Related papers: Root numbers of elliptic curves in residue charact…

A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational…

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x + b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ by \begin{equation*}…

Let $\{E_{(p,q)}\}$ be a family of elliptic curves over a rational field such that we have $E_{(p,q)} : y^2 = x^3 - p^2x + q^2$, where $p$ and $q$ are prime numbers greater than five. Earlier work showed that the elliptic curve $E_{(p,q)}$…

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…

We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient…

For a polarized complex Abelian variety A, of dimension g>1, we study the function N_A(t) counting the number of elliptic curves in A with degree bounded by t. We describe elliptic curves as solutions of Diophantine equations which, at…

Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we…

We determine all the possible torsion groups of elliptic curves over cyclic cubic fields, over non-cyclic totally real cubic fields and over complex cubic fields.

In 1990, Kraus classified all possible inertia images of the $\ell$-adic Galois representation attached to an elliptic curve over a non-archimedean local field. In previous work, the author computed explicitly the Galois representation of…

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form…

We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic…

We find a closed formula for the number $\operatorname{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k=\mathbb{F}_q$ of odd characteristic. These numbers $\operatorname{hyp}(g)$ are expressed as a polynomial in $q$ with…

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the…

If $E$ is an elliptic curve defined over a quadratic field $K$, and the $j$-invariant of $E$ is not 0 or 1728, then $E(\mathbf{Q}^{\ab})$ has infinite rank. If $E$ is an elliptic curve in Legendre form, $y^2 = x(x-1)(x-\lambda)$, where…

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given…

In this article we present a characterization of elliptic curves defined over a finite field Fq which possess a rational subgroup of order three. There are two posible cases depending on the rationality of the points in these groups. We…

We describe moduli spaces of logarithmic rank $2$ connections on elliptic curves with $n \geq 1$ poles and generic residues. In particular, we generalize a previous work by the first and second named authors. Our main approach is to analyze…

In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of…

We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also…

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal…