Eisenstein series

Last updated December 05, 2024

Eisenstein series, named after German mathematician Gotthold Eisenstein,^[1] 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

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:^[2]

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 $τ$ in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at $τ = i \infty$ . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its $\mathbb {Z}$ -covariance. Explicitly if $\mathbb {Z}$ and $ad - bc = 1$ then

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

(Proof)

{\begin{aligned}G_{2k}\left({\frac {a\tau +b}{c\tau +d}}\right)&=\sum _{(m,n)\in \mathbb {Z} ^{2}\setminus \{(0,0)\}}{\frac {1}{\left(m+n{\frac {a\tau +b}{c\tau +d}}\right)^{2k}}}\\&=\sum _{(m,n)\in \mathbb {Z} ^{2}\setminus \{(0,0)\}}{\frac {(c\tau +d)^{2k}}{(md+nb+(mc+na)\tau )^{2k}}}\\&=\sum _{\left(m',n'\right)=(m,n){\begin{pmatrix}d\ \ c\\b\ \ a\end{pmatrix}} \atop (m,n)\in \mathbb {Z} ^{2}\setminus \{(0,0)\}}{\frac {(c\tau +d)^{2k}}{\left(m'+n'\tau \right)^{2k}}}\end{aligned}}

If $ad - bc = 1$ then

{\begin{pmatrix}d&c\\b&a\end{pmatrix}}^{-1}={\begin{pmatrix}\ a&-c\\-b&\ d\end{pmatrix}}

so that

(m,n)\mapsto (m,n){\begin{pmatrix}d&c\\b&a\end{pmatrix}}

is a bijection $\mathbb {Z}$ , i.e.:

\sum _{\left(m',n'\right)=(m,n){\begin{pmatrix}d\ \ c\\b\ \ a\end{pmatrix}} \atop (m,n)\in \mathbb {Z} ^{2}\setminus \{(0,0)\}}{\frac {1}{\left(m'+n'\tau \right)^{2k}}}=\sum _{\left(m',n'\right)\in \mathbb {Z} ^{2}\setminus \{(0,0)\}}{\frac {1}{(m'+n'\tau )^{2k}}}=G_{2k}(\tau )

Overall, if $ad - bc = 1$ 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

\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.

Note that $k \geq 2$ is necessary such that the series converges absolutely, whereas $k$ needs to be even otherwise the sum vanishes because the $(- m, - n)$ and $(m, n)$ terms cancel out. For $k = 1$ the series converges but it is not a modular form.

Relation to modular invariants

The modular invariants $g 2$ and $g 3$ of an elliptic curve are given by the first two Eisenstein series:^[3]

{\begin{aligned}g_{2}&=60G_{4}\\g_{3}&=140G_{6}.\end{aligned}}

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^[4] can be written as a polynomial in $G 4$ and $G 6$ . Specifically, the higher order $G 2 k$ can be written in terms of $G 4$ and $G 6$ through a recurrence relation. Let $d k = (2 k + 3) k! G 2 k + 4$ , so for example, $d 0 = 3 G 4$ and $d 1 = 5 G 6$ . 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.

The $d k$ occur in the series expansion for the Weierstrass's elliptic functions:

{\begin{aligned}\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}.\end{aligned}}

Fourier series

Define $q = e 2π iτ$ . (Some older books define $q$ to be the nome $q = e π iτ$ , but $q = e 2 π iτ$ is now standard in number theory.) Then the Fourier series of the Eisenstein^[5] 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 coefficients $c 2 k$ are given by

{\begin{aligned}c_{2k}&={\frac {(2\pi i)^{2k}}{(2k-1)!\zeta (2k)}}\\[4pt]&={\frac {-4k}{B_{2k}}}={\frac {2}{\zeta (1-2k)}}.\end{aligned}}

