Special Points on Fibered Powers of Elliptic Surfaces

TL;DR

This paper characterizes subvarieties in fibered powers of elliptic surfaces containing dense torsion points with complex multiplication, linking Manin-Mumford, André-Oort, and Pink's conjectures, and introduces a new height inequality.

Contribution

It provides a new height inequality and characterizes special subvarieties with dense torsion points in fibered powers of elliptic surfaces.

Findings

Characterization of subvarieties with dense torsion points

Introduction of a new height inequality

Alternative proof of a case of Gubler's Bogomolov result

Abstract

Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the Manin-Mumford and Andr\'e-Oort Conjecture and is related to a conjecture of Pink. The main technical tool is a new height inequality. We also use it to give another proof of a case of Gubler's result on the Bogomolov Conjecture over function fields.