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Categories: Mathematical series | Modular forms | Analytic number theory | Fractals

Eisenstein series

In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

Contents

1 Eisenstein series for the Modular group

2 Recurrence relation

3 Fourier series

4 Generalizations

5 References

Eisenstein series for the Modular group

Let $τ$ be a complex number with strictly positive imaginary part. Define the Eisenstein series

G 2 k (τ)

for each integer $k > 1$ by:

$G_{2k}(\tau) = \sum_{ (m,n) \neq (0,0)} \frac{1}{(m+n\tau )^{2k}}$

It is a remarkable fact that the Eisenstein series is a modular form. Explicitly

$G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)$

such that

$a,b,c,d \in \mathbb{Z}$

and satisfy

a d - b c = 1

and therefore is a modular form of weight $2 k$ .

Recurrence relation

Any holomorphic modular form for the modular group can be written as a polynomial in $G 4$ and $G 6$ . Specifically, the higher order $G 2 k$ 's can be written in terms of $G 4$ and $G 6$ through a recurrence relation. Let $d k = (2 k + 3) k! G 2 k + 4$ . Then the $d k$ satisfy the relation

$\sum_{k=0}^n {n \choose k} d_k d_{n-k} = \frac{2n+9}{3n+6}d_{n+2}$

for all $n\ge 0$ . Here, ${n \choose k}$ is the binomial coefficient and $d 0 = 3 G 4$ and $d 1 = 5 G 6$ .

The $d k$ occur in the series expansion for the Weierstrass's elliptic functions:

$\wp(z) =\frac{1}{z^2} + z^2 \sum_{k=0}^\infty \frac {d_k z^{2k}}{k!} =\frac{1}{z^2} + \sum_{k=1}^\infty (2k+1) G_{2k+2} z^{2k}$

Fourier series

Define the nome $q = e i πτ$ . Then the Fourier series of the Eisenstein series is

$G_{2k}(\tau) = 2\zeta(2k) \left(1+c_{2k}\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{2n} \right)$

where the Fourier coefficients $c 2 k$ are given by

$c_{2k} = \frac{(2\pi i)^{2k}}{(2k-1)! \zeta(2k)} = \frac {-4k}{B_{2k}}$ .

Here, B_n are the Bernoulli numbers, $ζ(z)$ is Riemann's zeta function and the sigma function $σ p (n)$ is the sum of the $p$ th powers of the divisors of $n$ . Note the summation over q can be resummed as a Lambert series.

When working with the q-series, the alternate notation

$E_{2k}(q)=\frac{G_{2k}(\tau)}{2\zeta (2k)}= 1-\frac {4k}{B_{2k}}\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{2n}$

is frequently introduced.

Generalizations

Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion.

Defining O_K to be the ring of integers of an algebraic number field K, one then defines the Hilbert-Blumenthal modular group as PSL(2,O_K). One can then associate an Eisenstein series to every cusp of the Hilbert-Blumenthal modular group.

References

Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0

Categories: Mathematical series | Modular forms | Analytic number theory | Fractals

Last updated: 10-10-2005 06:47:06

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12-19-2008 14:25:18
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