Let $X$, $Y$ be compact Riemann surfaces and $\Theta_{X}$, $\Theta_{Y}$ the theta divisors on $J(X)$, $J(Y)$ respectively. Where $J(X):=$ Jacobian of the Riemann surface $X$.

My question is what does it mean an analytic isomorphism $\varphi:J(X)\longrightarrow J(Y)$ such that $\varphi^*(\Theta_{Y})=\Theta_{X}?$

I understand that $\varphi:J(X)\longrightarrow J(Y)$ is analytic isomorphism, that is biholomorphic between $J(X)$ and $J(Y)$ as Riemann surface, but I do not understand how to define the pull-back $\varphi^*$ such that $\varphi^*(\Theta_{Y})=\Theta_{X}?$

Any help is welcome, including books references.

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    $\begingroup$ $J(X)$ is not a Riemann Surface. It has dimension $g$, the genus of $X$. Given any line bundle on $Y$ and a morphism $f:X\to Y$, $f^*L$ makes sense as a line bundle on $X$. $\endgroup$ – Mohan May 10 '17 at 0:14
  • $\begingroup$ @Mohan the analytic isomorphism $\varphi:J(X)\longrightarrow J(Y)$ it is about what structure in $J(X)$? $\endgroup$ – Manoel May 14 '17 at 19:58
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    $\begingroup$ The Jacobian is an abelian variety and any such isomorphism, upto a translate is also a group isomorphism. I am not sure I understand your question correctly. $\endgroup$ – Mohan May 14 '17 at 20:59

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