The Mordell-Lang Question for Endomorphisms of Semiabelian Varieties

TL;DR

This paper explores the Mordell-Lang conjecture for semiabelian varieties under endomorphisms, establishing conditions where the conjecture holds and providing counterexamples when these conditions are not met.

Contribution

It extends the Mordell-Lang conjecture to settings involving algebraic endomorphisms, identifying specific cases where the conjecture remains valid.

Findings

The conjecture holds if the semiabelian variety is simple.

The conjecture holds for the product of a one-dimensional semiabelian variety with itself.

The conjecture holds when the subvariety is a connected one-dimensional algebraic subgroup.

Abstract

The Mordell-Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell-Lang conjecture remains true in this setting if either (1) the semiabelian variety is simple, (2) the semiabelian variety is $A^2$, where $A$ is a one-dimensional semiabelian variety, (3) the subvariety is a connected one-dimensional algebraic subgroup, or (4) each endomorphism has diagonalizable Jacobian at the origin. We also give examples showing that the conclusion fails if we make slight modifications to any of these hypotheses.