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A telescopic curve is a certain algebraic curve defined by $m-1$ equations inthe 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 witha 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\[R\langle\langle u\rangle\rangle=\left\{\sum_{k_1,\dots,k_g\ge0}\zeta_{k_1,\dots,k_g}\frac{u_1^{k_1}\cdotsu_g^{k_g}}{k_1!\cdots k_g!}\;\middle|\;\zeta_{k_1,\dots,k_g}\in R\right\}.\] Ifthe 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 $f(u)$ issaid to be Hurwitz integral over $R$. In this paper, we show that the sigmafunction $\sigma(u)$ associated with a telescopic curve is Hurwitz integralover the ring generated by the coefficients of the defining equations of thecurve and $\frac{1}{2}$ over $\mathbb{Z}$, and its square $\sigma(u)^2$ isHurwitz integral over the ring generated by the coefficients of the definingequations of the curve over $\mathbb{Z}$. Our results are a generalization ofthe results of Y. \^Onishi for the $(n,s)$ curves to the telescopic curves.

Hurwitz integrality of the power series expansion of the sigma function for telescopic curves