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Arithmetical Power Series Expansion of the Sigma Function for a Plane Curve

Part of: Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  22 June 2018

Yoshihiro Ônishi
Yoshihiro Ônishi*
Affiliation:
Department of Mathematics, Faculty of Sciences and Engineering, Meijo University, 1-501, Shiogama-guchi, Tempaku-ku, Nagoya 468-0051, Japan ([email protected])
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Abstract

The Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n, s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve) y3 + (μ1x + μ4)y2 + (μ2x2 + μ5x + μ8)y = x4 + μ3x3 + μ6x2 + μ9x + μ12 (where μj are constants) are given.

Keywords

abelian functionshigher genus curvesgeneralized sigma functionsHurwitz integrality

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 14K20: Analytic theory; abelian integrals and differentials 14K25: Theta functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 995 - 1022
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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