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1 results

  • 6 - σ-Functions: Old and New Results

  • Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
  • Book:
    Integrable Systems and Algebraic Geometry
    Published online:
    19 March 2020
    Print publication:
    02 March 2020, pp 175-214

  • Summary

    We are considering multi-variable sigma function of genus g hyperelliptic curve as a function of two groups of variables - jacobian variables and parameters of the curve. In theta-functional representation of sigma-function the second group arises as periods of first and second kind differentials of the curve. We develop representation of periods in terms of theta-constants, for the first kind, period generalizations of Rosenhain type formulae are obtained whilst for the second kind, period theta-constant expressions are presented which are explicitly related to thefixed co-homology basis. We describe a method of constructing differentiation operators for hyperelliptic analogues ofζ- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we give the detailed construction of these operators in the case genus 1 and 2.