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This article describes holomorphic Eisenstein series; for the non-holomorphic case see real analytic Eisenstein series

Eisenstein series, named after German mathematician Gotthold Eisenstein, 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.

[edit] Eisenstein series for the modular group

The real part of G_6 as a function of q on the unit disk.

The imaginary part of G_6 as a function of q on the unit disk.

Let $τ$ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series $G 2 k (τ)$ of weight $2 k,$ where $k\geq 2$ is an integer, by the following series:

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

This series absolutely converges to a holomorphic function of $τ$ in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at $\tau=i\infty.$ It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its $SL_2(\mathbb{Z})-$ invariance. Explicitly if $a,b,c,d \in \mathbb{Z}$ and $a d - b c = 1$ then

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

and $G 2 k$ is therefore a modular form of weight $2 k$ . Note that it is important to assume that $k\geq 2,$ otherwise it would be illegitimate to change the order of summation, and the $SL_2(\mathbb{Z})$ -invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for $k = 1,$ although it would only be a near modular form.

[edit] Relation to modular invariants

The modular invariants $g 2$ and $g 3$ of an elliptic curve are given by the first two terms of the Eisenstein series as

g 2 = 60 G 4

and

g 3 = 140 G 6

The article on modular invariants provides expressions for these two functions in terms of theta functions.

[edit] 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:

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

[edit] Fourier series

G_4

G_6

G_8

G_10

G_12

G_14

Define $q = e 2π i τ$ . (Some older books define q to be the nome $q = e i πτ$ , but $q = e 2π i τ$ is now standard in number theory.) Then the Fourier series of the Eisenstein series is

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

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

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

Here, B_n are the Bernoulli numbers, ζ(z) is Riemann's zeta function and σ_p(n) is the divisor sum function, the sum of the p^th powers of the divisors of n. In particular, one has

$G_4(\tau)=\frac{\pi^4}{45} \left[ 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n} \right]$

and

$G_6(\tau)=\frac{2\pi^6}{945} \left[ 1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n} \right]$

Note the summation over q can be resummed as a Lambert series; that is, one has

$\sum_{n=1}^{\infty} q^n \sigma_a(n) = \sum_{n=1}^{\infty} \frac{n^a q^n}{1-q^n}$

for arbitrary complex |q| ≤ 1 and a. When working with the q-expansion of the Eisenstein series, the alternate notation

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

is frequently introduced.

[edit] Identities involving Eisenstein series

[edit] Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group $SL_2(\mathbb{Z}).$ Since the space of modular forms of weight $2 k$ has dimension 1 for $2 k = 4,6,8,10,14$ different products of Eisenstein series having those weights have to be proportional. Thus we obtain the identities:

$E_4^2 = E_8, \quad E_4 E_6 = E_{10} \quad E_4 E_{10} = E_{14}, \quad E_6 E_8 = E_{14}.$

Using the q-expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:

$(1+240\sum_{n=1}^\infty \sigma_3(n) q^n)^2 = 1+480\sum_{n=1}^\infty \sigma_7(n) q^n,$

hence

$\sigma_7(n)=\sigma_3(n)+120\sum_{m=1}^{n-1}\sigma_3(m)\sigma_3(n-m),$

and similarly for the others. Perhaps, even more interestingly, the theta function of an eight-dimensional even unimodular lattice Γ is a modular form of weight 4 for the full modular group, which gives the following identities:

$\theta_{\Gamma}(\tau)=1+\sum_{n=1}^\infty r_{\Gamma}(2n) q^{n} = E_4(\tau), \quad r_{\Gamma}(n) = 240\sigma_3(n)$

for the number $r Γ (n)$ of vectors of the squared length 2n in the root lattice of the type E₈.

Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer n as a sum of two, four, and eight squares in terms of the divisors of n.

[edit] Ramanujan identities

Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation. Let

$L(q)=1-24\sum_{n=1}^\infty \frac {nq^n}{1-q^n}=E_2(\tau)$

and

$M(q)=1+240\sum_{n=1}^\infty \frac {n^3q^n}{1-q^n}=E_4(\tau)$

and

$N(q)=1-504\sum_{n=1}^\infty \frac {n^5q^n}{1-q^n}=E_6(\tau),$

then

$q\frac{dL}{dq} = \frac {L^2-M}{12}$

and

$q\frac{dM}{dq} = \frac {LM-N}{3}$

and

$q\frac{dN}{dq} = \frac {LN-M^2}{2}.$

These identities, like the identities between the series, yield arithmetical convolution identities involving the sum-of-divisor function. Following Ramanujan, to put these identities in the simplest form it is necessary to extend the domain of σ_p(n) to include zero, by setting

$\sigma_p(0) = \frac12\zeta(-p).\;$ E.g.:

$\begin{align} \sigma(0) &= -\frac{1}{24}\\ \sigma_3(0) &= \frac{1}{240}\\ \sigma_5(0) &= -\frac{1}{504} \end{align}$

Then, for example

$\sum_{k=0}^n\sigma(k)\sigma(n-k)=\frac5{12}\sigma_3(n)-\frac12n\sigma(n).$

Other identities of this type, but not directly related to the preceding relations between L, M and N functions, have been proved by Ramanujan and Melfi, as for example

$\sum_{k=0}^n\sigma_3(k)\sigma_3(n-k)=\frac1{120}\sigma_7(n)$

$\sum_{k=0}^n\sigma(2k+1)\sigma_3(n-k)=\frac1{240}\sigma_5(2n+1)$

$\sum_{k=0}^n\sigma(3k+1)\sigma(3n-3k+1)=\frac19\sigma_3(3n+2).$

For a comprehensive list of convolution identities involving sum-of-divisors functions and related topics see

S. Ramanujan, On certain arithmetical functions, pp 136-162, reprinted in Collected Papers, (1962), Chelsea, New York.
Heng Huat Chan and Yau Lin Ong, On Eisenstein Series, (1999) Proceedings of the Amer. Math. Soc. 127(6) pp.1735-1744
G. Melfi, On some modular identities, in Number Theory, Diophantine, Computational and Algebraic Aspects: Proceedings of the International Conference held in Eger, Hungary. Walter de Grutyer and Co. (1998), 371-382.

[edit] 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 a totally real 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.

[edit] 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
Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), America Mathematical Society, Providence, RI ISBN 0-8218-3160-7 (See chapter 3)
Serre, Jean-Pierre, A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.

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