Harmonic differentials and infinite geodesics joining two cusps on a Riemann Surface

Thérèse Falliero

Abstract. Let M be an hyperbolic Riemann surface. The harmonic dual form to an infinite geodesic joining two cusps on M, is obtained in two different ways. First of all, using the degeneration of hyperbolic Eisenstein series, it is made explicit in terms of Eisentein series. Secondly, generalizing the construction of Kudla and Millson to the case of an infinite geodesic joining two cusps, we give an automorphic realization of this harmonic form.

Introduction

In this paper, we answer in particular, some questions in [K-M] (p.194 and 208) and in [G] (p. 562-09). In the first part we recall the definitions and results we need, we also give the example of the principal congruence subgroup of level 2, Γ(2) (see proposition 1.1). In the second we obtain the harmonic dual 1-form to an infinite geodesic joining two cusps, using the degeneration of hyperbolic Eisenstein series (see proposition 2.3). Moreover, in the third part we answer a question that P.Gérardin asks in [G] (see theorem 3.2): can we extend the Kudla and Millson’s construction to the case of ]p, q[, the geodesic joining the cusps p and q and orthogonal to the horocycles at p and at q ?

I hope this work could be generalized to the higher dimensional case ([KM1], [KM2]).

Acknowledgments. This work is a continuation of the second chapter of my Ph.D.Thesis and was written while at the Max-Planck-Institut für Mathematik in Bonn. The second section of the paper was begun after the submission of my thesis. I would also like to thank Professor S.Kudla for his answer about whether he and Millson had written something on the previous question, and for giving me some references.

1. Preliminary definitions and results

1.1. Definitions. In the following M will be a hyperbolic Riemann surface, that is to say, M has for universal covering space the upper half plane H. The fundamental group of M is identified to a Fuchsian group of the first kind, denoted Γ. We’ll suppose to have only two cusps on M, p and q; it is easy to verify that all the results are true if there are another cusps than p and q.

In [K-M], Kudla and Millson construct the harmonic 1-form dual to a simple closed geodesic on M. Let us recall the definition:

Definition 1.1. Let η be a simple closed geodesic or an infinite geodesic joining p and q. An 1-form α is dual to η if for any closed 1-form with compact support, ω

∫_{M} ω ∧ α = ∫_{η} ω .

Or equivalently: for any closed oriented cycle c′, we have

∫_{c′} α = c′ . η ,

(here c′.η denotes the intersection number of the cycles c′ and η.)

We are interesting in the 1-form dual to ]p, q[ and which is harmonic. We begin by giving, in an elementary way, the unicity of the cohomology class and the existence of such a form. For this, recall the following results:

Lemma 1.1. Let α be a 1-form such that 1. ∫_{M} α ∧ β = 0 for each closed 1-form β, or

2. for any simple closed curve c, ∫_{c} α = 0. then there exists a function f with α = df.

and the theorem in [S]

Theorem 1.1. (theorem 3, p.106) Let X be a compact Riemann surface. Let p and q be two distincts points on X connected by a simple, oriented curve L, and X^{∗} be the region obtained by cutting along L. There exists a unique differential of the first kind dF (p, q) which is regular in X apart from simple poles at p and q with residues 1 and -1. The periods of F (p, q) on X^{∗} are all real, and, for every loop which crosses L exactly once from right to left, the imaginary part of the period is 2π.

Then the 1/2π imaginary part of dF (p, q) gives the dual form to the curve L.

The authors of [K-M] obtain the harmonic dual 1-form to a simple closed geodesic η as the analytic continuation at s = 1 of the hyperbolic Eisentein series associated to η: η̂^{s}. This hyperbolic Eisenstein series is defined by taking the sum over the set H(η) of all the oriented geodesics Y of H whose projection is η, of the 1-form:

ω_{Y}^{s} = ||ω_{Y} ||^{s−1}ω_{Y} ,

where ω_{Y} = ∗P_{Y}^{∗}V_{Y}, P_{Y} is the orthogonal projection on Y, V_{Y} the volume form of Y and ||ω_{Y} || denotes the pointwise norm of ω_{Y}. The 1-form ω_{Y}^{s} is normalised so that the integral over any fiber of P_{Y} is 1:

∫ \frac{1}{k(s − 1)} ω_{Y}^{s} = 1 , k(s − 1) = \frac{Γ(1/2)Γ(s/2)}{Γ(1/2 + s/2)} .

