Quantum Ergodicity of Eisenstein series for Arithmetic 3-Manifolds

Shin-ya Koyama

Abstract. We prove the quantum ergodicity for Eisenstein series for P SL(2, O_{K}), where O_{K} is the integer ring of an imaginary quadratic field K of class number one.

1. Introduction. Luo and Sarnak [LS] proved the quantum ergodicity of Eisenstein series for P SL(2, Z). It is stated as follows:

Theorem 1.1. Let A, B be compact Jordan measurable subsets of P SL(2, Z)\H^{2}, then

lim_{t→∞} \frac{μ_{t}(A)}{μ_{t}(B)} = \frac{Vol(A)}{Vol(B)},

where μ_{t} = |E(z, \frac{1}{2} +it)|^{2}dV with E(z, s) being the Eisenstein series for P SL(2, Z), and dV is the volume element of the upper half plane H^{2}.

In this paper we will generalize Theorem 1.1 to three dimensional cases X = P SL(2, O_{K})\H^{3}, where O_{K} is the integer ring of an imaginary quadratic field K of class number one, and H^{3} is the three dimensional upper half space. Our main theorem is analogously described as follows:

Theorem 1.2. Let A, B be compact Jordan measurable subsets of X, then

lim_{t→∞} \frac{μ_{t}(A)}{μ_{t}(B)} = \frac{Vol(A)}{Vol(B)},

Partially supported by Japan Association for Mathematical Sciences Email: koyama@math.keio.ac.jp, FAX: +81-45-566-1644

where μ_{t} = |E(v, 1 + it)|^{2}dV with E(v, s) being the Eisenstein series for X, and dV is the volume element of H^{3}.

Indeed we show that as t → ∞,

μ_{t}(A) ∼ \frac{2Vol(A)}{ζ_{K}(2)} log t,

where ζ_{K}(s) is the Dedekind zeta function.

In two dimensional cases numerical examples [HR] suggested that the quantum ergodicity would hold. For higher dimensional cases no numerical examples are known. Theorem 1.2 is the first result along this direction.

The author would like to express his thanks to Professor Peter Sarnak, who introduced the author to the subject.

2. Three-Dimensional Settings. In this section we introduce some notation on the three-dimensional hyperbolic space.

A point in the hyperbolic three-dimensional space H^{3} is denoted by v = z + yj, z = x_{1} +x_{2}i ∈ C, y > 0. We fix an imaginary quadratic field K whose class number is one. Denote its discriminant by D_{K} and integer ring O = O_{K}. Put D = |D_{K}|. We often regard O as a lattice in R^{2}, which is denoted by L with the fundamental domain F_{L} ⊂ R^{2}. Also put ω = ω_{K} = D^{−1/2}, the inverse different of K. The group Γ = P SL(2, O) acts on H^{3} and the quotient space X = Γ\H^{3} is a three dimensional arithmetic hyperbolic orbifold. The Laplacian on X is defined by

Δ = −y^{2} (\frac{d^{2}}{dx_{1}^{2}} + \frac{d^{2}}{dx_{2}^{2}} + \frac{d^{2}}{dy^{2}} ) + y \frac{d}{dy} .

It has a self-adjoint extension on L^{2}(X). It is known that the spectra of Δ is composed of both discrete and continuous ones. The eigenfunction for a discrete spectrum is called a cusp form. We denote it by ϕ_{j}(v) with eigenvalue λ_{j} (0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ ⋯). We put λ_{j} = 1 + r_{j}^{2}. We shall assume the ϕ_{j}(v)’s to be chosen so that they are eigenfunctions of the ring of Hecke operators and are L^{2}-normalized. The Fourier development of ϕ_{j}(v) is given in [S] (2.20):

ϕ_{j}(v) = ∑_{n∈O^{∗}/∼} ρ_{j}(n)yK_{ir_{j}}(2π|n|y)e(⟨n, z⟩),  (2.1)

where n ∼ m means that they generate the same ideal in O, and ⟨n, z⟩ is the standard inner product in R^{2} with K_{ν} being the K-Bessel function.

