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

TL;DR

This paper proves the Hurwitz integrality of the sigma function's power series coefficients for certain plane curves, revealing a specific structure where the square of the sigma function is integral at prime 2.

Contribution

It establishes the Hurwitz integrality of sigma function expansions for plane telescopic curves and clarifies the structure of non-integrality at prime 2.

Findings

Sigma function coefficients are Hurwitz integral for the plane curve.

The square of the sigma function is Hurwitz integral at prime 2.

Provides computational examples for a genus three trigonal curve.

Abstract

This paper shows Hurwitz integrality of the coefficients of expansion at the origin of the sigma function \(\sigma(u)\) associated to a certain plane curve which should be called a plane telescopic curve. For the prime \(2\), the expansion of \(\sigma(u)\) is not Hurwitz integral, but \(\sigma(u)^2\) is. This paper clarifies the precise structure of this phenomenon. Throughout the paper, computational examples for the trigonal genus three curve (\((3,4)\)-curve) \(y^3+(\mu_1x+\mu_4)y^2+(\mu_2x^2+\mu_5x+\mu_8)y=x^4+\mu_3x^3+\mu_6x^2+\mu_9x+\mu_{12}\) (\(\mu_j\) are constants) are given.