nLab Weierstrass sigma-function

Idea

A modular form.

Properties

Relation to Eisenstein series

Let $G_{2k}$ be the Eisenstein series, then

$$\frac{x}{e^{x/2} - e^{-x/2}} \prod_{n\geq 1} \frac{(1-q^n)^2}{(1-q^n e^x)(1-q^n e^{-x})} = \exp\left( \sum_{k \geq 2} 2 G_k \frac{x^k}{k!} \right)$$

(Ando-Hopkins-Rezk 10, prop. 10.9)

Related concepts

• The Weierstrass $\sigma$-function is proportional to the (inverse of the) characteristic series of the Witten genus (Ando-Basterra 00, section 5.1)

References

Named after Karl Weierstrass.

An introductory review is in

• Richard Hain, section 5.1 of Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)

A textbook account includes for instance

• Joseph Silverman, The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer, 1986

Relation to the Witten genus is discussed for instance in

• Matthew Ando, Maria Basterra, The Witten genus and equivariant elliptic cohomology (arXiv:0008192)

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)