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

Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight , where is an integer, by the following series:

:$G_(\tau) = \sum_ \frac.$

This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its -invariance. Explicitly if and then

:$G_ \left( \frac \right) = (c\tau +d)^ G_(\tau)$

Relation to modular invariants

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

:$\begin g_2 &= 60 G_4 \\ g_3 &= 140 G_6 .\end$

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 and . Specifically, the higher order can be written in terms of and through a recurrence relation. Let , so for example, and . Then the satisfy the relation

:$\sum_^n d_k d_ = \fracd_$

for all . Here, is the binomial coefficient.

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

:$\begin \wp(z) &=\frac + z^2 \sum_^\infty \frac \\ &=\frac + \sum_^\infty (2k+1) G_ z^. \end$

Fourier series

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

:$G_(\tau) = 2\zeta(2k) \left(1+c_\sum_^\infty \sigma_(n)q^n \right)$

where the coefficients are given by

:\begin
c_ &= \frac \\ pt&= \frac = \frac 2 .
\end

Here, are the Bernoulli numbers, is Riemann's zeta function and is the divisor sum function, the sum of the th powers of the divisors of . In particular, one has

:\begin
G_4(\tau)&=\frac \left( 1+ 240\sum_^\infty \sigma_3(n) q^ \right) \\ ptG_6(\tau)&=\frac \left( 1- 504\sum_^\infty \sigma_5(n) q^n \right).
\end

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

:$\sum_^ q^n \sigma_a(n) = \sum_^ \frac$

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

:$\begin E_(\tau)&=\frac\\ &= 1+\frac \sum_^ \frac \\ &= 1- \frac\sum_^ \sigma_(n)q^n \\ &= 1 - \frac \sum_ n^ q^. \end$

Identities involving Eisenstein series

As theta functions

Given , let

:$\begin E_4(\tau)&=1+240\sum_^\infty \frac \\ E_6(\tau)&=1-504\sum_^\infty \frac \\ E_8(\tau)&=1+480\sum_^\infty \frac \end$

and define the Jacobi theta functions which normally uses the nome ,

:$\begin a&=\theta_2\left(0; e^\right)=\vartheta_(0; \tau) \\ b&=\theta_3\left(0; e^\right)=\vartheta_(0; \tau) \\ c&=\theta_4\left(0; e^\right)=\vartheta_(0; \tau) \end$

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

:\begin
E_4(\tau)&= \tfrac\left(a^8+b^8+c^8\right) \\ ptE_6(\tau)&= \tfrac\sqrt \\ ptE_8(\tau)&= \tfrac\left(a^+b^+c^\right) = a^8b^8 +a^8c^8 +b^8c^8
\end

Basic algebra immediately implies

:$E_4^3-E_6^2 = \tfrac(abc)^8$

an expression related to the modular discriminant,

:$\Delta = g_2^3-27g_3^2 = (2\pi)^ \left(\tfraca b c\right)^8$

The third symmetric relation, on the other hand, is a consequence of and .

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group . Since the space of modular forms of weight has dimension 1 for , different products of Eisenstein series having those weights have to be equal up to a scalar multiple. In fact, we obtain the identities:

:$E_4^2 = E_8, \quad E_4 E_6 = E_, \quad E_4 E_ = E_, \quad E_6 E_8 = E_.$

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

:$\left(1+240\sum_^\infty \sigma_3(n) q^n\right)^2 = 1+480\sum_^\infty \sigma_7(n) q^n,$

hence

:$\sigma_7(n)=\sigma_3(n)+120\sum_^\sigma_3(m)\sigma_3(n-m),$

and similarly for the others. The theta function of an eight-dimensional even unimodular lattice is a modular form of weight 4 for the full modular group, which gives the following identities:

:$\theta_\Gamma (\tau)=1+\sum_^\infty r_(2n) q^ = E_4(\tau), \qquad r_(n) = 240\sigma_3(n)$

for the number of vectors of the squared length in the root lattice of the type .

Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer ' as a sum of two, four, or eight squares in terms of the divisors of .

Using the above recurrence relation, all higher can be expressed as polynomials in and . For example:

:$\begin E_ &= E_4^2 \\ E_ &= E_4\cdot E_6 \\ 691 \cdot E_ &= 441\cdot E_4^3+ 250\cdot E_6^2 \\ E_ &= E_4^2\cdot E_6 \\ 3617\cdot E_ &= 1617\cdot E_4^4+ 2000\cdot E_4 \cdot E_6^2 \\ 43867 \cdot E_ &= 38367\cdot E_4^3\cdot E_6+5500\cdot E_6^3 \\ 174611 \cdot E_ &= 53361\cdot E_4^5+ 121250\cdot E_4^2\cdot E_6^2 \\ 77683 \cdot E_ &= 57183\cdot E_4^4\cdot E_6+20500\cdot E_4\cdot E_6^3 \\ 236364091 \cdot E_ &= 49679091\cdot E_4^6+ 176400000\cdot E_4^3\cdot E_6^2 + 10285000\cdot E_6^4 \end$

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

: $\left(\frac\right)^2=-\frac\det \beginE_4&E_6&E_8\\ E_6&E_8&E_\\ E_8&E_&E_\end$

where

:$\Delta=(2\pi)^\frac$

is the modular discriminant.

Ramanujan identities

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

:$\begin L(q)&=1-24\sum_^\infty \frac &&=E_2(\tau) \\ M(q)&=1+240\sum_^\infty \frac &&=E_4(\tau) \\ N(q)&=1-504\sum_^\infty \frac &&=E_6(\tau), \end$

then

:$\begin q\frac &= \frac \\ q\frac &= \frac \\ q\frac &= \frac . \end$

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 to include zero, by setting

:$\begin\sigma_p(0) = \tfrac12\zeta(-p) \quad\Longrightarrow\quad \sigma(0) &= -\tfrac\\ \sigma_3(0) &= \tfrac\\ \sigma_5(0) &= -\tfrac. \end$

Then, for example

:$\sum_^n\sigma(k)\sigma(n-k)=\tfrac5\sigma_3(n)-\tfrac12n\sigma(n).$

Other identities of this type, but not directly related to the preceding relations between , and functions, have been proved by Ramanujan and Giuseppe Melfi, as for example

:$\begin \sum_^n\sigma_3(k)\sigma_3(n-k)&=\tfrac1\sigma_7(n) \\ \sum_^n\sigma(2k+1)\sigma_3(n-k)&=\tfrac1\sigma_5(2n+1) \\ \sum_^n\sigma(3k+1)\sigma(3n-3k+1)&=\tfrac19\sigma_3(3n+2). \end$

Generalizations

Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion.

Defining to be the ring of integers of a totally real algebraic number field , one then defines the Hilbert–Blumenthal modular group as . One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.

References

Further reading

* Translated into English as
*
*
*
* {{cite book, authorlink=Jean-Pierre Serre, last=Serre, first=Jean-Pierre, title=A Course in Arithmetic, url=https://archive.org/details/courseinarithmet00serr, url-access=registration, edition=transl., series=Graduate Texts in Mathematics 7, publisher=Springer-Verlag, location=New York & Heidelberg, date=1973, isbn=9780387900407

Mathematical series
Modular forms
Analytic number theory
Fractals