A conjecture concerning minus parts in the spirit of Gross
by
Cornelius Greither
Universität der Bundeswehr München, Germany
Coauthors: R. Kucera

There are many conjectures postulating a precise link between the leading term of an L-function at zero and an arithmetic expression of the type ``class number times regulator'', Dirichlet's analytic class number formula being a prototype. In the eighties, Gross came up with a conjecture in which the regulator is not a complex or p-adic number but an element of a finite abelian group. This conjecture just has received renewed interest, triggered by very general ideas of David Burns. In particular, a ``minus conjecture'' emerges, which should be obtainable, roughly speaking, by dividing a conjectural equation for a complex (CM) field K by the corresponding equation for the real subfield K⁺. However, this division process often does not make sense (all quantities involved may be zero), and we are interested in direct proofs of the ``minus conjecture'' which do not use the conjectural equations for either K or K<sup>+</sup>. Some (very partial) results will be presented. Part of this is joint work with Radan Kucera. Actually this ``minus conjecture'' is a special case of the so-called Gross conjecture on tori, on which not much seems to be known.

Date received: April 30, 2004