${\mathfrak J}_{r}$ is $3(r-1)$; which is strictly smaller that that of ${\mathfrak H}_{r}$. So the density property of the ${\rm Sp}_{2r}({\mathbb Q})$-orbit of $z_{0}$ does not ensure intersection of it with ${\mathfrak J}_{r}$ (note that the density asserts that the jacobian variety of any closed Riemann surface of genus $r$ is very near, in some sense, to be isogenous to $E_{1} \times \cdots \times E_{r}$).

In Theorem \[coro2\] we observe that, for $r \geq 4$, $$e(r) \leq \left\{ \begin{array}{ll}
1+2^{(r-2)/2}r, & \mbox{for $r$ even,} \\
1+2^{(r-3)/2}(r-1), & \mbox{for $r$ odd.}
\end{array}
\right.$$

We believe these upper bounds are sharp.

The proof of Theorem \[coro2\] is based in an explicit construction. Given $(2s-3)$ elliptic curves, where $s \geq 3$, we explicitly construct a Riemann surface of genus $1+2^{s-2}(s-2)$ whose jacobian variety is isogenous to the product of $s(s-1)/2$ elliptic curves and some other jacobian varieties of elliptic/hyperelliptic curves (Theorem \[construccion\]). Such a Riemann surface happens to be one of the two irreducible component of a certain fiber product of $s$ elliptic curves (these two components being isomorphic). The construction also permits to obtain a $2$-dimensional family of Riemann surfaces of genus $g \in \{5,9\}$ whose jacobian varieties are isogenous to the 