Optimal curves of low genus over finite fields

TL;DR

This paper improves bounds on the number of rational points for low genus curves over finite fields with specific discriminants by analyzing optimal curves, advancing understanding in algebraic geometry and finite field theory.

Contribution

It provides new bounds for low genus curves over finite fields with certain discriminants through the study of optimal curves, enhancing existing theoretical results.

Findings

Improved Hasse-Weil-Serre bounds for specific discriminants

Identification of optimal curves achieving these bounds

Enhanced understanding of curve properties over finite fields

Abstract

The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.