Nonholomorphic Eisenstein series

Eisenstein Series

4.10 Nonholomorphic Eisenstein series

4.9.4.Formula (1.2) is a special case of the Lipschitz formulaforτ∈ H and q=e^2πiτ,

d∈Z

1

(τ+d)^s = (−2πi)^s Γ(s)

∞ m=1

m^s−1q^m, Re(s)>1.

Prove the Lipschitz formula by applying Poisson summation to the function f ∈ L¹(R) given by

f(x) =

x^s−1e^2πixτ ifx >0,

0 ifx≤0.

(A hint for this exercise is at the end of the book.)

4.9.5.Ifg is a holomorphic function of the complex variablesin some right half plane then itsinverse Mellin transformis

f(t) = 1 2πi

σ+i∞ s=σ−i∞

g(s)t^−sds

for positivet-values such that the integral converges absolutely. Complex con- tour integration shows that this is independent ofσ.

(a) Let t=e^x,s=σ+ 2πiy,f(t) = e^−σxf_σ(x), andg(s) = g_σ(y). Show that

g(s) = _∞

t=0

f(t)t^sdt

t ⇐⇒ gσ(y) = _∞

x=−∞

fσ(x)e^2πixydx and that

f(t) = 1 2πi

σ+i∞ s=σ−i∞

g(s)t^−sds ⇐⇒ fσ(x) = _∞

y=−∞

gσ(y)e^−2πiyxdy.

The right side conditions are equivalent by Fourier inversion, and so the left side conditions are equivalent as well. This is theMellin inversion formula.

(b) Evaluate the integral σ+i∞

s=σ−i∞Γ(s)x^−sds for any σ > 0. (A hint for this exercise is at the end of the book.)

4.9.6.Verify the calculations leading up to formula (4.38).

E_k^v(τ, s) =_N

(c,d)≡v(N) gcd(c,d)=1

y^s

(cτ +d)^k|cτ+d|^2s, τ =x+iy∈ H.

This converges absolutely on the right half plane {s : Re(k+ 2s) > 2}. The convergence is uniform on compact subsets, making E^v_k(τ, s) an ana- lytic function ofson this half plane. To rewrite the series in intrinsic form, letδ=_{a b}

c_v d_v

∈SL2(Z) with (cv, dv) a lift ofv toZ², and recall the positive part of the parabolic subgroup of SL2(Z),P+ ={[¹_{0 1}ⁿ] :n∈Z}. Extend the weight-koperator to functions ofτ andsby the definition

(f[γ]_k)(τ, s) =j(γ, τ)^−kf(γ(τ), s), γ∈SL₂(Z).

Then

E^v_k(τ, s) =N

γ∈(P+∩Γ(N))\Γ(N)δ

Im(τ)^s[γ]k.

This formula makes it clear that (E^v_k[γ]k)(τ, s) =E_k^vγ(τ, s) for allγ∈SL2(Z) as in Proposition 4.2.1, so that in particular

E^v_k[γ]k =E_k^v for allγ∈Γ(N).

As earlier, the corresponding nonnormalized series G^v_k(τ, s) =

(c,d)≡v(N)

y^s

(cτ+d)^k|cτ+d|^2s (4.39) will be easier to analyze. Relations like (4.5) and (4.6) expressing eachG^v_k as a linear combination of theE_k^v and vice versa still hold (Exercise 4.10.1(a)), soG^v_k transforms under SL₂(Z) in the same way as E_k^v (Exercise 4.10.1(b)).

The discussion leading to relations (4.20) and (4.21) generalizes to show that meromorphically continuing either sort of Eisenstein series meromorphically continues the other (Exercise 4.10.1(c)). This section will show that G^v_k as a function of s has a meromorphic continuation to the entire s-plane. For s-values to the left of the original half plane of convergence Re(k+ 2s)>2, the continuation is no longer defined by the sum, but the transformation law continues to hold by the Uniqueness Theorem from complex analysis. The natural Eisenstein series to consider is

G^v_k(τ,0) =G^v_k(τ, s)|s=0.

Fork ≥3 the parameter s simply appears and disappears, but for k≤ 2 it gives something new. Hecke’s method for weights 1 and 2, obtaining the func- tionsg₁^v(τ) andg^v₂(τ) from before, is presented in [Sch74], and a generalization of Hecke’s method due to Shimura is presented in [Miy89] and [Hid93]. The method here uses a Mellin transform and Poisson summation, extending the ideas of the previous section. See for example Rankin [Ran39], Selberg [Sel40], and Godement [God66].

