Eisenstein series - Alchetron, The Free Social Encyclopedia

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 τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G_2k(τ) of weight 2k, where k ≥ 2 is an integer, by the following series:

G 2 k ( τ ) = ∑ ( m , n ) ∈ Z 2 ∖ ( 0 , 0 ) 1 ( m + n τ ) 2 k .

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

G 2 k ( a τ + b c τ + d ) = ( c τ + d ) 2 k G 2 k ( τ )

and G_2k 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 modular invariants g₂ and g₃ of an elliptic curve are given by the first two Eisenstein series:

g 2 = 60 G 4 g 3 = 140 G 6 .

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 G₄ and G₆. Specifically, the higher order G_2k can be written in terms of G₄ and G₆ through a recurrence relation. Let d_k = (2k + 3)k! G_{2k + 4}. Then the d_k satisfy the relation

∑ k = 0 n ( n k ) d k d n − k = 2 n + 9 3 n + 6 d n + 2

for all n ≥ 0. Here, (n
k) is the binomial coefficient and d₀ = 3G₄ and d₁ = 5G₆.

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

℘ ( z ) = 1 z 2 + z 2 ∑ k = 0 ∞ d k z 2 k k ! = 1 z 2 + ∑ k = 1 ∞ ( 2 k + 1 ) G 2 k + 2 z 2 k .

Fourier series

Define q = e^2πiτ. (Some older books define q to be the nome q = e^πiτ, but q = e^2πiτ is now standard in number theory.) Then the Fourier series of the Eisenstein series is

G 2 k ( τ ) = 2 ζ ( 2 k ) ( 1 + c 2 k ∑ n = 1 ∞ σ 2 k − 1 ( n ) q n )

where the coefficients c_2k are given by

c 2 k = ( 2 π i ) 2 k ( 2 k − 1 ) ! ζ ( 2 k ) = − 4 k B 2 k = 2 ζ ( 1 − 2 k ) .

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

G 4 ( τ ) = π 4 45 ( 1 + 240 ∑ n = 1 ∞ σ 3 ( n ) q n ) G 6 ( τ ) = 2 π 6 945 ( 1 − 504 ∑ n = 1 ∞ σ 5 ( n ) q n ) .

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

∑ n = 1 ∞ q n σ a ( n ) = ∑ n = 1 ∞ n a q n 1 − q n

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

E 2 k ( τ ) = G 2 k ( τ ) 2 ζ ( 2 k ) = 1 + 2 ζ ( 1 − 2 k ) ∑ n = 1 ∞ n 2 k − 1 q n 1 − q n = 1 − 2 ⋅ 2 k B 2 k ∑ n = 1 ∞ σ 2 k − 1 ( n ) q n = 1 − 4 k B 2 k ∑ d , n ≥ 1 n 2 k − 1 q n d .

As theta functions

Given q = e^2πiτ, let

E 4 ( τ ) = 1 + 240 ∑ n = 1 ∞ n 3 q n 1 − q n E 6 ( τ ) = 1 − 504 ∑ n = 1 ∞ n 5 q n 1 − q n E 8 ( τ ) = 1 + 480 ∑ n = 1 ∞ n 7 q n 1 − q n

and define

a = θ 2 ( 0 ; e π i τ ) = ϑ 10 ( 0 ; τ ) b = θ 3 ( 0 ; e π i τ ) = ϑ 00 ( 0 ; τ ) c = θ 4 ( 0 ; e π i τ ) = ϑ 01 ( 0 ; τ )

where θ_m and ϑ_ij are alternative notations for the Jacobi theta functions. Then,

E 4 ( τ ) = 1 2 ( a 8 + b 8 + c 8 ) , E 6 ( τ ) = 1 2 ( − 3 a 8 ( b 4 + c 4 ) + b 12 + c 12 ) = 1 2 ( a 8 + b 8 + c 8 ) 3 − 54 ( a b c ) 8 2 ,

thus,

E 4 3 − E 6 2 = 27 4 ( a b c ) 8

an expression related to the modular discriminant,

Δ = g 2 3 − 27 g 3 2 = ( 2 π ) 12 ( 1 2 a b c ) 8

Also, since E₈ = E2
4 and a⁴ − b⁴ + c⁴ = 0, this implies

E 8 ( τ ) = 1 2 ( a 16 + b 16 + c 16 ) .