Then, for Re s > 1:

(1)  η̂^{s} = \frac{1}{k(s − 1)} ∑_{Y ∈H(η)} ω_{Y}^{s}

= \frac{Γ(1/2 + s/2)}{Γ(1/2)Γ(s/2)} ∑_{Γ_{1}\Γ} γ^{∗} \frac{dx_{2}}{(ch x_{2})^{s}} ,

where Γ_{1} denote the stabilizer group in Γ of some geodesic Y ∈ H(η) and any point P ∈ H is located by its Fermi coordinates (x_{1}, x_{2}) associated to the geodesic Y.

Recall now of the definition of the classical Eisentein series.

The stability group of a cusp 𝔞 is an infinite cyclic group generated by a parabolic motion,

Γ_{𝔞} = {γ ∈ Γ : γ 𝔞 = 𝔞} = ⟨γ_{𝔞}⟩ ,

say. There exists σ_{𝔞} ∈ SL_{2}(ℝ) such that

σ_{𝔞}∞ = 𝔞, σ_{𝔞}^{−1}γ_{𝔞}σ_{𝔞} = \binom{1 1}{1} .

We call σ_{𝔞} a scaling matrix of the cusp 𝔞, it is determinated up to composition with a translation from the right side. The Eisenstein series for the cusp 𝔞 is then defined by:

E_{𝔞}(z, s) = ∑_{Γ_{𝔞}\Γ} y(σ_{𝔞}^{−1}γz)^{s} ,

where s is a complex variable, Re s > 1.

An Eisenstein series of weight 2 associated to a cusp 𝔞, is for us, a 1-form defined, for Re s > 1, by:

E_{𝔞}(s, z) = ∑_{γ∈Γ_{𝔞}\Γ} y(σ_{𝔞}^{−1}γz)^{s−1} d(σ_{𝔞}^{−1}γz) = \frac{E_{𝔞,1}(s, z)}{y} dz ,

with

E_{𝔞,1}(s, z) = y ∑_{γ∈Γ_{𝔞}\Γ} y(σ_{𝔞}^{−1}γz)^{s−1} ((σ_{𝔞}^{−1}γ)′z) .

By definition, a standard horocyclic Eisenstein series, called for sake of convenience, an horocyclic Eisenstein series is

𝔞̂^{s}(z) = Im(E_{𝔞}(s, z)) .

Remark 1.1. 1. In the notations of [Fay], E_{𝔞,1}(s, z) ∈ F_{1}, where F_{1} is the space of

all functions f on H with f(γz) = (\frac{cz + d}{cz̄ + d}) f(z), γ ∈ Γ.

2. A standard horocyclic Eisenstein series is associated to a horocycle of length 1: if |ξ| is the length of the horocycle and ξ̂^{s} is the horocyclic Eisenstein series associated to ξ then the 1-form |ξ|^{−s}ξ̂^{s} depends only on the cusp (for more details see [G]).

1.2. Example. Now we are going to treat the explicit example of the geodesic ]0, i∞[ on M = Γ(2)\H where Γ(2) is the principal congruence subgroup of level 2:

Γ(2) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), a = d = 1 mod 2, b = c = 0 mod 2\} .

In this case comparing the Hecke summation process and a result in [M], we obtain:

E_{∞}(1, z) − E_{0}(1, z) = θ(0, z)^{4}dz ,

where θ(0, z)^{4} = (∑_{n∈ℤ} e^{iπn^{2}z})^{2} = ∑_{n∈ℕ} r_{4}(n)e^{iπnz} and r_{4}(n) is the number of representations of n as a sum of 4 squares. Let c_{1} and c_{2} be horocycles associated respectively to the cusps 0 and ∞, which cross exactly once from right to left ]0, i∞[, we easily verify that

∫_{c_{2}} − Re(θ(0, z)^{4} dz) = −\frac{1}{2} ∫_{(1+iy_{o},1−iy_{o})}(1 + ∑_{n≥1} r_{4}(n)e^{−nπy_{o}} cos nπx) dx
= 1

and, with σ : z → −1/z

∫_{c_{1}} − Re(θ(0, z)^{4} dz) = ∫_{σ(c_{2})} Re(θ(0, z)^{4} dz)
= ∫_{c_{2}} Re(θ(0, σ(z))^{4} dσ(z)) = ∫_{c_{2}} − Re(θ(0, z)^{4} dz)
= 1

So, it follows

Proposition 1.1. The 1-form, − Re(E_{∞}(1, z) − E_{0}(1, z)) = − Re(θ(0, z)^{4} dz) is the harmonic dual form to ]0, i∞[.

