On the Coates–Sinnott Conjecture

In [5], Coates and Sinnott formulated a far reaching conjecture linking the values Θ F/k,S (1 — n) for even integers n ≥ 2 of an S ‐imprimitive, Galois‐equivariant L ‐function Θ F/k,S associated to an abelian extension F/k of totally real number fields to the annihilators over the group ring ℤ[ G ( F/k )] of the even Quillen K ‐groups K 2 n –2 ( O F ) associated to the ring of integers O F of the top field F . In the same paper, Coates and Sinnott essentially prove the ‐adic étale cohomological version of their conjecture, in which K 2 n –2 ( OF ) is replaced by H 2 et ( OF [1 / ], ℤ( n )), for all primes > 2, under the hypothesis that k = ℚ. Refinements of this result for k = ℚ, involving Fitting ideals rather than annihilators of H 2 et ( OF [1/], ℤ ( n )), were obtained in particular cases by Cornacchia–Østvaer [7] and in general by Kurihara [14]. More recently, Burns and Greither [3] proved the same type of refinements (involving Fitting ideals of étale cohomology groups) for arbitrary totally real base fields k , but working under the very strong hypothesis that the Iwasawa μ ‐invariants μ F, vanish for all odd primes . In this paper, we study a class of abelian extensions of an arbitrary totally real base field k including, for example, subextensions of real cyclotomic extensions of type k (ζ   p   m) + / k , where p is an odd prime. For this class of extensions, we prove similar refinements of the étale cohomological version of the Coates–Sinnott conjecture, under no vanishing hypotheses for the Iwasawa μ‐invariants in question. Our methods of proof are different from the ones employed in [3], [14] and [7]. We build upon ideas developed by Greither in [10] and Wiles in [23] and [22], in the context of Brumer's Conjecture. If the Quillen–Lichtenbaum Conjecture is proved (and a proof seems tobe within reach), then we have canonical ℤ [ G ( F/k )]‐module isomorphismsfor all n ≥ 2, all i = 1,2, and all primes > 2, and all these results will yield proofs of the original K ‐theoretic version of the Coates–Sinnott Conjecture, in the cases and under the various hypotheses mentioned above (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)