论文标题
完美的正方形表示有限场扩展的椭圆曲线上合理点的数量
Perfect squares representing the number of rational points on elliptic curves over finite field extensions
论文作者
论文摘要
令$ q $是质数$ p $和$ e（{\ mathbb f} _q）$的完美功率，是$ {\ mathbb f} _q $的椭圆曲线，公式给出了$ y^2 = x^3+ax+ax+b $。对于一个正整数$ n $，我们用$ \＃e（{\ mathbb f} _ {q^n}）$ $ e $（包括infinity）上的理性点$ {\ mathbb f} _ {q^n} $。在温和的技术条件下，我们表明序列$ \ lbrace \＃e（{\ Mathbb f} _ {q^n}）\ rbrace_ {n> 0} $最多包含$ 10^{200} $ PERSCT PERSECT SQUARES。如果不满足温和的条件，则$ \＃E（{\ Mathbb f} _ {q^n}）$是无限多个$ n $的完美正方形，包括所有$ 24 $的倍数。我们的证明使用了子空间定理的定量版本。我们还找到了所有此类序列的所有完美广场，$ Q <50 $和$ n \ leq 1000 $。
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 $\lbrace \# E({\mathbb F}_{q^n}) \rbrace_{n>0}$ 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 $n\leq 1000$.
