The Number of Rational Points On Genus 4 Hyperelliptic Supersingular Curves in Characteristic 2

TL;DR

This paper classifies the possible numbers of rational points on genus 4 hyperelliptic supersingular curves over finite fields of characteristic 2, specifically for fields with odd degree extension.

Contribution

It provides a complete classification of point counts for genus 4 hyperelliptic supersingular curves over finite fields of order 2^n with n odd.

Findings

Identifies all possible point counts for the specified curves.

Provides a classification that was previously unknown.

Focuses on curves over fields of characteristic 2 with odd degree extensions.

Abstract

One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.