Related papers: Quartic equations and 2-division on elliptic curve…

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having…

An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.

Let G be the separable Galois group of a finite field F of characteristic p, and X/F an imaginary hyperelliptic curve such that G acts transitively on its set W(X) of Weierstrass points. The existence of a G-invariant 2-torsion point on the…

We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and…

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…

In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ of $K$ by solving $S$-unit equations. We give examples of…

Quantum curves arise from Seiberg-Witten curves associated to 4d $\mathcal{N}=2$ gauge theories by promoting coordinates to non-commutative operators. In this way the algebraic equation of the curve is interpreted as an operator equation…

A central problem in arithmetic geometry is to construct non-torsion rational points on elliptic curves. We study a canonical quadratic point $\xi_C \in {\rm Jac}(C)$ attached to a smooth non-hyperelliptic curve of genus 4 and use it to…

The {\it Weierstrass semigroup} of pole orders of meromorphic functions in a point $p$ of a smooth algebraic curve $C$ is a classical object of study; a celebrated problem of Hurwitz is to characterize which semigroups ${\rm S} \subset…

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic…

We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements…

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…

We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…

A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $\psi: C \to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree…

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we…

For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang…

The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…

There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion…

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…

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…