For a Maass-Hecke cusp form ϕ_{j}(v) with its Fourier development given by (2.1), we have the Rankin-Selberg convolution L-function L(s, ϕ_{j} × ϕ_{j}) and the second symmetric power L-function L^{(2)}(s, ϕ_{j}) which satisfy the following:

L(s, ϕ_{j} × ϕ_{j}) = ζ_{K}(2s) ∑_{n∈O^{∗}/∼} \frac{|λ_{j}(n)|^{2}}{N(n)^{s}}

L^{(2)}(s, ϕ_{j}) = ∑_{n∈O^{∗}/∼} \frac{c_{j}(n)}{N(n)^{s}} = ζ_{K}(s)^{−1}L(s, ϕ_{j} × ϕ_{j}),

with ρ_{j}(n) = \sqrt{\frac{sinh πr_{j}}{r_{j}}} v_{j}(n), v_{j}(n) = v_{j}(1)λ_{j}(n) and c_{j}(n) = ∑_{l^{2}k=n} λ_{j}(k^{2}). It is known that the both functions converge in Re(s) > 1. The functional equation of L(s, ϕ_{j} × ϕ_{j}) is inherited from Eisenstein series by our unfolding the integral. We compute that

∫_{X} |ϕ_{j}(v)|^{2}E(v, 2s)dv = |ρ_{j}(1)|^{2} \frac{L(s, ϕ_{j} × ϕ_{j})}{ζ_{K}(2s)} \frac{Γ(s + ir_{j})Γ(s − ir_{j})Γ(s)^{2}}{8π^{2s}Γ(2s)}

is invariant under changing the variable s to 1−s. We normalize such that ∥ϕ_{j}∥ = 1 with respect to the Petersson inner product

⟨f, g⟩ = \frac{1}{vol(X)} ∫_{X} f(v)\overline{g(v)}dv.

The residue R_{j} of L(s, ϕ_{j} × ϕ_{j}) at its unique simple pole s = 1 is equal to

\frac{8πζ_{K}(2)}{|v_{j}(1)|^{2}}Res_{s=2}E(v, s) = \frac{8πζ_{K}(2)Vol(F_{L})}{|v_{j}(1)|^{2}Vol(X)},  (2.4)

where Res_{s=2}E(v, s) = Vol(F_{L})/Vol(X) is known by Sarnak [S] Lemma 2.15.

3. Proofs. In this section we prove Theorem 1.2. We first define the Eisenstein series by

E(v, s) = ∑_{Γ_{∞}\Γ} y(γv)^{s},  (3.1)

where y(v) = y for v = z + jy ∈ H^{3} and Re(s) > 2. Here the group Γ_{∞} is given by

Γ_{∞} = \{( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) : n ∈ O\} .

The Fourier development of E(v, s) is known by Asai [A] and Elstrodt et al. [E]:

E(v, s) = y^{s} + y^{2−s} \frac{ξ_{K}(s − 1)}{ξ_{K}(s)}

+ \frac{2}{ξ_{K}(s)} ∑_{n∈O^{∗}/∼} |n|^{s−1}σ_{2(1−s)}(n)e^{4πiRe(nωz)}K_{s−1}(4π|nω|y)y, (3.2)

where σ_{s}(n) = ∑_{d|n} |d|^{s} and ξ_{K}(s) = (\frac{\sqrt{D}}{2π})^{s}Γ(s)ζ_{K}(s).

Our goal is to prove the equidistribution of the measure μ_{t} = |E(v, 1+it)|^{2}dV (v), where dV (v) = \frac{dx_{1}dx_{2}dy}{y^{3}} . We consider its inner product with various functions spanning L^{2}(X). We begin with inner products with Maass cusp forms ϕ_{j}. Proposition 3.1. For any fixed ϕ_{j},

lim_{t→∞} ∫_{X} ϕ_{j}dμ_{t} = 0

Proof. Set

J_{j}(t) = ∫_{X} ϕ_{j}dμ_{t} = ∫_{X} ϕ_{j}(v)E(v, 1 + it)E(v, 1 − it)\frac{dx_{1}dx_{2}dy}{y^{3}}  (3.3)

with z = x_{1} + x_{2}i. To investigate this we first consider

I_{j}(s) = ∫_{X} ϕ_{j}(v)E(v, 1 + it)E(v, s)\frac{dx_{1}dx_{2}dy}{y^{3}} .  (3.4)

All of the above integrals converge since ϕ_{j} is a cusp form. We unfold the integral (3.4) to get

I_{j}(s) = ∫_{0}^{∞} ∫_{F_{L}} ϕ_{j}(v)E(v, 1 + it)y^{s} \frac{dx_{1}dx_{2}dy}{y^{3}} .  (3.5)

