Related papers: Local properties of J-complex curves in Lipschitz-…

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the…

We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz…

In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and,…

We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\widehat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it…

Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. In this paper, we further investigate properties of reducible $J-$holomorphic subvarieties. We offer an upper bound of the total genus of…

We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and generalized Hessian curves.

We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements…

We generalize two classical formulas for complete intersection curves by introducing the the complete intersection discrepancy of a curve as a correction term. The first is a well-known multiplicity formula in singularity theory, due to…

For a given constant $\lambda > 0$ and a bounded Lipschitz domain $D \subset \mathbb{R}^n$ ($n \geq 2$), we establish that almost-minimizers of the functional $$ J(\mathbf{v}; D) = \int_D \sum_{i=1}^{m} \left|\nabla v_i(x) \right|^p+…

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated,…

This is the first paper of a series of three. Here we give an abstract definition of the relative compactified Jacobian of a family of reduced curves. We prove that, under some mild assumptions on the family of curves, the fibres of the…

It is shown that higher degree exact differential forms on compact Riemannian $n$-manifolds possess continuous primitives whose uniform norm is controlled by their $L^n$ norm. A contact sub-Riemannian analogue is proven, with differential…

In all possible cases, we prove that local embeddings between two curve complexes whose complexities do not increase from domain to codomain are induced by surface homeomorphism. This is our first main result. From this we can deduce our…

Topological properties of the jacobian curve ${\mathcal J}_{\mathcal{F},\mathcal{G}}$ of two foliations $\mathcal{F}$ and $\mathcal{G}$ are described in terms of invariants associated to the foliations. The main result gives a decomposition…

We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\ell$-adic representations, Selmer groups, rational points and…

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…

We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary,…

We study genus one curves that arise as 2-, 3- and 4-coverings of elliptic curves. We describe efficient algorithms for testing local solubility and modify the classical formulae for the covering maps so that they work in all…

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of…

We prove that the boundary of an almost minimizer of the intrinsic perimeter in a plentiful group can be approximated by intrinsic Lipschitz graphs. Plentiful groups are Carnot groups of step~$2$ whose center of the Lie algebra is generated…