# Proof of Serge Lang's Heights Conjecture and an almost optimal Bound for the Torsion of Elliptic Curves

## TL;DR

This paper proves Serge Lang's Heights Conjecture effectively and offers an improved, nearly optimal polynomial bound for the torsion of elliptic curves based on the degree of the ground field.

## Contribution

It provides a complete effective proof of Serge Lang's Heights Conjecture and enhances the Mazur-Merel torsion bound with a smaller polynomial estimate.

## Key findings

- Effective proof of Serge Lang's Heights Conjecture
- New proof and improvement of Mazur-Merel torsion bound
- Torsion bound expressed as a small polynomial in the degree of the field

## Abstract

This paper focuses on the proof of Serge Lang's Heights Conjecture in a form that is completely effective. As a complementary result the author provides a new proof of Mazur-Merel theorem about a bound for the torsion of elliptic curves in terms of the degree of the ground field and improves the known result by providing a small polynomial bound in terms of this degree. (Submitted)