Values at s=-1 of L-functions for multiquadratic extensions of number fields, and annililation of the tame kernel

2006/06/06 by Jonathan W. Sands, Sands, Jonathan W., Lloyd D. Simons +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.NT

paper · pdf · doi:10.48550/arxiv.math/0606139

15 pages

arxiv created 2006/06/06 · arxiv updated 2009/12/01

Abstract

Suppose that EE 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 ring of integers Ø_E, assume the truth of the Birch-Tate conjecture (which is almost fully established) relating the order of the tame kernel K₂(Ø_E) 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(\Oc_EE) by a generalized Stickelberger element in the group ring of the Galois group of EE/F. Annihilation of the odd part of this group is proved unconditionally.

Related

• Values at s=-1 of L-functions for relative quadratic extensions of number fields, and the Fitting ideal of the tame kernel
• Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel
• An application of Birch-Tate formula to tame kernels of real quadratic number fields
• Integrality of Stickelberger elements and annihilation of natural Galois modules
• Equivariant algebraic K-theory and Artin L-functions