Eisenstein series

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

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 $\tau$ 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 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 $2k$.

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

$g_{2}=60G_{4}$

and

$g_{3}=140G_{6}$

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 $n\geq 0$. Here, ${n \choose k}$ is the binomial coefficient and $d_{0}=3G_{4}$ and $d_{1}=5G_{6}$.

The $d_{k}$ occur in the series expansion for the Weierstrass's elliptic functions:

$\wp (z)={\frac {1}{z^{2}}}+z^{2}\sum _{k=0}^{\infty }{\frac {d_{k}z^{2k}}{k!}}={\frac {1}{z^{2}}}+\sum _{k=1}^{\infty }(2k+1)G_{2k+2}z^{2k}$

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 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

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

Here, B_n are the Bernoulli numbers, $\zeta (z)$ is Riemann's zeta function and the sigma function $\sigma _{p}(n)$ is the sum of the $p$th powers of the divisors of $n$. In particular, on 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-series expressions for the Eisenstein series, the alternate notation

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

is frequently introduced.

Ramanujan identities

Ramanujan gave several interesting identities between the first few terms. Let

$L(q)=1-24\sum _{n=1}^{\infty }{\frac {nq^{n}}{1-q^{n}}}$

and

$M(q)=1+240\sum _{n=1}^{\infty }{\frac {n^{3}q^{n}}{1-q^{n}}}$

and

$N(q)=1-504\sum _{n=1}^{\infty }{\frac {n^{5}q^{n}}{1-q^{n}}}$

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 yield correspondent arithmetical convolution identities involving the sum-of-divisor function, as 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 L, M and N functions, have been proved by Ramanujan and Melfi, as for example

$\sum _{k=0}^{n}\sigma _{3}(k)\sigma _{3}(n-k)={\frac {1}{120}}\sigma _{7}(n)$

$\sum _{k=0}^{n}\sigma (2k+1)\sigma _{3}(n-k)={\frac {1}{240}}\sigma _{5}(2n+1)$

$\sum _{k=0}^{n}\sigma (3k+1)\sigma (3n-3k+1)={\frac {1}{9}}\sigma _{3}(3n+2).$

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.

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.

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)
• Tomio Kubota, Elementary Theory of Eisenstein Series, Kodansha and J. Wiley (1973).