Decomposable abelian $G$-curves and special subvarieties

Abstract

We consider families of abelian Galois coverings of the line. When the Jacobian of the general element is totally decomposable, i.e., is isogenous to a product of elliptic curves, we prove that they yield special subvarieties of $\A_g$ if and only if a numerical condition holds, which in the general case is only known to be sufficient.

Significantly rewritten. The main theorem and its proof have been fixed, and the exposition has been improved