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.

Eisenstein series for the modular group

The real part of Template:Math as a function of Template:Mvar on the unit disk. Negative numbers are black.

The imaginary part of Template:Math as a function of Template:Mvar on the unit disk.

Let Template:Mvar be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series Template:Math of weight Template:Math, where Template:Math is an integer, by the following series:

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

This series absolutely converges to a holomorphic function of Template:Mvar in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at Template:Math. It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its Template:Math-covariance. Explicitly if Template:Math and Template:Math then

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

Note that Template:Math is necessary such that the series converges absolutely, whereas Template:Math needs to be even otherwise the sum vanishes because the Template:Math and Template:Math terms cancel out. For Template:Math the series converges but it is not a modular form.

Relation to modular invariants

The modular invariants Template:Math and Template:Math of an elliptic curve are given by the first two Eisenstein series:

$\begin{array}{cc}{g}_2& =60G_4\\ g_3& =140G_6.\end{array}$

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 Template:Math and Template:Math. Specifically, the higher order Template:Math can be written in terms of Template:Math and Template:Math through a recurrence relation. Let Template:Math, so for example, Template:Math and Template:Math. Then the Template:Mvar satisfy the relation

$\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})d_k{d}_{n-k}=\frac{2n+9}{3n+6}{d}_{n+2}$

for all Template:Math. Here, $(\genfrac{}{}{0ex}{}{n}{k})$ is the binomial coefficient.

The Template:Math occur in the series expansion for the Weierstrass's elliptic functions:

$\begin{array}{cc}\mathrm{\wp }(z)& =\frac{1}{z^2}+z^2\sum _{k=0}^{\mathrm{\infty }}\frac{d_k{z}^{2k}}{k!}\\ & =\frac{1}{z^2}+\sum _{k=1}^{\mathrm{\infty }}(2k+1)G_{2k+2}{z}^{2k}.\end{array}$

Fourier series

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Define Template:Math. (Some older books define Template:Mvar to be the nome Template:Math, but Template:Math is now standard in number theory.) Then the Fourier series of the Eisenstein series is

$G_{2k}(\tau )=2\zeta (2k)(1+c_{2k}\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{2k-1}(n)q^n)$

where the coefficients Template:Math are given by

$\begin{array}{cc}{c}_{2k}& =\frac{(2\pi i{)}^{2k}}{(2k-1)!\zeta (2k)}\\ & =\frac{-4k}{B_{2k}}=\frac{2}{\zeta (1-2k)}.\end{array}$

Here, Template:Math are the Bernoulli numbers, Template:Math is Riemann's zeta function and Template:Math is the divisor sum function, the sum of the Template:Mvarth powers of the divisors of Template:Mvar. In particular, one has

$\begin{array}{cc}{G}_4(\tau )& =\frac{{\pi }^4}{45}(1+240\sum _{n=1}^{\mathrm{\infty }}{\sigma }_3(n)q^n)\\ G_6(\tau )& =\frac{2{\pi }^6}{945}(1-504\sum _{n=1}^{\mathrm{\infty }}{\sigma }_5(n)q^n).\end{array}$

The summation over Template:Mvar can be resummed as a Lambert series; that is, one has

$\sum _{n=1}^{\mathrm{\infty }}{q}^n{\sigma }_a(n)=\sum _{n=1}^{\mathrm{\infty }}\frac{n^a{q}^n}{1-q^n}$

for arbitrary complex Template:Math and Template:Mvar. When working with the [[q-expansion|Template:Mvar-expansion]] of the Eisenstein series, this alternate notation is frequently introduced:

$\begin{array}{cc}{E}_{2k}(\tau )& =\frac{G_{2k}(\tau )}{2\zeta (2k)}\\ & =1+\frac{2}{\zeta (1-2k)}\sum _{n=1}^{\mathrm{\infty }}\frac{n^{2k-1}{q}^n}{1-q^n}\\ & =1-\frac{4k}{B_{2k}}\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{2k-1}(n)q^n\\ & =1-\frac{4k}{B_{2k}}\sum _{d,n\ge 1}{n}^{2k-1}{q}^{nd}.\end{array}$

Identities involving Eisenstein series

As theta functions

Source:

Given Template:Math, let

$\begin{array}{cc}{E}_4(\tau )& =1+240\sum _{n=1}^{\mathrm{\infty }}\frac{n^3{q}^n}{1-q^n}\\ E_6(\tau )& =1-504\sum _{n=1}^{\mathrm{\infty }}\frac{n^5{q}^n}{1-q^n}\\ E_8(\tau )& =1+480\sum _{n=1}^{\mathrm{\infty }}\frac{n^7{q}^n}{1-q^n}\end{array}$

and define the Jacobi theta functions which normally uses the nome Template:Math,

$\begin{array}{cc}a& ={\theta }_2(0;e^{\pi i\tau })={\vartheta }_{10}(0;\tau )\\ b& ={\theta }_3(0;e^{\pi i\tau })={\vartheta }_{00}(0;\tau )\\ c& ={\theta }_4(0;e^{\pi i\tau })={\vartheta }_{01}(0;\tau )\end{array}$

where Template:Math and Template:Math are alternative notations. Then we have the symmetric relations,

$\begin{array}{cc}{E}_4(\tau )& =\frac{1}{2}(a^8+b^8+c^8)\\ E_6(\tau )& =\frac{1}{2}\sqrt{\frac{{(a^8+b^8+c^8)}^3-54(abc{)}^8}{2}}\\ E_8(\tau )& =\frac{1}{2}(a^{16}+b^{16}+c^{16})=a^8{b}^8+a^8{c}^8+b^8{c}^8\end{array}$

Basic algebra immediately implies

$E_4^3-E_6^2=\frac{27}{4}(abc{)}^8$

an expression related to the modular discriminant,

$\mathrm{\Delta }=g_2^3-27g_3^2=(2\pi {)}^{12}{\left(\frac{1}{2}abc\right)}^8$

The third symmetric relation, on the other hand, is a consequence of Template:Math and Template:Math.