2. The harmonic dual form and degeneration

The aim of this section is to find the harmonic dual form to an infinite geodesic joining two cusps on a Riemann surface using the degeneration of hyperbolic Eisenstein series. To do this we use the following results:

Consider a family of compact Riemann surfaces (S_{l})_{l>0} degenerating as l → 0, l > 0 to S_{0}, the geodesic c_{l} (of length l), non homologous to zero, on S_{l} being pinched to form a pair of cusps on S_{0}.

We note p and q the two cusps of S_{0} arising from pinching c_{l}: p the limit of the right side of the c_{l}-collar and q the limit of the left side of the c_{l}-collar. Without loss of generality we can represent p and q by ∞ and 0 respectively. We denote for simplicity ĉ_{l}^{s} = l̂^{s} and the complex 1-form of type (1, 0) associated:

θ_{l}(s, z) = 1/2(il̂^{s}(z) − ∗l̂^{s}(z)) .

There exist homeomorphisms f_{l} from S_{l}\c_{l} to S_{0}, with f_{l} tending to an isometry C^{∞}- uniformly on the complementary in S_{l} of a neighbourhood of c_{l}. We have the results (for the details see [F]):

Theorem 2.1. 1. For Re s > 1, the family of 1-forms \frac{1}{l^{s}} θ_{l}(s, f_{l}^{−1}(.)) converge uni-

formly on compact subsets of S_{0} to \frac{1}{2k(s − 1)}[E_{∞}(s, .) − E_{0}(s, .)].

2. The family of 1-forms (\frac{1}{l}θ_{l}(1, f_{l}^{−1}(.)))_{l>0} converge uniformly on compact subsets

of S_{0} to \frac{1}{2π} [E_{∞}(1, .) − E_{0}(1, .)].

3. For s ∈ \{Re s > 1, s = 1\} the family of 1-forms \frac{1}{l^{s}} l̂^{s}(s, f_{l}^{−1}(.)) converge uniformly

on compact subsets of S_{0} to \frac{1}{k(s − 1)}[ ˆ∞^{s} − ˆ0^{s}].

First of all, as an application and using a classical result on the degeneration of the canonical period matrix we obtain the asymptotic behavior of the periods of the holomorphic 1-form θ_{l}(1, z). Suppose S_{0} is of genus g and let {b_{j}, c_{j}}_{j=2}^{g+1} be a geodesic canonical homology base of the complementary of the cuspidal regions:

We put π_{l} = f_{l}^{−1} and we denote by b_{i}(l) (resp. c_{i}(l)) the unique geodesic in the homology class of π_{l}(b_{i}) (resp. π_{l}(c_{i})) for i = 2, ..., g + 1. We complete this family with b_{1}(l) and c_{1}(l) = c_{l} in order to form a geodesic canonical homology base on S_{l}: {b_{i}(l), c_{i}(l)}_{i=1}^{g+1}.

We obtain the following two propositions (see [F]):

Proposition 2.1. For A belonging to {b_{i}, c_{i}, i = 2, ..., g + 1}

lim_{l→0} ∫_{A(l)} \frac{θ_{l}(1, z)}{l} = \frac{1}{2π} ∫_{A} E_{∞}(1, z) − E_{0}(1, z) .

and

Proposition 2.2. For l → 0, ∫_{b_{1}(l)} θ_{l}(1, z) = O(1) and ∫_{c_{l}} θ_{l}(1, z) ∼ −\frac{l}{2π}.

Secondly, at the end of [K-M], there is left the question of the harmonic dual form to an infinite geodesic joining two cusps. To explicit this form, we decompose the harmonic 1-form ∗l̂ on the following base of harmonic differentials: {b̂_{i}(l) , i = 1, ..., g + 1, ĉ_{i}(l) , i = 1, ..., g + 1 }, where b̂_{i}(l) is the harmonic dual form to b_{i}(l), (resp. ĉ_{i}(l) is the harmonic dual form to c_{i}(l)), so:

∗l̂= ∑_{i=1}^{g+1} e_{i} ĉ_{i}(l) + ∑_{i=1}^{g+1} d_{i} b̂_{i}(l)

with:

∀i = 1, ..., g + 1  d_{i} = ∫_{c_{i}(l)} − ∗ l̂= 2 ∫_{c_{i}(l)} θ_{l}
∀i = 2, ..., g + 1  e_{i} = ∫_{b_{i}(l)} ∗l̂= −2 ∫_{b_{i}(l)} θ_{l}

