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

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.