Ranks of abelian varieties and the full Mordell-Lang conjecture in dimension one

TL;DR

This paper proves that for certain large fields, the group of rational points on a non-zero abelian variety has infinite rank, advancing the understanding of the Mordell-Lang conjecture in dimension one.

Contribution

It establishes the infinitude of the rational rank of abelian varieties over ample fields, specifically addressing the 1-dimensional case of the relative Mordell-Lang conjecture.

Findings

Rational rank of A(F) is infinite for large fields F.

Deduction of the 1-dimensional case of the relative Mordell-Lang conjecture.

Utilizes a result of R"ossler to achieve the proof.

Abstract

Let $A$ be a non-zero abelian variety over a field $F$ that is not algebraic over a finite field. We prove that the rational rank of the abelian group $A(F)$ is infinite when $F$ is large in the sense of Pop (also called ample). The main ingredient is a deduction of the 1-dimensional case of the relative Mordell-Lang conjecture from a result of R\"ossler.