Mathematical Research Letters

Volume 8, Issue 4, July 2001 pp. 437-442.

The Bloch-Kato conjecture for adjoint motives of modular forms

Authors: Fred Diamond, Matthias Flach, and Li Guo
Author institution: Brandeis University, California Institute of Technology, and University of Rutgers at Newark

Summary: The Tamagawa number conjecture of Bloch and Kato describes the behavior at integers of the $L$-function associated to a motive over ${\mathbf Q}$. Let $f$ be a newform of weight $k\geq 2$, 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 $\lambda$ be a finite prime of $K$. We verify the $\lambda$-part of the Bloch-Kato conjecture for $L(A,0)$ and $L(A,1)$ when $\lambda\nmid Nk!$ and the mod $\lambda$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over ${\mathbf Q}\left (\sqrt{(-1)^{(\ell-1)/2}\ell}\right )$ where $\lambda\mid \ell$.

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