On the Sato-Tate conjecture for QM-curves of genus two

Abstract

An abelian surface A A is called a QM-abelian surface if its endomorphism ring includes an order of an indefinite quaternion algebra, and a curve C C of genus two is called a QM-curve if its jacobian variety is a QM-abelian surface. We give a computational result about the distribution of the arguments of the eigenvalues of the Frobenius endomorphisms of QM-abelian surfaces modulo good primes, which supports an analogue of the Sato-Tate Conjecture for such abelian surfaces. We also make some remarks on the field of definition of QM-curves and their endomorphisms.