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

DOI:
https://doi.org/10.1090/mcom/3342

Published electronically:
May 4, 2018

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Abstract | References | Similar Articles | Additional Information

Abstract: Given an elliptic curve over a finite field we study the finite extensions of such that the number of -rational points on attains the Hasse upper bound. We obtain an upper bound on the degree for ordinary using an estimate for linear forms in logarithms, which allows us to compute the pairs of isogeny classes of such curves and degree for small . Using a consequence of Schmidt's Subspace Theorem, we improve the upper bound to for sufficiently large . We also show that there are infinitely many isogeny classes of ordinary elliptic curves with .

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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

Email:
a.s.i.anema@22gd7.nl

DOI:
https://doi.org/10.1090/mcom/3342

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