Multi-variable sigma-functions: old and new results

Victor Buchstaber; Victor Enolski; Dmitry Leykin

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arXiv · 1810.11079

Multi-variable sigma-functions: old and new results

Victor Buchstaber·Victor Enolski·Dmitry Leykin

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Abstract

We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables -jacobian variables and parameters of the curve. In the 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 periods, generalizations of Rosenhain type formulae are obtained, whilst for the second kind periods theta-constant expressions arepresented which are explicitly related to the fixed co-homology basis.We describe a method of constructing differentiation operators for hyperelliptic analogues of $\zeta$- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we gave the detailed construction of these operators in the cases of genus 1 and 2.

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Victor Buchstaber, Victor Enolski, Dmitry Leykin. 2018-10-25. Multi-variable sigma-functions: old and new results. https://arxiv.org/abs/1810.11079

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