Unlikely Intersections of Curves with Algebraic Subgroups in Semiabelian Varieties

TL;DR

This paper proves finiteness of unlikely intersections between a curve and algebraic subgroups of codimension at least 2 in semiabelian varieties over algebraic numbers, extending previous results in special cases.

Contribution

It establishes the finiteness conjecture for unlikely intersections in general semiabelian varieties over algebraic numbers, broadening prior partial results.

Findings

Finiteness of unlikely intersections in semiabelian varieties.

Extension of previous results from toric and abelian cases.

Supports Pink-Zilber conjectures in this setting.

Abstract

Let $G$ be a semiabelian variety and $C$ a curve in $G$ that is not contained in a proper algebraic subgroup of $G$. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of $C$ with subgroups of codimension at least $2$. In this note, we establish this assertion for general semiabelian varieties over $\bar{\mathbb{Q}}$. This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case.