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arXiv · 1711.08395

Theory of Heat Equations for Sigma Functions

J. C. Eilbeck·J. Gibbons·Y. Ônishi·S. Yasuda
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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.

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J. C. Eilbeck, J. Gibbons, Y. Ônishi, S. Yasuda. 2017-11-22. Theory of Heat Equations for Sigma Functions. https://arxiv.org/abs/1711.08395

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