# On the linear bounds on genera of pointless hyperelliptic curves

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.
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Type de document :
Pré-publication, Document de travail
2017
Domaine :

https://hal.archives-ouvertes.fr/hal-01494793
Contributeur : Ivan Pogildiakov <>
Soumis le : vendredi 24 mars 2017 - 03:08:42
Dernière modification le : samedi 25 mars 2017 - 01:10:45

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• HAL Id : hal-01494793, version 1
• ARXIV : 1703.08312

### Citation

Ivan Pogildiakov. On the linear bounds on genera of pointless hyperelliptic curves. 2017. <hal-01494793>

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