Related papers: On the order sequence of an embedding of the Ree c…

In 2017, D. Skabelund constructed a maximal curve over $\mathbb{F}_{q^4}$ as a cyclic cover of the Suzuki curve. In this paper we explicitly determine the structure of the Weierstrass semigroup at any point $P$ of the Skabelund curve. We…

In this article we explicitly determine the structure of the Weierstrass semigroups $H(P)$ for any point $P$ of the Giulietti-Korchm\'aros curve $\mathcal{X}$. We show that as the point varies, exactly three possibilities arise: One for the…

Let $E$ be an elliptic curve over $\mathbb{Q}$ described by $y^2= x^3+ Kx+ L$ where $K, L \in \mathbb{Q}$. A set of rational points $(x_i,y_i) \in E(\mathbb{Q})$ for $i=1, 2, \cdots, k$, is said to be a sequence of consecutive cubes on $E$…

In this lecture we give a brief introduction to Weierstrass points of curves and computational aspects of $q$-Weierstrass points on superelliptic curves.

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

Let $E$ be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let $P=(x_1/z_1^2,y_1/z_1^3)$ be a rational point of infinite order on $E$, where $x_1,y_1,z_1$ are coprime integers. We show that the integer…

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the…

We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2. We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending on the…

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves.

Let $C$ be an elliptic curve defined over $\mathbb Q$ by the equation $y^2=x^3+Ax+B$ where $A,B\in\mathbb Q$. A sequence of rational points $(x_i,y_i)\in C(\mathbb Q),\,i=1,2,\ldots,$ is said to form a sequence of consecutive squares on $C$…

Let $\mathbb F_q$ be a finite field with $q$ elements, where $q$ is a power of an odd prime $p$. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve $y^q - y= x \cdot F(x) - \lambda,$ where $F(x)$…

In this paper, we study configurations of three rational points on the Hermitian curve over $\mathbb{F}_{q^2}$ and classify them according to their Weierstrass semigroups. For $q>3$, we show that the number of distinct semigroups of this…

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over…

For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for…

In this paper, we study the number of $\mathbb F_{q^n}$-rational points on the affine curve $\mathcal{X}_{d,a,b}$ given by the equation $$ y^d=ax\text{Tr}(x)+b,$$ where $\text{Tr}$ denote the trace function from $\mathbb F_{q^n}$ to…

The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter…

Let $X\to \mathbb P^2$ be the elliptic Calabi-Yau threefold given by a general Weierstrass equation. We answer the enumerative question of how many discrete rational curves lie over lines in the base, proving part of a conjecture by Huang,…

We give a characterization of the Deligne-Lusztig curve associated to the Suzuki group based on the genus and the number of rational points of the curve.

The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon…

This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular…