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

• Autores: Ernst Kani
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 67, Fasc. 1, 2016, págs. 21-54
• Idioma: inglés
• DOI: 10.1007/s13348-015-0148-9
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• Resumen

  • 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.