Eisenstein series - AbsoluteAstronomy.com

Eisenstein series, named after German

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Gotthold Eisenstein, are particular modular form

Modular form

In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...

s with infinite series expansions that may be written down directly. Originally defined for the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

, Eisenstein series can be generalized in the theory of automorphic form

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

Eisenstein series for the modular group

Let

\tau

be a complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

with strictly positive imaginary part. Define the holomorphic Eisenstein series

G_{2k}(\tau)

of weight

2k,

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

Absolute convergence

In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

to a holomorphic function of

\tau

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

Modular form

. Indeed, the key property is its

SL_2(\mathbb{Z})-

invariance. Explicitly if

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

and

ad-bc=1

then

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

and

G_{2k}

is therefore a modular form of weight

2k

. 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 quasimodular form.

Relation to modular invariants

The modular invariants

g_2

and

g_3

of an elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

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.

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_{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}

is the binomial coefficient

Binomial coefficient

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

and

d_0=3G_4

and

d_1=5G_6

. The

d_k

occur in the series expansion for the Weierstrass's elliptic functions

Weierstrass's elliptic functions

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass...

\wp(z)

Fourier series

Define

q=e^{2\pi i\tau}

. (Some older books define q to be the nome

q=e^{i\pi\tau}

, but

q=e^{2\pi i\tau}

is now standard in number theory.) Then the Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

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_{2k}

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 number

Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....

s, ζ(z) is Riemann's zeta function and σ_p(n) is the divisor sum function

Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...

, 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

Complex number

|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.

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular form

Modular form

s for the full modular group

SL_2(\mathbb{Z}).

Since the space of modular forms of weight

2k

has dimension 1 for

2k=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: NEWLINE

NEWLINE

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

NEWLINE for the number

r_{\Gamma}(n)

of vectors of the squared length 2n in the root lattice of the type E₈

E8 lattice

In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...

. Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character

Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...

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.

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

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

identities involving the sum-of-divisor function

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.: NEWLINE

NEWLINE

\begin{align}

NEWLINE

NEWLINE \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

Giuseppe Melfi

Giuseppe Melfi is an Italo-Swiss mathematician. He got his PhD in mathematics in 1997 at the University of Pisa. After some years spent at the University of Lausanne, he works now at the University of Neuchâtel, where is a lecturer...

, 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 seeNEWLINE

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
Giuseppe Melfi
Giuseppe Melfi is an Italo-Swiss mathematician. He got his PhD in mathematics in 1997 at the University of Pisa. After some years spent at the University of Lausanne, he works now at the University of Neuchâtel, where is a lecturer...

, 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.

NEWLINE

Generalizations

Automorphic form

In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...

s generalize the idea of modular forms for general Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s; and Eisenstein series generalize in a similar fashion. Defining O_K to be the ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

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

Cusp form

In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...

of the Hilbert-Blumenthal modular group.

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