Fred Diamond , Matthias Flach

The Tamagawa number conjecture of Bloch and Kato describes the behavior at integers of the L-function associated to a motive over Q. Let f be a newform of weight k ≥ 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 λ be a finite prime of K. We verify the λ-part of the Bloch-Kato conjecture for L(A, 0) and L(A, 1) when λ Nk! and the mod λ representation associated to f is absolutely irreducible when restricted to the Galois group over Q (√ (−1)(`−1)/2` ) where λ | `. This is a summary of results on the Tamagawa number conjecture of Bloch and Kato [B-K] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated L-function to arithmetic invariants of the motive. We prove in [D-F-G] that it holds up to powers of certain “bad primes.” The strategy for achieving this is essentially due to Wiles [Wi], as completed with Taylor in [T-W]. 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 [Wi] and [T-W] in the context of modular forms of weight 2, where it was used to prove results in the direction of the Fontaine-Mazur conjecture [F-M]. 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 `adic and de Rham realizations of the motive provided by theorems of Faltings [Fa] or Tsuji [Ts], 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. Suppose f is a newform of weight k ≥ 2, level N and character ψ−1 with coefficients in the ring of integers O of a number field K. For ∗ = B, dr or λ (a prime of K), we letM∗ denote the corresponding realization of the motive attached to f (see [De2]). This is a two-dimensional K∗-subspace (where KB = Kdr = K) of 1991 Mathematics Subject Classification. Primary 11F67, 11F80, 11G40; Secondary 14G10, 14F, 19F27.