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Zbl 1121.11045
Diamond, Fred; Flach, Matthias; Guo, Li
The Tamagawa number conjecture of adjoint motives of modular forms. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 37, No. 5, 663-727 (2004). ISSN 0012-9593

From the text: Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the $\lambda$-part of the Tamagawa number conjecture of Bloch and Kato for $L(A,0)$ and $L(A,1)$. Here $\lambda$ is any prime of $K$ not dividing $Nk$!, and such that the mod $\lambda$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over ${\Bbb Q(\sqrt{(-1)^{(t-1)/2}l})}$, where $\lambda\mid l$. The method also establishes modularity of all lifts of the mod $\lambda$ representation which are crystalline of Hodge-Tate type $(0,k-1)$. This paper concerns the Tamagawa number conjecture of {\it S. Bloch} and {\it K. Kato} [in: The Grothendieck Festschrift. Vol. I, Prog. Math. 86, 333--400 (1990; Zbl 0768.14001)] for adjoint motives of modular forms of weight $k\geq 2$. The conjecture relates the value at 0 of the associated $L$-function to arithmetic invariants of the motive. We prove that it holds up to powers of certain `bad primes'. The strategy for achieving this is essentially due to {\it A. Wiles} [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11029)], as completed with {\it R. Taylor} in and [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)]. The Taylor-Wiles construction yields a formula relating the size of a certain module measuring congruences between modular forms to that of a certain Galois cohomology group. This was carried out in [A. J. Wiles, loc. cit.] and [R. L. Taylor and A. J. Wiles, loc. cit.] in the context of modular forms of weight 2, where it was used to prove results in the direction of the Fontaine-Mazur conjecture {\it J.-M. Fontaine} and {\it B. Mazur} in [Coates, John (ed.) et al., Elliptic curves, modular forms, \& Fermat's last theorem; Cambridge, MA: International Press. Ser. Number Theory 1, 41--78 (1995; Zbl 0839.14011)]. While it was no surprise that the method could be generalized to higher weight modular forms and that the resulting formula would be related to the Bloch-Kato conjecture, there remained many technical details to verify in order to accomplish this. In particular, the very formulation of the conjecture relies on a comparison isomorphism between the $l$-adic and de Rham realizations of the motive provided by theorems of {\it G. Faltings} [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf. Baltimore 1988, 25-80 (1989; Zbl 0805.14008)] or {\it T. Tsuji} [Invent. Math. 137, 233--411 (1999; Zbl 0945.14008)], and verification of the conjecture requires the careful application of such a theorem. We also need to generalize results on congruences between modular forms to higher weight, and to compute certain local Tamagawa numbers.

MSC 2000:: *11G40 L-functions of varieties over global fields
11F11 Modular forms, one variable
11F67 Special values of automorphic L-series, etc
11G18 Arithmetic aspects of modular and Shimura varieties

Citations: Zbl 0768.14001; Zbl 0823.11029; Zbl 0823.11030; Zbl 0839.14011; Zbl 0805.14008; Zbl 0945.14008

Cited in: Zbl 1151.11323 Zbl 1070.11025 Zbl 1162.11355

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