Related papers: Explicit n-descent on elliptic curves, I. Algebra

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use…

We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves, considered over anticyclotomic $\mathbb{Z}_p$-extensions in both the definite and indefinite settings. The results in this paper lie at the…

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…

Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves…

We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.

We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new…

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the…

I provide a systematic construction of points (defined over number fields) on Legendre elliptic curves over $\mathbb{Q}$: for any odd integer $n\geq 3$ my method constructs $n$ points on the Legendre curve and I show that rank of the…

Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ defined over a number field $K$ with $p(x)$ integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the $2$-Selmer group of the Jacobian of…

We characterise genus 3 complex smooth hyperelliptic curves that contain two additional involutions as curves that can be build from five points in $\mathbb{P}^1$ with a distinguished triple. We are able to write down explicit equations for…

We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.

We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (arXiv:2009.08622v2) in the case…

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in…

This paper gives additional background in algebraic geometry as an accompaniment to the article, ``Formal Groups, Elliptic Curves, and some Theorems of Couveignes'' [arXiv:math.NT/9708215]. Section 1 discusses the addition law on elliptic…

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of…

In this paper, we study the $p$-Selmer groups in the family of $p$-twists of an elliptic curve $E$ over a number field $K$. We prove that if $E/K$ is an elliptic curve over a number field $K$, and if $d$ is congruent to the dimension of the…

Given a principally polarized abelian variety $(A,\Theta)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its N\'eron-Severi group.

We generalize a result of Frey [Fre88] on the Selmer group of twists of elliptic curves over Q with Q-rational torsion points to elliptic curves defined over number fields of small degree K with a K-rational point. We also provide examples…

Given a subgroup $\Gamma$ of rational points on an elliptic curve $E$ defined over ${\mathbf Q}$ of rank $r \ge 1$ and any sufficiently large $x \ge 2$, assuming that the rank of $\Gamma$ is less than $r$, we give upper and lower bounds on…