Modular endomorphism algebras
(Un exposé de Eknath Ghate au séminaire de théorie
des nombres de Chevaleret le 3 juin 2002)
Résumé :
We study the endomorphism algebra of the motive
attached to a non-CM elliptic modular cusp form. We prove that this
algebra has the structure of a crossed product algebra over a
number field. As a consequence we obtain the Tate conjecture for the
motive. We then investigate the Brauer class of this algebra.
We show that in many cases it is locally at $p$ determined by the
$p$-adic valuations of the Fourier coefficients of the form.
This is joint work with Alexander Brown.