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 G₆ as a function of q on the unit disk. Negative numbers are black.

The imaginary part of G₆ as a function of q on the unit disk.

Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G_2k(τ) of weight 2k, where k ≥ 2 is an integer, by the following series:

[math]\displaystyle{ G_{2k}(\tau) = \sum_{ (m,n)\in\Z^2\setminus\{(0,0)\}} \frac{1}{(m+n\tau )^{2k}}. }[/math]

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∞. It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its SL(2, [math]\displaystyle{ \mathbb{Z} }[/math])-covariance. Explicitly if a, b, c, d ∈ [math]\displaystyle{ \mathbb{Z} }[/math] and ad − bc = 1 then

[math]\displaystyle{ G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau) }[/math]

(Proof)

[math]\displaystyle{ \begin{align} G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right) &= \sum_{(m,n) \in \Z^2 \setminus \{(0,0)\}} \frac{1}{\left(m+n\frac{a\tau+b}{c\tau+d}\right)^{2k}} \\ &= \sum_{(m,n) \in \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 \Z^2 \setminus \{(0,0)\}} \frac{(c\tau+d)^{2k}}{\left(m'+n'\tau\right)^{2k}} \end{align} }[/math]

If ad − bc = 1 then

[math]\displaystyle{ \begin{pmatrix}d & c\\b & a\end{pmatrix}^{-1} = \begin{pmatrix}\ a & -c\\-b & \ d\end{pmatrix} }[/math]

so that

[math]\displaystyle{ (m,n) \mapsto (m,n)\begin{pmatrix}d & c\\b & a\end{pmatrix} }[/math]

is a bijection [math]\displaystyle{ \mathbb{Z} }[/math]² → [math]\displaystyle{ \mathbb{Z} }[/math]², i.e.:

[math]\displaystyle{ \sum_{\left(m',n'\right) = (m,n)\begin{pmatrix}d \ \ c\\b \ \ a\end{pmatrix}\atop (m,n)\in \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) }[/math]

Overall, if ad − bc = 1 then

[math]\displaystyle{ G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{2k} G_{2k}(\tau) }[/math]

and G_2k is therefore a modular form of weight 2k. Note that it is important to assume that k ≥ 2, otherwise it would be illegitimate to change the order of summation, and the SL(2, [math]\displaystyle{ \mathbb{Z} }[/math])-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 ≥ 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 = 2 the series converges but it is not a modular form.

Relation to modular invariants

The modular invariants g₂ and g₃ of an elliptic curve are given by the first two Eisenstein series:

[math]\displaystyle{ \begin{align} g_2 &= 60 G_4 \\ g_3 &= 140 G_6 .\end{align} }[/math]

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 G₄ and G₆. Specifically, the higher order G_2k can be written in terms of G₄ and G₆ through a recurrence relation. Let d_k = (2k + 3)k! G_{2k + 4}, so for example, d₀ = 3G₄ and d₁ = 5G₆. Then the d_k satisfy the relation

[math]\displaystyle{ \sum_{k=0}^n {n \choose k} d_k d_{n-k} = \frac{2n+9}{3n+6}d_{n+2} }[/math]

for all n ≥ 0. Here, (nk) is the binomial coefficient.

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

[math]\displaystyle{ \begin{align} \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{align} }[/math]

Fourier series

G₄

G₆

G₈

G₁₀

G₁₂

G₁₄

Define q = e^2πiτ. (Some older books define q to be the nome q = e^πiτ, but q = e^2πiτ is now standard in number theory.) Then the Fourier series of the Eisenstein series is

[math]\displaystyle{ G_{2k}(\tau) = 2\zeta(2k) \left(1+c_{2k}\sum_{n=1}^\infty \sigma_{2k-1}(n)q^n \right) }[/math]

where the coefficients c_2k are given by

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

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 pth powers of the divisors of n. In particular, one has

[math]\displaystyle{ \begin{align} 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{align} }[/math]

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

[math]\displaystyle{ \sum_{n=1}^{\infty} q^n \sigma_a(n) = \sum_{n=1}^{\infty} \frac{n^a q^n}{1-q^n} }[/math]

for arbitrary complex |q| < 1 and a. When working with the q-expansion of the Eisenstein series, this alternate notation is frequently introduced:

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Identities involving Eisenstein series

As theta functions

Source:

Given q = e^2πiτ, let

[math]\displaystyle{ \begin{align} E_4(\tau)&=1+240\sum_{n=1}^\infty \frac {n^3q^n}{1-q^n} \\ E_6(\tau)&=1-504\sum_{n=1}^\infty \frac {n^5q^n}{1-q^n} \\ E_8(\tau)&=1+480\sum_{n=1}^\infty \frac {n^7q^n}{1-q^n} \end{align} }[/math]

and define the Jacobi theta functions which normally uses the nome e^πiτ,

[math]\displaystyle{ \begin{align} 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{align} }[/math]

where θ_m and ϑ_ij are alternative notations. Then we have the symmetric relations,

