Abstract
We will give a complete classification of non-rigid families of abelian varieties by means of the endomorphism algebra of the variation of Hodge structure. As a consequence, we can obtain several conditions of rigidity for abelian schemes. For example, we show that an abelian scheme which has no isotrivial factor is rigid if the relative dimension is less than 8. Moreover, examples of nonrigid abelian schemes are obtained as Kuga fiber spaces associated to symplectic representations classified by Satake.
