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On a conjecture concerning minus parts in the style of Gross

C. Greither, Radan Kučera
2008 Acta Arithmetica  
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
more » ... lly, 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 F K + where F is an arbitrary imaginary quadratic field. Our methods involve a good deal of explicit calculation; among other things, we use p-adic Γ-functions and the Gross-Koblitz formula.
doi:10.4064/aa132-1-1 fatcat:4k7orjf4gndovllpxaiwa653ly