genus 3 curves over finite fields

Kristin Lauter (Microsoft)

More than half a century ago, Andre Weil proved a formula

for the number of rational points, N(C), on a smooth projective

algebraic curve C of genus g over a finite field F_q. This

formula, along with his proof of what is referred to as the

Riemann hypothesis for curves, provides upper (resp. lower)

bounds on the maximum (resp. minimum) number of rational points

possible. There are many cases in which the Weil upper and

lower bounds cannot be attained. Some are trivial: for example

when the bound is not an integer. Also, when the field size, q,

is small with respect to the genus, g, the lower bound will be

negative and thus cannot be attained. In 1983, Serre made a

non-trivial improvement to the Weil bound. Since then there

has been considerable interest in determining the actual maximum

and minimum.

It follows from Honda-Tate theory that for genus 1 and any q, the

difference between the Serre bound and the actual maximum is either

0 or 1. For genus 2, Serre determined the actual maximum for all q,

and showed that the difference from the Serre bound is always less

than or equal to 3. For genus 3, he determined the maximum for

q <= 25. This talk will be devoted to showing that for genus 3 and

all q, either the maximum or the minimum is within 3 of the Serre

upper or lower bound. The techniques used include the classification

of Hermitian forms over rings and glueing of finite group schemes.