nLab
Eisenstein series

Contents

Idea

The Eisenstein series G 2kG_{2k} for k2k \geq 2 are series expressions for certain modular forms.

They appear as the coefficients of the exponential series-expression of the Weierstrass sigma-function, and hence of the Hirzebruch series of the Witten genus.

Moreover, a combination of G 2G_2 and G 4G_4 expresses the j-invariant which characterizes elliptic curves.

Definition

G 2k(τ)(m,n) 2\(0,0)1(m+nτ) 2k. G_{2k}(\tau) \coloneqq \underset{(m,n) \in \mathbb{Z}^2\backslash (0,0)}{\sum} \frac{1}{(m+n \tau)^{2k}} \,.

Properties

Relation to the jj-invariant

With

g 260G 4 g_2 \coloneqq 60 G_4
g 3140G 6 g_3 \coloneqq 140 G_6
Δg 2 327g 3 2 \Delta \coloneqq g_2^3 - 27 g_3^2

the j-invariant is

j=1728g 2 3Δ. j = 1728 \frac{g_2^3}{\Delta} \,.

Relation to Weierstrass σ\sigma-function and Witten genus

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)

Relation to Bernoulli numbers

The qq-independent term in G kG_{k} is proportional to the kkth Bernoulli number B kB_k

G k=B k2k+ n=1 σ k1(n)q n, G_k = - \frac{B_k}{2k} + \sum_{n= 1}^\infty \sigma_{k-1}(n)q^n \,,

where

σ k1(n)= d|nd k1. \sigma_{k-1}(n) = \sum_{d|n} d^{k-1} \,.

Reducing to this constant term reduces the above exponential characteristic series for the Witten genus to that of the A-hat genus.

References

Last revised on March 26, 2014 at 08:44:49. See the history of this page for a list of all contributions to it.