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@Felipe, As much as I understood from Silverman, if you have maps between function fields, then we have map between curves. This induces maps $\phi_*$ and $\phi^*$ on the divisors.
@Felipe, I made the example explicit. Could you please show me how can I use the generic point technique to compute $K^{\mu}$? Thanx. (My other problem is that I'm not sure if I know the definition of the function field "of" an Ab variety, my guess is the function field whose Jacobian is that variety :? ).
@Felipe I read in this paper: arxiv.org/abs/math/0205314 at top of page 2: A curve with large automorphism group always has orbit genus 0. Large means |Aut(C)| > |4(g-1)|.
@Inkspot I don't get "a general curve will have no morphisms to any other curve of positive genus" part. If $F$ is a subfield of $F$ then there's morphism from the defining curve of $F$ to the defining curve of $E$. Isn't in second chapter of Silverman? By random, I mean given function field $F$. Is there a way to generate a subfield of it, of given genus. As much as I understand you say there's no subfield other than rational subfields (of genus zero)?
