The Bloch-Kato conjecture for adjoint motives of modular forms

Abstract

The Tamagawa number conjecture of Bloch and Kato describes the behavior at integers of the L-function associated to a motive over Q. Let f be a newform of weight k 2, level N with coefficients in a number field K. Let M be the motive associated to f and let A be the adjoint motive of M . Let be a finite prime of K. We verify the -part of the Bloch-Kato conjecture for L(A, 0) and L(A, 1) when Nk! and the mod representation associated to f is absolutely irreducible when restricted to the Galois group over Q (-1) ( -1)/2 where | .