be found, for instance, in [@Earle; @E-S; @HN; @Nakajima; @Paulhus1; @Paulhus2; @RR].

The space of isomorphism classes of principally polarized abelian varieties of dimension $r$ is the upper half Siegel space ${\mathfrak H}_{r}$; of complex dimension $r(r+1)/2$. The rational symplectic group ${\rm Sp}_{2r}({\mathbb Q})$ acts on ${\mathfrak H}_{r}$ as a group of holomorphic automorphism and, moreover, if $z \in {\mathfrak H}_{r}$ and $T \in {\rm Sp}_{2r}({\mathbb Q})$, then $T(z)$ and $z$ represent isogenous principally polarized abelian varieties.

If $E_{1},\ldots, E_{r}$ are elliptic curves and $z_{0}$ represents the abelian variety $E_{1} \times \cdots \times E_{r}$, then the ${\rm Sp}_{2r}({\mathbb Q})$-orbit of $z_{0}$ is dense in ${\mathfrak H}_{r}$. As for $r \in \{2,3\}$, the jacobian locus ${\mathfrak J}_{r}$ in ${\mathfrak H}_{r}$ (i.e. the principally polarized abelian varieties obtained as jacobian of Riemann surfaces) is an open dense set, the density of the previous orbit asserts that there are infinitely many closed Riemann surfaces of genus $r$ whose jacobian variety is isogenous to $E_{1}\times \cdots \times E_{r}$; in particular, $e(r)=r$. In Theorems \[r=2\] and \[coro1\] we construct explicit equations for such Riemann surfaces (for $r=2$ this was already done by Gaudry and Schost in [@Gaudry]). If $r \geq 4$, then the dimension of 