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From this question, we know that every map of the form $J(C) \to J(C)$ for a curve $C$ and it's jacobian $J(C)$ comes from a correspondence between $C\times C$ and in fact we can take this correspondence as a divisor on $C\times C$.

Now, let us take a map $f: C\to D$ of curves and consider the induced map $\pi: J(C) \to J(D) \to J(C)$ which is a projection onto a factor. What correspondence does it correspond to, or more precisely, what is the divisor $X \subset C\times C$ with maps $\alpha,\beta: X \to C$ such that $\pi = \beta_*\alpha^*$?

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    $\begingroup$ $X=$ the inverse image of the diagonal in $D\times D$ by the map $(f,f):C\times C\rightarrow D\times D$. $\endgroup$
    – abx
    Commented Apr 28, 2020 at 13:45
  • $\begingroup$ Thanks, I was thinking about that too. $\endgroup$
    – Asvin
    Commented Apr 28, 2020 at 13:46
  • $\begingroup$ Note that $X$ is not unique: you can add fibres of the projections and principal divisors on $C\times C$ without changing the induced $\pi$. But @abx's $X$ is a good choice. $\endgroup$
    – Ben Smith
    Commented Apr 28, 2020 at 15:11

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