Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel

2007/08/06 by Jonathan W. Sands, Sands, Jonathan W.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math.KT #math.NT #msc:11R42 #msc:11R70 #msc:19F27

paper · pdf · doi:10.48550/arxiv.0708.0817

arxiv created 2007/08/06 · arxiv updated 2009/12/01

Abstract

Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K₂(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z₂-extension of F, and describe a class of biquadratic extensions of F=Q that satisfy this condition.

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