Related papers: Relationships Between Hyperelliptic Functions of G…

This article is devoted to an investigation of a reality condition of a hyperelliptic loop soliton of higher genus. In the investigation, we have a natural extension of Jacobi am-function for an elliptic curves to that for a hyperelliptic…

We give a precise formula for the value of the canonical Green's function at a pair of Weierstrass points on a hyperelliptic Riemann surface. Further we express the 'energy' of the Weierstrass points in terms of a spectral invariant…

We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the…

The mathematical pendulum is traditionally solved using a Jacobi elliptic functions. We solve it here using the Weierstrass elliptic function. Every initial condition of the pendulum produces an elliptic curve and a point which by the…

In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary…

Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame function is applicable to diverse areas such as boundary value problems in ellipsoidal geometry,…

A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related…

We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit…

A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…

An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.

We describe a class of spectral curves and find explicit formulas for Darboux coordinates for hyperelliptic Hitchin systems corresponding to classical simple Lie groups. We consider in detail the systems with classical rank 2 gauge groups…

This paper presents method for obtaining high-degree compression functions using natural symmetries in a given model of an elliptic curve. Such symmetries may be found using symmetry of involution $[-1]$ and symmetry of translation morphism…

In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein…

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other…

We study the integrable system of first order differential equations $\omega_i(v)'=\alpha_i\,\prod_{j\neq i}\omega_j(v)$, $(1\!\leq i, j\leq\! N)$ as an initial value problem, with real coefficients $\alpha_i$ and initial conditions…

Exploiting the generalization of the Weierstrass $\wp$ function to genus $2$ given by Komori, we give the exact connection of the related monodromy problem for genus $2$ and the classical weak $n$-point correlation functions. We also…

Assuming complex functions defined on complex curves satisfy recursion relations with respect to number of parameters, we express the corresponding cohomology theory via generalizations of holomorphic connections. In examples provided, the…

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity. We call them Baker functions. The construction is based on the Abel-Jacobi map, which allows us to identify the…

We introduce a class of functions which constitutes an obvious elliptic generalization of multiple polylogarithms. A subset of these functions appears naturally in the \epsilon-expansion of the imaginary part of the two-loop massive sunrise…

Let $E$ be an elliptic curve defined over a field $K$ (with $char(K)\neq 2$) given by a Weierstrass equation and let $P=(x,y)\in E(K)$ be a point. Then for each $n$ $\geq 1$ and some $\gamma \in K^{\ast }$ we can write the $x$- and…