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.

Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G2k(τ) of weight 2k, where k ≥ 2 is an integer, by the following series:

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, )-invariance. Explicitly if a, b, c, d and adbc = 1 then

The modular invariants g2 and g3 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.

Any holomorphic modular form for the modular group can be written as a polynomial in G4 and G6. Specifically, the higher order G2k can be written in terms of G4 and G6 through a recurrence relation. Let dk = (2k + 3)k! G2k + 4, so for example, d0 = 3G4 and d1 = 5G6. Then the dk satisfy the relation

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

Define q = e. (Some older books define q to be the nome q = eπ, but q = e2π is now standard in number theory.) Then the Fourier series of the Eisenstein series is

Here, Bn are the Bernoulli numbers, ζ(z) is Riemann's zeta function and σp(n) is the divisor sum function, the sum of the pth powers of the divisors of n. In particular, one has

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

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

where θm and ϑij are alternative notations for the Jacobi theta functions. Then,

Eisenstein series form the most explicit examples of modular forms for the full modular group SL(2, ). 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:

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

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:

for the number rΓ(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 E2k can be expressed as polynomials in E4 and E6. For example:

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

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

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 σp(n) to include zero, by setting

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,[2][3] as for example

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

Defining OK 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,OK). One can then associate an Eisenstein series to every cusp of the Hilbert–Blumenthal modular group.