Perfect squares representing the number of rational points on elliptic curves over finite field extensions

摘要

Let q be a perfect power of a prime number p and E(F-q) be an elliptic curve over F-q given by the equation y(2) = x(3)+Ax+B. For a positive integer n we denote by #E(F-qn) the number of rational points on E (including infinity) over the extension F-qn. Under a mild technical condition, we show that the sequence {#E(F-qn)}(n0) contains at most 10(200) perfect squares. If the mild condition is not satisfied, then #E(F-qn) is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range q 50 and n = 1000. (C) 2020 Elsevier Inc. All rights reserved.