A functorial approach to regular homomorphisms

Classically, regular homomorphisms have been defined as a replacement for Abel-Jacobi maps for smooth varieties over an algebraically closed field.In this work, we interpret regular homomorphisms as morphisms from the functor of families of algebraically trivial cycles to abelian varieties and thereby define regular homomorphisms in the relative setting, for example, for families of schemes parameterized by a smooth variety over a given field.We establish existence and base change results, extending several results in the literature over algebraically closed fields.We also discuss the connection with intermediate Jacobians and Abel-Jacobi maps in the complex setting.