A characterization of certain Shimura curves in the moduli stack of abelian varieties

Abstract

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the Arakelov inequalities say that 2deg(F(1,0)) is bounded from above by g=rank(F(1,0))(2q-2+s). We show that for s>0 families reaching this bound are isogenous to the g-fold product of a modular family of elliptic curves, and a constant abelian variety. For s=0, if the flat part of the VHS is defined over the rational numbers, the family is isogeneous to the g-fold product of a family h:Z-->Y, and a constant abelian variety. In this case, h:Z-->Y is obtained from the corestriction of a quaternion algebra A, defined over a totally real numberfield F, and ramified over all infinite places but one. In case the flat part of the VHS is not defined over the rational numbers, we determine the structure of the VHS.