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I am trying to find an hyperelliptic curve (say of genus 2) together with an endomorphism $\phi$ of its Jacobian with the property that it sends Mumford reduced divisors of the exceptional form $P-\mathcal{O}$ to divisors of the same form (and not $P+Q-2\mathcal{O}$).

For instance, if $f$ is a morphism from the hyperelliptic curve to itself, then $\phi=f_*$ will have this property. However, such a $f$ must be an automorphism, and therefore $\phi$ has finite order.

My question is: can I find such a $\phi$ which is not a root of unity?

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I may not have understood what you wrote, but: would such a phi, restricted to the image of Abel-Jacobi, yield a map from the curve to itself? Such a map cannot have degree greater than 1 once the curve has genus at least 2 (e.g. by Riemann-Hurwitz.) –  JSE Mar 22 '11 at 0:55
    
Ok great thanks (I am not an algebraic geometer, as you can see, but I get it this far). What about higher genus, say $g=3$, with exceptional divisors (not of the form $P+Q+R-3\mathcal{O}$) being sent to exceptional divisors? –  Francesco Sica Mar 30 '11 at 1:13

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