Let α be a closed 1-form on S_{0} with compact support and take the corresponding α_{l} on S_{l}, i.e. α_{l}(π_{l}(w)) = α(w), w ∈ S_{0}. The 1-form α_{l} is closed on S_{l}. Therefore:

∫_{M} α_{l} ∧ ∗l̂ = ∑_{i=1}^{g+1} e_{i} ∫_{M} α_{l} ∧ ĉ_{i}(l) + ∑_{i=1}^{g+1} d_{i} ∫_{M} α_{l} ∧ b̂_{i}(l)
= ∑_{i=1}^{g+1} e_{i} ∫_{c_{i}(l)} α_{l} + ∑_{i=1}^{g+1} d_{i} ∫_{b_{i}(l)} α_{l} .

So

∫_{S_{l}} α_{l} ∧ \frac{∗l̂}{l} = \frac{d_{1}}{l} ∫_{b_{1}(l)} α_{l} + \frac{e_{1}}{l} ∫_{c_{1}(l)} α_{l} + ∑_{i=2}^{g+1} \frac{e_{i}}{l} ∫_{c_{i}(l)} α_{l} + ∑_{i=2}^{g+1} \frac{d_{i}}{l} ∫_{b_{i}(l)} α_{l} .

As α is with compact support, for l sufficiently small ∫_{c_{1}(l)} α_{l} = 0, it remains:

∫_{S_{l}} α_{l} ∧ \frac{∗l̂}{l} = \frac{d_{1}}{l} ∫_{]p,q[} α + ∑_{i=2}^{g+1} \frac{e_{i}}{l} ∫_{c_{i}} α + ∑_{i=2}^{g+1} \frac{d_{i}}{l} ∫_{b_{i}} α . (]p, q[=]i∞, 0[) (2)

Moreover we have the following behaviors (propositions 2.1 and 2.2), for i = 2, ..., g +1:

lim_{l→0} \frac{d_{i}}{l} = \frac{1}{π} ∫_{c_{i}} E_{∞} − E_{0}, lim_{l→0} \frac{e_{i}}{l} = −\frac{1}{π} ∫_{b_{i}} E_{∞} − E_{0} ,

and

lim_{l→0} \frac{d_{1}}{l} = −\frac{1}{π} .

If we let l goes to 0 in (2), we obtain:

∫_{S_{0}} α∧(−\frac{1}{π} Re[E_{∞} − E_{0}]) = \frac{1}{π} ∫_{]0,i∞[} α −∑_{i=2}^{g+1} (\frac{1}{π} ∫_{b_{i}} E_{∞} − E_{0} ∫_{c_{i}} α − \frac{1}{π} ∫_{c_{i}} E_{∞} − E_{0} ∫_{b_{i}} α) .

Proposition 2.3. Setting:

Θ(z) = − Re[E_{∞}(1, z)−E_{0}(1, z)]+∑_{i=2}^{g+1} ∫_{b_{i}}(E_{∞}(1, .)−E_{0}(1, .)) ĉ_{i}(z)−∑_{i=2}^{g+1} ∫_{c_{i}}(E_{∞}(1, .)−E_{0}(1, .)) b̂_{i}(z) ,

we have, for any closed 1-form with compact support on S_{0}, α:

∫_{S_{0}} α ∧ Θ = ∫_{]0,i∞[} α ,

so Θ is the dual harmonic form to ]0, i∞[.

3. Automorphic realization

3.1. Fourier expansion of the Eisenstein series. We begin by recalling the Fourier expansion of the classical Eisenstein series (see for example [I]).

Let 𝔞, 𝔟 be cusps of Γ and σ = Re s > 1, we have, uniformly in z ∈ H, the implied constant depending on s and the group Γ,

E_{𝔞}(σ_{𝔟}z, s) = δ_{𝔞𝔟} y^{s} + ϕ_{𝔞𝔟}(s)y^{1−s} + O((1 + y^{−σ})e^{−2πy}) ,

where δ_{𝔞𝔟} = 1 if 𝔞, 𝔟 are equivalent, otherwise it vanishes and ϕ_{𝔞𝔟}(s) is a function defined with Kloosterman sums and Gamma factors.

3.2. Eisenstein series of weight 2 . We’ll need to investigate the Fourier expansion of E_{𝔞} for 𝔞 = ∞ and 0 to obtain:

Proposition 3.1. We have the following asymptotic behaviors:

1. at 𝔞 = ∞, E_{∞}(1, z) = (1 + O(1)1/y + O(e^{−2πy})) dz and E_{0}(1, z) = O(1/y) dz, (*)

2. at 𝔞 = 0,

E_{∞}(1, z) = O(1/y) dz and E_{0}(1, z) = (\frac{1}{ez^{2}} + O(1/y)) dz.

