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Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points

Published: 14 March 2024 Publication History

Abstract

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-5 curves having more rational points than a specified bound, where “non-special curve” means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic 3.

References

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Kudo, M.; Harashita, S. (16 Apr 2020): Superspecial trigonal curves of genus 5. Experimental Mathematics. Published online: https://doi.org/10.1080/10586458.2020.1723745 Preprint ver: : Superspecial trigonal curves of genus 5. arXiv:1804.11277 [math.AG].
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        Published In

        cover image Journal of Symbolic Computation
        Journal of Symbolic Computation  Volume 122, Issue C
        May 2024
        183 pages

        Publisher

        Academic Press, Inc.

        United States

        Publication History

        Published: 14 March 2024

        Author Tags

        1. 14G05
        2. 14G15
        3. 14H10
        4. 14H45
        5. 14H50
        6. 14Q05
        7. 68W30

        Author Tags

        1. Algebraic curves
        2. Rational points
        3. Curves of genus five
        4. Non-hyperelliptic and non-trigonal curves

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