statement (order of vanishing) rather generally, and we give a proof of the Minus Conjecture for some specific classes of absolutely abelian extensions K/Q, for which K^+/Q is l-elementary and ramified in at most two primes. The field K is assumed to be of the form FK^+ where F is an arbitrary imaginary quadratic field. Our methods involve a good deal of explicit calculation; among other things, we use p-adic Gamma-functions and the Gross-Koblitz formula.
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