On the Tamagawa Number Conjecture for motives attached to modular forms

by Gealy, Matthew Thomas

We carry out certain automorphic and l-adic computations, the former extending results of Beilinson and Scholl, and the latter using ideas of Kato and Kings, related to explicit motivic cohomology classes on modular varieties. Under mild local and global conditions on a modular form, these give exactly the coordinates of the Deligne and l-adic realizations of said motivic cohomology class in the eigenspace attached to the modular form (Theorem 4.1.1). Assuming Kato's Main Conjecture and a Leopoldt-type conjecture, we deduce (a weak version of) the Tamagawa Number Conjecture for the motive attached to a modular form, twisted by a negative integer.