The moduli spaces of Jacobians isomorphic to a product of two elliptic curves

Abstract

The purpose of this paper is to study the moduli spaces of curves C of genus 2 with the property that their Jacobians \(J_C\) are isomorphic to a product surface \(E_1\times E_2\). Theorem 1 shows that the set of such curves is the union of infinitely many closed subvarieties T(d), \(d\ge 3\), of the moduli space \(M_2\). Each T(d) is a curve except for finitely many d’s for which T(d) is empty. The precise list of the exceptional d’s is given in Theorem 5 and depends on the validity of a conjecture due to Euler and Gauss. Each T(d) is the union of finitely many irreducible components \(H'(q)\), where q runs over the equivalence classs of certain binary quadratic forms of discriminant \(-16d\); cf. Theorems 2 and 3. The birational structure of the curve \(H'(q)\) (which can be viewed a “generalized Humbert variety”) is determined in Theorem 4. It turns out that \(H'(q)\) is a quotient of the modular curve \(X_0(d)\) modulo certain Atkin–Lehner involutions.