Explicit computation of the modular parametrization of elliptic curves over function fields by Drinfeld modular curves

TL;DR

This paper provides an explicit method to compute the modular parametrization of elliptic curves over function fields via Drinfeld modular curves, including formulas for evaluating at cusps and practical examples.

Contribution

It introduces a new explicit computational approach for modular parametrizations of elliptic curves over function fields using Drinfeld modular curves.

Findings

Derived a formula for evaluating modular parametrizations at cusps

Developed an explicit method for computation of these values

Presented examples in characteristic 2 and 3

Abstract

Let q be a prime power and E a non-isotrivial elliptic curve over Fq(T) given by a Weierstrass model. We survey the construction, with an explicit point of view, of the modular parametrization of E by the associated Drinfeld modular curve. We then prove a formula that allows us to evaluate this modular parametrization at cusps and we produce an explicit method to compute these values. Finally we illustrate our results with several examples in characteristic 2 and 3.