Related papers: Elliptic fibrations and $3 \cdot 2^k$

Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in…

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…

In 2006, Elkies presented an elliptic curve with 28 independent rational points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to 28 and analytic rank at most 28. We prove similar results for a previously unpublished…

We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series.

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's…

Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)\in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve…

In this short note we determine the set $\Phi^\infty(7)$ of Abelian groups that appear as torsion groups of infinitely many elliptic curves (up to $\overline \mathbb Q$-isomorphism) over number fields of degree 7.

We investigate the average rank in the family of quadratic twists of a given elliptic curve defined over $\mathbb{Q}$, when the curves are ordered using the canonical height of their lowest non-torsion rational point.

This paper classifies Lagrangian fibrations over surfaces with compact total spaces up to fiberwise symplectomorphism identical on the base.

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi--Yau 4-folds. We find 914,164 weights with degree $d\le150$ whose maximal Newton polyhedra are…

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which…

We describe a simple, but effective, method for deriving families of elliptic curves, with high rank, all of whose members have the same torsion subgroup structure.

The aim of this paper is to introduce a new family of elliptic curves with positive ranks. These elliptic curves have been constructed with certain rational numbers, namely a, b, and c as sides of Heron triangles having rational areas $k$.…

For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(\mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the…

The classification of elliptic curves E over the rationals Q is studied according to their torsion subgroups E_{tors}(Q) of rational points. Explicit criteria for the classification are given when E_{tors}(Q) are cyclic groups with even…

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over $\mathbb{F}_p$ relates to the class number of the quadratic imaginary number field $\Q(\sqrt{-p})$. In this sense,…

In this paper, for every prime $p$ and every $0\le n\le \infty$, we classify the structure of the torsion subgroup of the group of $\mathbb{Q}_p(\mu_{p^n})$-rational points of elliptic curves over $\mathbb{Q}_p$ with good reduction, where…

In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be…

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the…

Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where $p$ is an odd prime. We study the arithmetic of $E$ over extensions $\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$. The rank of $E$…