Eisenstein series
Template:Short description Template:Distinguish Template:About Template:Cleanup MOS Eisenstein series, named after German mathematician Gotthold Eisenstein,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 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:<ref>Template:Cite journal</ref>
- <math>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 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-invariance. Explicitly if Template:Math and Template:Math then
- <math>G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)</math>
Template:Hidden \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 Template:Math then
- <math>\begin{pmatrix}d & c\\b & a\end{pmatrix}^{-1} = \begin{pmatrix}\ a & -c\\-b & \ d\end{pmatrix}</math>
so that
- <math>(m,n) \mapsto (m,n)\begin{pmatrix}d & c\\b & a\end{pmatrix}</math>
is a bijection Template:Math, i.e.:
- <math>\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 Template:Math then
- <math>G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{2k} G_{2k}(\tau)</math>
and Template:Math is therefore a modular form of weight Template:Math. }}
Note that it is important to assume that Template:Math to ensure absolute convergence of the series, as otherwise it would be illegitimate to change the order of summation in the proof of Template:Math-invariance. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for Template:Math, although it would only be a quasimodular form. It is also necessary that the weight be even, as otherwise the sum vanishes because the Template:Math and Template:Math terms cancel each other.
Relation to modular invariants
The modular invariants Template:Math and Template:Math of an elliptic curve are given by the first two Eisenstein series:<ref>Template:Cite journal</ref>
- <math>\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<ref>Template:Cite journal</ref> 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
- <math>\sum_{k=0}^n {n \choose k} d_k d_{n-k} = \frac{2n+9}{3n+6}d_{n+2}</math>
for all Template:Math. Here, <math>n \choose k</math> is the binomial coefficient.
The Template:Math occur in the series expansion for the Weierstrass's elliptic functions:
- <math>\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
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<ref>Template:Cite journal</ref> series is
- <math>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 Template:Math are given by
- <math>\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, 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
- <math>\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 Template:Mvar can be resummed as a Lambert series; that is, one has
- <math>\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 Template:Math and Template:Mvar. When working with the [[q-expansion|Template:Mvar-expansion]] of the Eisenstein series, this alternate notation is frequently introduced:
- <math>\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:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Given Template:Math, let
- <math>\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 Template:Math,
- <math>\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 Template:Math and Template:Math are alternative notations. Then we have the symmetric relations,
- <math>\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>E_4^3-E_6^2 = \tfrac{27}{4}(abc)^8 </math>
an expression related to the modular discriminant,
- <math>\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 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:<ref>Template:Cite journal</ref>
- <math>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 Template:Mvar-expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:
- <math>\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>\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 Template:Math is a modular form of weight 4 for the full modular group, which gives the following identities:
- <math> \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 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:
- <math>\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> \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> \Delta=(2\pi)^{12}\frac{E_4^3-E_6^2}{1728}</math>
is the modular discriminant.<ref>Template:Cite arXiv The paper uses a non-equivalent definition of <math>\Delta</math>, but this has been accounted for in this article.</ref>
Ramanujan identities
Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation.<ref>Template:Cite journal</ref> Let
- <math>\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>\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 Template:Math to include zero, by setting
- <math>\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>\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 Template:Mvar, Template:Mvar and Template:Mvar functions, have been proved by Ramanujan and Giuseppe Melfi,<ref>Template:Cite book</ref><ref>Template:Cite book</ref> as for example
- <math>\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 semisimple Lie groups, where the modular group is replaced by an arithmetic group. Robert Langlands generalised the theory of Eisenstein series to this setting.<ref>Template:Cite book</ref>
When the Lie group is of type A1 the theory resembles the classical case. For example Hilbert modular forms are well-studied.
References
Further reading
- Template:Cite book Translated into English as Template:Cite book
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