SPECTRAL DEFORMATIONS AND EISENSTEIN SERIES ASSOCIATED WITH MODULAR SYMBOLS

YIANNIS N. PETRIDIS

Abstract. We give a new proof of the analytic continuation of the Eisenstein series associated with modular symbols. We study its singularities, analytic properties, and functional equation using perturbation theory of the Laplace operator on character varieties.

1. Introduction

Let f(z) be a holomorphic cusp form of weight 2 for the cofinite discrete subgroup Γ of SL_{2}(R). In [5], [6] Goldfeld introduced Eisenstein series associated with modular symbols. It is defined as

(1.1)  E^{∗}(z, s) = ∑_{γ∈Γ_{∞}\Γ}⟨γ, f⟩ℑ(γz)^{s},

where for γ ∈ Γ the modular symbol is given by

(1.2)  ⟨γ, f⟩ = −2πi ∫_{z_{0}}^{γz_{0}} f(τ) dτ.

Here z_{0} is an arbitrary point in ℍ. The aim is to study the distribution of the modular symbols. Goldfeld conjectured in [5] that

(1.3)  ∑_{c^{2}+d^{2}≤X}⟨γ, f⟩ ∼ R(i)X,

where R(z) is the residue at s = 1 of E^{∗}(z, s), and we sum over the elements in Γ with lower row (c, d). In fact, he conjectured corresponding statements for the more general Eisenstein series associated with modular symbols E^{m,n}(z, s), see (1.12). If we take f(z) to be a Hecke eigenform for Γ_{0}(N) and E_{f} is the elliptic curve over Q corresponding to it by the Eichler-Shimura theory, then

⟨γ, f⟩ = n_{1}(f, γ)Ω_{1}(f) + n_{2}(f, γ)Ω_{2}(f),

where n_{i} ∈ Z and Ω_{i} are the periods of E_{f}. The conjecture n_{i} ≪ N^{k} for |c| ≤ N^{2} and some fixed k (Goldfeld’s conjecture) is equivalent to Szpiro’s conjecture D ≪ N^{C} for some C, where D is the discriminant of E_{f}. This has been the motivation to look at the distribution of modular symbols.

In [14] the analytic continuation of the Eisenstein series has been proved and in [15] it is proved that the analytic continuation of them on the line ℜ(s) = 1/2 has poles at s_{j}, where s_{j}(1 − s_{j}) are the eigenvalues of the Laplace operator on Γ \ ℍ. A functional equation was also found. The action of Hecke operators on the Eisenstein series was studied in [4].

1991 Mathematics Subject Classification. Primary 11F67; Secondary 11F72, 11M36.

One of the problems is that the Eisenstein series is not a modular form in the classical setting, i.e., it is not invariant or transform nicely under the action of Γ. In fact, it transforms as

(1.4)  E^{∗}(γz, s) = E^{∗}(z, s) − ⟨γ, f⟩E(z, s),

where E(z, s) is the standard nonholomorphic Eisenstein series for Γ.

We study in this paper a new approach to this Eisenstein series. We consider Eisenstein series with characters depending on a parameter ϵ and we notice that the Eisenstein series with modular symbols is their derivative when ϵ = 0. We define

(1.5)  E_{ϵ}(z, s) = ∑_{γ∈Γ_{∞}\Γ} χ_{ϵ}(γ)ℑ(γz)^{s},

where χ_{ϵ} is a one-parameter family of characters of the group defined by

(1.6)  χ_{ϵ}(γ) = exp (−2πiϵ ∫_{z_{0}}^{γz_{0}} f(τ) dτ) .

This series is defined formally, because the character χ is not unitary. In practice one substitutes with a unitary character, by considering the real and imaginary part of the holomorphic differential f(τ)dτ. In this case convergence is guaranteed for ℜ(s) > 1 by comparison with the standard Eisenstein series. The Eisenstein series with character transform as

(1.7)  E_{ϵ}(γz, s) = χ̄_{ϵ}(γ)E_{ϵ}(z, s)

They satisfy functional equations

(1.8)  E_{ϵ}(z, s) = ϕ_{ϵ}(s)E_{ϵ}(z, 1 − s).

In the domain of absolute convergence we see that

(1.9)  \frac{d}{dϵ}_{|ϵ=0}E_{ϵ}(z, s) = E^{∗}(z, s),

by termwise differentiation. Also we differentiate (1.8) to get the functional equation

(1.10)  E^{∗}(z, s) = \frac{dϕ(s)}{dϵ}_{|ϵ=0}E(z, 1 − s) + ϕ(s)E^{∗}(z, 1 − s).

Here ϕ(s) is the (standard) scattering function for ϵ = 0.

Our first theorem describes the analytic proporties of E^{∗}(z, s). It gives a new proof of the main result in [14] and another result in [15]. Theorem 1. (a) The Eisenstein series associated with modular symbols E^{∗}(z, s) has a meromorphic continuation in the whole complex plane and satisfies the functional equation (1.10).

