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

2015/10/11 by Ônishi, Yoshihiro
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1510.03002

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) \(y³+(μ₁x+μ₄)y²+(μ₂x²+μ₅x+μ₈)y=x⁴+μ₃x³+μ₆x²+μ₉x+μ₁₂\) (\(μ_j\) are constants) are given.

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