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The Tamagawa number conjecture of adjoint motives of modular forms. (English) Zbl 1121.11045

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 \({\mathbb Q(\sqrt{(-1)^{(t-1)/2}l})}\), 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 S. Bloch and 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 A. Wiles [Ann. Math. (2) 141, No. 3, 553–572 (1995; Zbl 0823.11029)], as completed with 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 J.-M. Fontaine and 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 G. Faltings [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf. Baltimore 1988, 25-80 (1989; Zbl 0805.14008)] or 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:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F11 Holomorphic modular forms of integral weight
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G18 Arithmetic aspects of modular and Shimura varieties
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