arXiv subjects

Antigona Pajaziti

Publications and source records attributed to Antigona Pajaziti.

3 recordsLinked to original sources

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↗

arXiv2025

The entanglement of radicals

In this work we achieve a full understanding of the so-called entanglement of radicals, showing that over any field there are extremely few additive relations among radicals. Our results complete a famous theorem by Kneser from 1975 on the linear independence of radicals and solve a problem discussed by Lenstra in 2006.

math.NT↗

arXiv2025

Divisibility of orders of reductions of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set of primes $p$ of density $1$ is determined by the torsion subgroups of elliptic curves that are $\mathbb Q$-isogenous to $E$. In this work, we give explicit families of elliptic curves $E$ over $\mathbb Q$ together with integers $m_E$ such that the congruence class of $|\widetilde{E}_p(\mathbb{F}_p)|$ modulo $m_E$ can be computed explicitly. In addition, we can estimate the density of primes $p$ for which each congruence class occurs. These include elliptic curves over $\mathbb Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $\mathbb Q$-isogeny class of $E$. We also exhibit elliptic curves over $\mathbb Q(t)$ for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than $1$ are divisible by given small integers.

math.NT↗