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

TL;DR

This paper proves that the sigma function of telescopic algebraic curves has a power series expansion with integer-like coefficients, extending previous results for special cases like hyperelliptic curves.

Contribution

It establishes Hurwitz integrality of the sigma function for telescopic curves, generalizing prior results for (n,s) curves and providing new insights into their algebraic properties.

Findings

Sigma function is Hurwitz integral over a specific ring.

Sigma function squared is Hurwitz integral over the ring.

Results extend previous work from (n,s) curves to telescopic curves.

Abstract

A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. The sigma function $\sigma(u)$ associated with the telescopic curve of genus $g$ is a holomorphic function on $\mathbb{C}^g$. For a subring $R$ of $\mathbb{C}$ and variables $u={}^t(u_1,\dots, u_g)$, let \[R\langle\langle u \rangle\rangle=\left\{\sum_{i_1,\dots,i_g\ge0}\kappa_{i_1,\dots,i_g}\frac{u_1^{i_1}\cdots u_g^{i_g}}{i_1!\cdots i_g!}\;\middle|\;\kappa_{i_1,\dots,i_g}\in R\right\}.\] If the power series expansion of a holomorphic function $f(u)$ on $\mathbb{C}^g$ around the origin belongs to $R\langle\langle u \rangle\rangle$, then $f(u)$ is said to be Hurwitz integral over $R$. In this paper, we show that the sigma function $\sigma(u)$ associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and $\frac{1}{2}$ over $\mathbb{Z}$. Further, we show that $\sigma(u)^2$ is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over $\mathbb{Z}$. Our results are a generalization of the results of Y. \^Onishi for $(n,s)$ curves to telescopic curves.