Theory of Heat Equations for Sigma Functions

Abstract

We consider the heat equations satisfied by the sigma function associated with a planar curve, extending and developing earlier pioneering work of Buchstaber and Leykin. These heat equations lead to useful {\em linear} recursive relations for the coefficients of power series expansion of the sigma function. In particular we exhibit explicit results for curves of genus 3, and give a new constructive proof of an explicit expression for the main matrix in the theory for {\em any} hyperelliptic curve. We also state and prove a new explicit formula for the eigenvalues of the linear operators associated with this matrix, as well as other practical formulae.

47 pages

References in corpus (1)

• Universal elliptic functions

Cited by in corpus (6)

• Lie Algebras of Heat Operators in Nonholonomic Frame
• Differentiation of Genus $4$ Hyperelliptic Functions
• Sigma functions and Lie algebras of Schrödinger operators
• Analytical and number-theoretical properties of the two-dimensional sigma function
• The sigma function over a family of cyclic trigonal curves with a singular fiber
• Hyperelliptic sigma functions and Adler-Moser polynomials