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
τ
{\displaystyle \tau }
be a complex number with strictly positive imaginary part . Define the Eisenstein series
G
2
k
(
τ
)
{\displaystyle G_{2k}(\tau )}
for each integer
k
>
1
{\displaystyle k>1}
by
G
2
k
(
τ
)
=
∑
(
m
,
n
)
≠
(
0
,
0
)
1
(
m
+
n
τ
)
2
k
.
{\displaystyle 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
∈
Z
{\displaystyle a,b,c,d\in \mathbb {Z} }
and
a
d
−
b
c
=
1
{\displaystyle ad-bc=1}
then
G
2
k
(
a
τ
+
b
c
τ
+
d
)
=
(
c
τ
+
d
)
2
k
G
2
k
(
τ
)
{\displaystyle G_{2k}\left({\frac {a\tau +b}{c\tau +d}}\right)=(c\tau +d)^{2k}G_{2k}(\tau )}
and
G
2
k
{\displaystyle G_{2k}}
is therefore a modular form of weight
2
k
{\displaystyle 2k}
.
Relation to modular invariants
The modular invariants
g
2
{\displaystyle g_{2}}
and
g
3
{\displaystyle g_{3}}
of an elliptic curve are given by the first two terms of the Eisenstein series as
g
2
=
60
G
4
{\displaystyle g_{2}=60G_{4}}
and
g
3
=
140
G
6
{\displaystyle 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
{\displaystyle G_{4}}
and
G
6
{\displaystyle G_{6}}
. Specifically, the higher order
G
2
k
{\displaystyle G_{2k}}
's can be written in terms of
G
4
{\displaystyle G_{4}}
and
G
6
{\displaystyle G_{6}}
through a recurrence relation. Let
d
k
=
(
2
k
+
3
)
k
!
G
2
k
+
4
{\displaystyle d_{k}=(2k+3)k!G_{2k+4}}
. Then the
d
k
{\displaystyle d_{k}}
satisfy the relation
∑
k
=
0
n
(
n
k
)
d
k
d
n
−
k
=
2
n
+
9
3
n
+
6
d
n
+
2
{\displaystyle \sum _{k=0}^{n}{n \choose k}d_{k}d_{n-k}={\frac {2n+9}{3n+6}}d_{n+2}}
for all
n
≥
0
{\displaystyle n\geq 0}
. Here,
(
n
k
)
{\displaystyle {n \choose k}}
is the binomial coefficient and
d
0
=
3
G
4
{\displaystyle d_{0}=3G_{4}}
and
d
1
=
5
G
6
{\displaystyle d_{1}=5G_{6}}
.
The
d
k
{\displaystyle d_{k}}
occur in the series expansion for the Weierstrass's elliptic functions :
℘
(
z
)
=
1
z
2
+
z
2
∑
k
=
0
∞
d
k
z
2
k
k
!
=
1
z
2
+
∑
k
=
1
∞
(
2
k
+
1
)
G
2
k
+
2
z
2
k
{\displaystyle \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
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
. (Some older books define q to be the nome
q
=
e
i
π
τ
{\displaystyle q=e^{i\pi \tau }}
, but
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
is now standard in number theory.) Then the Fourier series of the Eisenstein series is
G
2
k
(
τ
)
=
2
ζ
(
2
k
)
(
1
+
c
2
k
∑
n
=
1
∞
σ
2
k
−
1
(
n
)
q
n
)
{\displaystyle 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
{\displaystyle c_{2k}}
are given by
c
2
k
=
(
2
π
i
)
2
k
(
2
k
−
1
)
!
ζ
(
2
k
)
=
−
4
k
B
2
k
{\displaystyle c_{2k}={\frac {(2\pi i)^{2k}}{(2k-1)!\zeta (2k)}}={\frac {-4k}{B_{2k}}}}
.
Here, B n are the Bernoulli numbers ,
ζ
(
z
)
{\displaystyle \zeta (z)}
is Riemann's zeta function and the sigma function
σ
p
(
n
)
{\displaystyle \sigma _{p}(n)}
is the sum of the
p
{\displaystyle p}
th powers of the divisors of
n
{\displaystyle n}
. In particular, on has
G
4
(
τ
)
=
π
4
45
[
1
+
240
∑
n
=
1
∞
σ
3
(
n
)
q
n
]
{\displaystyle G_{4}(\tau )={\frac {\pi ^{4}}{45}}\left[1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}\right]}
and
G
6
(
τ
)
=
2
π
6
945
[
1
−
504
∑
n
=
1
∞
σ
5
(
n
)
q
n
]
{\displaystyle 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
∑
n
=
1
∞
q
n
σ
a
(
n
)
=
∑
n
=
1
∞
n
a
q
n
1
−
q
n
{\displaystyle \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
2
k
(
τ
)
=
G
2
k
(
τ
)
2
ζ
(
2
k
)
=
1
−
4
k
B
2
k
∑
n
=
1
∞
σ
2
k
−
1
(
n
)
q
n
{\displaystyle 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
∑
n
=
1
∞
n
q
n
1
−
q
n
{\displaystyle L(q)=1-24\sum _{n=1}^{\infty }{\frac {nq^{n}}{1-q^{n}}}}
and
M
(
q
)
=
1
+
240
∑
n
=
1
∞
n
3
q
n
1
−
q
n
{\displaystyle M(q)=1+240\sum _{n=1}^{\infty }{\frac {n^{3}q^{n}}{1-q^{n}}}}
and
N
(
q
)
=
1
−
504
∑
n
=
1
∞
n
5
q
n
1
−
q
n
{\displaystyle N(q)=1-504\sum _{n=1}^{\infty }{\frac {n^{5}q^{n}}{1-q^{n}}}}
then
q
d
L
d
q
=
L
2
−
M
12
{\displaystyle q{\frac {dL}{dq}}={\frac {L^{2}-M}{12}}}
and
q
d
M
d
q
=
L
M
−
N
3
{\displaystyle q{\frac {dM}{dq}}={\frac {LM-N}{3}}}
and
q
d
N
d
q
=
L
N
−
M
2
2
{\displaystyle 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
∑
k
=
0
n
σ
(
k
)
σ
(
n
−
k
)
=
5
12
σ
3
(
n
)
−
1
2
n
σ
(
n
)
.
{\displaystyle \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
∑
k
=
0
n
σ
3
(
k
)
σ
3
(
n
−
k
)
=
1
120
σ
7
(
n
)
{\displaystyle \sum _{k=0}^{n}\sigma _{3}(k)\sigma _{3}(n-k)={\frac {1}{120}}\sigma _{7}(n)}
∑
k
=
0
n
σ
(
2
k
+
1
)
σ
3
(
n
−
k
)
=
1
240
σ
5
(
2
n
+
1
)
{\displaystyle \sum _{k=0}^{n}\sigma (2k+1)\sigma _{3}(n-k)={\frac {1}{240}}\sigma _{5}(2n+1)}
∑
k
=
0
n
σ
(
3
k
+
1
)
σ
(
3
n
−
3
k
+
1
)
=
1
9
σ
3
(
3
n
+
2
)
.
{\displaystyle \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).