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

•  Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1967 (USA)
•  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 $g/2$.

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