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# 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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https://hal.archives-ouvertes.fr/hal-01494793
Contributor : Ivan Pogildiakov Connect in order to contact the contributor
Submitted on : Friday, March 24, 2017 - 3:08:42 AM
Last modification on : Monday, November 15, 2021 - 7:30:02 PM
Long-term archiving on: : Sunday, June 25, 2017 - 12:21:56 PM

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### Identifiers

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

### Citation

Ivan Pogildiakov. On the linear bounds on genera of pointless hyperelliptic curves. {date}. ⟨hal-01494793⟩

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