The Arithmetic of Qm-abelian Surfaces through Their Galois Representations

This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with an explicit example. 1. Abelian surfaces with quaternionic multiplication Fix Q an algebraic closure of the field Q of rational numbers and let K ⊂ Q be a number field. Let A be an abelian surface defined over K. Due to Albert’s classification of involuting division algebras (cf. [11]), there is a limited number of possible structures for the algebra of endomorphisms EndQ(A)⊗Q of A. We focus our attention on the quaternionic case. While the existing literature concerning the theme mainly restrict to abelian surfaces with multiplication by a maximal order in a division quaternion algebra, in this note we consider quaternionic multiplication in a wider sense: we shall assume that EndQ(A)⊗Q is an arbitrary indefinite quaternion algebra B over Q, including the split case B = M2(Q), which is recurringly encountered in the modular setting. This allows the abelian surface to be isogenous to the product of two isogenous elliptic curves without CM. Moreover, we will let O = EndQ(A) be an arbitrary order in B, although our main results restrict to so called hereditary orders. We need to be careful on the exact order EndQ(A) ⊂ B of endomorphisms of A since we are interested on properties of A that heavily depend on its isomorphism class and badly behave up to isogeny. Let then B = ( Q ) = Q + Qi + Qj + Qij, ij = −ji, i = a, j = b with a, b ∈ Q, be a quaternion algebra and let tr : B → Q and n : B → Q, denote the reduced trace and norm, respectively. The algebra B is said to be indefinite if the archimedean place of Q is unramified: B ⊗ R ≃ M2(R). Equivalently, B is indefinite if either a > 0 or b > 0. Supported by European Research Networks at the Université de Paris 13 and the Institut de Matématiques de Jussieu and by a MECD postdoctoral grant at the Centre de Recerca Matemática from Ministerio de Educación y Cultura; Partially supported by DGICYT Grant BFM2003-06768-C02-02. 1991 Mathematics Subject Classification. 11G18, 14G35.