Computing binary curves of genus five. (English) Zbl 1533.11121
Summary: Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in \(\mathbb{P}^4\). We present and explain algorithms we used to determine, up to isomorphism over \(\mathbb{F}_2\), all genus 5 curves defined over \(\mathbb{F}_2\), and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over \(\mathbb{F}_2\) of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over \(\mathbb{F}_2\) weighted by the size of their \(\mathbb{F}_2\)-automorphism groups.
MSC:
| 11G20 | Curves over finite and local fields |
| 14H10 | Families, moduli of curves (algebraic) |
| 14H40 | Jacobians, Prym varieties |
| 14H37 | Automorphisms of curves |
References:
| [1] | Arbarello, E.; Cornalba, M.; Griffiths, P.; Harris, J., Geometry of Algebraic Curves, vol. 1. Grundlehren der Mathematischen Wissenschaften (1984), Springer: Springer Berlin |
| [2] | Bergström, J., Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves. Doc. Math., 259-296 (2009) · Zbl 1211.14030 |
| [3] | Brock, B. W.; Granville, A., More points than expected on curves over finite field extensions. Finite Fields Appl., 1, 70-91 (2001) · Zbl 1023.11029 |
| [4] | Conrad, K., Groups of order 12 (2023) |
| [5] | Dupuy, T.; Kedlaya, K.; Roe, D.; Vincent, C., Isogeny classes of abelian varieties over finite fields in the LMFDB, 375-448 · Zbl 07912227 |
| [6] | Faber, X.; Grantham, J., Binary curves of small fixed genus and gonality with many rational points. J. Algebra, 24-46 (2022) · Zbl 1508.11065 |
| [7] | van der Geer, G.; Howe, E.; Lauter, K.; Ritzenthaler, C., Tables of curves with many points (2009), Retrieved 14.8.2023 |
| [8] | Hartshorne, R., Algebraic Geometry (1977), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0367.14001 |
| [9] | The L-functions and modular forms database (2023) |
| [10] | The Sage Developers, SageMath (2020), the Sage Mathematics Software System (Version 9.2) |
| [11] | Wennink, T., Counting the number of trigonal curves of genus 5 over finite fields. Geom. Dedic., 31-48 (2020) · Zbl 1457.14066 |
| [12] | Xarles, X., A census of all genus 4 curves over the field with 2 elements (2020) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
