The maximum or minimum number of points on
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.