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arXiv · 1510.03002

Hurwitz integrality of power series expansion of the sigma function for a plane curve

Yoshihiro Ônishi
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Abstract

This paper shows Hurwitz integrality of the coefficients of expansion at the origin of the sigma function \(σ(u)\) associated to a certain plane curve which should be called a plane telescopic curve. For the prime \(2\), the expansion of \(σ(u)\) is not Hurwitz integral, but \(σ(u)^2\) is. This paper clarifies the precise structure of this phenomenon. Throughout the paper, computational examples for the trigonal genus three curve (\((3,4)\)-curve) \(y^3+(μ_1x+μ_4)y^2+(μ_2x^2+μ_5x+μ_8)y=x^4+μ_3x^3+μ_6x^2+μ_9x+μ_{12}\) (\(μ_j\) are constants) are given.

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Yoshihiro Ônishi. 2015-10-11. Hurwitz integrality of power series expansion of the sigma function for a plane curve. https://arxiv.org/abs/1510.03002

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