You are trying to relate the periods of the curve (which are analytic invariants), to algebraic invariants, so the most you can get is some power series. Suppose you get from Gamma to the period matrix of the Jacobian: tau, then:
In genus 1 - which you are not interested in - you have the j-invariant, which is a function of tau.
In genus 2 you have the Igusa invariants.
In genus 3 you don't have a formula (too complicated), but you have an algorithm: there are 28 tangents to the theta divisor at the 28 odd 2-torsion points, these are the 28 bitangents of the canonical curve, and you can (effectively) reconstruct the curve from the bitangents.
In higher genera you can start in a similar way: map the 4-torsion points to some grassmanian (which is an embedding of A_g plus some level: Grushevsky and Salvati-Mani), but I'm not aware of a reconstruction algorithm.
No comments:
Post a Comment