Related papers: Elliptic curves with rational subgroups of order t…
In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there…
Let K be a finite field. We know that a half of elements of K* is a square. So it is natural to ask how many of them appear as x-coordinate of points on an elliptic curve over K. We consider a specific class of elliptic curves over finite…
Given an elliptic curve ${\mathcal E}$ over a field $K$ it is a challenging problem to write down explicit elements of its endomorphism ring ${\rm End}({\mathcal E});$ the problem amounts to find all possible solutions to a functional…
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we…
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.
In this article, we study the family of elliptic curves $E/\mathbb{Q}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set…
We study and partially classify cubic rational expressions $g(x)/h(x)$ over a finite field $\mathbb{F}_q$, up to pre- and post-composition with independent M\"obius transformations. In particular, we obtain a full classification when $q$ is…
Let $E$ be an elliptic curve defined over $\Q$, and let $G$ be the torsion group $E(K)_{tors}$ for some cubic field $K$ which does not occur over $\Q$. In this paper, we determine over which types of cubic number fields (cyclic cubic,…
In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in…
The number $N_9(5)$, the maximal number of $\mathbb{F}_9$-rational points on curves over $\mathbb{F}_9$ of genus $5$ is unknown, but it is known that $32 \le N_9(5)\le 35$. In this paper, we enumerate hyperelliptic curves and trigonal…
We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of…
We bound the j -invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit (p3) has no non-trivial rational point if p is a sufficiently large…
For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex numbers, we classify all translates of elliptic curves in $E^3$ such that the $x$-coordinates satisfy a linear equation. This classification enables us to establish a…
We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from…
This is an extended version of an invited lecture I gave at the Journees Arithmetiques in St. Etienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective)…
Recently Fukasawa, Homma and Kim introduced and studied certain projective singular curves over $\mathbb {F}_q$ with many extremal properties. Here we extend their definition to more general non-rational curves.
In this article, we determine all intermediate modular curves $X_\Delta(N)$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$.
This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$,…
We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite…
We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…