A variant of the Mordell-Lang conjecture

TL;DR

This paper investigates a new version of the Mordell-Lang conjecture focusing on semiabelian varieties that include the additive group, extending the classical understanding of intersections with finite rank subgroups.

Contribution

It introduces a variant of the Mordell-Lang conjecture specifically for products of abelian varieties and the additive group, expanding the scope of the original conjecture.

Findings

Proposes a new conjectural framework for semiabelian varieties involving the additive group.

Provides evidence or partial results supporting the variant conjecture.

Highlights differences from the classical Mordell-Lang conjecture in this setting.

Abstract

The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semiabelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite rank subgroup $\Gamma \le G(\mathbb{k})$ is a finite union of cosets of subgroups of $\Gamma$. We explore a variant of this conjecture when $G$ is a product of an abelian variety $A$ defined over $\mathbb{k}$ with the additive group $\mathbb{G}_a$.