  We show that the image of the Abel-Jacobi map admits functorially a model
over the field of definition, with the property that the Abel-Jacobi map is
equivariant with respect to this model. The cohomology of this abelian variety
over the base field is isomorphic as a Galois representation to the deepest
part of the coniveau filtration of the cohomology of the projective variety.
Moreover, we show that this model over the base field is dominated by the
Albanese variety of a product of components of the Hilbert scheme of the
projective variety, and thus we answer a question of Mazur. We also recover a
result of Deligne on complete intersections of Hodge level one.
