Modularity of Abelian Surfaces with Quaternionic Multiplication

Abstract

We prove that any abelian surface defined over Q of GL2-type having quaternionic multiplication and good reduction at 3 is modular. We generalize the result to higher dimensional abelian varieties with “sufficiently many endomorphisms” 1 Statement of the theorem In this brief article we interest ourselves in the relation to classical modular forms of the following geometric object: A an abelian surface defined over Q such that EndQ(A) = Q( √ d) (d a non-square integer) and End0Q(A) = B where B/Q is an indefinite rational quaternion algebra. We borrow from P. Clark’s note (see [C]) the name “premodular” QM-surfaces over Q for these abelian surfaces. The condition of having real multiplication defined over Q is necessary (and sufficient) in order to obtain a two dimensional Galois representation of the full Gal(Q/Q) on the Tate modules of A . This condition has to be imposed because there are examples of QM-abelian surfaces defined over Q that are not premodular (and a fortiori they are not modular, see [DR2]). Observe that the action of the quaternion algebra can not be defined over Q because Q ⊆ R. In fact (we keep from now on the premodularity condition) the minimal field K such that EndK(A) = B is an imaginary quadratic field (cf. [DR1]).