The Lifted Root Number Conjecture for small sets of places and an application to CM-extensions

In this paper we study a famous conjecture
which relates the leading terms at zero of Artin L-functions
attached to a finite Galois extension L/K of number fields to
natural arithmetic invariants. This conjecture is called the
Lifted Root Number Conjecture (LRNC) and has been introduced by
K.W. Gruenberg, J. Ritter and A. Weiss; it depends on a set S of
primes of L which is supposed to be sufficiently large. We
formulate a LRNC for small sets S which only need to contain the
archimedean primes. We apply this to CM-extensions which we
require to be (almost) tame above a fixed odd prime p. In this
case the conjecture naturally decomposes into a plus and a minus
part, and it is the minus part for which we prove the LRNC at p
for an infinite class of relatively abelian extensions. Moreover,
we show that our results are closely related to the Rubin-Stark
conjecture.