On elements of large order of elliptic curves and multiplicative dependent images of rational functions over finite fields

TL;DR

This paper investigates the large order elements in elliptic curves and the multiplicative order of images of rational functions over finite fields, revealing that such elements are abundant under certain conditions.

Contribution

It establishes bounds and abundance results for elements of large order on elliptic curves and rational functions over finite fields, extending understanding of their distribution.

Findings

Most points on reduced elliptic curves have large order

Images of rational functions often have large multiplicative order

Results hold for almost all primes with finitely many exceptions

Abstract

Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant $C$ such that for almost all primes, for all but at most $C$ pairs of points on the reduction of $E_1 \times E_2$ modulo $p$ having equal $x$ coordinate, at least one among $P_1$ and $P_2$ has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo $p$ of a finite set of rational functions have large multiplicative orders