(b) For ℜs ≥ 1/2 the poles of E^{∗}(z, s) are simple and contained in the set

{1/2} ∪ ⋃_{j} {s_{j}, s̄_{j}} .

(c) At a cuspidal eigenvalue s_{j}(1 − s_{j}) of Δ corresponding to the cusp forms ϕ_{l}(z), l = 1, … N, the residue of E^{∗}(z, s) is equal to

∑_{l=1}^{N} \frac{c}{π^{s_{j}}} L(f ⊗ ϕ_{l}, s_{j} + 1/2)Γ(s_{j} − 1/2)ϕ_{l}(z).

Here c is a certain constant, Γ(s) is the Gamma function and L(f ⊗ ϕ_{l}, s) is the Rankin-Selberg convolution of f(z) with ϕ_{l}(z).

We prove Theorem 1(a) in Section 2, (b) in Section (4), and (c) in Section 5. It follows that the scattering function ϕ^{∗}(s) identified in [14, 0.3] using Kloost-

erman sums is given simply by

(1.11)  ϕ^{∗}(s) = dϕ_{ϵ}(s)/dϵ

at ϵ = 0 and its functional equation [14, Th. 0.2] follows by the standard functional equation for the scattering matrix ϕ(s)ϕ(1 − s) = 1 by differentiation. This gives the following theorem, see also [2, Theorem 1]. Theorem 2. Let i and j be cusps. The entries of dϕ_{ϵ}(s)/dϵ at ϵ = 0 are given by

ϕ_{ij}^{∗}(s) = −2πi ∫_{j}^{i} f(τ) dτ · ϕ_{ij}(s).

Since we are interested in the analytic continuation of Eisenstein series, we follow the method of Colin de Verdière [3], which is the shortest known method. We notice that at every step we can differentiate in the parameter ϵ and that everything remains meromorphic in s ∈ ℂ. Remark 3. Our method also allows to prove the meromorphic continuation of more general Eisenstein series of the form

(1.12)  E^{m,n}(z, s) = ∑_{Γ_{∞}\Γ}⟨γ, f⟩^{m}\overline{⟨γ, g⟩}^{n}ℑ(γ · z)^{s}

for two cusp forms f, g of weight 2, which are relevant to the distribution of modular symbols, as explained in [14, p. 165]. See (2.3).

A corollary of our method is to show the following theorem: Theorem 4. Assume that s_{j}(1 − s_{j}) has multiplicity one and the corresponding Maaß cusp form is ϕ_{j}(z). If the value of the L-series L(f ⊗ϕ_{j}, s_{j} +1/2) is nonzero, then the perturbed Eisenstein series E_{ϵ}(z, s) has a pole close to s_{j}.

The hypothesis

(1.13)  L(f ⊗ ϕ_{l}, s_{j} + 1/2) ≠ 0

is the Phillips-Sarnak condition and appeared in [20], [17]. See Remark 12.

We also study the behavior of E^{∗}(z, s) on vertical lines. We get the following theorem. Theorem 5. The Eisenstein series associated with modular symbols E^{∗}(z, s) is bounded on vertical lines with σ > 1/2. More precisely, for z ∈ K, a compact set, and for s bounded away from the poles of ϕ(s) on (1/2, 1] we have

(1.14)  E^{∗}(z, s) ≪_{K,σ} 1.

Remark 6. In fact Theorem 5 allows to improve the asymptotic formula (1.3), i.e., gives an estimate for the remainder of the form O(X^{a}), with a < 1, using standards techniques in analytic number theory (contour integration). See the forthcoming article of Goldfeld and O’Sullivan [7]. Remark 7. We introduce sums over closed geodesics γ with length l(γ) as follows:

π_{ϵ}(x) = ∑_{l(γ)≤x} χ_{ϵ}(γ).

Then

\frac{d}{dϵ}_{|ϵ=0}π_{ϵ}(x) = −2πi ∑_{l(γ)≤x} ∫_{γ} f.

The asymptotic behavior of the sums π_{ϵ}(x) can be understood using the Selberg trace formula. To estimate their derivative one should differentiate the trace formula in ϵ. On the other hand, to understand geodesics in homology classes as in [21], one integrates the trace formula over the character variety.

The study of E^{∗}(z, s) using perturbed Eisenstein series is a new application of the spectral deformations used in [17], [18] and [16]. Our contribution is to put the Eisenstein series with modular symbols into this framework. We avoid completely the Kloosterman sums with modular symbols introduced and used in [5], [14].

