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
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, ℤ)-invariance. Explicitly if a, b, c, d ∈ ℤ and ad − bc = 1 then
and G2k is therefore a modular form of weight 2k. Note that it is important to assume that k ≥ 2, otherwise it would be illegitimate to change the order of summation, and the SL(2, ℤ)-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 k = 1, although it would only be a quasimodular form.
Relation to modular invariants
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
for all n ≥ 0. Here, (n
k) is the binomial coefficient.
The dk occur in the series expansion for the Weierstrass's elliptic functions:
where the coefficients c2k are given by
The summation over q can be resummed as a Lambert series; that is, one has
Identities involving Eisenstein series
As theta functions
Given q = e2πiτ, let
where θm and ϑij are alternative notations for the Jacobi theta functions. Then,
an expression related to the modular discriminant,
Also, since E8 = E2
4 and a4 − b4 + c4 = 0, this implies
Products of Eisenstein series
Eisenstein series form the most explicit examples of modular forms for the full modular group SL(2, ℤ). 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
is the modular discriminant.
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
Then, for example
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, as for example
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.
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