Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields

TL;DR

This paper explicitly constructs models for Hilbert modular surfaces with square discriminants, linking them to moduli spaces of elliptic K3 surfaces and genus-2 curves covering elliptic curves, extending classical and recent work.

Contribution

It provides explicit rational models for Hilbert modular surfaces with square discriminants and describes moduli spaces for genus-2 curves covering elliptic curves with degrees 2 to 11.

Findings

Explicit birational models for moduli spaces of genus-2 and elliptic curves.

Construction of families of reducible Jacobians.

Analysis of models from an arithmetic geometry perspective.

Abstract

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, V\"olklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the moduli space of pairs (C,E) of a genus 2 curve C and elliptic curve E with a map of degree n from C to E, as well as a tautological family over the base, for 2 <= n <= 11. We also analyze the resulting models from the point of view of arithmetic geometry, and produce several interesting curves on them.