An elliptic curve analogue of Pillai’s lower bound on primitive roots

Abstract

Let /Q be an elliptic curve. For a prime of good reduction, let ( , ) be the smallest non-negative integer that gives the -coordinate of a point of maximal order in the group (F ). We prove unconditionally that ( , ) > 0.72 log log for infinitely many , and ( , ) > 0.36 log under the assumption of the Generalized Riemann Hypothesis. These can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime. IntroductionLet /F be an elliptic curve. Recall that there exist unique positive integers , such that (F ) Z/ Z × Z/ Z and | . Here is the maximal order of a point of (F ). In order to find a point on /F of maximal order, a natural strategy is to compute the orders of points with -coordinates 0, 1, 2, . . . and continue until the desired point is found. In practice, this works fairly well. A natural question is how long this process takes in the worst case.Along these lines, fix an elliptic curve /Q and let be a prime of good reduction. Let ( , ) denote the minimal -coordinate of a point of maximal order in the reduction /F . The goal of this note is to prove the following two lower bounds on ( , ). Theorem 1.1 Let /Q be an elliptic curve. There are infinitely many primes such that ( , ) > 0.72 log log . Theorem 1.2 Let /Q be an elliptic curve. Under GRH, there are infinitely many primes such that ( , ) > 0.36 log .These results can be viewed as elliptic curve analogues of lower bounds on the least primitive root ( ) of a prime . Pillai [14] proved that there is a positive constant such that ( ) > log log for infinitely many . Using Linnik's theorem in Pillai's proof, Fridlender [6] and Salié [15] improved the result to the following. We include a proof since it inspired the result of the present paper.