Jacobian varieties with many elliptic curves

2015/07/28 by Rubén A. Hidalgo, Hidalgo, Ruben A.
Mathematics · #30F10 #30F20 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1507.07822

openalex publication_date 2015/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties. The first ones, provided by Ekedahl and Serre, are examples for which the isogenous decomposition has all factors being elliptic curves. It is well known that given two elliptic curves E₁ and E₂, there is a closed Riemann surface X of genus two, with equations in terms of the elliptic curves, and whose jacobian variety JX is isogenous to E₁ × E₂. In this paper, given s ≥ 3 elliptic curves E₁,…, E_s, we provide an explicit construction of a closed Riemann surface X of genus g=1+2^s-2(s-2), with JX isogenous to E₁ × ⋯ × E_s × A, where A is the product of some elliptic curves and jacobian varieties of hyperelliptic Riemann surfaces, all of them explicitly in terms of the given elliptic curves. In particular, for s=3, this provides explicit Riemann surface of genus three whose jacobian variety is isogenous to E₁ × E₂ × E₃, for given elliptic curves.

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