Pointless Hyperelliptic Curves

TL;DR

This paper investigates the existence of hyperelliptic curves over finite fields that lack rational points, focusing on various genus and field size pairs, to understand their distribution and properties.

Contribution

It explores the existence of hyperelliptic curves with no rational points over finite fields for different genus and field size combinations, providing new insights into their distribution.

Findings

Identification of conditions for existence/non-existence of such curves

Results on the distribution of hyperelliptic curves without rational points

New theoretical bounds or classifications

Abstract

In this paper we consider the question of whether there exists a hyperelliptic curve of genus $g$ which is defined over $\FF_q$ but has no rational points over $\FF_q$ for various pairs $(g,q)$.