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On the size of the Jacobians of curves over finite fields

On the size of the Jacobians of curves over finite fields Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field $$ \mathbb{F}_q $$ of q elements, we obtain the asymptotic formula q g+o(g) for the size of set of the $$ \mathbb{F}_q $$ -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

On the size of the Jacobians of curves over finite fields

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References (13)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-008-0006-4
Publisher site
See Article on Publisher Site

Abstract

Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field $$ \mathbb{F}_q $$ of q elements, we obtain the asymptotic formula q g+o(g) for the size of set of the $$ \mathbb{F}_q $$ -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Dec 5, 2008

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