An abelian variety isogenous to a product of elliptic curves is said to be completely decomposable. In 1993 Ekedahl and Serre posed the following question: ""For which g does there exist a curve of genus g whose Jacobian is completely decomposable?"". For fields of characteristic zero this is expected to be a finite set. This question remains open, but Ekedahl and Serre produced a list of integers g for which they could exhibit a curve over C whose Jacobian is completely decomposable. Their list was later extended by others to include every g