Related papers: Generalizing quadratic $\mathbb{R}$-Algebraic sets…

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…

Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings…

In this paper we generalize Poletsky's classical theorem to a situation where the kernel of Poisson functional is not upper semicontinuous. We give a characterization of thinness of a subset at a point in $\C^n$ in term of analytic discs.

Let $C$ be a $4$-cover of an elliptic curve $E$, written as a quadric intersection in $\mathbb{P}^3$. Let $E'$ be another elliptic curve with $4$-torsion isomorphic to that of $E$. We show how to write down the $4$-cover $C'$ of $E'$ with…

Let $E_\lambda$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-\lambda)$. We recently proved that, given $n$ linearly independent points $P_1(\lambda), \dots,P_n(\lambda)$ on $E_\lambda$ with coordinates in…

Let $X \subset \mathbb{C}^n$ be an algebraic variety, and let $\Lambda \subset \mathbb{C}^n$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^n / \Lambda$,…

Let $S$ be a smooth irreducible curve defined over $\overline{\mathbb{Q}}$, let $\mathcal{A}$ be an abelian scheme over $S$ and $\mathcal{C}$ a curve inside $\mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In this paper we prove…

Let $G$ be a connected semi-simple algebraic group of adjoint type over an algebraically closed field, and let $\overline{G}$ be the wonderful compactification of $G$. For a fixed pair $(B, B^-)$ of opposite Borel subgroups of $G$, we look…

We prove the following local version of Blaschke--Kakutani's characterization of ellipsoids: Let $V$ be a finite-dimensional real vector space, $B\subset V$ a convex body with 0 in its interior, and ${2\le k<\dim V}$ an integer. Suppose…

Let $K$ be a compact convex body in $\mathbb R^n.$ For any affine line $L,$ denote $\widehat{\chi}_K(L)=\int_{L}\chi_K(x)dl(x),$ where $dl$ is the arc length measure, the $X$-ray transform of the characteristic function $\chi_K,$ i.e., the…

Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of ${\mathbb R}^4$. This includes the ellipsoidal…

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to…

We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Ka\v{s}in decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's…

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach…

Motivated by relating the representation theory of the split real and $p$-adic forms of a connected reductive algebraic group $G$, we describe a subset of $2^r$ orbits on the complex flag variety for a certain symmetric subgroup. (Here $r$…

We consider a product $X=E_1\times\cdots\times E_d$ of elliptic curves over a finite extension $K$ of $\mathbb{Q}_p$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has…

A discrete group $\G$ is called rigidly symmetric if the projective tensor product between the convolution algebra $\ell^1(\G)$ and any $C^*$-algebra $\A$ is symmetric. We show that in each topologically graded $C^*$-algebra over a rigidly…

Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of…

The main result states that a connected conic singular sub-manifold of a Riemannian manifold, compact when the ambient manifold is non-Euclidean, is Lipschitz Normally Embedded: the outer and inner metric space structures are metrically…

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…