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

摘要

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 × 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.