2. Proof of the analytic continuation of E^{∗}(z, s)

We first notice that E^{∗}(z, s) is linear in the differential f(τ)dτ. So we can consider separately the real and imaginary part of f(τ)dτ. Let w_{i} be either of the two. We let

(2.1)  χ_{ϵ}^{i}(γ) = exp (−2πiϵ ∫_{z_{0}}^{γz_{0}} w_{i}) ,

which is now a unitary character of Γ. We define Eisenstein series

(2.2)  E_{ϵ}(z, s, w_{i}) = ∑_{γ∈Γ_{∞}\Γ} χ_{ϵ}^{i}(γ)ℑ(γ · z)^{s}

for ℜ(s) > 1. More generally one can define Eisenstein series depending on a vector of parameters ϵ⃗ = (ϵ_{1}, ϵ_{2}, …, ϵ_{m}) and a vector of real-valued harmonic 1-forms (w_{1}, w_{2}, …, w_{m}) as

(2.3)  Eϵ⃗(z, s, w⃗) = ∑_{γ∈Γ_{∞}\Γ} ∏_{i=1}^{m} χ_{ϵ_{i}}^{i}(γ)ℑ(γ · z)^{s}.

The Eisenstein series (1.12) are linear combinations of their derivatives in ϵ_{i}, when we set ϵ⃗ = 0. For simplicity we will restrict our attention to one parameter and one cusp. We drop the subscript i. The generalization to many cusps can proceed as in [12]. Using the identification of harmonic cuspical cohomology with cohomology with compact support, see [1], one can assume that w is a compactly supported form. We consider the space L^{2}(Γ \ ℍ, χ̄_{ϵ}) of L^{2} functions which transform as

h(γ · z) = χ̄_{ϵ}(γ)h(z), γ ∈ Γ

under the action of the group. We introduce unitary operators

U_{ϵ} : L^{2}(Γ \ ℍ) → L^{2}(Γ \ ℍ, χ̄_{ϵ})

given by

(U_{ϵ}h)(z) = exp (2πiϵ ∫_{z_{0}}^{z} w) h(z).

We set

L_{ϵ} = U_{ϵ}^{−1}ΔU_{ϵ}.

The operators L_{ϵ} on L^{2}(Γ \ ℍ) and Δ on L^{2}(Γ \ ℍ, χ_{ϵ}) are unitarily equivalent. Also L_{ϵ} = Δ outside the support of w. The cusp C is isometric to [b, ∞) × ℝ/ℤ with the metric (dx^{2} + dy^{2})/y^{2}, where b is sufficiently large. We can assume that supp(w) ∩ C = ∅. We let h(y) ∈ C^{∞}(ℝ^{+}) be a function which is 0 for y ≤ b + 1 and 1 for y ≥ b + 2. Let

Ω_{ϵ} = {s ∈ ℂ, ℜ(s) > 1/2, s(1 − s) ∉ Spec(L_{ϵ})} .

Lemma 8. For s ∈ Ω_{ϵ} we can find a unique D_{ϵ}(z, s) such that

(L_{ϵ} + s(1 − s))D_{ϵ}(z, s) = 0

and

D_{ϵ}(z, s) − h(y)y^{s} ∈ L^{2}(Γ \ ℍ).

Moreover the functions

s → D_{ϵ}(z, s), s → \frac{d}{dϵ}D_{ϵ}(z, s)

are holomorphic in s ∈ Ω_{ϵ} and the function

ϵ → D_{ϵ}(z, s)

is real analytic. Remark 9. The functions D_{ϵ}(z, s) are not the Eisenstein series themselves. These are E_{ϵ}(z, s) = U_{ϵ}D_{ϵ}(z, s).

Proof. We write D_{ϵ}(z, s) = h(y)y^{s} + g_{ϵ}(z, s) with g_{ϵ} ∈ L^{2}(Γ \ ℍ). We set

(2.4)  H_{ϵ}(z, s) = −(L_{ϵ} + s(1 − s))(h(y)y^{s})

and we see that H_{ϵ} has compact support and depends holomorphically on s ∈ ℂ and real analytically in ϵ. The same is true for Ḣ_{ϵ}(z, s) = −L̇_{ϵ}(h(y)y^{s}). For notational convenience we put a dot to denote differentiation with respect to the parameter ϵ. As long as s(1−s) in not in the spectrum of L_{ϵ}, the equation (L_{ϵ}+s(1−s))g_{ϵ}(z, s) = H_{ϵ}(z, s) can be inverted to give

g_{ϵ}(z, s) = (L_{ϵ} + s(1 − s))^{−1}H_{ϵ}(z, s)

and g_{ϵ} ∈ H^{2}(Γ \ ℍ), the second Sobolev space. The resolvent is holomorphic outside the spectrum of L_{ϵ} and depends real analytically on the parameter ϵ, see [10, p. 66-67]. □

We define pseudo-Laplacian operators associated with L_{ϵ} exactly as in [11] and [3]. We set

H_{a} = {f ∈ H^{1}(Γ \ ℍ), f_{0}|_{(a,∞)} = 0}

where f_{0} is the zero Fourier coefficient at the cusp. We take a ≥ b+2. The operator L_{ϵ,a} is the Friedrichs extension of the restriction to H^{1}(Γ\ℍ) of the quadratic form q(f) = ∫ ||∇U_{ϵ}f||^{2} to H_{a}. Intuitively we map f to L^{2}(Γ \ ℍ, χ̄_{ϵ}) and we know that L_{ϵ} is unitarily equivalent to Δ on this space. As in [19], [11] the operators L_{ϵ,a} have compact resolvents and depend real analytically on ϵ. Consequently this is true for their resolvents R_{a,ϵ}(s) = (L_{ϵ,a} + s(1 − s))^{−1} by standard perturbation theory, [10, p. 66–67]. We define

