Open Access

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

Research in the Mathematical Sciences20152:24

DOI: 10.1186/s40687-015-0042-9

Received: 9 June 2015

Accepted: 30 September 2015

Published: 24 November 2015

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, Völklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the moduli space of pairs (CE) 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 \(2 \le n \le 11\). We also analyze the resulting models from the point of view of arithmetic geometry, and produce several interesting curves on them.

Keywords

K3 surfaces Moduli spaces Hilbert modular surfaces Genus-2 curves Jacobians Elliptic curves