# Improved upper bounds for the number of points on curves over finite fields

• [1] Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967 (USA)
• [2] Microsoft Research, One Microsoft Way, Redmond, WA 98052 (USA)
• Volume: 53, Issue: 6, page 1677-1737
• ISSN: 0373-0956

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## Abstract

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We give new arguments that improve the known upper bounds on the maximal number ${N}_{q}\left(g\right)$ of rational points of a curve of genus $g$ over a finite field ${𝔽}_{q}$, for a number of pairs $\left(q,g\right)$. Given a pair $\left(q,g\right)$ and an integer $N$, we determine the possible zeta functions of genus-$g$ curves over ${𝔽}_{q}$ with $N$ points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-$g$ curve over ${𝔽}_{q}$ with $N$ points must have a low-degree map to another curve over ${𝔽}_{q}$, and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown values of ${N}_{q}\left(g\right)$, namely: ${N}_{4}\left(5\right)=17$, ${N}_{4}\left(10\right)=27$, ${N}_{8}\left(9\right)=45$, ${N}_{16}\left(4\right)=45$, ${N}_{128}\left(4\right)=215$, ${N}_{3}\left(6\right)=14$, ${N}_{9}\left(10\right)=54$, and ${N}_{27}\left(4\right)=64$. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-$4$ curves over ${𝔽}_{8}$ having exactly $27$ rational points. Furthermore, we show that there is an infinite sequence of $q$’s such that for every $g$ with $0<g<{log}_{2}q$, the difference between the Weil-Serre bound on ${N}_{q}\left(g\right)$ and the actual value of ${N}_{q}\left(g\right)$ is at least $g/2$.

## How to cite

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Howe, Everett W., and Lauter, Kristin E.. "Improved upper bounds for the number of points on curves over finite fields." Annales de l’institut Fourier 53.6 (2003): 1677-1737. <http://eudml.org/doc/116083>.

@article{Howe2003,
abstract = {We give new arguments that improve the known upper bounds on the maximal number $N_q(g)$ of rational points of a curve of genus $g$ over a finite field $\{\mathbb \{F\}\}_q$, for a number of pairs $(q,g)$. Given a pair $(q,g)$ and an integer $N$, we determine the possible zeta functions of genus-$g$ curves over $\{\mathbb \{F\}\}_q$ with $N$ points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-$g$ curve over $\{\mathbb \{F\}\}_q$ with $N$ points must have a low-degree map to another curve over $\{\mathbb \{F\}\}_q$, and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown values of $N_q(g)$, namely: $N_4(5) = 17$, $N_4(10) = 27$, $N_8(9) = 45$, $N_\{16\}(4) = 45$, $N_\{128\}(4) = 215$, $N_3(6) = 14$, $N_9(10) = 54$, and $N_\{27\}(4) = 64$. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-$4$ curves over $\{\mathbb \{F\}\}_8$ having exactly $27$ rational points. Furthermore, we show that there is an infinite sequence of $q$’s such that for every $g$ with $0&lt;g&lt;\log _2 q$, the difference between the Weil-Serre bound on $N_q(g)$ and the actual value of $N_q(g)$ is at least $g/2$.},
affiliation = {Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967 (USA); Microsoft Research, One Microsoft Way, Redmond, WA 98052 (USA)},
author = {Howe, Everett W., Lauter, Kristin E.},
journal = {Annales de l’institut Fourier},
keywords = {curve; rational point; zeta function; Weil bound; Serre bound; Oesterlé bound},
language = {eng},
number = {6},
pages = {1677-1737},
publisher = {Association des Annales de l'Institut Fourier},
title = {Improved upper bounds for the number of points on curves over finite fields},
url = {http://eudml.org/doc/116083},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Howe, Everett W.
AU - Lauter, Kristin E.
TI - Improved upper bounds for the number of points on curves over finite fields
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 6
SP - 1677
EP - 1737
AB - We give new arguments that improve the known upper bounds on the maximal number $N_q(g)$ of rational points of a curve of genus $g$ over a finite field ${\mathbb {F}}_q$, for a number of pairs $(q,g)$. Given a pair $(q,g)$ and an integer $N$, we determine the possible zeta functions of genus-$g$ curves over ${\mathbb {F}}_q$ with $N$ points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-$g$ curve over ${\mathbb {F}}_q$ with $N$ points must have a low-degree map to another curve over ${\mathbb {F}}_q$, and often this is enough to give us a contradiction. In particular, we are able to provide eight previously unknown values of $N_q(g)$, namely: $N_4(5) = 17$, $N_4(10) = 27$, $N_8(9) = 45$, $N_{16}(4) = 45$, $N_{128}(4) = 215$, $N_3(6) = 14$, $N_9(10) = 54$, and $N_{27}(4) = 64$. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-$4$ curves over ${\mathbb {F}}_8$ having exactly $27$ rational points. Furthermore, we show that there is an infinite sequence of $q$’s such that for every $g$ with $0&lt;g&lt;\log _2 q$, the difference between the Weil-Serre bound on $N_q(g)$ and the actual value of $N_q(g)$ is at least $g/2$.
LA - eng
KW - curve; rational point; zeta function; Weil bound; Serre bound; Oesterlé bound
UR - http://eudml.org/doc/116083
ER -

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