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.

Isogenous components of Jacobian surfaces

“…Hence, we can get an explicit equation of L n in terms of the Igusa invariants J 2 , J 4 , J 6 , J 10 ; see [45] for L 2 , [42] for L 3 , and [32] for L 5 . There is a more recent paper on the subject [25] where results of [32,42] are confirmed and equations for n > 5 are studied. 4.1.…”

“…We let R := (27v + 4v 2 − u 2 v + 4u 3 − 18uv) = 0. For 4u − v − 9 = 0 the degree 3 coverings are given by 25) and the elliptic curves have equations:…”

“…Reducible Jacobian varieties have been studied extensively since the XIX-century, most notably by Friecke, Clebch, and Bolza. In the late XX-century they became the focus of many mathematicians through the work of Frey [14,16], Shaska and Volklein [32,42,45], Kumar [25] and many others. If A/k is a 2-dimensional reducible Jacobian variety defined over a field k, then there is a degree n 2 isogeny to a product A ∼ = E 1 × E 2 , where E i , i = 1, 2 are 1dimensional.…”

Graded rings of paramodular forms of levels 5 and 7

“…In this section we give a more careful study of modular forms for the group G considered earlier. The structure of M * (G) is surely well-known (and for example the underlying surface was considered in detail in [22], section 3) but because of the frequent need to refer to it we give a complete account. Note that a related problem was solved in [24] for the group of pairs (M 1 , M 2 ) with (M −1 1 ) T ≡ M 2 mod 3 by means of invariant theory (Molien series).…”

Explicit moduli spaces for congruences of elliptic curves

“…In our terminology, it is part of the definition of an elliptic surface that it has a section. As we describe below, some of the cases in the next two theorems were already treated in [6], [10], [16].…”