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:
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 -covariance. Explicitly if and then
Note that is necessary such that the series converges absolutely, whereas needs to be even otherwise the sum vanishes because the and terms cancel out. For the series converges but it is not a modular form.
Relation to modular invariants
The modular invariants and of an elliptic curve are given by the first two Eisenstein series:
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
for all . Here, is the binomial coefficient.
The occur in the series expansion for the Weierstrass's elliptic functions:
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
where the coefficients are given by
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
The summation over can be resummed as a Lambert series; that is, one has
for arbitrary complex and . When working with the -expansion of the Eisenstein series, this alternate notation is frequently introduced:
Identities involving Eisenstein series
As theta functions
Given , let
and define the Jacobi theta functions which normally uses the nome ,
where and are alternative notations. Then we have the symmetric relations,
Basic algebra immediately implies
an expression related to the modular discriminant,
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:
Using the -expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:
hence
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:
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:
Many relationships between products of Eisenstein series can be written in an elegant way using Hankel determinants, e.g. Garvan's identity
where
is the modular discriminant.
Ramanujan identities
Srinivasa Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation. Let
then
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
Then, for example
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
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
Mathematical series
Modular forms
Analytic number theory
Fractals