Ultraelliptic Integrals and Two-Dimensional Sigma Functions

T. Ayano, V. M. Buchstaber

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)157-173
Number of pages17
JournalFunctional Analysis and its Applications
Issue number3
Publication statusPublished - 1 Jul 2019
Externally publishedYes


  • 4-periodic meromorphic functions
  • Abel-Jacobi inversion problem
  • functions on sigma divisor


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