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.
Highlights
In algebraic geometry and number theory, one is frequently interested in abelian varieties with extra endomorphisms, and their moduli spaces
The study of elliptic curves and their moduli spaces has been extremely influential in the last century
We must show that the branch locus of the map Y−(D) → HD corresponds to a divisor Z in the moduli space of MLD of K3 surfaces lattice polarized by LD, such that the rank of the K3 surface corresponding to a generic point on Z jumps to 19, and the discriminant of the Picard group is D / 2 or 2D
Summary
In algebraic geometry and number theory, one is frequently interested in abelian varieties with extra endomorphisms, and their moduli spaces. We must show that the branch locus of the map Y−(D) → HD corresponds to a divisor Z in the moduli space of MLD of K3 surfaces lattice polarized by LD, such that the rank of the K3 surface corresponding to a generic point on Z jumps to 19, and the discriminant of the Picard group is D / 2 or 2D. Proof By [24], the generic point on any component of a modular curve FN corresponds to an abelian surface A whose ring of endomorphisms is a quaternionic order R of discriminant N 2. A different proof was given by Kani and Schanz [21], who corrected a typo in [13], and described the connection with the moduli space of pairs of elliptic curves E1, E2 with an isomorphism on the n-torsion E1[n] ∼= E2[n].
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