Shimura Varieties, Kummer Varieties, and Rational Curves

TL;DR

This paper investigates the presence of rational curves on Kummer varieties associated with products of elliptic curves and certain abelian varieties, revealing conditions under which these varieties lack rational curves.

Contribution

It establishes that for a very general product of seven or more elliptic curves, the Kummer variety contains no rational curves, and extends this to certain abelian varieties parametrized by Shimura varieties.

Findings

Rational curves on Kummer varieties of large products of elliptic curves project trivially onto factors.

Very general abelian varieties in certain Shimura families have Kummer varieties without rational curves.

The results connect the geometry of Kummer varieties with the structure of Shimura varieties.

Abstract

For a very general product $A$ of seven or more elliptic curves, every rational curve on the Kummer variety of $A$ projects trivially onto the Kummer variety of at least one of its factors. As a consequence, a very general member of certain families of abelian varieties parametrized by connected Shimura varieties of unitary type has the property that its Kummer variety has no rational curves.