VALUES AT s = −1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL
نویسنده
چکیده
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 SE denote the primes of E lying above those in S, and let OS E denote the ring of SE -integers of E. We then compare the Fitting ideal of K2(O E ) 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 Z2extension of F , and describe a class of biquadratic extensions of F = Q that satisfy this condition. 2000 MSC: Primary 11R42; Secondary 11R70, 19F27. Typeset by AMS-TEX 1
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