(2.5)  F_{ϵ}(z, s) = h(y)y^{s} + (L_{ϵ,a} + s(1 − s))^{−1}(H_{ϵ}(z, s))

and we see that F_{ϵ} is meromorphic in s and the same applies to

(2.6)  Ḟ_{ϵ}(z, s) = (L_{ϵ,a} + s(1 − s))^{−1}(Ḣ_{ϵ}(z, s)) + Ṙ_{a,ϵ}(s)H_{ϵ}(z, s).

We notice that L_{ϵ,a} does not change the nonzero Fourier coefficients and it removes the zero Fourier coefficient at height y = a. For b < y < a we see that (L_{ϵ} + s(1 − s))F_{ϵ}(z, s) = 0. Consequently the zero Fourier coefficient F_{0,ϵ}(z, s) of F_{ϵ}(z, s) is of the form

F_{0,ϵ}(z, s) = A_{ϵ}(s)y^{s} + B_{ϵ}(s)y^{1−s}

for some holomorphic functions A_{ϵ}(s), B_{ϵ}(s), s ≠ 1/2. The real analyticity of the expansion in ϵ is also obvious, as follows from the definition of the Fourier coefficients. By looking at height y = a in (2.5) we get

(2.7)  A_{ϵ}(s)a^{s} + B_{ϵ}(s)a^{1−s} = a^{s},

from which it follows that A_{ϵ}(s) and B_{ϵ}(s) are not identically 0 in s. We modify the functions F_{ϵ}(z, s) to relate them to the functions D_{ϵ}(z, s) as follows. We define

F̃_{ϵ}(z, s) = F_{ϵ}(z, s) + χ_{[a,∞)}(y)(A_{ϵ}(s)y^{s} + B_{ϵ}(s)y^{1−s} − y^{s}).

We notice that (L_{ϵ} + s(1 − s))F̃_{ϵ}(z, s) = 0 for y ≥ a. If ℜ(s) > 1/2 and s(1 − s) ∉ Spec(L_{ϵ}), all terms are in L^{2}(Γ \ ℍ) with the exception of h(y)y^{s} + χ_{[a,∞)}(y)(A_{ϵ}(s)y^{s} − y^{s}), so

F̃_{ϵ}(z, s) − A_{ϵ}(s)h(y)y^{s} ∈ L^{2}(Γ \ ℍ)

and, therefore,

(2.8)  F̃_{ϵ}(z, s) = A_{ϵ}(s)D_{ϵ}(z, s)

by Lemma 8. Similarly we see that F̃_{ϵ}(z, s) − χ_{[a,∞)}(y)B_{ϵ}(s)y^{1−s} ∈ L^{2}(Γ \ ℍ) for ℜ(s) < 1/2 and s(1 − s) ∉ Spec(L_{ϵ}), so

(2.9)  F̃_{ϵ}(z, s) = B_{ϵ}(s)D_{ϵ}(z, 1 − s).

From (2.8), (2.9) we get the analytic continuation of D_{ϵ}(z, s) and its functional equation. As in [3] we see that D_{ϵ}(z, s) does not have poles on ℜ(s) = 1/2 (using the Maaß-Selberg relations). The scattering matrix is

(2.10)  ϕ_{ϵ}(s) = \frac{B_{ϵ}(s)}{A_{ϵ}(s)}.

We mention the various formulas for the derivatives:

\frac{dF̃_{ϵ}(z, s)}{dϵ} = Ḟ_{ϵ}(z, s) + χ_{[a,∞)}(y)(Ȧ_{ϵ}(s)y^{s} + Ḃ_{ϵ}(s)y^{1−s}),

(2.11)  Ḋ_{ϵ}(z, s) = \frac{dA_{ϵ}^{−1}(s)}{dϵ}F̃_{ϵ}(z, s) + A_{ϵ}^{−1}(s)\frac{dF̃_{ϵ}(z, s)}{dϵ},
(2.12)  Ė_{ϵ}(z, s) = U̇_{ϵ}D_{ϵ}(z, s) + U_{ϵ}Ḋ_{ϵ}(z, s).

3. Proof of Theorem 2

We discuss the case of one cusp first. By Remark 61 in [9, p. 218] we know that ϕ_{ϵ}(s) = ϕ_{−ϵ}(s). As a result ϕ̇_{0}(s) = 0, being the derivative of an even function at ϵ = 0. Consequently, by (1.11) we have ϕ^{∗}(s) = 0.