Denote the conjugate of v = z +yj ∈ H^{3} by \overline{v} = z −yj. As is well-known in the two dimensional case, the space of the Maass cusp forms is expressed as a direct sum of spaces of even and odd cusp forms. Here even (resp. odd) cusp forms are ones satisfying ϕ_{j}(1 − \overline{v}) = ϵϕ_{j}(v) with ϵ = 1 (resp. −1). Since E(v, s) = E(1 − \overline{v}, s), it follows that I_{j}(s) ≡ 0 if ϕ_{j} odd. So we may assume that ϕ_{j} is even. In this case the Fourier development (2.1) is written as

ϕ_{j}(v) = y ∑_{n∈O^{∗}/∼} ρ_{j}(n)K_{ir_{j}}(2π|n|y) cos(2πi⟨n, z⟩),  (3.6)

where 1 + r_{j}^{2} = λ_{j}. Normalizing the coefficients by ρ_{j}(n) = ρ_{j}(1)λ_{j}(n), the multiplicative relations are satisfied by λ_{j}(n). These amounts to

L(ϕ_{j}, s) := ∑_{n∈O^{∗}/∼} \frac{λ_{j}(n)}{N(n)^{s}} = ∏_{(p):prime ideal} (1 − \frac{λ_{j}(p)}{N(p)^{s}} + \frac{1}{N(p)^{2s}} )^{−1} . (3.7)

By substituting (3.2) and (3.6) into (3.5) we have

I_{j}(s) = ∫_{0}^{∞} ∫_{F_{L}} (y ∑_{n∈O^{∗}/∼} ρ_{j}(n)K_{ir_{j}}(2π|n|y) cos(2π⟨n, z⟩))

(y^{1+it} + y^{1−it} \frac{ξ_{K}(it)}{ξ_{K}(1 + it)}

+\frac{2y}{ξ_{K}(1 + it)} ∑_{m∈O^{∗}/∼} |m|^{it}σ_{−2it}(m)e^{4πiRe(mωz)}K_{it}(4π|m|ωy))

y^{s} \frac{dx_{1}dx_{2}dy}{y^{3}} . (3.8)

Now we have

∫_{F_{L}} cos(2πi⟨nω, z⟩)dv = \{ \begin{matrix}0 & n ∈ O − \{0\} \\ 1 & n = 0 \end{matrix} .

In the expansion of (3.8), we appeal to the formula cos x cos y = \frac{1}{2}(cos(x + y) + cos(x − y)). Only the terms with n = m remain as follows:

I_{j}(s) = \frac{2}{ξ_{K}(1 + it)} ∫_{0}^{∞} ∑_{n∈O^{∗}/∼} |n|^{it}σ_{−2it}(n)K_{it}(2π|n|y)ρ_{j}(n)K_{ir_{j}}(2π|n|y)y^{s} \frac{dy}{y}

= \frac{2}{ξ_{K}(1 + it)} ∑_{n∈O^{∗}/∼} \frac{|n|^{it}σ_{−2it}(n)ρ_{j}(n)}{|n|^{s}} ∫_{0}^{∞} K_{it}(2πy)K_{ir_{j}}(2πy)y^{s} \frac{dy}{y} .

An evaluation of the integral involving Bessel functions [GR] yields

I_{j}(s) = \frac{2π^{−s}}{ξ_{K}(1 + it)} \frac{Γ(\frac{s+ir_{j}+it}{2})Γ(\frac{s+ir_{j}−it}{2})Γ(\frac{s−ir_{j}+it}{2})Γ(\frac{s−ir_{j}−it}{2})}{Γ(s)}R(s)

with

R(s) = ∑_{n∈O^{∗}/∼} \frac{|n|^{it}σ_{−2it}(n)ρ_{j}(n)}{|n|^{s}}.

We compute R(s) as follows:

R(s) = \frac{1}{ρ_{j}(1)} ∏_{(p):prime ideal} ∑_{k=0}^{∞} \frac{λ_{j}(p^{k})|p|^{ikt}σ_{−2it}(p^{k})}{|p|^{ks}}

= \frac{1}{ρ_{j}(1)} ∏_{(p)} ∑_{k=0}^{∞} \frac{λ_{j}(p^{k})|p|^{ikt}}{|p|^{ks}} ∑_{l=0}^{k} |p|^{−2itl}

= \frac{1}{ρ_{j}(1)} ∏_{(p)} ∑_{k=0}^{∞} \frac{λ_{j}(p^{k})|p|^{ikt}}{|p|^{ks}} \frac{1 − |p|^{−2it(k+1)}}{1 − |p|^{−2it}}

