Universal elliptic functions

Abstract

For the elliptic curve defined by the most general form $y^2 + (\mu_1 x + \mu_3) y = x^3 + \mu_2 x^2 + \mu_4 x + \mu_6$, we show the power series expansion of Weierstsass sigma function $\sigma(u)$ at the origin is of Hurwitz integral over $\mathbb{Z}[\mu_1/2, \mu_2, \mu_3, \mu_4, \mu_6]$. Namely, the coefficient $c_n$ of any term $c_n u^n/n!$ of the expansion belongs to $\mathbb{Z}[\mu_1/2, \mu_2, \mu_3, \mu_4, \mu_6]$. The last section contains several first terms of $n$-plication equation of the curve.