We include a detailed proof of ϕ_{ϵ}(s) = ϕ_{−ϵ}(s) to facilitate the understanding of the multiple-cusp case. Using the Bruhat decomposition Γ_{∞} \ Γ/Γ_{∞} we can write the zero Fourier coefficient as

y^{s} + ∑_{γ∈Γ_{∞}\Γ/Γ_{∞}} ∑_{m∈ℤ} ∫_{0}^{1} χ_{ϵ}(γ)\frac{y^{s}}{|cz + cm + d|^{2s}} dx

since χ_{ϵ}(γS^{m}) = χ_{ϵ}(γ) as χ_{ϵ} is a character with χ_{ϵ}(S) = 1. Here S is the standard parabolic generator. So

ϕ_{ϵ}(s)y^{1−s} = ∑_{γ∈Γ_{∞}\Γ/Γ_{∞}} \frac{χ_{ϵ}(γ)}{|c|^{2s}} ∫_{−∞}^{∞} \frac{y^{s}}{|x^{2} + y^{2}|^{s}} dx.

The integral can be evaluted in terms of the Gamma function to be y^{1−s}\sqrt{π}Γ(s − 1/2)/Γ(s), see [8, 8.380.3]. To show that ϕ_{ϵ}(s) = ϕ(s, χ) = ϕ(s, χ̄) = ϕ_{−ϵ}(s) it suffices to notice that we can take as coset representatives in the Bruhat decomposition γ^{−1}, where γ ∈ Γ_{∞} \ Γ/Γ_{∞} and that γ^{−1} has lower left entry −c. The same calculation works for the case of many cusps and the diagonal entries ϕ_{ii}(s, χ) of the scattering matrix Φ_{ϵ}(s). We get ϕ̇_{ii}(s) = 0. Remark 10. In general, for a group Γ with many cusps, Φ_{ϵ}(s) = Φ_{−ϵ}(s)^{T}. By differentiation we get that Φ^{∗}(s) is skew-symmetric, which already gives Proposition 4.2 in [14].

In the case of many cusps the ij-entry of the scattering matrix is given by

(3.1)  ϕ_{ij,ϵ}(s) = \sqrt{π} \frac{Γ(s − 1/2)}{Γ(s)} ∑_{γ∈Γ_{∞}\Γ/Γ_{∞}} \frac{χ_{ϵ}(σ_{i}^{−1}σ_{j}γ)}{|c|^{2s}}.

Here σ_{j} maps the j cusp to i∞ and σ_{i} maps the i cusp to i∞. Since χ_{ϵ} is a character we can write χ_{ϵ}(σ_{i}^{−1}σ_{j})χ_{ϵ}(γ) and differentiate to get

ϕ_{ij}^{∗}(s) = (−2πi ∫_{z_{0}}^{σ_{i}^{−1}σ_{j}z_{0}} f(τ) dτ) ϕ_{ij}(s) + ϕ̇_{ii}(s).

But we have shown that ϕ̇_{ii}(s) = 0. Since ϕ^{∗}(s) does not depend on z_{0}, we take z_{0} to be the j cusp.

4. Poles of E^{∗}(z, s).

It is clear from (2.6) and (2.8) that we get potential poles for E^{∗}(z, s) at the eigenvalues of L_{ϵ,a} or at the poles of the scattering matrix. The eigenvalues of L_{ϵ,a} come in two types, [3]. Any cusp form for ϵ = 0 is an eigenfunction of L_{0,a} for all a, as its zero Fourier coefficient is identically zero. These are the eigenvalues of Type (I). The eigenvalues of Type (II) correspond to noncuspidal eigenfunctions: For ℜ(s_{j}) ≥ 1/2, s_{j} ≠ 1/2 we must have

(4.1)  a^{s_{j}} + ϕ_{0}(s_{j})a^{1−s_{j}} = 0.

For s_{j} = 1/2 the condition is ϕ_{0}(1/2) = −1 and ϕ′_{0}(1/2) = −2 log a. For details see [3, p. 93]. Recall that ϕ_{0}(s) is the (standard) scattering function ϕ(s). It is inconvenient to work with L_{ϵ,a}, so we try to characterize the poles of E^{∗}(z, s) in terms of the eigenvalues of Δ: Lemma 11. If s_{j} does not correspond to a cuspidal eigenvalue of Δ on L^{2}(Γ \ ℍ) and ϕ_{0}(s) does not have a pole at s_{j}, then E^{∗}(z, s) is regular at s_{j}.

Proof. If A_{0}(s_{j}) = 0, then (2.7) implies that B_{0}(s_{j}) ≠ 0. But then ϕ_{0}(s) has a pole at s_{j}. Consequently A_{0}(s_{j}) ≠ 0 for all a large, so we do not get a pole from the contribution of A_{ϵ}^{−1} and dA_{ϵ}^{−1}/dϵ in (2.11). We also need to arrange that we do not get a pole from the resolvent of of L_{0,a} for some a, see (2.6). We need to exclude eigenvalues of type (II). Then all the formulas become regular at s_{j}. Notice that, by the second Neumann series for the resolvent [10, p. 66–67],

(4.2)  Ṙ_{ϵ}(z) = −R_{ϵ}(z)L_{ϵ}R_{ϵ}(z),

so the derivative of the resolvent in (2.6) is regular away from the eigenvalues of L_{ϵ}. However, the conditions (4.1) are satisfied for a discrete set of values of positive a. Once an a is chosen (sufficiently large) not satisfying (4.1), for small enough ϵ, L_{ϵ} do not have eigenvalue close to s_{j}. □

This lemma proves part (b) in Theorem 1.

