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

2007/02/13 by Jonathan W. Sands, Sands, Jonathan W.
Computer Science · Mathematics · #11R42 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R42

paper · pdf · doi:10.48550/arxiv.math/0702346

24 pages

arxiv created 2007/02/13 · openalex publication_date 2007/02/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix a relative quadratic extension E/F of totally real number rields and let G denote the Galois group of order 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 and let O_E^S denote the ring of S_E-integers of E. Assume the truth of the 2-part of the Birch-Tate conjecture relating the order of the tame kernel K₂(O_E^S) to the value of the Dedekind zeta function of E at s=-1, and assume the same for F as well. We then prove that the Fitting ideal of K₂(O_E^S) as a Z[G]-module is equal to a generalized Stickelberger ideal. Equality after tensoring with Z[1/2][G] holds unconditionally.

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