Jacobian varieties with many elliptic curves

TL;DR

This paper constructs explicit examples of closed Riemann surfaces whose jacobian varieties decompose into products of multiple elliptic curves and other jacobians, extending previous work to higher numbers of elliptic factors.

Contribution

It provides an explicit construction of Riemann surfaces of genus depending on the number of elliptic curves, with jacobians isogenous to a product of these elliptic curves and additional jacobians.

Findings

Explicit construction for genus g=1+2^{s-2}(s-2) surfaces

Decomposition of jacobians into multiple elliptic factors

Construction for genus three with three elliptic curves

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 $E_{1}$ and $E_{2}$, 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 $E_{1} \times E_{2}$. In this paper, given $s \geq 3$ elliptic curves $E_{1},\ldots, E_{s}$, we provide an explicit construction of a closed Riemann surface $X$ of genus $g=1+2^{s-2}(s-2)$, with $JX$ isogenous to $E_{1} \times \cdots \times E_{s} \times A$, 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 $E_{1} \times E_{2} \times E_{3}$, for given elliptic curves.