Mathematical Research Letters

Volume 15, Issue 2, March 2008 pp. 251-263.

Which quaternion algebras act on a modular abelian variety?

Authors: Victor Rotger
Author institution: Universitat Politècnica de Catalunya

Summary: Let $A/\Q $ be a modular abelian variety. We establish criteria to prevent a given quaternion algebra $B$ over a totally real number field to be the endomorphism algebra of $A$ over $\qbar $. We accomplish this by analyzing the representation of $\Gal(\qbar /\Q )$ on the points of $N$-torsion of $A$ at primes $N$ which ramify in $B$.

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