arXiv subjects

Leolin Nkuete

Publications and source records attributed to Leolin Nkuete.

2 recordsLinked to original sources

arXiv2026

Hopf-Galois Structures on Extensions Arising from Division Points on Elliptic Curves

We provide a general construction of infinite Hopf-Galois extensions from a direct system of finite Hopf-Galois extensions. We then apply this framework to extensions generated by division points on elliptic curves. Specifically, for a number field $K$ and an elliptic curve $E/K$ of rank at least $1$ with $P\in E(K)$ of infinite order, we consider for each integer $n\geq 2$ the extension $L_n=K(Q_n)$, where $Q_n\in E(\overline{K})$ is such that $nQ_n=P$. We show that the finite-level extensions $L_n/K$ need not be Hopf-Galois, and we give sufficient conditions for them to be so. We then prove that the infinite extension $L_\infty=\bigcup_{n\geq 2}L_n$ of $K$ is potentially Hopf-Galois, meaning it becomes Hopf-Galois after a finite extension of the base field.

math.NT↗

arXiv2026

Maximal curves of genus 5 over finite fields

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 $q=5^7$.

math.NT↗