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

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.