On the classification of complex tori arising from real Abelian surfaces

Let A′ be an Abelian surface over ℝ and denote by A its complexification. We define an intrinsic volume vol(A) of A and show that there are seven possibilities with respect to the rank of End(A) and if vol(A) is rational or not. We prove that each possibility determines the Picard number and the endomorphism algebra of A′ and A respectively.