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Let $q$ be a perfect power of a prime number $p$ and $E({\mathbb F}_q)$ be anelliptic curve over ${\mathbb F}_q$ given by the equation $y^2=x^3+Ax+B$. For apositive integer $n$ we denote by $ \# E({\mathbb F}_{q^n})$ the number ofrational points on $E$ (including infinity) over the extension ${\mathbbF}_{q^n}$. Under a mild technical condition, we show that the sequence $\lbrace\# E({\mathbb F}_{q^n}) \rbrace_{n>0}$ contains at most $10^{200}$ perfectsquares. 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 alsofind all the perfect squares for all such sequences in the range $q < 50$ and$n\leq 1000$.

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