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Let $Y_i=J(C_i)=\textrm{Pic}^0(C_i)$ be Jacobians of two smooth projective curves $C_1,C_2$. Suppose there is a morphism $Y_1\to Y_2$. Does it come from a morphism $C_1\to C_2$?

I ask this because I have a curve $D$ mapping to a Jacobian $Y=J(C)$ for some curve $C$. Hence there is a map from the Jacobian $J(\tilde D)$ of the normalization of $D$ to $Y$, namely the dotted arrow below:

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So I was wondering whether this map has to come from a map of curves $\tilde D\to C$. I know this is not enough motivation, but maybe it is well-known, or there is some functorial issue that I am missing. So, thanks for any help!

Brenin
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    In general, no. Consider $C_1=C_2$ of genus $>1$. There are lot of endomorphisms (e.g. multiplication by an integer) on $J(C)$, but there are only finitely many non-constant endomorphisms on $C$ (they are automorphisms). – Cantlog Nov 20 '13 at 20:58
  • @Cantlog: Thanks for your comment. – Brenin Nov 22 '13 at 23:45

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