On degenerate sigma-functions of genus two

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 Schrödinger-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.

20 pages, 2 figures

References in corpus (3)

• On higher genus Weierstrass sigma-function
• Identities for hyperelliptic P-functions of genus one, two and three in covariant form
• An Extended Abel-Jacobi Map

Cited by in corpus (3)

• Sato Grassmannian and Degenerate Sigma Function
• Solution of the Jacobi inversion problem on non-hyperelliptic curves
• Abelian function fields on Jacobian varieties