Ultraelliptic Integrals and Two-Dimensional Sigma Functions

Abstract

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u₁, u₃) ∈ ℂ²: σ(u) = 0}, where σ(u) is the two-dimensional sigma function. We show that G(z) = F(ξ(z)), where z is a local coordinate in a neighborhood of a point of the smooth curve W and ξ(z) is the smooth function in this neighborhood given by the equation σ(u₁, ξ(u₁)) = 0. We obtain differential equations for the functions F(z), G(z), and ξ(z), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function G(z) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.