On the linear bounds on genera of pointless hyperelliptic curves

TL;DR

This paper establishes new linear bounds on the minimum genus of pointless hyperelliptic curves over finite fields, improving previous results for odd characteristic fields and providing bounds for even characteristic fields.

Contribution

It introduces explicit constructions of hyperelliptic curves to derive improved linear bounds on the genus of pointless curves over finite fields.

Findings

New bounds depend linearly on the size of the finite field q.

Improved bounds for odd characteristic fields over previous results.

Derived bounds for even characteristic fields.

Abstract

An irreducible smooth projective curve over $\mathbb{F}\_q$ is called \textit{pointless} if it has no $\mathbb{F}\_q$-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field $\mathbb{F}\_q$. Using some explicit constructions of hyperelliptic curves, we establish two new bounds that depend linearly on the number $q$. In the case of odd characteristic this improves upon a result of R. Becker and D. Glass. We also provide a similar new bound when $q$ is even.