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Endomorphism algebras of motives attached to elliptic modular forms

Alexander F. Brown; Eknath P. Ghate — 2003

Annales de l’institut Fourier

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...

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