University of Chicago Algebraic Geometry Seminar

THE MAXIMUM OR MINIMUM NUMBER OF POINTS ON GENUS 3 CURVES OVER FINITE FIELDS

Kristin Lauter

(Microsoft)

2:30 pm, Thursday, June 7th

Eckhart 206

More than half a century ago, André 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 of q elements. 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 is small with respect to the genus, 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 at most 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.