= \frac{1}{ρ_{j}(1)(1 − |p|^{−2it})} ∏_{(p)} (∑_{k=0}^{∞} λ_{j}(p^{k})|p|^{−k(s−it)} − |p|^{−2it} ∑_{k=0}^{∞} λ_{j}(p^{k})|p|^{−k(s+it)})

= \frac{1}{ρ_{j}(1)(1 − |p|^{−2it})}

∏_{(p)} (\frac{1}{1 − λ_{j}(p)|p|^{−(s−it)} + |p|^{−2(s−it)}} − \frac{|p|^{−2it}}{1 − λ_{j}(p)|p|^{−(s+it)} + |p|^{−2(s+it)}} )

= \frac{1}{ρ_{j}(1)}

∏_{(p)} \frac{1 − |p|^{−2s}}{(1 − λ_{j}(p)|p|^{−(s−it)} + |p|^{−2(s−it)})(1 − λ_{j}(p)|p|^{−(s+it)} + |p|^{−2(s+it)})}

= \frac{1}{ρ_{j}(1)} \frac{L(ϕ_{j}, \frac{s−it}{2})L(ϕ_{j}, \frac{s+it}{2})}{ζ_{K}(s)}.  (3.9)

Therefore

J_{j}(t) = I_{j}(1 − it)
= \frac{2π^{−1+it}}{ξ_{K}(1 + it)} \frac{Γ(\frac{1+ir_{j}}{2} )Γ(\frac{1+ir_{j}−2it}{2})Γ(\frac{1−ir_{j}}{2} )Γ(\frac{1−ir_{j}−2it}{2})}{Γ(1 − it)}R(1 − it). (3.10)

By Stirling’s formula |Γ(σ + it)| ∼ e^{−πt/2}|t|^{σ− \frac{1}{2}} , we see

the gamma factors in (3.10) ≪ |t|^{−1}  (3.11)

as t → ∞. It is known that the Dedekind zeta function in (3.10) is estimated as

t^{−ϵ} ≪ |ζ_{K}(1 + it)| ≪ t^{ϵ}.  (3.12)

Estimating the automorphic L-functions in (3.10) was recently succeeded by Sarnak and Petridis [SP]. They proved there exists δ > 0 such that for any ϵ > 0,

L(ϕ_{j}, \frac{1}{2} + it) ≪_{j,ϵ} |t|^{1−δ+ϵ}  (3.13)

as |t| → ∞. The estimates (3.11)-(3.13) yields

J_{j}(t) ≪ |t|^{−δ+ϵ}.  (3.14)

This implies Proposition 3.1. □

We now turn to inner products of μ_{t} with incomplete Eisenstein series. Let h(y) be a rapidly decreasing function at 0 and ∞, that is h(y) = O_{N}(y^{N}) as y → ∞ or 0 and N ∈ Z. Let H(s) be its Mellin transform

H(s) = ∫_{0}^{∞} h(y)y^{−s} \frac{dy}{y} .

Clearly H(s) is entire in s and is of Schwartz class in t for each vertical line σ + it. The inversion formula gives

h(y) = \frac{1}{2πi} ∫_{(σ)} H(s)y^{s}ds

for any σ ∈ R. For such an h we form the convergent series

F_{h}(v) = ∑_{γ∈Γ_{∞}\Γ} h(y(γv)) = \frac{1}{2πi} ∫_{(3)} H(s)E(v, s)ds,

which we call incomplete Eisenstein series.

Proposition 3.2. For incomplete Eisenstein series F (v), we have

∫_{X} F (v)dμ_{t}(v) ∼ \frac{2}{ζ_{K}(2)} (∫_{X} F (v)dV (v)) log t

as t → ∞. Proof. Incomplete Eisenstein series decreases rapidly as y → ∞ and belongs to C^{∞}(X). Hence

∫_{X} F_{h}(v)dμ_{t}(v) = ∫_{X} F_{h}(v)|E(v, 1 + it)|^{2} \frac{dzdy}{y^{3}}

= \frac{1}{2πi} ∫_{X} ∫_{(3)} H(s)E(v, s)ds|E(v, 1 + it)|^{2} \frac{dzdy}{y^{3}}

= \frac{1}{2πi} ∫_{0}^{∞} ∫_{(3)} H(s)y^{s}ds ∫_{F_{L}} |E(v, 1 + it)|^{2} \frac{dzdy}{y^{3}}