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

{\begin{aligned}G_{4}(\tau )&={\frac {\pi ^{4}}{45}}\left(1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}\right)\\[4pt]G_{6}(\tau )&={\frac {2\pi ^{6}}{945}}\left(1-504\sum _{n=1}^{\infty }\sigma _{5}(n)q^{n}\right).\end{aligned}}

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, this alternate notation is frequently introduced:

{\begin{aligned}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}}}\\&=1-{\frac {4k}{B_{2k}}}\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}\\&=1-{\frac {4k}{B_{2k}}}\sum _{d,n\geq 1}n^{2k-1}q^{nd}.\end{aligned}}

Identities involving Eisenstein series

As theta functions

Source:^[6]

Given $q = e 2 π iτ$ , let

{\begin{aligned}E_{4}(\tau )&=1+240\sum _{n=1}^{\infty }{\frac {n^{3}q^{n}}{1-q^{n}}}\\E_{6}(\tau )&=1-504\sum _{n=1}^{\infty }{\frac {n^{5}q^{n}}{1-q^{n}}}\\E_{8}(\tau )&=1+480\sum _{n=1}^{\infty }{\frac {n^{7}q^{n}}{1-q^{n}}}\end{aligned}}

and define the Jacobi theta functions which normally uses the nome $e π iτ$ ,

{\begin{aligned}a&=\theta _{2}\left(0;e^{\pi i\tau }\right)=\vartheta _{10}(0;\tau )\\b&=\theta _{3}\left(0;e^{\pi i\tau }\right)=\vartheta _{00}(0;\tau )\\c&=\theta _{4}\left(0;e^{\pi i\tau }\right)=\vartheta _{01}(0;\tau )\end{aligned}}

where $θ m$ and $ϑ ij$ are alternative notations. Then we have the symmetric relations,

{\begin{aligned}E_{4}(\tau )&={\tfrac {1}{2}}\left(a^{8}+b^{8}+c^{8}\right)\\[4pt]E_{6}(\tau )&={\tfrac {1}{2}}{\sqrt {\frac {\left(a^{8}+b^{8}+c^{8}\right)^{3}-54(abc)^{8}}{2}}}\\[4pt]E_{8}(\tau )&={\tfrac {1}{2}}\left(a^{16}+b^{16}+c^{16}\right)=a^{8}b^{8}+a^{8}c^{8}+b^{8}c^{8}\end{aligned}}

Basic algebra immediately implies

E_{4}^{3}-E_{6}^{2}={\tfrac {27}{4}}(abc)^{8}

an expression related to the modular discriminant,

\Delta =g_{2}^{3}-27g_{3}^{2}=(2\pi )^{12}\left({\tfrac {1}{2}}abc\right)^{8}

The third symmetric relation, on the other hand, is a consequence of $E 8 = E 24$ and $a 4 - b 4 + c 4 = 0$ .

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group $\mathbb {Z}$ . 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 equal up to a scalar multiple. In fact, we obtain the identities:^[7]

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:

\left(1+240\sum _{n=1}^{\infty }\sigma _{3}(n)q^{n}\right)^{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. 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 ),\qquad r_{\Gamma }(n)=240\sigma _{3}(n)

for the number $r Γ (n)$ of vectors of the squared length $2 n$ in the root lattice of the type $E 8$ .

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, or eight squares in terms of the divisors of $n$ .

Using the above recurrence relation, all higher $E 2 k$ can be expressed as polynomials in $E 4$ and $E 6$ . For example:

{\begin{aligned}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{aligned}}

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

\left({\frac {\Delta }{(2\pi )^{12}}}\right)^{2}=-{\frac {691}{1728^{2}\cdot 250}}\det {\begin{vmatrix}E_{4}&E_{6}&E_{8}\\E_{6}&E_{8}&E_{10}\\E_{8}&E_{10}&E_{12}\end{vmatrix}}

where

\Delta =(2\pi )^{12}{\frac {E_{4}^{3}-E_{6}^{2}}{1728}}

is the modular discriminant.^[8]

