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Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?

Does the situation change in positive characteristic?

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What exactly do you mean by isogeny? Are you asking for a holomorphic maps between the tori? Or surjective group homomorphisms with finite kernel (in which case you are looking at finite-index subgroups of a lattice). – J. Martel May 6 '13 at 19:03
Being principal, jacobians do not embed as finite-index sublattices of other jacobians (i.e. are not finitely-covered by any other jacobian). – J. Martel May 6 '13 at 19:06
I thought those two notions of isogeny were equivalent over the complex numbers. Don't elliptic curves provide a counterexample to your second comment? I wanted to prove that given a hyperbolic complete curve $X$, there are only finitely many hyperbolic curves it maps to. By looking at Jacobians, we are led to the above question (with the guess that the answer is no!) – Raju May 6 '13 at 20:26
@RK I think Martel's second comment is wrong, too. I don't know the answer to the question you asked, but if you want to see a proof of the statement in your comment, this is de Franchis's theorem. – Felipe Voloch May 6 '13 at 23:20
You are right (and i had forgotten) that the two notions of isogeny are the same (after a translation along the jacobian) -- see the first section of Milne's notes on abelian varieties. Now i'm concerned with what type of isogenies we are interested in, i.e. do we expect them to preserve the polarization, to preserve up to a multiple $d$, or not. If we don't care about polarizations, then we're just talking about self-coverings of tori (of which there are many). If we demand that the polarization be preserved, then we're talking about the (finite) automorphism group of a ppav. – J. Martel May 6 '13 at 23:24

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