= \frac{1}{2πi} ∫_{0}^{∞} ∫_{(3)} H(s)y^{s}ds (|y^{1+it} + y^{1−it} \frac{ξ_{K}(it)}{ξ_{K}(1 + it)}|^{2}

+ |\frac{2y}{ξ_{K}(1 + it)}|^{2} ∑_{n∈O^{∗}/∼} |σ_{−2it}(n)K_{it}(4π|n|ωy)|^{2}) \frac{dy}{y^{3}}

= F_{1}(t) + F_{2}(t),

where we put

F_{1}(t) = \frac{1}{2πi} ∫_{0}^{∞} ∫_{(3)} H(s)y^{s}ds |y^{1+it} + y^{1−it} \frac{ξ_{K}(it)}{ξ_{K}(1 + it)}|^{2} \frac{dy}{y^{3}} .

Since |\frac{ξ_{K}(it)}{ξ_{K}(1+it)} | = 1, we have

F_{1}(t) = 2 ∫_{0}^{∞} h(y)\frac{dy}{y} + (a rapidly decreasing function of t).  (3.15)

Whereas

F_{2}(t) = \frac{2}{πi|ξ_{K}(1 + it)|^{2}} ∫_{(3)} H(s) ∑_{n∈O^{∗}/∼} \frac{|σ_{−2it}(n)|^{2}}{|n|^{s}} ∫_{0}^{∞} |K_{it}(4πωy)|^{2}y^{s} \frac{dy}{y} ds.

(3.16)

The series is computed as follows:

∑_{n∈O^{∗}/∼} \frac{|σ_{a}(n)|^{2}}{|n|^{s}} = ∏_{(p): prime ideal} ∑_{k=0}^{∞} \frac{σ_{a}(p^{k})σ_{−a}(p^{k})}{|p|^{ks}}

= ∏_{(p)} ∑_{k=0}^{∞} \frac{1}{|p|^{ks}} (\frac{1 − |p|^{a(k+1)}}{1 − |p|^{a}} ) (\frac{1 − |p|^{−a(k+1)}}{1 − |p|^{−a}} )^{2}

= ∏_{(p)} \frac{1}{(1 − |p|^{a})(1 − |p|^{−a})}

∑_{k=0}^{∞} (2|p|^{−ks} − |p|^{(a−s)k+a} + |p|^{(−a−s)k−a})

= ∏_{(p)} \frac{1}{(1 − |p|^{a})(1 − |p|^{−a})}

(\frac{2}{1 − |p|^{−s}} − \frac{|p|^{a}}{1 − |p|^{a−s}} − \frac{|p|^{−a}}{1 − |p|^{−a−s}} )

= ∏_{(p)} \frac{1 + p^{−s}}{(1 − p^{−s})(1 − p^{−(s−a)})(1 − p^{−(s+a)})}

= \frac{ζ_{K}(\frac{s}{2})^{2}ζ_{K}(\frac{s−a}{2})ζ_{K}(\frac{s+a}{2})}{ζ_{K}(s)}.  (3.17)

The y-integral in (3.16) is evaluated in terms of Γ function as before. We obtain

F_{2}(t) = \frac{2}{πi|ξ_{K}(1 + it)|^{2}} ∫_{(3)} H(s) ∑_{n∈O^{∗}/∼} \frac{|σ_{−2it}(n)|^{2}}{|n|^{s}} ∫_{0}^{∞} |K_{it}(4πωy)|^{2}y^{s} \frac{dy}{y} ds

= \frac{2}{πi|ξ_{K}(1 + it)|^{2}} ∫_{(3)} \frac{H(s)ζ_{K}(\frac{s}{2})^{2}|ζ_{K}(\frac{s}{2} + it)Γ(\frac{s}{2} + it)|^{2}Γ(\frac{s}{2})^{2}}{(4πω)^{s}ζ_{K}(s)Γ(s)} ds

= \frac{2}{πi|ξ_{K}(1 + it)|^{2}} ∫_{(3)} B(s)ds,  (3.18)

where we put

B(s) = \frac{H(s)ζ_{K}(\frac{s}{2})^{2}|ζ_{K}(\frac{s}{2} + it)Γ(\frac{s}{2} + it)|^{2}Γ(\frac{s}{2})^{2}}{(4πω)^{s}ζ_{K}(s)Γ(s)} .  (3.19)

By Stirling’s formula to estimate the gamma factors and from the fact that H(σ+it) is rapidly decreasing in t, we can shift the integral in (3.18) to Re(s) = 1:

F_{2}(t) = \frac{4Res_{s=2}B(s)}{|ξ_{K}(1 + it)|^{2}} + \frac{2}{πi|ξ_{K}(1 + it)|^{2}} ∫_{(1)} B(s)ds.  (3.20)

The second term in (3.20) is evaluated by Heath-Brown [H] as

ζ_{K}(\frac{1}{2} + it) ≪ t^{\frac{1}{3} +ϵ}

for any fixed ϵ > 0. We find that

\frac{2}{πi|ξ_{K}(1 + it)|^{2}} ∫_{(1)} B(s)ds ≪_{ϵ} t^{− \frac{1}{3} +ϵ}.

