On the equivariant Tamagawa number conjecture
For a finite Galois extension $K/Q$ of number fields with Galois group $G$ and a motive $M = M' otimes h^0(Spec(K))(0)$ 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 $K_0([G];R)$ 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 $h^0(Spec(K))(0)$ for an $A_4$-extension $K/Q$ and, $h^1 (E imes Spec(L))(1)$ for an $S_3$-extension $L/Q$ and an elliptic curve $E/Q$.
Advisor:Matthias Flach; Dinakar Ramakrishnan; Mladen Dimitrov; Eric Wambach
School:California Institute of Technology
School Location:USA - California
Source Type:Master's Thesis
Date of Publication:05/08/2006