nLab Eisenstein series

Idea

The Eisenstein series $G_{2k}$ for $k \geq 2$ 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 $G_2$ and $G_4$ expresses the j-invariant which characterizes elliptic curves.

Definition

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

Properties

Relation to the $j$-invariant

With

$$g_2 \coloneqq 60 G_4$$

$$g_3 \coloneqq 140 G_6$$

$$\Delta \coloneqq g_2^3 - 27 g_3^2$$

the j-invariant is

$$j = 1728 \frac{g_2^3}{\Delta} \,.$$

Relation to Weierstrass $\sigma$-function and Witten genus

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

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

Relation to Bernoulli numbers

The $q$-independent term in $G_{k}$ is proportional to the $k$th Bernoulli number $B_k$

$$G_k = - \frac{B_k}{2k} + \sum_{n= 1}^\infty \sigma_{k-1}(n)q^n \,,$$

where

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

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

References

• Wikipedia, Eisenstein series

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