This corresponds to the bound (3.14).

Next we deal with the residue term in (3.20), which is more complicated. Write B(s) as ζ_{K}(\frac{s}{2})^{2}G(s) where G(s) is holomorphic at s = 2. Put

ζ_{K}(s/2) = \frac{A_{−1}}{s − 2} + A_{0} + O(s − 2) (s → 2).

In the expansion of

B(s) = (\frac{A_{−1}}{s − 2} + A_{0} + O(s − 2))^{2} (G(2) + G′(2)(s − 2) + O(s − 2)^{3}) ,

the coefficient of (s − 2)^{−1} gives the residue

Res_{s=2}B(s) = G(2)A_{−1} (2A_{0} + A_{−1} \frac{G′}{G} (2)) .

A simple calculation gives

G(2) = \frac{H(2)|ζ_{K}(1 + it)Γ(1 + it)|^{2}Γ(\frac{1}{2})^{2}}{(4πω)^{2}ζ_{K}(2)} = \frac{H(2)|ξ_{K}(1 + it)|^{2}}{4ζ_{K}(2)}

and

\frac{G′}{G} (2) = \frac{H′}{H} (2) + \frac{ζ′_{K}(1 + it)}{2ζ_{K}(1 + it)} + \frac{ζ′_{K}(1 − it)}{2ζ_{K}(1 − it)} + \frac{Γ′(1 + it)}{2Γ(1 + it)} + \frac{Γ′(1 − it)}{2Γ(1 − it)} + C

with C being independent of t. The Weyl-Hadamard-De La Vallée Poussin bound [T, (6.15.3)] and its generalization to Dirichlet L-functions by Landau, we have

\frac{ζ′_{K}(1 + it)}{ζ_{K}(1 + it)} ≪ \frac{log t}{log log t}.

This together with \frac{Γ′}{Γ} (1 + it) ∼ log t gives

Res_{s=2}B(s) = \frac{H(2)|ξ_{K}(1 + it)|^{2}}{2ζ_{K}(2)} log t + O (\frac{log t}{log log t}) .

Finally the first term of (3.20) is evaluated as

\frac{4Res_{s=2}B(s)}{|ξ_{K}(1 + it)|^{2}} = \frac{2H(2)}{ζ_{K}(2)} log t + O(1).

Taking into account that

H(2) = ∫_{0}^{∞} h(y)\frac{dy}{y^{3}} = ∫_{X} F_{h}(z)\frac{dzdy}{y^{3}}

we reach the conclusion. □ Proposition 3.3. Let F be a continuous function of compact support in X. Then

∫_{X} F (v)dμ_{t}(v) ∼ \frac{2}{ζ_{K}(2)} (∫_{X} F (v)dV (v)) log t

as t → ∞. Proof. The space of all incomplete Eisenstein series and cusp forms is dense in the space of continuous functions vanishing in the cusp. For any ϵ > 0, we can find G = G_{1} + G_{2} with G_{1} the finite sum of cusp forms and G_{2} in the space of incomplete Eisenstein series, such that ∥G − F ∥_{∞} < ϵ. The difference H = G − F is sufficiently small and rapidly decreasing in the cusp. Namely, it is majorized in terms of another incomplete Eisenstein series

H_{1}(v) = ∑_{γ∈Γ_{∞}\Γ} h_{1}(y(γv))

as

H_{1}(v) ≥ |H(v)|

satisfying

∫_{X} H_{1}(v)dV (v) < C(K)ϵ

with some constant C(K) depending only on the field K. Hence the conclusion. □ Propositions 2.3 implies Theorem 1.1 by standard approximation arguments.

References

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[HR] D. Hejhal and B. Rackner, On the topography of Maass waveforms for P SL(2, Z), Experimental Math. 1 (1992), 275-305.

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Department of Mathematics, Keio University, 3 Hiyoshi, Kohoku, Yokohama 223-8522, Japan