Related papers: Mordell-Weil Lattice via String Junctions

In the framework of algebraic topology the closed sequence of 4-dimensional polyhedra(algebraic polytopes) was defined. These polytopes were determined by the second coordination sphere of 8-dimensional lattice E8. The ordered…

We find new examples of complex surfaces with countably many non-isomorphic algebraic structures. Here is one such example: take an elliptic curve $E$ in $\mathbb P^2$ and blow up nine general points on $E$. Then the complement $M$ of the…

We investigate a system of 1,3, and 7-branes in type IIB string theory. We identify the 1-brane with the vortex of the Abelian Higgs model. We claim that in the S-dual, the theory on the 3-brane confines electric flux which we identify with…

We construct infinitely many abelian surfaces A defined over the rational numbers such that, for a prime ell <= 7, the ell-torsion subgroup of A is not isomorphic as a Galois module to the ell-torsion subgroup of its dual. We do this by…

We analyze the brane content and charges in all of the orientifold string theories on space-times of the form E x R^8, where E is an elliptic curve with holomorphic or anti-holomorphic involution. Many of these theories involve "twistings"…

The combinatorial type of an elliptic K3 surface with a zero section is the pair of the ADE -type of singular fibers and the torsion part of the Mordell-Weil group. We determine the set of connected components of the moduli of elliptic K3…

We consider the countably many families $\mathcal{L}_d$, $d\in\mathbb{N}_{\geq 2}$, of K3 surfaces admitting an elliptic fibration with positive Mordell--Weil rank. We prove that the elliptic fibrations on the very general member of these…

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is…

We explore the connection between super $\mathcal{W}$-algebras ($\mathcal{SW}$-algebras) and $\mathrm{G}$-structures with torsion. The former are realised as symmetry algebras of strings with $\mathcal{N}=(1,0)$ supersymmetry on the…

A referee found an error in the proof of the Theorem 2 that we could not fix. More precisely, the proof of Lemma 2.1 is incorrect. Hence the fact that integer cohomology of complement of toric Weyl arrangements is torsion free is still a…

Given a finite lattice $L$ that can be embedded in the recursively enumerable (r.e.) Turing degrees $\mathcal{R}_{\mathrm{T}}$, it is not known how one can characterize the degrees $\mathbf{d}\in\mathcal{R}_{\mathrm{T}}$ below which $L$ can…

We demonstrate that the functorial properties of the symplectic field theory under strong cobordisms and surgery cobordisms can produce finite algebraic (planar) torsions from simple examples, which gives a unified treatment of most of the…

In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski…

We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the…

The classical Mordell-Weil theorem implies that an abelian variety $A$ over a number field $K$ has only finitely many $K$-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\rm…

We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for some…

Suppose $E$ is an elliptic curve defined over $\Q$. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of…

We study the distribution of ranks of elliptic curves in quadratic twist families using Iwasawa-theoretic methods, contributing to the understanding of Goldfeld's conjecture. Given an elliptic curve $ E/\mathbb{Q} $ with good ordinary…

We study the Diophantine equation $a^5+b^5=c^5+d^5$ under the linear slicing constraint $(c+d)-(a+b)=h$. We first prove the necessary congruence $30\mid h$. After symmetrization, the associated discriminant equation defines, for each fixed…

In this paper we study the 6d localized charged matter spectrum of F-theory directly on a singular elliptic Calabi-Yau 3-fold, i.e. without smoothing via resolution or deformation of the entire fibration. Given only the base surface,…