5. Residues of E^{∗}(z, s) at cuspidal eigenvalues

Let s_{j} be such that s_{j}(1 − s_{j}) is a cuspidal eigenvalue of Δ. The formula for L_{ϵ} has been used in [16], [18] and is given by

(5.1)  L_{ϵ}u = Δu − 4πiϵ⟨du, w⟩ − 4π^{2}ϵ^{2}|w|_{H}^{2}u + 2πiϵ(δw)u.

Here δ(p dx + q dy) = −y^{2}(p_{x} + q_{y}), ⟨p dx + q dy, f dx + g dy⟩ = y^{2}(pf̄ + qḡ) and |p dx + q dy|_{H}^{2} = y^{2}(|p|^{2} + |q|^{2}). The difference in the signs is due to the fact that we use −w in the formula in [16]. We have

(L_{ϵ} + s(1 − s))D_{ϵ}(z, s) = 0

away from the poles of D_{ϵ}(z, s). We differentiate it and evaluate at ϵ = 0 to get

(L_{0} + s(1 − s))Ḋ_{0}(z, s) = −L̇_{0}D_{0}(z, s).

It follows from (2.8), (2.10) that the zero Fourier coefficient of D_{ϵ}(z, s) is

y^{s} + ϕ_{ϵ}(s)y^{1−s}.

Since ϕ̇_{0}(s) = 0, by Theorem 2, Ḋ_{0}(z, s) is in H_{a}. We can substitute L_{0} with L_{0,a} to get

(L_{0,a} + s(1 − s))Ḋ_{0}(z, s) = −L̇_{0}D_{0}(z, s).

As in the previous section we can assume that we chose a in such a way that L_{0,a} does not have a Type (II) eigenvalue at s_{j}(1 − s_{j}). For s(1 − s) ∉ Spec(L_{0}), ℜ(s) > 1/2 we can introduce the resolvent to get

(5.2)  Ḋ_{0}(z, s) = −R_{0}(s)L̇_{0}D_{0}(z, s),

where R_{0}(s) = (L_{0} + s(1 − s))^{−1}. The residue A(z) of Ḋ_{0}(z, s) at s_{j}, which is the same as the residue of Ė_{0}(z, s) by (1.9) and (2.12) is the residue of R_{0,a}(s) at s_{j} applied to −L̇_{0}D_{0}(z, s). We recall that D_{0}(z, s) is regular at s_{j}. The resolvent kernel for L_{0,a} has an expansion at s_{j} of the form

r_{L_{0,a}}(z, z′, s) = \frac{1}{s(1 − s) − s_{j}(1 − s_{j})} ∑_{l=1}^{N} ϕ_{l}(z)ϕ_{l}(z′) + analytic at s_{j},

where ϕ_{l}(z), l = 1, …, N are an orthonormal basis of cusp forms at s_{j}. As a result the residue A(z) is given by

A(z) = \frac{4πi}{2s_{j} − 1} ∑_{l=1}^{N} ϕ_{l}(z) ∫_{Γ\ℍ} ϕ_{l}(z′)⟨dD_{0}(z′, s_{j}), w⟩dμ(z′)

−\frac{2πi}{2s_{j} − 1} ∑_{l=1}^{N} ϕ_{l}(z) ∫_{Γ\ℍ} ϕ_{l}(z′)(δw)D_{0}(z, s_{j})dμ(z′), (5.3)

where dμ(z) is the invariant hyperbolic measure dx dy/y^{2}. This residue is independent of a. We can change w in its cohomology class, even though U_{ϵ} depends on it. Consequently we can approximate α, the real or the imaginary part of the differential f(z) dz by a family w_{l}, supported in a compact set K_{l} with Γ\ℍ = ∪K_{l} and with convergence in the Sobolev space H^{1}. Then

lim_{l} ∫_{Γ\ℍ} ϕ_{j}(z)⟨dD_{0}(z, s_{j}), w_{l}⟩dμ(z) = ∫_{Γ\ℍ} ϕ_{j}(z)⟨dD_{0}(z, s_{j}), α⟩dμ(z),

and

lim_{l} ∫_{Γ\ℍ} ϕ_{j}(z)(δw_{l})D_{0}(z, s_{j})dμ(z) = ∫_{Γ\ℍ} ϕ_{j}(z)(δα)D_{0}(z, s_{j})dμ(z).