Working in dimensionl= 2, view elements ofR²as row vectors and define a modified theta function whose argument is now a matrix,

ϑ(γ) =

n∈Z²

e^−π|nγ|2, γ∈GL2(R).

Again the second variable from the general notation for ϑis omitted since l is fixed at 2. When γ = It^1/2 where I is the identity matrix and t > 0, this is ϑ(it) from the previous section. To see how a more general matrix γ affects the transformation law for ϑ, compute that for any function f ∈ L¹(R²), any matrix γ ∈ SL₂(R), and any positive number r, the Fourier transform of the function ϕ(x) = f(xγr) is ˆϕ(x) = r⁻²fˆ(xγ^−T/r) where γ^−T denotes the inverse-transpose ofγ(Exercise 4.10.2). In particular, again letting f(x) = e^−π|x|2 be the Gaussian, Poisson summation shows that for γ∈SL₂(R),r

f(nγr) =r⁻¹

f(nγ^−T/r). LetS=₀₋₁

1 0

. ThenSγ^−T = γS for all γ ∈ SL2(R) and |xS| = |x| for all x ∈ R², so f(nSγ^−T/r) = f(nγ/r). Asnvaries throughZ²so doesnS, and thus the Poisson summation isr

f(nγr) =r⁻¹

f(nγ/r). That is,

rϑ(γr) = (1/r)ϑ(γ/r), γ∈SL₂(R), r >0. (4.40) For anyγ∈SL₂(R), the Mellin transform of

n∈Z²

e^−π|nγ|2t=ϑ(γt^1/2)− 1 fort >0 is

g(s, γ) = _∞

t=0

n∈Z²

e^−π|nγ|2tt^sdt t =

_∞

t=0

(ϑ(γt^1/2)−1)t^sdt

t . (4.41) Again ϑ(γt^1/2) converges to 1 as t → ∞, so the transformation law (4.40) shows that ast→0,ϑ(γt^1/2) grows as 1/t, and therefore the integral converges at its left endpoint if Re(s)>1. Also as before, the integral converges at its right end for all values of s, and rapid convergence of the sum lets it pass through the integral to yield after a change of variable,

g(s, γ) =

n∈Z²

(π|nγ|²)^−s _∞

t=0

e^−tt^sdt

t =π^−sΓ(s)

n∈Z²

|nγ|^−2s

for Re(s)>1. To connect this with Eisenstein series, for any pointτ =x+iy∈ Hletγτ be the corresponding matrix

γτ = 1

√y y x

0 1

∈SL2(R).

(This matrix, which transforms i to τ, appeared in Chapter 2, in Exer- cise 2.1.3(c) and again in the proof of Corollary 2.3.4.) For n= (c, d) ∈ Z² compute that|nγ_τ|²=|(cy, cx+d)/√y|²=|cτ+d|²/y. Therefore the Mellin transform is essentially the Eisenstein series,

g(s, γ_τ) =π^−sΓ(s)

(c,d) y^s

|cτ+d|^2s =π^−sΓ(s)G₀(τ, s), Re(s)>1, whereG0(τ, s) is (4.39) with weightk= 0 and levelN = 1. (In this case there is only onev∈(Z/NZ)², so it is suppressed from the notationG^v₀.)

The second integral in (4.41) provides the meromorphic continuation and functional equation ofG₀. As in the previous section, the transformation law forϑshows that part of the integral is

1 t=0

(ϑ(γt^1/2)−1)t^sdt t =

1 t=0

ϑ(γt^1/2)t^sdt t −1

s

= _∞

t=1

ϑ(γt^−1/2)t^−sdt t −1

s

= _∞

t=1

ϑ(γt^1/2)t^1−sdt t −1

s

= _∞

t=1

(ϑ(γt^1/2)−1)t^1−sdt t −1

s − 1 1−s. Combining this with the remainder of the integral gives

g(s, γ) = _∞

t=1

(ϑ(γt^1/2)−1)(t^s+t^1−s)dt t −1

s − 1

1−s, Re(s)>1.

