Maximal curves of genus 5 over finite fields
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912592516087808 |
|---|---|
| author | Nkuete, Leolin Pajaziti, Antigona Suluyer, Hamide Uysal, Rabia Gülşah |
| author_facet | Nkuete, Leolin Pajaziti, Antigona Suluyer, Hamide Uysal, Rabia Gülşah |
| contents | A maximal curve over a finite field $\mathbb F_q$ is a curve whose number of points reaches the upper Hasse-Weil-Serre bound. We define the discriminant of $\mathbb F_q$ as $d(\mathbb F_q):= \lfloor2\sqrt{q}\rfloor^2-4q$, which arises as the discriminant of the characteristic polynomial of the Frobenius for a maximal elliptic curve defined over $\mathbb F_q$. In this article we investigate the existence of a maximal curve of genus $5$ defined over a finite field $\mathbb F_q$ of discriminant $-19$. Using the knowledge on the automorphism group of such a curve, we prove that such curve does not exist when $q\equiv 2,3,4 \mod 5$. In the case $q\equiv 1\mod 5$ we give models of the potential maximal curve. Finally, for the case $q\equiv 0\bmod 5$, we prove that such a curve might exist only for $q=5^7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal curves of genus 5 over finite fields Nkuete, Leolin Pajaziti, Antigona Suluyer, Hamide Uysal, Rabia Gülşah Number Theory Algebraic Geometry 14H25 (Primary) 11G20 (Secondary) A maximal curve over a finite field $\mathbb F_q$ is a curve whose number of points reaches the upper Hasse-Weil-Serre bound. We define the discriminant of $\mathbb F_q$ as $d(\mathbb F_q):= \lfloor2\sqrt{q}\rfloor^2-4q$, which arises as the discriminant of the characteristic polynomial of the Frobenius for a maximal elliptic curve defined over $\mathbb F_q$. In this article we investigate the existence of a maximal curve of genus $5$ defined over a finite field $\mathbb F_q$ of discriminant $-19$. Using the knowledge on the automorphism group of such a curve, we prove that such curve does not exist when $q\equiv 2,3,4 \mod 5$. In the case $q\equiv 1\mod 5$ we give models of the potential maximal curve. Finally, for the case $q\equiv 0\bmod 5$, we prove that such a curve might exist only for $q=5^7$. |
| title | Maximal curves of genus 5 over finite fields |
| topic | Number Theory Algebraic Geometry 14H25 (Primary) 11G20 (Secondary) |
| url | https://arxiv.org/abs/2509.14871 |