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

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 (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.

36 pages. Final version

References in corpus (3)

• Descent via (3,3)-isogeny on Jacobians of genus 2 curves
• Real multiplication through explicit correspondences
• Algebraic models and arithmetic geometry of Teichmüller curves in genus two

Cited by in corpus (7)

• Computing the geometric endomorphism ring of a genus 2 Jacobian
• Machine learning for moduli space of genus two curves and an application to isogeny based cryptography
• Isogenous components of Jacobian surfaces
• Modularity of Landau-Ginzburg models
• On pairs of 17-congruent elliptic curves
• Families of (3,3)-split Jacobians
• Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)