[math]\displaystyle{ \begin{align} 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^8b^8 +a^8c^8 +b^8c^8 \end{align} }[/math]

Basic algebra immediately implies

[math]\displaystyle{ E_4^3-E_6^2 = \tfrac{27}{4}(abc)^8 }[/math]

an expression related to the modular discriminant,

[math]\displaystyle{ \Delta = g_2^3-27g_3^2 = (2\pi)^{12} \left(\tfrac{1}{2}a b c\right)^8 }[/math]

The third symmetric relation, on the other hand, is a consequence of E₈ = E24 and a⁴ − b⁴ + c⁴ = 0.

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group SL(2, [math]\displaystyle{ \mathbb{Z} }[/math]). 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 equal up to a scalar multiple. In fact, we obtain the identities:

[math]\displaystyle{ 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}. }[/math]

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

[math]\displaystyle{ \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, }[/math]

hence

[math]\displaystyle{ \sigma_7(n)=\sigma_3(n)+120\sum_{m=1}^{n-1}\sigma_3(m)\sigma_3(n-m), }[/math]

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:

[math]\displaystyle{ \theta_\Gamma (\tau)=1+\sum_{n=1}^\infty r_{\Gamma}(2n) q^{n} = E_4(\tau), \qquad r_{\Gamma}(n) = 240\sigma_3(n) }[/math]

for the number r_Γ(n) of vectors of the squared length 2n in the root lattice of the type E₈.

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_2k can be expressed as polynomials in E₄ and E₆. For example:

[math]\displaystyle{ \begin{align} 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{align} }[/math]

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

[math]\displaystyle{ \left(\frac{\Delta}{(2\pi)^{12}}\right)^2=-\frac{691}{1728^2\cdot250}\det \begin{vmatrix}E_4&E_6&E_8\\ E_6&E_8&E_{10}\\ E_8&E_{10}&E_{12}\end{vmatrix} }[/math]

where

[math]\displaystyle{ \Delta=(2\pi)^{12}\frac{E_4^3-E_6^2}{1728} }[/math]

is the modular discriminant.

Ramanujan identities

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

[math]\displaystyle{ \begin{align} 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^3q^n}{1-q^n}&&=E_4(\tau) \\ N(q)&=1-504\sum_{n=1}^\infty \frac {n^5q^n}{1-q^n}&&=E_6(\tau), \end{align} }[/math]

then

[math]\displaystyle{ \begin{align} 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{align} }[/math]

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

[math]\displaystyle{ \begin{align}\sigma_p(0) = \tfrac12\zeta(-p) \quad\Longrightarrow\quad \sigma(0) &= -\tfrac{1}{24}\\ \sigma_3(0) &= \tfrac{1}{240}\\ \sigma_5(0) &= -\tfrac{1}{504}. \end{align} }[/math]

Then, for example

[math]\displaystyle{ \sum_{k=0}^n\sigma(k)\sigma(n-k)=\tfrac5{12}\sigma_3(n)-\tfrac12n\sigma(n). }[/math]

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, as for example

[math]\displaystyle{ \begin{align} \sum_{k=0}^n\sigma_3(k)\sigma_3(n-k)&=\tfrac1{120}\sigma_7(n) \\ \sum_{k=0}^n\sigma(2k+1)\sigma_3(n-k)&=\tfrac1{240}\sigma_5(2n+1) \\ \sum_{k=0}^n\sigma(3k+1)\sigma(3n-3k+1)&=\tfrac19\sigma_3(3n+2). \end{align} }[/math]

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.

Further reading

• Akhiezer, Naum Illyich (1970) (in Russian). Elements of the Theory of Elliptic Functions. Moscow.  Translated into English as Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs 79. Providence, RI: American Mathematical Society. 1990. ISBN 0-8218-4532-2. 
• Apostol, Tom M. (1990). Modular Functions and Dirichlet Series in Number Theory (2nd ed.). New York, NY: Springer. ISBN 0-387-97127-0. https://archive.org/details/modularfunctions0000apos. 
• Chan, Heng Huat; Ong, Yau Lin (1999). "On Eisenstein Series". Proc. Amer. Math. Soc. 127 (6): 1735–1744. doi:10.1090/S0002-9939-99-04832-7. https://www.ams.org/proc/1999-127-06/S0002-9939-99-04832-7/S0002-9939-99-04832-7.pdf. 
• Iwaniec, Henryk (2002). Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics 53 (2nd ed.). Providence, RI: American Mathematical Society. ch. 3. ISBN 0-8218-3160-7. 
• Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics 7 (transl. ed.). New York & Heidelberg: Springer-Verlag. ISBN 9780387900407. https://archive.org/details/courseinarithmet00serr.