Related papers: Building Abelian Functions with Generalised Baker-…
An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular…
In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of…
There are two famous Abel Theorems. Most well-known is his description of abelian (analytic) functions on a one dimensional compact complex torus. The other collects together those complex tori, with their prime degree isogenies, into one…
In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta…
We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special…
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a…
In this short survey we give a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. We describe how to determine such fundamental theta functions and describe the components…
We present an extension of the deformation method applied to self-dual solutions of generalized Abelian Higgs-Chern-Simons models. Starting from a model defined by a potential $V(| \phi |)$ and a non-canonical kinetic term $\omega(| \phi |)…
We derive a set of bilinear functional equations of Hirota type for the partition functions of the $sl(2)$ related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota…
We extend the group law of curves of degree three by chords and tangents to the Jacobi variety of plane curves of degree n>4 by replacing points by point groups and lines by algebraic curves. The curves are nonsingular or have simple…
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to…
One of the main purposes of this article is to give functional equations and differential equations between Bernstein basis functions and generating functions of B-spline curves. Using these equations, very useful formulas containing the…
We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations…
We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of…
In this article, a generalized Kleinian sigma function for an affine (3,4,5) space curve of genus 2 was constructed as the simplest example of the sigma function for an affine space curve, and in terms of the sigma function, the Jacobi…
We construct Lie algebras of vector fields on universal bundles $\mathcal{E}^2_{N,0}$ of symmetric squares of hyperelliptic curves of genus $g=1,2,\dots$, where $g=\left[\frac{N-1}{2}\right], \ N=3,4,\ldots$. For each of these Lie algebras,…
Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational article is to introduce the notion of a curved operad, and to develop the operadic calculus…
Necessary and sufficient conditions are given for boundedness of Hausdorff operators on generalized Hardy spaces $H^p_E(G)$, real Hardy space $H^1_{\mathbb{R}}(G)$, $BMO(G)$, and $BMOA(G)$ for compact Abelian group $G$. Surprisingly, these…
We define a new pair of dual bases that generalize the immaculate and dual immaculate bases to the colored algebras $QSym_A$ and $NSym_A$. The colored dual immaculate functions are defined combinatorially via tableaux, and we present…
This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…