New bounds on the maximum number of points on genus-4 curves over small finite fields

TL;DR

This paper establishes new bounds and exact values for the maximum number of points on genus-4 curves over small finite fields with fewer than 100 elements, advancing understanding in algebraic geometry and finite field theory.

Contribution

It provides new upper and lower bounds for N_q(4) and determines exact values for 17 previously unknown cases over small finite fields.

Findings

New bounds on N_q(4) for q<100

Exact values determined for 17 prime powers

Improved understanding of genus-4 curves over small fields

Abstract

For prime powers q<100, we compute new upper and lower bounds on N_q(4), the maximal number of points on a genus-4 curve over a finite field with q elements. We determine the exact value of N_q(4) for 17 prime powers q for which the value was previously unknown.