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arXiv · 1709.01352

Elliptic curves maximal over extensions of finite base fields

Ane Anema
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Abstract

Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite 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. We obtain an upper bound on the degree $n$ for $E$ ordinary using an estimate for linear forms in logarithms, which allows us to compute the pairs of isogeny classes of such curves and degree $n$ for small $q$. Using a consequence of Schmidt's Subspace Theorem, we improve the upper bound to $n\leq 11$ for sufficiently large $q$. We also show that there are infinitely many isogeny classes of ordinary elliptic curves with $n=3$.

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Ane Anema. 2017-09-05. Elliptic curves maximal over extensions of finite base fields. https://arxiv.org/abs/1709.01352

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