The intersection of a curve with a union of translated codimension 2 subgroups in a power of an elliptic curve

TL;DR

This paper proves finiteness results for algebraic points on a curve within a power of an elliptic curve, relating to conjectures in number theory and algebraic geometry.

Contribution

It establishes finiteness of algebraic points on a curve intersecting unions of translated codimension 2 subgroups in an elliptic curve power, advancing Pink-Zilber and Mordell-Lang conjectures.

Findings

Finite set of algebraic points satisfying certain conditions

Implications for Pink-Zilber Conjecture

Connections to Mordell-Lang and Bogomolov Theorems

Abstract

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the Pink-Zilber Conjeture and the Mordel-Lang plus Bogomolov Theorem for curves.