Integral points on punctured abelian varieties

TL;DR

This paper proves that, under certain conditions, the punctured abelian variety with no rational points has no integral points over almost all cyclic degree  number fields, advancing understanding of integral points in number theory.

Contribution

It establishes the non-existence of integral points on punctured abelian varieties over most cyclic degree  fields under mild mod  representation conditions.

Findings

No integral points over 100% of cyclic degree  fields.

Conditions on mod  representations are sufficient.

Results apply to abelian varieties with trivial rational points.

Abstract

Let $A$ be an abelian variety over the rationals, and suppose $A(\mathbb{Q})=0$. Let $\ell$ be a rational prime. Subject to a mild condition on the mod $\ell$ representation of $A$, we show that the punctured variety $A-0$ has no integral points over 100% of cyclic degree $\ell$ number fields.