Degree of irrationality of a product of two elliptic curves

Abstract

In this short paper, we prove that, for any two complex elliptic curves \(E\) and \(F\), the product \(E\times F\) has degree of irrationality \(3\). The upper bound is obtained from compatible cyclic plane-cubic models of \(E\) and \(F\): three diagonal bihomogeneous sections define a dominant rational map \(E\times F\dashrightarrow\mathbb P^2\) of degree \(3\). The lower bound follows by excluding dominant rational maps of degrees \(1\) and \(2\) from an abelian surface to \(\mathbb P^2\).