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Mathematics of Computation

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Elliptic curves maximal over extensions of finite base fields

Author: A. S. I. Anema
Journal: Math. Comp. 88 (2019), 453-465
MSC (2010): Primary 11G20; Secondary 11J86, 11J87, 11N36
Published electronically: May 4, 2018
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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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Additional Information

A. S. I. Anema
Affiliation: Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands

Received by editor(s): September 5, 2017
Received by editor(s) in revised form: November 22, 2017
Published electronically: May 4, 2018
Additional Notes: This research was performed by the author at the University of Groningen for his Ph.D. thesis and was financially supported by Discrete, Interactive and Algorithmic Mathematics, Algebra and Number Theory (DIAMANT), a mathematics cluster funded by the Netherlands Organisation for Scientific Research (NWO)
Article copyright: © Copyright 2018 American Mathematical Society

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