\in \Delta_{1}$ there exist infinitely many values $\mu \in \Delta_{1}$ (in fact a dense subset) so that $E_{\mu}$ and $E_{\lambda}$ are isogenous. In particular, given $s \geq 2$ elliptic curves $E_{1},\ldots, E_{s}$, there are infinitely many tuples $(\lambda_{1},\ldots,\lambda_{s}) \in \Delta_{s}$ so that $E_{\lambda_{j}}$ and $E_{j}$ are isogenous for each $j=1,\ldots,s$. For $s=2$, we may even replace “isogenous" by “isomorphic".

Genus two case: the known situation
-----------------------------------

As already noted in the introduction, given any pair $(E_{1},E_{2})$ of elliptic curves, there is a closed Riemann surface $S$ of genus two whose jacobian variety $JS$ is isogenous to $E_{1} \times E_{2}$. The following describes explicit equations for one of them (an argument is provided in Section \[Sec:r=2\]). This is not new and it can be tracked back to, for instance, [@Earle; @Gaudry; @HN; @Hermite].

\[r=2\] Let $E_{1}$ and $E_{2}$ two elliptic curves. Choose $(\lambda_{1},\lambda_{2}) \in \Delta_{2}$ so that $E_{j}$ is isomorphic (or isogenous) to $E_{\lambda_{j}}$, for $j=1,2$, and set $$\eta_{1}=\frac{\lambda_{1}-1}{\lambda_{2}-1}, \quad \eta_{2}=\frac{\lambda_{2}(\lambda_{1}-1)}{\lambda_{1}(\lambda_{2}-1)}.$$

If $S$ is the genus two Riemann surface defined by the hyperelliptic curve $$y^{2}=(x^{2}-1)(x^{2}-\eta_{1})(x^{2}-\eta_{2}),$$ then $JS$ is isogenous to $E_{1} \times E_{2}$.

The constructed genus two Riemann surface $S$ in Theorem \[r=2\] admits a non-hyperelliptic involution $\alpha$ so that $S/\langle 