Mathematical Research Letters

Volume 8 (2001)

Number 4

The Bloch-Kato conjecture for adjoint motives of modular forms

Pages: 437 – 442



Fred Diamond (Brandeis University)

Matthias Flach (California Institute of Technology)

Li Guo (University of Rutgers at Newark)


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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