Sharp Bound on the Number of Rational Points on Curves with Genus 1

An explicit formula for the maximal number of rational places in a global function field over Fb with genus 1 can be found, e.g., in [1]. Let μ = ⌊2$ \sqrt{{b}}$⌋ and b = pr with p prime. Then we have

Nb(1) = b + μ + \begin{displaymath}\begin{cases}0 & \textrm{if $r$\ is odd, $r\geq3$, and $p$\ divides $\mu$,}\\ 1 & \textrm{otherwise.}\end{cases}\end{displaymath}

References

[1]W. C. Waterhouse.
Abelian varieties over finite fields.
Annales Scientifiques de l’École Normale Supérieure. Quatrième Série, 2:521–560, 1969.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Sharp Bound on the Number of Rational Points on Curves with Genus 1.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_FG1.html

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