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

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.