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

Abstract

A telescopic curve is an 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. If all the coefficients of the power series expansion of an entire function around the origin in the form of the Hurwitz series are contained in a ring, then the function is said to be Hurwitz integral over the ring. In this paper, we show that the sigma function associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and 1/2 over the ring of integers. Further, we show that the square of the sigma function associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over the ring of integers. Our results are a generalization of the results of Y. Ônishi for (𝑛, 𝑠) curves to telescopic curves.