Explicit height bounds and the explicit Mordell-Lang Conjecture

TL;DR

This paper reviews key theorems and conjectures in Diophantine Geometry, providing new explicit bounds for torsion anomalous points and heights of rational points on certain curves, advancing the effective Mordell-Lang Conjecture.

Contribution

It introduces new explicit height bounds in the elliptic case and clarifies their implications for the effective Mordell-Lang Conjecture.

Findings

New bounds for torsion anomalous points

Explicit bounds on Neron-Tate height of rational points

Implications for the effective Mordell-Lang Conjecture

Abstract

We give an overview of some landmark theorems and recent conjectures in Diophantine Geometry. In the elliptic case, we prove some new bounds for torsion anomalous points and we clarify the implications of several height bounds on the effective Mordell-Lang Conjecture. We also explicitly bound the Neron-Tate height of the rational points of a family of curves of increasing genus.