Ramanujan identities

Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation.^[9] Let

{\begin{aligned}L(q)&=1-24\sum _{n=1}^{\infty }{\frac {nq^{n}}{1-q^{n}}}&&=E_{2}(\tau )\\M(q)&=1+240\sum _{n=1}^{\infty }{\frac {n^{3}q^{n}}{1-q^{n}}}&&=E_{4}(\tau )\\N(q)&=1-504\sum _{n=1}^{\infty }{\frac {n^{5}q^{n}}{1-q^{n}}}&&=E_{6}(\tau ),\end{aligned}}

then

{\begin{aligned}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{aligned}}

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

{\begin{aligned}\sigma _{p}(0)={\tfrac {1}{2}}\zeta (-p)\quad \Longrightarrow \quad \sigma (0)&=-{\tfrac {1}{24}}\\\sigma _{3}(0)&={\tfrac {1}{240}}\\\sigma _{5}(0)&=-{\tfrac {1}{504}}.\end{aligned}}

Then, for example

\sum _{k=0}^{n}\sigma (k)\sigma (n-k)={\tfrac {5}{12}}\sigma _{3}(n)-{\tfrac {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 Giuseppe Melfi,^[10]^[11] as for example

{\begin{aligned}\sum _{k=0}^{n}\sigma _{3}(k)\sigma _{3}(n-k)&={\tfrac {1}{120}}\sigma _{7}(n)\\\sum _{k=0}^{n}\sigma (2k+1)\sigma _{3}(n-k)&={\tfrac {1}{240}}\sigma _{5}(2n+1)\\\sum _{k=0}^{n}\sigma (3k+1)\sigma (3n-3k+1)&={\tfrac {1}{9}}\sigma _{3}(3n+2).\end{aligned}}

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.

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References

↑ "Gotthold Eisenstein - Biography". Maths History. Retrieved 2023-09-05.
↑ Gekeler, Ernst-Ulrich (2011). "PARA-EISENSTEIN SERIES FOR THE MODULAR GROUP GL(2, 𝔽q[T])". Taiwanese Journal of Mathematics. 15 (4): 1463–1475. doi: 10.11650/twjm/1500406358 . ISSN 1027-5487. S2CID 119499748.
↑ Obers, N. A.; Pioline, B. (2000-03-07). "Eisenstein Series in String Theory". Classical and Quantum Gravity. 17 (5): 1215–1224. arXiv: hep-th/9910115 . Bibcode:2000CQGra..17.1215O. doi:10.1088/0264-9381/17/5/330. ISSN 0264-9381. S2CID 250864942.
↑ Mertens, Michael H.; Rolen, Larry (2015). "Lacunary recurrences for Eisenstein series". Research in Number Theory. 1. arXiv: 1504.00356 . doi: 10.1007/s40993-015-0010-x . ISSN 2363-9555.
↑ Karel, Martin L. (1974). "Fourier Coefficients of Certain Eisenstein Series". Annals of Mathematics. 99 (1): 176–202. doi:10.2307/1971017. ISSN 0003-486X. JSTOR 1971017.
↑ "How to prove this series identity involving Eisenstein series?". Mathematics Stack Exchange. Retrieved 2023-09-05.
↑ Dickson, Martin; Neururer, Michael (2018). "Products of Eisenstein series and Fourier expansions of modular forms at cusps". Journal of Number Theory. 188: 137–164. arXiv: 1603.00774 . doi:10.1016/j.jnt.2017.12.013. S2CID 119614418.
↑ Milne, Steven C. (2000). "Hankel Determinants of Eisenstein Series". arXiv: math/0009130v3 . The paper uses a non-equivalent definition of $\Delta$ , but this has been accounted for in this article.
↑ Bhuvan, E. N.; Vasuki, K. R. (2019-06-24). "On a Ramanujan's Eisenstein series identity of level fifteen". Proceedings - Mathematical Sciences. 129 (4): 57. doi:10.1007/s12044-019-0498-4. ISSN 0973-7685. S2CID 255485301.
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Eisenstein series

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