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

Abstract

Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer $n$ we denote by $\#E({\mathbb{F}}_{q^n})$ the number of rational points on $E$ (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence $\{\#E({\mathbb{F}}_{q^n}){\}}_{n>0}$ contains at most $10^{200}$ perfect squares. If the mild condition is not satisfied, then $\#E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many $n$ including all the multiples of $24$. 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 $n\le 1000$.

15 pages