On the Coates-Sinnott conjecture for CM fields

Abstract

Let k be a totally real number field and F be a totally real or CM field (a quadratic extension of a totally real field) that is a finite abelian extension over k. Let FΣ be the maximal algebraic extension of F unramified outside the Archimedean primes and all primes above p, and let GΣ(F)=Gal(FΣ/F). The cohomological Coates–Sinnott conjecture for an odd prime number p asserts that a certain element constructed by the special values of the partial zeta functions annihilates the Galois cohomology group H2(GΣ (F), ℤp(n+1)), where n is a positive integer. If we assume that the Quillen–Lichtenbaum conjecture holds, then the Galois cohomology group is isomorphic to the p-part of the K-group K2n(풪F), where 풪F is the ring of the algebraic integers of F. Popescu proved the cohomological Coates–Sinnott conjecture for totally real fields F and odd prime numbers p satisfying p ∤ [F:F∩k∞], where k∞ is the cyclotomic ℤp-extension of k. In this paper, we will give a simple proof of the conjecture for a totally real or CM field F and odd prime numbers p satisfying p ∤ [F:F∩k∞].