Torsion bounds for elliptic curves and Drinfeld modules

TL;DR

This paper establishes optimal upper bounds on the number of torsion points on elliptic curves and Drinfeld modules over finitely generated fields, using Galois representation techniques to unify the bounds for both structures.

Contribution

It provides a unified framework to derive asymptotically optimal torsion bounds for elliptic curves and Drinfeld modules based on adelic Galois representation openness.

Findings

Derived optimal bounds on torsion points as a function of field extension degree

Unified approach applicable to both elliptic curves and Drinfeld modules

Utilized adelic openness of Galois images for bounding torsion

Abstract

We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.