New examples of maximal curves with low genus

TL;DR

This paper explores algebraic curves over finite fields, providing explicit equations for maximal or minimal curves with low genus, and analyzes their Jacobian decompositions and automorphism groups.

Contribution

It introduces new explicit equations for maximal and minimal curves over finite fields with genus 4, 5, and 10, expanding known examples and understanding of their automorphisms.

Findings

Explicit equations for maximal/minimal curves over finite fields with genus 4, 5, 10

Identification of primes p where maximality/minimality holds

Descriptions of automorphism groups in some cases

Abstract

We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements are obtained for infinitely many $p$'s. Lists of small $p$'s for which maximality holds are provided. In some cases we describe the automorphism group of the curve.