The Bloch-kato Conjecture for Adjoint Motives of Modular Forms (to Appear in Math. Res. Letters)

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 λ be a finite prime of K. We verify the λ-part of the Bloch-Kato conjecture for L(A, 0) and L(A, 1) when λ Nk! and the mod λ representation associated to f is absolutely irreducible when restricted to the Galois group over Q q (−1)(`−1)/2` where λ | `.