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.

References in corpus (1)

• On Algebraic Expressions of Sigma Functions for (n,s) Curves

Cited by in corpus (5)

• Elliptic formal group laws, integral Hirzebruch genera and Krichever genera
• Theory of Heat Equations for Sigma Functions
• Analytical and number-theoretical properties of the two-dimensional sigma function
• Some New Addition Formulae for Weierstrass Elliptic Functions
• Properties of the series solution for Painleve I