On the equivariant Tamagawa number conjecture

Abstract

For a finite Galois extension K/Q of number fields with Galois group G and a motive M = M' h°(Spec(K))(O) with coefficients in Q[G], the equivariant Tamagawa number conjecture relates the special value L* (M, 0) of the motivic L-function to an element of Ko(Z[G];R) constucted via complexes associated to M. The conjecture for nonabelian groups G is very much unexplored. In this thesis, we will develop some techniques to verify the conjecture for Artin motives and motives attached to elliptic curves. In particular, we consider motives h°(Spec(K))(0) for an A4-extension K/Q and, hi (E x Spec(L))(1) for an S3-extension L/Q and an elliptic curve E/Q.