This integral is entire ins, making the right side holomorphic everywhere in thes-plane except for simple poles ats= 0 ands= 1. And the right side is invariant unders→1−s. Specializing toγ=γ_τ, the functionπ^−sΓ(s)G₀(τ, s) for Re(s) > 1 has a meromorphic continuation to the full s-plane that is invariant unders→1−s.

Suitably modified, this argument extends to higher weights and levels. Let N≥1 and letGbe the group (Z/NZ)². Rather than taking one vectorv∈G, consider any function

a:G−→C.

Of course, the functionacould simply pick off a vector, but thinking about the entire group symmetrizes the result nicely. LetµN =e^2πi/N, letS =₀₋₁

1 0

, and let,be the usual inner product onR². TheFourier transform ofais a function ˆa:G−→C,

ˆ

a(v) = 1 N

w∈G

a(w)µ⁻_N ^w,vS, v∈G.

The resulting Fourier series reproduces the original function (Exercise 4.10.3) a(u) = 1

N

v∈G

ˆ

a(v)µ_N^u,vS, u∈G,

and ˆˆa=a(Exercise 4.10.3 again).

Letkbe a positive integer and leth_k be the harmonic polynomial h_k(c, d) = (−i)^k(c+id)^k, (c, d)∈R².

Associate a theta function to each vectorv∈(Z/NZ)², ϑ^v_k(γ) =

n∈Z²

hk((v/N+n)γ)e^{−π|(v/N+n)γ|2}, γ∈GL2(R). (4.42)

Again letting f(x) = e^−π|x|2, define the Schwartz function f_k(x) to be a modified Gaussian,

f_k(x) =h_k(x)f(x).

Thusϑ^v_k(γ) =

n∈Z²fk((v/N +n)γ). For γ ∈SL2(R) andr > 0, compute that rϑ^v_k(γr) = r

fk((v/N +n)γr) = r

ϕk(v/N +n) where ϕk(x) = fk(xγr). The Schwartz function has Fourier transform ˆfk = (−i)^kfk (Exer- cise 4.10.4(a)), and so ˆϕk(x) = (−i)^kr⁻²fk(xγ^−Tr⁻¹). Poisson summation overnS and the relationsSγ^−T =γS andS^T =−S give

r ϑ^v_k(γr) = (−i)^kr⁻¹

n∈Z²

fk(nSγ^−Tr⁻¹)e^2πinS,v/N

= (−i)^kr⁻¹

n∈Z²

f_k(nγSr⁻¹)e^{2πin,−vS/N}.

Letting z(x) = c +id ∈ C for any x = (c, d) ∈ R² makes h_k(x) = (−i)^k(z(x))^k= (z(xS))^k and thush_k(xS) = (−i)^kh_k(x) (Exercise 4.10.4(b)).

Consequentlyf_k(xS) = (−i)^kf_k(x) as well, and now r ϑ^v_k(γr) = (−1)^kr⁻¹

n∈Z²

f_k(nγr⁻¹)µ⁻_N^n,vS

= (−1)^kr⁻¹

w∈G

n∈Z² n≡w(N)

fk(nγr⁻¹)µ⁻_N^w,vS

= (−1)^kr⁻¹

w∈G

n∈Z²

fk((w/N+n)γN r⁻¹)µ⁻_N^w,vS

= (−1)^kr⁻¹

w∈G

ϑ^w_k(γN r⁻¹)µ⁻_N ^w,vS.

Thinking here of ϑ^v as a function of v, the sum is N times this function’s Fourier transform, giving r ϑ^v_k(γr) = (−1)^kN r⁻¹ϑ^v_k(γN r⁻¹). Since r > 0 is arbitrary, replace it byN^1/2rto get

r ϑ^v_k(γN^1/2r) = (−1)^kr⁻¹ϑ^v_k(γN^1/2r⁻¹), γ∈SL2(R), r >0. (4.43) Thus (Exercise 4.10.5)

Proposition 4.10.1.For any function a : G−→ C, the associated sum of theta functions

Θ^a_k(γ) =

v∈G

(a(v) + (−1)^kˆa(−v))ϑ^v_k(γN^1/2), γ∈GL₂(R) satisfies the transformation law

r Θ^a_k(γr) =r⁻¹Θ_k^a(γr⁻¹), γ∈SL₂(R), r >0.

