Английская Википедия:Eisenstein series
Шаблон:Short descriptionШаблон:Distinguish Шаблон:About
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 Шаблон:Mvar be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series Шаблон:Math of weight Шаблон:Math, where Шаблон:Math is an integer, by the following series:[2]
- <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 Шаблон:Mvar in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at Шаблон:Math. It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its Шаблон:Math-covariance. Explicitly if Шаблон:Math and Шаблон:Math then
- <math>G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)</math>
Шаблон: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 Шаблон: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 Шаблон: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 Шаблон:Math then
- <math>G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{2k} G_{2k}(\tau)</math>
and Шаблон:Math is therefore a modular form of weight Шаблон:Math. Note that it is important to assume that Шаблон:Math, otherwise it would be illegitimate to change the order of summation, and the Шаблон: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 Шаблон:Math, although it would only be a quasimodular form. }}
Note that Шаблон:Math is necessary such that the series converges absolutely, whereas Шаблон:Math needs to be even otherwise the sum vanishes because the Шаблон:Math and Шаблон:Math terms cancel out. For Шаблон:Math the series converges but it is not a modular form.
Relation to modular invariants
The modular invariants Шаблон:Math and Шаблон:Math of an elliptic curve are given by the first two Eisenstein series:[3]
- <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[4] can be written as a polynomial in Шаблон:Math and Шаблон:Math. Specifically, the higher order Шаблон:Math can be written in terms of Шаблон:Math and Шаблон:Math through a recurrence relation. Let Шаблон:Math, so for example, Шаблон:Math and Шаблон:Math. Then the Шаблон: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 Шаблон:Math. Here, Шаблон:Math is the binomial coefficient.
The Шаблон: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 Шаблон:Math. (Some older books define Шаблон:Mvar to be the nome Шаблон:Math, but Шаблон:Math is now standard in number theory.) Then the Fourier series of the Eisenstein[5] 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 Шаблон: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, Шаблон:Math are the Bernoulli numbers, Шаблон:Math is Riemann's zeta function and Шаблон:Math is the divisor sum function, the sum of the Шаблон:Mvarth powers of the divisors of Шаблон: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 Шаблон: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 Шаблон:Math and Шаблон:Mvar. When working with the [[q-expansion|Шаблон: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:[6]
Given Шаблон: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 Шаблон: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 Шаблон:Math and Шаблон: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 Шаблон:Math and Шаблон:Math.
Products of Eisenstein series
Eisenstein series form the most explicit examples of modular forms for the full modular group Шаблон:Math. Since the space of modular forms of weight Шаблон:Math has dimension 1 for Шаблон:Math, different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In fact, we obtain the identities:[7]
- <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 Шаблон: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 Шаблон: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 Шаблон:Math of vectors of the squared length Шаблон:Math in the [[E8 lattice|root lattice of the type Шаблон:Math]].
Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer Шаблон:Mvar' as a sum of two, four, or eight squares in terms of the divisors of Шаблон:Mvar.
Using the above recurrence relation, all higher Шаблон:Math can be expressed as polynomials in Шаблон:Math and Шаблон: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.[8]
Ramanujan identities
Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation.[9] 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 Шаблон: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 Шаблон:Mvar, Шаблон:Mvar and Шаблон:Mvar functions, have been proved by Ramanujan and Giuseppe Melfi,[10][11] 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 Lie groups; and Eisenstein series generalize in a similar fashion.
Defining Шаблон:Math to be the ring of integers of a totally real algebraic number field Шаблон:Mvar, one then defines the Hilbert–Blumenthal modular group as Шаблон:Math. One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.
References
Further reading
- Шаблон:Cite book Translated into English as Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite arXiv The paper uses a non-equivalent definition of <math>\Delta</math>, but this has been accounted for in this article.
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
