Endomorphism algebras of motives attached to elliptic modular forms

Abstract

We study the endomorphism algebra of the motive attached to a non-CM elliptic modular cusp form. We prove that this algebra has a sub-algebra isomorphic to a certain crossed product algebra X. The Tate conjecture predicts that X is the full endomorphism algebra of the motive. We also investigate the Brauer class of X. For example we show that if the nebentypus is real and p is a prime that does not divide the level, then the local behaviour of X at a place lying above p is essentially determined by the corresponding valuation of the p-th Fourier coefficient of the form.