Since α is harmonic, δα = 0. Since the modular symbols is linear, while ⟨·, ·⟩ is antilinear in the second variable, we take α = \overline{f(z)dz}. By linearity we are left to compute

∫_{Γ\ℍ} ϕ_{j}(z)⟨dD_{0}(z, s_{j}), \overline{f(z)dz}⟩dμ(z) = ∫_{Γ\ℍ} ϕ_{j}(z)y^{2}f(z)Ez̄(z, s_{j})dμ(z),

since dD_{0}(z, s) = ∂_{z}D_{0}(z, s)dz + ∂z̄D_{0}(z, s)dz̄ and ⟨f_{1}dz + f_{2}dz̄, g_{1}dz + g_{2}dz̄⟩ = 2y^{2}(f_{1}\overline{g_{1}} + f_{2}\overline{g_{2}}).

5.1. Relation with Rankin-Selberg convolutions. We analyze the integral

(5.4)  I(s) = ∫_{Γ\ℍ} ϕ_{j}(z)y^{2}f(z)Ez̄(z, s)dμ(z)

where E(z, s) = D_{0}(z, s). For ℜ(s) sufficiently large we can differentiate the series

E(z, s) = ∑_{γ∈Γ_{∞}\Γ} ℑ(γz)^{s}

to get

Ez̄(z, s) = \frac{is}{2} ∑_{γ∈Γ_{∞}\Γ} ℑ(γz)^{s−1}\overline{(cz + d)}^{−2}.

We can unfold as in the Rankin-Selberg method for modular forms of different weight to get

\frac{is}{2} ∑_{γ∈Γ_{∞}\Γ} ∫_{Γ\ℍ} ϕ_{j}(z)\frac{y^{2}}{|cz + d|^{4}} f(z)(cz + d)^{2} \frac{y^{s−1}}{|cz + d|^{2s−2}} dμ(z),

since Ez̄(z, s) has weight (0, 2). We then get

I(s) = \frac{is}{2} ∫_{Γ_{∞}\ℍ} ϕ_{j}(z)y^{2}f(z)y^{s−1}dμ(z).

We write Fourier expansions for ϕ_{j}(z) and f(z):

(5.5)  ϕ_{j}(z) = ∑_{n≠0} a_{n}y^{1/2}K_{s_{j}−1/2}(2π|n|y)e^{2πinx},

where K_{ν}(y) is the MacDonald Bessel function, and f(z) = ∑_{n>0} b_{n}e^{2πinz}. Using [8, 6.621.3, p. 733] we get

I(s) = \frac{is}{2} ∑_{n>0} \frac{a_{−n}b_{n}}{(2πn)^{s+1/2}} \frac{\sqrt{π}2^{−1/2}}{2^{s}} \frac{Γ(s + s_{j})Γ(s − s_{j} + 1)}{Γ(s + 1)}.

We denote the Rankin-Selberg convolution of f and ϕ_{j} as L(f ⊗ ϕ_{j}, s). We use the duplication formula for the Gamma function, plug in s = s_{j} and multiply by 4πi/(2s_{j} − 1) to get

\frac{−\sqrt{π}}{2π^{s_{j}}(2s_{j} − 1)}L(f ⊗ ϕ_{j}, s_{j} + 1/2)Γ(s_{j} + 1/2)

which gives Th. 1(c) and agrees up to a constant with the result in [15].

6. Proof of Theorem 4

If the value L(f ⊗ϕ_{j}, s_{j} +1/2) ≠ 0, then Ḋ_{0}(z, s) has definitely a pole at s_{j}. Since D_{0}(z, s) is regular at s_{j}, the functions D_{ϵ}(z, s) should have poles s_{j}(ϵ) converging to s_{j}, as ϵ → 0. Remark 12. According to [20] a pole of the perturbed Eisenstein series can occur if a cusp form eigenvalue becomes a scattering pole. This is so because for small ϵ the total multiplicity of the singular set in a small disc around s_{j} remains constant. Our result creates the scattering pole out of the Phillips-Sarnak condition (1.13) without the use of the singular set. If we can show that a pole of D_{ϵ}(z, s) close to the unitary axis forces a Type (II) eigenvalue for L_{ϵ,a} for some a, then we would have a new proof of the destruction of cusp forms under (1.13).

7. Proof of Theorem 5

For simplicity we assume we have only one cusp and we fix σ = ℜs > 1/2. It follows from the Maaß-Selberg relations, as in [13, Lemma 8.8], that the scattering function ϕ_{ϵ}(s) is bounded for ℜs ≥ 1/2 and away from the finite number of poles in the interval (1/2, 1]. For the McDonald-Bessel function K_{s}(x) the integral representation

K_{s}(x) = ∫_{0}^{∞} e^{−x cosh t} cosh(st) dt,

see [8, 8.432.1, p. 968], gives

(7.1)  |K_{s}(x)| ≤ e^{−x/2}K_{σ}(2)

for x ≥ K. This together with the polynomial bound on the Fourier coefficients of Eisenstein series gives

(7.2)  E_{ϵ}(z, s) ≪_{z} y^{σ} + |ϕ(s)|y^{1−σ} + O_{σ}(1) = O_{z,σ}(1).

