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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.

  • Give some examples of how the geometry of the moduli spaces has 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 invariants for genus 2?
  • Which moduli related to higher genus curves are constructed over the integers?

Any orientation, examples or references would be appreciated; I am entirely new to this area.

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Welcome to MathOverflow. That's a lot of questions, but regarding the use of Siegel modular forms, here is a relevant reference, which you might already know about: "Jacobians among abelian threefolds: a formula of Klein and a question of Serre", by Lachaud, Ritzenthaler and Zykin. – Barinder Banwait Aug 15 '11 at 21:05
There is a relation between the geometry of the moduli spaces and distribution of rational points on higher genus curves via the Lang-Vojta conjectures, cf. Caporaso-Harris-Mazur. – Jason Starr Aug 16 '11 at 13:44
Faltings's proof of the Mordell conjecture takes place in the moduli space of abelian varieties. – Felipe Voloch Jul 7 '14 at 18:00

1 Answer 1

There is some connection between the minimal height of curves and their moduli height. For example one can show that for a genus $g \geq 2$ curve $C$ we have that $$\mathcal H (C) < c \cdot \bar H (C), $$ where $\mathcal H (C)$ is the moduli height, $\bar H (C)$ the minimal height, and $c$ a constant.

You can check this paper for some of the details.

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