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

Abstract

Let E1 and E2 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 E1×E2 modulo p having equal x coordinate, at least one among P1 and P2 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.