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Weierstrass sigma-function

Contents

Idea

A modular form.

Properties

Relation to Eisenstein series

Let G 2kG_{2k} be the Eisenstein series, then

xe x/2e x/2 n1(1q n) 2(1q ne x)(1q ne x)=exp( k22G kx kk!) \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)

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

Last revised on April 17, 2017 at 06:43:04. See the history of this page for a list of all contributions to it.