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Values at s = −1 of L ‐functions for multi‐quadratic extensions of number fields and annihilation of the tame kernel
Author(s) -
Sands Jonathan W.,
Simons Lloyd D.
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm074
Subject(s) - mathematics , algebraic number field , cyclotomic field , kernel (algebra) , annihilation , conjecture , galois module , pure mathematics , quadratic equation , order (exchange) , galois group , field (mathematics) , ring (chemistry) , ring of integers , group (periodic table) , combinatorics , physics , particle physics , geometry , quantum mechanics , chemistry , organic chemistry , finance , economics
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 E , assume the truth of the 2‐primary part of the Birch–Tate conjecture relating the order of the tame kernel K 2 ( 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 K 2 ( ℰ ) 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.