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 $\mathrm{\#}E({\mathbb{F}}_{q^n})$ 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 ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ 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 $n\le 1000$.

令q为质数p的完美幂，并且$Ë（{\mathbb{F}}_q）$ 成为椭圆曲线 ${\mathbb{F}}_q$ 由等式给出 ${ÿ}^2=X^3+\mathrm{一种}X+乙$。对于正整数n，我们用$\mathrm{＃}Ë（{\mathbb{F}}_{q^{ñ}}）$扩展上E（包括无穷大）上的有理点数${\mathbb{F}}_{q^{ñ}}$。在温和的技术条件下，我们证明了该序列${\{\mathrm{＃}Ë（{\mathbb{F}}_{q^{ñ}}）\}}_{ñ>0}$最多包含10 ^200个完美正方形。如果不满足轻度条件，则$\mathrm{＃}Ë（{\mathbb{F}}_{q^{ñ}}）$是无限个n的完美平方，包括12的所有倍数。我们的证明使用了子空间定理的定量形式。我们还找到了范围内所有此类序列的所有理想平方$q<50$ 和 $ñ\le 1000$。