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Journal of the London Mathematical Society Advance Access published online on October 15, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm074
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© 2007 London Mathematical Society

Values at s = –1 of L-functions for multi-quadratic extensions of number fields and annihilation of the tame kernel

Jonathan W. Sands

Department of Mathematics
University of Vermont
Burlington VT 05405
USA

Lloyd D. Simons

Department of Mathematics
Saint Michael's College
Colchester VT 05439
USA
lsimons{at}smcvt.edu

Suppose that E is a totally real number field which is the composite of all of its subfields E that are relative quadratic extensions of a base field F. For each such E with a ring of integers OE, assume the truth of the 2-primary part of the Birch–Tate conjecture relating the order of the tame kernel K2(OE) to the value of the Dedekind zeta function of E at s=–1, and assume the same for F as well. Excluding a certain rare situation, we prove the annihilation of K2(OE) by a generalized Stickelberger element in the group ring of the Galois group of E/F. Annihilation of the odd part of this group is proved unconditionally. This result on the odd part establishes a special case of a conjecture stated by Snaith.

2000 Mathematics Subject Classification 11R42 (primary); 11R70, 19F27 (secondary).

The first author is partially supported by NSA grant MDA 904-03-1-0003.

Received June 5, 2006; revised January 9, 2007;
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