Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study thefinite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of$\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. Weobtain an upper bound on the degree $n$ for $E$ ordinary using an estimate forlinear forms in logarithms, which allows us to compute the pairs of isogenyclasses of such curves and degree $n$ for small $q$. Using a consequence ofSchmidt's Subspace Theorem, we improve the upper bound to $n\leq 11$ forsufficiently large $q$. We also show that there are infinitely many isogenyclasses of ordinary elliptic curves with $n=3$.
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机译：给定一个在有限域$ \ mathbb {F} _q $上的椭圆曲线$ E $，我们研究$ \ mathbb {F} _q $的有限扩展$ \ mathbb {F} _ {q ^ n} $，使得$ \ mathbb {F} _ {q ^ n} $ $ E $上的理性点达到Hasse上限。使用对数的线性形式的估计来获得$ E $普通度的度数$ n $的上限，这使我们能够计算出这些曲线的同构异类对和小$ q $的度数$ n $。利用施密特子空间定理的结果，我们将上限$ n \ leq 11 $改进为足够大的$ q $。我们还表明，具有$ n = 3 $的普通椭圆曲线有无限多个同构类。
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