Resumen de The moduli spaces of Jacobians isomorphic to a product of two elliptic curves
The purpose of this paper is to study the moduli spaces of curves C of genus 2 with the property that their Jacobians ?? are isomorphic to a product surface ?1×?2 . Theorem 1 shows that the set of such curves is the union of infinitely many closed subvarieties T(d), ?≥3 , of the moduli space ?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 ?′(?) , where q runs over the equivalence classs of certain binary quadratic forms of discriminant −16? ; cf. Theorems 2 and 3. The birational structure of the curve ?′(?) (which can be viewed a “generalized Humbert variety”) is determined in Theorem 4. It turns out that ?′(?) is a quotient of the modular curve ?0(?) modulo certain Atkin–Lehner involutions.
