On degenerate sigma-functions of genus two

TL;DR

This paper derives explicit formulas for degenerate genus 2 sigma-functions using genus 1 functions and elementary functions, and applies these to solve inversion problems, analyze Schrödinger operators, and construct meromorphic functions.

Contribution

It provides explicit expressions for degenerate genus 2 sigma-functions and applies them to solve inversion problems and construct special functions, advancing understanding of complex algebraic curves.

Findings

Explicit formulas for genus 2 degenerate sigma-functions

Solution to generalized Jacobi inversion problems on elliptic curves

Construction of three-periodic meromorphic functions

Abstract

We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus $1$ sigma-function and elementary functions as solutions of a system of linear PDEs satisfied by the sigma-function. By way of application we derive a solution for a class of generalized Jacobi inversion problems on elliptic curves, a family of Schr\"{o}dinger-type operators on a line with common spectrum consisting of a point and two segments, explicit construction of a field of three-periodic meromorphic functions. Generators of rank $3$ lattice in $\mathbb{C}^2$ are given explicitly.