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:

E 4 2 = E 8 , E 4 E 6 = E 10 , E 4 E 10 = E 14 , E 6 E 8 = E 14 .

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

( 1 + 240 ∑ n = 1 ∞ σ 3 ( n ) q n ) 2 = 1 + 480 ∑ n = 1 ∞ σ 7 ( n ) q n ,

hence

σ 7 ( n ) = σ 3 ( n ) + 120 ∑ m = 1 n − 1 σ 3 ( m ) σ 3 ( n − m ) ,

and similarly for the others. Perhaps, even more interestingly, 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:

θ Γ ( τ ) = 1 + ∑ n = 1 ∞ r Γ ( 2 n ) q n = E 4 ( τ ) , r Γ ( n ) = 240 σ 3 ( n )

for the number r_Γ(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 {{mvar|n}' as a sum of two, four, or eight squares in terms of the divisors of n.

Using the above recurrence relation, all higher E_2k can be expressed as polynomials in E₄ and E₆. For example:

E 8 = E 4 2 E 10 = E 4 ⋅ E 6 691 ⋅ E 12 = 441 ⋅ E 4 3 + 250 ⋅ E 6 2 E 14 = E 4 2 ⋅ E 6 3617 ⋅ E 16 = 1617 ⋅ E 4 4 + 2000 ⋅ E 4 ⋅ E 6 2 43867 ⋅ E 18 = 38367 ⋅ E 4 3 ⋅ E 6 + 5500 ⋅ E 6 3 174611 ⋅ E 20 = 53361 ⋅ E 4 5 + 121250 ⋅ E 4 2 ⋅ E 6 2 77683 ⋅ E 22 = 57183 ⋅ E 4 4 ⋅ E 6 + 20500 ⋅ E 4 ⋅ E 6 3 236364091 ⋅ E 24 = 49679091 ⋅ E 4 6 + 176400000 ⋅ E 4 3 ⋅ E 6 2 + 10285000 ⋅ E 6 4

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

Δ 2 = − 691 1728 2 ⋅ 250 det | E 4 E 6 E 8 E 6 E 8 E 10 E 8 E 10 E 12 |

where

Δ = E 4 3 − E 6 2 1728

is the modular discriminant.

Ramanujan identities

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

L ( q ) = 1 − 24 ∑ n = 1 ∞ n q n 1 − q n = E 2 ( τ ) M ( q ) = 1 + 240 ∑ n = 1 ∞ n 3 q n 1 − q n = E 4 ( τ ) N ( q ) = 1 − 504 ∑ n = 1 ∞ n 5 q n 1 − q n = E 6 ( τ ) ,

then

q d L d q = L 2 − M 12 q d M d q = L M − N 3 q d N d q = L N − M 2 2 .

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 σ_p(n) to include zero, by setting

σ p ( 0 ) = 1 2 ζ ( − p ) ⟹ σ ( 0 ) = − 1 24 σ 3 ( 0 ) = 1 240 σ 5 ( 0 ) = − 1 504 .

Then, for example

∑ k = 0 n σ ( k ) σ ( n − k ) = 5 12 σ 3 ( n ) − 1 2 n σ ( n ) .

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

∑ k = 0 n σ 3 ( k ) σ 3 ( n − k ) = 1 120 σ 7 ( n ) ∑ k = 0 n σ ( 2 k + 1 ) σ 3 ( n − k ) = 1 240 σ 5 ( 2 n + 1 ) ∑ k = 0 n σ ( 3 k + 1 ) σ ( 3 n − 3 k + 1 ) = 1 9 σ 3 ( 3 n + 2 ) .

Generalizations

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

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

References

Eisenstein series Wikipedia

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