Related papers: On cubic elliptic varieties

We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of…

In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we…

The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture…

We study automorphism groups of fibered surfaces for finite cyclic covering fibrations of an elliptic surface. We estimate the order of a finite subgroup of automorphism groups in terms of the genus of the fiber, the genus of the base…

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.

Let X be a smooth projective variety. Starting with a finite set of cycles on powers X^m of X, we consider the Q-vector subspaces of the Q-linear Chow groups of the X^m obtained by iterating the algebraic operations and pullback and push…

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma = +1or $-1.Here the Mordell-Weil group and the…

Let $X$ be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations $\pi_i$, $i=1,2$, defined over a number field $k$. We prove that there is an elliptic curve $C\subset X$ such that the generic rank over $k$ of $X$…

We prove that, on average, elliptic curves over Q have finitely many primes p for which they possess a p-adic point of order p. We include a discussion of applications to companion forms and the deformation theory of Galois representations.

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We…

Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve. If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$. In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture.…

Let k be a number field, let E/k be an elliptic curve, and let S be a finite set of places of k contianing the archimedean places. Let F be an algebraic closure of k. We prove that if a point P in E(F) is nontorsion, then there are only…

In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over…

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 study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is…

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield…

We present here quantitative versions in 1 dimension of Faltings'theorem according to which the set of the K-rational points (where K is a given number field) of an abelian variety A definied over K, which are close (with respect to a…

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its…

Let $X$ be a rational surface obtained by blowing up at a configuration $\mathcal{C}$ of infinitely near points over a Hirzebruch surface $\mathbb{F}_\delta$. We prove that there exist two positive integers $a \leq b$ such that the cone of…

Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $\sigma\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$…