Similarly ∂_{x}E_{ϵ}(z, s) ≪_{z} O(1). The estimates can clearly be made uniform in z on compact sets. For ∂_{y}E_{ϵ}(z, s) we study K′_{s}(x). Differentiating the integral in (7.1) we get

|K′_{s}(x)| ≤ e^{−x/2}|K′_{σ}(2)|.

This implies that

(7.3)  ∂_{y}E_{ϵ}(z, s) ≪_{z} |s|y^{σ−1} + |ϕ(s)||1 − s|y^{−σ} + O_{z}(1).

The estimates (7.2) and (7.3) show that both E_{ϵ}(z, s) and (1/|t|)dE_{ϵ}(z, s) are bounded on vertical lines for σ > 1/2. By (5.1) and (5.2) we get

Ḋ_{0}(z, s) = −R_{0}(s)(−4πi⟨dD_{0}(z, s), w⟩ + 2πi(δw)D_{0}(z, s)).

The bounds for D_{0}(z, s) = E_{0}(z, s) and its differential together with the fact that L̇ has compact support give a polynomial bound for the L^{2}-norm of L̇D_{0}(z, s) in the t aspect. Since

||R(z)|| ≤ \frac{1}{dist (z, spec A)}

for the resolvent of a general self-adjoint operator A on a Hilbert space and dist (s(1− s), spec L_{0,a}) ≥ |t|(2σ − 1) we get

Ḋ_{0}(z, s) ≪_{σ} \frac{|s|}{|t||2σ − 1|}.

We last notice that the above estimate, together with (7.2) and (2.12) finish the proof of the theorem.

Acknowledgments The author would like to acknowledge the financial support of the Max-Planck-Institut für Mathematik during the completion of this project. The author would like to thank G. Chinta for helpful discussions, C. O’Sullivan for motivating Theorem 5 and P. Sarnak for a critical reading of the manuscript.

References

[1] A. Borel; Stable real cohomology of arithmetic groups. II. Manifolds and Lie groups (Notre Dame, Ind., 1980), pp. 21–55, Progr. Math., 14, Birkhäuser, Boston, Mass., 1981.

[2] G. Chinta, D. Goldfeld; Grössencharakter L-functions of real quadratic fields twisted by modular symbols, Invent. Math 144, 3 (2001) 435–449.

[3] Y. Colin de Verdière; Pseudo-laplaciens. II. Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 87–113.

[4] N. Diamantis; C. O’Sullivan; Hecke theory of series formed with modular symbols and relations among convolution L-functions. Math. Ann. 318 (2000), no. 1, 85–105.

[5] D. Goldfeld; Zeta functions formed with modular symbols. Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), 111–121, Proc. Sympos. Pure Math., 66, Part 1, Amer. Math. Soc., Providence, RI, 1999.

[6] D. Goldfeld; The distribution of modular symbols. Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997), 849–865, de Gruyter, Berlin, 1999.

[7] D. Goldfeld; C. O’Sullivan; in preparation.

[8] I. D. Gradshteyn, I. M. Ryzhik; Tables of Integrals, Series, and Products, 5th ed., ed. Alan Jeffrey, Academic Press, Boston, 1994.

[9] D. Hejhal; The Selberg trace formula for PSL(2, R). Vol. 2. Lecture Notes in Mathematics, 1001. Springer-Verlag, Berlin, 1983. viii+806pp.

[10] T. Kato; Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976. xxi+619 pp.

[11] P. Lax, R. Phillips; Scattering theory for automorphic functions. Annals of Mathematics Studies, No. 87. Princeton Univ. Press, Princeton, N.J., 1976. x+300 pp.

[12] W. Müller; Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109 (1992), no. 2, 265–305.

[13] W. Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula. Math. Nachr. 111 (1983), 197–288.

[14] C. O’Sullivan; Properties of Eisenstein series formed with modular symbols. J. Reine Angew. Math. 518 (2000), 163–186.

[15] C. O’Sullivan; Ph. D. Thesis, Columbia University, 1998.

[16] Y. N. Petridis; Perturbation of scattering poles for hyperbolic surfaces and central values of L-series. Duke Math. J. 103 (2000), no. 1, 101–130.

[17] R. Phillips, P. Sarnak; Cusp forms for character varieties. Geom. Funct. Anal. 4 (1994), no. 1, 93–118.

[18] R. Phillips, P. Sarnak; The spectrum of Fermat curves. Geom. Funct. Anal. 1 (1991), no. 1, 80–146.

[19] R. Phillips, P. Sarnak; On cusp forms for co-finite subgroups of PSL(2, R). Invent. Math. 80 (1985), no. 2, 339–364.

[20] R. Phillips, P. Sarnak; Perturbation theory for the Laplacian on automorphic functions. J. Amer. Math. Soc. 5 (1992), no. 1, 1–32.

[21] R. Phillips, P. Sarnak; Geodesics in homology classes. Duke Math. J. 55 (1987), no. 2, 287–297.

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, Canada H3A 2K6

E-mail address: petridis@math.mcgill.ca