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group Template:Math. Since the space of modular forms of weight Template:Math has dimension 1 for Template:Math, different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In fact, we obtain the identities:

$E_4^2=E_8,E_4{E}_6=E_{10},E_4{E}_{10}=E_{14},E_6{E}_8=E_{14}.$

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

${(1+240\sum _{n=1}^{\mathrm{\infty }}{\sigma }_3(n)q^n)}^2=1+480\sum _{n=1}^{\mathrm{\infty }}{\sigma }_7(n)q^n,$

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. The theta function of an eight-dimensional even unimodular lattice Template:Math is a modular form of weight 4 for the full modular group, which gives the following identities:

${\theta }_{\mathrm{\Gamma }}(\tau )=1+\sum _{n=1}^{\mathrm{\infty }}{r}_{\mathrm{\Gamma }}(2n)q^n=E_4(\tau ),r_{\mathrm{\Gamma }}(n)=240{\sigma }_3(n)$

for the number Template:Math of vectors of the squared length Template:Math in the [[E8 lattice|root lattice of the type Template:Math]].

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

Using the above recurrence relation, all higher Template:Math can be expressed as polynomials in Template:Math and Template:Math. For example:

$\begin{array}{cc}{E}_8& =E_4^2\\ E_{10}& =E_4\cdot E_6\\ 691\cdot E_{12}& =441\cdot E_4^3+250\cdot E_6^2\\ E_{14}& =E_4^2\cdot E_6\\ 3617\cdot E_{16}& =1617\cdot E_4^4+2000\cdot E_4\cdot E_6^2\\ 43867\cdot E_{18}& =38367\cdot E_4^3\cdot E_6+5500\cdot E_6^3\\ 174611\cdot E_{20}& =53361\cdot E_4^5+121250\cdot E_4^2\cdot E_6^2\\ 77683\cdot E_{22}& =57183\cdot E_4^4\cdot E_6+20500\cdot E_4\cdot E_6^3\\ 236364091\cdot E_{24}& =49679091\cdot E_4^6+176400000\cdot E_4^3\cdot E_6^2+10285000\cdot E_6^4\end{array}$

Many relationships between products of Eisenstein series can be written in an elegant way using Hankel determinants, e.g. Garvan's identity

${\left(\frac{\mathrm{\Delta }}{(2\pi {)}^{12}}\right)}^2=-\frac{691}{1728^2\cdot 250}\det \left|\begin{array}{ccc}{E}_4& E_6& E_8\\ E_6& E_8& E_{10}\\ E_8& E_{10}& E_{12}\end{array}\right|$

where

$\mathrm{\Delta }=(2\pi {)}^{12}\frac{E_4^3-E_6^2}{1728}$

is the modular discriminant.

Ramanujan identities

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

$\begin{array}{cccc}L(q)& =1-24\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^n}& & =E_2(\tau )\\ M(q)& =1+240\sum _{n=1}^{\mathrm{\infty }}\frac{n^3{q}^n}{1-q^n}& & =E_4(\tau )\\ N(q)& =1-504\sum _{n=1}^{\mathrm{\infty }}\frac{n^5{q}^n}{1-q^n}& & =E_6(\tau ),\end{array}$

then

$\begin{array}{cc}q\frac{dL}{dq}& =\frac{L^2-M}{12}\\ q\frac{dM}{dq}& =\frac{LM-N}{3}\\ q\frac{dN}{dq}& =\frac{LN-M^2}{2}.\end{array}$

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 Template:Math to include zero, by setting

$\begin{array}{cc}{\sigma }_p(0)=\frac{1}{2}\zeta (-p)⟹\sigma (0)& =-\frac{1}{24}\\ {\sigma }_3(0)& =\frac{1}{240}\\ {\sigma }_5(0)& =-\frac{1}{504}.\end{array}$

Then, for example

$\sum _{k=0}^n\sigma (k)\sigma (n-k)=\frac{5}{12}{\sigma }_3(n)-\frac{1}{2}n\sigma (n).$

Other identities of this type, but not directly related to the preceding relations between Template:Mvar, Template:Mvar and Template:Mvar functions, have been proved by Ramanujan and Giuseppe Melfi, as for example

$\begin{array}{cc}\sum _{k=0}^n{\sigma }_3(k){\sigma }_3(n-k)& =\frac{1}{120}{\sigma }_7(n)\\ \sum _{k=0}^n\sigma (2k+1){\sigma }_3(n-k)& =\frac{1}{240}{\sigma }_5(2n+1)\\ \sum _{k=0}^n\sigma (3k+1)\sigma (3n-3k+1)& =\frac{1}{9}{\sigma }_3(3n+2).\end{array}$

Generalizations

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

Defining Template:Math to be the ring of integers of a totally real algebraic number field Template:Mvar, one then defines the Hilbert–Blumenthal modular group as Template:Math. One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.

Further reading

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