Modular curves (as moduli of elliptic curves with level structure) play a key role in the study of the arithmetic of elliptic curves. The higher genus curves have very different arithmetic, but I would like to know:
How have/can the moduli of higher genus curves (perhaps with level structure) be used to study the arithmetic of the curves (e.g. rational points, torsion in the Jacobian)?
This question is perhaps too broad: let me also give some more specific questions.
- Does the geometry of the moduli spaces have implications for the arithmetic of the curves?
- What role (and where can I find out about it) do Siegel modular forms play in the arithmetic of higher genus curves? (I have seen an example in notes of van der Geer using point counting on genus 2 curves to study the modular forms, what about the converse direction?)
- What has/can be gleaned from explicit examples (say for low genus, or hyperelliptics) like the Igusa quartic?
- Which moduli related to higher genus curves are constructed over the integers?
- What sensible questions can be asked about moduli and arithmetic of higher genus curves?
Any orientation, examples or references would be appreciated; I am entirely new to this area.