Related papers: Counting elliptic curves with prescribed level str…

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…

In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and…

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y^2=x^3+n, n \in Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds…

Rank computation of elliptic curves has deep relations with various unsolved questions in number theory, most notably in the congruent number problem for right-angled triangles. Similar relations between elliptic curves and Heron triangles…

Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of results of Hindry-Silverman, giving an upper bound for the number of k-rational torsion points, and a lower bound for the canonical height of…

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to…

For any elliptic curve E defined over the rationals with complex multiplication and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion…

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic…

We consider an algebraic variety and its foliation, both defined over a number field. We prove upper bounds for the geometric complexity of the intersection between a leaf of the foliation and a subvariety of complementary dimension (also…

This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this…

We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height…

The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…

We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_{\epsilon}(|{\rm Disc}(K)|^{1/2+\epsilon})$ by Brauer--Siegel).…

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

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,…

Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i…

We give new parametrisations of elliptic curves in Weierstrass normal form $y^2=x^3+ax^2+bx$ with torsion groups $\mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$ over $\mathbb{Q}$, and with $\mathbb{Z}/14\mathbb{Z}$ and…

Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}_{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}_{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.