Torelli's theorem states:



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> Let $R$, $R'$ be compact Riemann surfaces of genus $g$, $J(R)$, $J(R')$ their Jacobian varieties, $\Theta$, $\Theta'$ their respective theta divisors. The Riemann surfaces $R$ and $R'$ are isomorphic if and only if $(J(R), \Theta)$ and $(J(R'), \Theta')$ are isomorphic as principally polarized Abelian varieties. 
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In this theorem, $J(R)$ and $J(R')$ are required to be isomorphic not only as Abelian varieties but also as principally polarized Abelian varieties. 
It turns out that the condition for $J(R)$ and $J(R')$ 
to be isomorphic as Abelian varieties alone need not imply that $R$ and $R'$ are isomorphic. 


Where can I find an example that shows that $J(R)$ and $J(R')$ being isomorphic just as Abelian varieties, does not imply that $R$ and $R'$ are isomorphic? 



Isomorphic Jacobian Varieties Just Like Abelian Varieties — Torelli's Theorem
Consider the case of curves of genus $2$. If $\mathrm{A}$ is an abelian surface and $\mathrm{C}$ a smooth curve in $\mathrm{A}$ of genus $2$, then $\mathrm{A}\simeq\mathrm{J}(\mathrm{C})$ and $\mathrm{C}$ is the theta divisor of $\mathrm{J}(\mathrm{C})$. The special case $\mathrm{A}=\mathrm{E}\times\mathrm{E}$ (where $\mathrm{E}$ is an elliptic curve) was studied in [this paper](https://projecteuclid.org/euclid.jmsj/1260463292) by Hayashida. It is known that for a given abelian variety $\mathrm{A}$ there are only finitely many curves with Jacobian $\mathrm{A}$ (see [this paper](https://www.ias.ac.in/public/Volumes/pmsc/090/02/0125-0128.pdf) by Narasimhan and Nori; for surfaces this was proven much earlier by Hayashida and Nishi). The first paper I mentioned gives in fact formulae (depending on the nature of $\mathrm{End(E)}$) for the number of curves $\mathrm{C}$ with Jacobian $\mathrm{A}=\mathrm{E}\times\mathrm{E}$.


More explicit examples (in the sense that the equations for the curves $\mathrm{C}$ can be written down) were constructed by [Howe](https://www.ams.org/journals/proc/2001-129-06/S0002-9939-00-05887-1/S0002-9939-00-05887-1.pdf).


