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

Everett W. Howe^{[1]}; Kristin E. Lauter^{[2]}

- [1] Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967 (USA)
- [2] Microsoft Research, One Microsoft Way, Redmond, WA 98052 (USA)

Annales de l’institut Fourier (2003)

- Volume: 53, Issue: 6, page 1677-1737
- ISSN: 0373-0956

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topHowe, 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 $0<g<\log _2 q$,
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 $0<g<\log _2 q$,
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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