Generalised Howe curves of genus five attaining the Serre bound

TL;DR

This paper constructs genus 5 curves called generalized Howe curves that reach the Serre bound over specific finite fields, and analyzes their Jacobians and field conditions.

Contribution

It introduces new genus 5 curves attaining the Serre bound and provides a complete Jacobian decomposition under certain conditions.

Findings

Genus 5 generalized Howe curves attain the Serre bound over finite fields of size p, p^2, or p^3.

Complete Jacobian decomposition is achieved under specific assumptions.

Precise conditions on the finite field for attaining the bound are determined.

Abstract

We find that non-hyperelliptic generalised Howe curves and their twists of genus 5 attain the Hasse-Weil-Serre bound over some finite fields of order p, p^2 or p^3 for a prime p. We are able to decompose their Jacobians completely under certain assumptions and to determine the precise condition on the finite field over which they attain the Hasse-Weil-Serre bound.