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Journal of the London Mathematical Society Advance Access originally published online on August 9, 2009
Journal of the London Mathematical Society 2009 80(2):446-470; doi:10.1112/jlms/jdp036
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© 2009 London Mathematical Society

The lifted root number conjecture for small sets of places

A. Nickel

Institut für Mathematik
Universitt Augsburg
86135 Augsburg
Germany
Current address:
Université de Bordeaux 1
UFR de Mathématiques et Informatique
351, cours de la Libération
33405 Talence cedex
France

Let L/K be a finite Galois extension of number fields with Galois group G. The lifted root number conjecture (LRNC) by Gruenberg, Ritter and Weiss relates the leading terms at zero of Artin L-functions attached to L/K to natural arithmetic invariants. Burns used complexes arising from étale cohomology of the constant sheaf Z to define a canonical element T{Omega}(L/K) of the relative K-group K0(ZG, R). It was shown that the LRNC for L/K is equivalent to the vanishing of T{Omega}(L/K) and that this, in turn, is equivalent to the equivariant Tamagawa number conjecture for the pair (h0(Spec (L))(0), ZG). These conjectures make use of a finite G-invariant set S of places of L that is supposed to be sufficiently large. We formulate an LRNC for small sets S that only need to contain the archimedean primes and give an application to a special class of CM-extensions.

2000 Mathematics Subject Classification 11R33, 11R42.

Received September 18, 2008; revised March 24, 2009; published online August 9, 2009.


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