Diamond, Fred; Flach, Matthias; Guo, Li The Bloch-Kato conjecture for adjoint motives of modular forms. (English) Zbl 1022.11023 Math. Res. Lett. 8, No. 4, 437-442 (2001). Summary: The Tamagawa number conjecture of S. Bloch and K. Kato [Grothendieck Festschrift, Vol. 1, Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)] describes the behavior at integers of the \(L\)-functions associated to a motive over \(\mathbb{Q}\). 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 \(\mathbb{Q} (\sqrt{(-1)^{(\ell-1)/2} \ell}\) where \(\lambda\mid \ell\).See also the authors’ paper [Adjoint motives of modular forms and the Tamagawa number conjecture (preprint)]and L. Guo [Math. Ann. 297, 221–233 (1993; Zbl 0789.14018)]. Cited in 5 ReviewsCited in 5 Documents MSC: 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F80 Galois representations 19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) Keywords:\(\lambda\)-part; Taylor-Wiles construction; Fontaine-Mazur conjecture; comparison isomorphism; \(\ell\)-adic and de Rham realizations; Tamagawa number conjecture; Bloch-Kato conjecture; Galois group Citations:Zbl 0789.14018; Zbl 0768.14001 PDFBibTeX XMLCite \textit{F. Diamond} et al., Math. Res. Lett. 8, No. 4, 437--442 (2001; Zbl 1022.11023) Full Text: DOI