Adjoint motives of modular forms and the Tamagawa number conjecture

Paper

Abstract

Let

be a newform of weight

, level

with coefficients in a number field

, and

the adjoint motive of the motive

associated to

. We carefully discuss the construction of the realisations of

and

, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the

-part of the Tamagawa number conjecture of Bloch and Kato for

and

. Here

is any prime of

not dividing

, and so that the mod

representation associated to

is absolutely irreducible when restricted to the Galois group over

where

. The method also establishes modularity of all lifts of the mod

representation which are crystalline of Hodge-Tate type