arXiv subjects

Hamide Suluyer

Publications and source records attributed to Hamide Suluyer.

4 recordsLinked to original sources

arXiv2025

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↗

arXiv2024

Rational torsion on hyperelliptic jacobian varieties

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its Jacobian. Lepr\'{e}vost proved this conjecture with $\kappa=3$. In this work we prove that given an integer $N$ in the interval $[3g,4g+1]$, $g\ge 3$, satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order $N$. In particular, we establish the existence of such varieties for $N=4g+1$ when $g$ is odd and for $N=4g-1$ when $g$ is even. A few explicit applications of this result produce the first known infinite examples of torsion 13 when $g=3$, torsion 15 when $g=4$, and torsion $17,18,21$ when $g=5$. In fact, we show that infinitely many of the latter abelian varieties are absolutely simple.

math.NT↗

arXiv2024

Quadratic torsion orders on Jacobian varieties

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+ 2g -4$. For $N = 2g^{2} + 7g + 1$, we introduce a 1-parameter family of polynomials $f_{t}(x)$ of degree $2g+1$. For all but finitely many rational values of $t$, if the discriminant of $f_{t}(x)$ is nonzero, then the hyperelliptic curve defined by $y^{2} = f_{t}(x)$ has a rational point of order $N$ on its Jacobian.

math.NT↗

arXiv2023

Rational approximations, multidimensional continued fractions and lattice reduction

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.

math.NT↗