smaller dimensions. A non-simple abelian variety is then isogenous to a non-trivial product of abelian varieties of smallest positive dimension. Such a decomposition is not unique, but if we, moreover, require all the factors to be simple, then Poincaré reducibility theorem asserts that the decomposition is unique up to permutation of factors. Examples of (principally polarized) abelian varieties are given by the jacobian variety of a closed Riemann surface (the polarization is induced by the intersection form of the first integral homology group of the surface). In this paper we should restrict to these types of principally polarized abelian varieties and isogenous decompositions of them into a product of jacobian varieties.

Let $S$ be a closed Riemann surface and $JS$ be its jacobian variety. If $JS$ is a non-simple abelian variety, then it is isogenous to a product $A_{1}^{n_{1}} \times \cdots \times A_{s}^{n_{s}}$, where $A_{j}$ is an abelian variety of positive dimension, $n_{j} \geq 1$ is an integer and $s \geq 2$. If $g \leq 4$, then it possible to assume all the factors $A_{j}$ to be jacobian varieties. For $g \geq 5$, this seems to be a difficult problem to see if one may choose such a decomposition so 