The functional equation for the associated sum of Eisenstein series fol- lows as before. The constant terms of the seriesϑ^v_k(γ) are 0, so thatΘ^a_k(γr) converges to 0 rapidly asr→ ∞. Consider the Mellin transform

g^a_k(s, γ) = _∞

t=0

Θ^a_k(γt^1/2)t^sdt

t . (4.44)

The transformation law shows that the integral converges at t = 0 for all values ofs, and as before it converges at its right end for allsas well. Passing the sums through the integral, noting thathk(xr) =hk(x)r^k for r∈R, and changing variable gives

g^a_k(s, γ) =π^−k/2−sΓ(k/2 +s)N^s

·

v∈G

(a(v) + (−1)^kˆa(−v))

n≡v(N)

h_k(nγ)|nγ|^−k−2s.

Setting γ = γτ gives hk(nγ) = (z(nγS))^k = (cτ+d)^k/y^k/2 and gives

|nγ|^−k−2s=y^k/2+s/|cτ+d|^k+2s, with producty^s/((cτ+d)^k|cτ+d|^2(s−k/2)).

Thus for Re(s)>1,

g^a_k(s, γτ) =π^−k/2−sΓ(k/2 +s)N^sy^k/2G^a_k(τ, s−k/2) (4.45) whereG^a_k is the associated sum of Eisenstein series (4.39),

G^a_k(τ, s) =

v

(a(v) + (−1)^ka(ˆ −v))G^v_k(τ, s), Re(k/2 +s)>1. (4.46) Proposition 4.10.1 shows that part of the integral (4.44) is

1 t=0

Θ^a_k(γt^1/2)t^sdt t =

_∞

t=1

Θ^a_k(γt^−1/2)t^−sdt t =

_∞

t=1

Θ_k^a(γt^1/2)t^1−sdt t . Combining this with the remainder of the integral gives

g_k^v(s, γ) = _∞

t=1

Θ_k^a(γt^1/2)(t^s+t^1−s)dt t .

The integral is entire insand invariant unders→1−s. Similar arguments fork= 0 andk <0 (Exercise 4.10.6) show that in all cases,

Theorem 4.10.2.LetN be a positive integer and let G= (Z/NZ)². For any function a:G−→C, let G^a_k(τ, s)be the associated sum (4.46) of Eisenstein series. Then for any integerk and any pointτ =x+iy ∈ H, the function

(π/N)^−sΓ(|k|/2 +s)G^a_k(τ, s−k/2), Re(s)>1

has a continuation to the fulls-plane that is invariant under s→1−s. The continuation is analytic fork= 0 and has simple poles ats= 0,1 fork= 0.

The function in the theorem comes from canceling thes-independent terms in the Mellin transform (4.45). Exercise 4.10.8(b–c) shows that sums of only two Eisenstein series satisfy functional equations for the larger groupΓ₁(N).

The meromorphically continued Eisenstein series of this section have sev- eral applications. As already mentioned, continuing the series for weightsk= 1 andk= 2 tos= 0 recovers the Eisenstein series from earlier in the chapter.

Also, the Rankin–Selberg method integrates an Eisenstein series against the square of the absolute value of a cusp form, obtaining anL-function satisfying a functional equation. This idea was originally used to estimate the Fourier coefficients of the cusp form, but L-functions of the sort obtained by this method are now understood to be important in their own right. For more on this topic, see [Bum97].

A more subtle use of the meromorphically continued Eisenstein series is in the study of square-integrable functions called automorphic formson the quotient space SL₂(Z)\SL₂(R), the arithmetic quotient. The space of such functions decomposes as a discrete part and a continuous part, the former like Fourier series and the latter like Fourier transforms. The discrete part contains cusp forms and a little more, theresidual spectrum, so called because it consists of residues of Eisenstein series in the half plane to the right of the line of symmetry for the functional equation. The continuous part consists of integrals of Eisenstein series on the critical line against square-integrable functions on that line. Thus the explicit spectral decomposition cannot be stated at all without knowing the meromorphic continuation of Eisenstein se- ries. These ideas extend in interesting ways to matrix groups beyond SL2(R), where their original proofs often proceeded by establishing the meromorphic continuation en route.

Exercises

4.10.1.(a) Find relations like (4.5) and (4.6) expressing eachG^v_k as a linear combination of theE^v_k and vice versa.

(b) Verify thatG^v_k transforms under SL₂(Z) the same way asE_k^v.

