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Licensed Unlicensed Requires Authentication Published by De Gruyter June 16, 2009

The Minus Conjecture revisited

  • C. Greither and R. Kučera

Abstract

In an earlier paper we proved some results concerning Gross's conjecture on tori. This conjecture, which we call the Minus Conjecture, is closely related to a conjecture of Burns, which is now known to hold generally in the absolutely abelian setting; however Burns' conjecture does not directly imply the Minus Conjecture. The result proved in the earlier paper was concerned with imaginary absolutely abelian extensions K/ℚ of the form K = FK+, with F imaginary quadratic and K+/ℚ being tame, l-elementary and ramified at most at two primes. In the present paper we complement these results by proving the Minus Conjecture for extensions K/ℚ as above but without any restriction on the number s of ramified primes. The price we have to pay for this generality is that our proof only works if the odd prime l is large enough, more precisely if l ≧ 3(s + 1). There is one more restriction, namely . (A similar restriction was already needed in our previous paper.)

Received: 2006-12-19
Revised: 2008-04-23
Published Online: 2009-06-16
Published in Print: 2009-July

© Walter de Gruyter Berlin · New York 2009

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