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.

Eisenstein series for the Modular group

Let

\tau

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

G_{2k}(\tau)

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

ad-bc=1

and therefore is a modular form of weight

2k

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_{2k}

's can be written in terms of

G_4

and

G_6

through a recurrence relation. Let

d_k=(2k+3)k!G_{2k+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}

d_0=3G_4

and

d_1=5G_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}

Define the nome

q=e^{i\pi\tau}

. 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_{2k}

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,

\zeta(z)

\sigma_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.

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.

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

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