(c) Show that meromorphic continuation of either sort of Eisenstein series gives meromorphic continuation of the other.

4.10.2.Let f(x) be a function in L¹(R²). For any γ ∈ SL₂(R), and r >

0, show that the function ϕ(x) = f(xγr) has Fourier transform ˆϕ(x) =

r⁻²fˆ(xγ^−T/r) whereγ^−T denotes the inverse-transpose ofγ. (A hint for this exercise is at the end of the book.)

4.10.3.Show that a(u) = 1 N

v∈G

ˆ

a(v)µ_N^u,vS for u∈ G and that ˆˆa= a. (A hint for this exercise is at the end of the book.)

4.10.4.(a) Letf ∈ L¹(R^l), letj∈ {1, . . . , l}, and define g:R^l−→C, g(x) =x_jf(x)

wherexj is thejth component ofx. Suppose that g∈ L¹(R^l). Show that ˆ

g(x) =− 1 2πi

∂

∂x_j fˆ(x).

Applying this identity repeatedly shows that the Schwartz functionf_k(x, y) = (−i)^k(x+iy)^ke^−π(x2+y2)has Fourier transform

fˆ_k(x, y) = (−i)^k

− 1 2πi

k

∂

∂x+i ∂

∂y k

e^−π(x2+y2).

Switching to complex notationfk(z) = (−i)^kz^ke^−πz¯z, this is fˆk(z) = (−i)^k

− 1 πi

k

∂

∂z¯ k

e^−πzz¯. Compute that this relation is ˆfk = (−i)^kfk as desired.

(b) Show thathk(x) = (z(xS))^k andhk(xS) = (−i)^khk(x).

4.10.5.(a) Show that for allγ∈GL2(R),

v∈G

a(v)ϑ^v_k(γ) =

v∈G

ˆ

a(−v)ϑ^v_k(γ) and

v∈G

ˆ

a(−v)ϑ^v_k(γ) =

v∈G

a(v)ϑ^v_k(γ).

(A hint for this exercise is at the end of the book.) (b) Prove Proposition 4.10.1.

4.10.6.(a) Defineh0(x) = 1 for allx∈R². ForN ≥1 andk= 0 define theta series{ϑ^v₀:v∈G}by the formula in the section. Prove Theorem 4.10.2 when k= 0.

(b) When k < 0 define h_k(x) = ¯h_−k(x) for x ∈ R², where the overbar denotes complex conjugation. Prove Theorem 4.10.2 whenk <0. (A hint for this exercise is at the end of the book.)

4.10.7.What does Theorem 4.10.2 say when the function a picks off one vector, i.e.,a=a_v wherea_v(w) is 1 ifw=v and 0 otherwise?

4.10.8.(a) Let ψ and ϕ be primitive Dirichlet characters modulo u and v with (ψϕ)(−1) = (−1)^k anduv=N. Ifa: (Z/NZ)²−→C is the function

a(cv, d+ev) =ψ(c) ¯ϕ(d), a(x, y) = 0 otherwise, show that its Fourier transform is

ˆ

a(−cu,−(d+eu)) = (g( ¯ϕ)/v)g(ψ)ϕ(c) ¯ψ(d), a(x, y) = 0 otherwise.ˆ (Hints for this exercise are at the end of the book.)

(b) Define an Eisenstein series with parameter, G^ψ,ϕ_k (τ, s) =

u−1

c=0 v−1

d=0 u−1

e=0

ψ(c) ¯ϕ(d)G^(cv,d+ev)_k (τ, s).

Show that the sum (4.46) of Eisenstein series for the functionain this problem is

G^a_k(τ, s) =G^ψ,ϕ_k (τ, s) + (−1)^k(g( ¯ϕ)/v)g(ψ)G^ϕ,ψ_k (τ, s).

Thus the functional equations for the eigenspaces Ek(N, χ) involve only two series at a time.

(c) Define a function b : (Z/NZ)² −→ C and a series E_k^ψ,ϕ(τ, s) by the conditions

a= g( ¯ϕ)

v b, G^ψ,ϕ_k = g( ¯ϕ) v E^ψ,ϕ_k .

Show that the sum (4.46) of Eisenstein series forbis more nicely symmetrized than the one fora,

G^b_k(τ, s) =E_k^ψ,ϕ(τ, s) +ϕ(−1)E_k^ϕ,ψ(τ, s).