Proof.

To obtain the Fourier expansion of E_{∞} at 𝔞 = ∞, we use the formula (see for example [Fay]):

E_{∞,1} = y^{s} + y^{1−s}ψ_{0}(s) + ∑_{m≠0} ψ_{m}(s) W_{\frac{m}{|m|} ,s−1/2}(4π|m|y) e^{2πimx} ,

where W_{\frac{m}{|m|} ,s−1/2} are Whittaker functions and the Fourier coefficients are given by:

(n ≠ 0) ψ_{n}(s) = −π^{s} \frac{|n|^{s−1}}{Γ(s + \frac{n}{|n|})} ∑_{c>0} \frac{1}{c^{2s}} S_{1}(0, n, c)

ψ_{0}(s) = −π \frac{4^{s−1}Γ(2s − 1)}{Γ(s − 1)Γ(s + 1)}ϕ_{0}(s)

with S_{1}(0, n, c) Kloosterman sums and ϕ_{0}(s) = ∑_{c>0} \frac{1}{c^{2s}} ∑_{0≤d<c} 1 . Therefore, we have:

E_{∞}(s, z) = (y^{s−1} + y^{−s}ψ_{0}(s) + ∑_{n≠0} ψ_{n}(s) W_{\frac{m}{|m|} ,s−1/2}(4π|m|y)/y e^{2πimx}) dz .

The analytic continuation at s = 1 of ψ_{n}(s) is well known (see for example [I]). Here we only point out that s = 1 is a regular value for E_{∞}(s, z). In fact, the residue of ϕ_{0}(s) at s = 1 is given by lim_{s→1}(s − 1)ϕ_{0}(s) = 1/(π Vol(M)). As (1/Γ)(0) = 0, we deduce the analytic continuation at s = 1 of ψ_{0}(s). So lim_{s→1}(s − 1)E_{∞}(s, z) = lim_{s→1}(s − 1)ψ_{0}(s) = 0.

Moreover we make use of the asymptotic behavior to obtain the first equality in (*): W_{μ,s−1/2}(t) ≈ e^{−t/2} t^{μ}, as t → +∞.

Now, the Fourier expansion of E_{∞} at 0 is given in the form

E_{∞}(s, σ_{0}z) = ∑_{τ∈Γ_{∞}\Γσ_{0}} y(τz)^{s−1}(τ′z) dz

= ∑_{d>0} ∑_{c} ∑_{n∈ℤ} \frac{y^{s−1}(z)}{|d\sqrt{e} (z + n) − c/\sqrt{e}|^{2(s−1)}(d\sqrt{e} (z + n) − c/\sqrt{e})^{2}} dz

= (−y^{−s}π \frac{4^{s−1}Γ(2s − 1)}{Γ(s − 1)Γ(s + 1)} ∑_{d>0} \frac{1}{(d\sqrt{e})^{2s}} ∑_{c/\sqrt{e}(mod d\sqrt{e})} 1 + O(e^{−2πy})) dz

= (e^{s}y^{−s}ψ̃_{1}(s) + O(e^{−2πy})) dz

In a similar way we find the Fourier expansion of E_{0} at 0 and ∞.

E_{0}(s, σ_{0}z) = ∑_{τ∈Γ_{∞}\σ_{0}^{−1}Γσ_{0}} y(τz)^{s−1} (τ′z) dz

= (y^{s−1} + y^{−s} ∑_{b>0} \frac{1}{(be)^{2s}} ∑_{a(mod be)} 1 + O(e^{−2πy})) dz .

Now we remark that, with obvious notations,

E_{0}(s, z) = E_{∞}(s, σ_{0}z, σ_{0}Γσ_{0}^{−1}), z → ∞ .

3.3. The Kudla and Millson ’s construction of hyperbolic Eisenstein series.

In the Bourbaki seminar [G], P.Gérardin asks the following question: if η is the infinite geodesic ]p, q[, we can do the same construction as Kudla and Millson, i.e., the general term of the series (1) retains a meaning, does this series converge for Re s > 1, does it define an analytic function in s and admit an anlytic continuation at s = 1 which gives the harmonic dual form to η? Without loss of generality we suppose the two cusps p and q to be 0 and ∞ respectively and, as the lift of the geodesic, we take the imaginary axis. So the problem reduces to study the following series:

η̂^{s}(z) = \frac{1}{k(s − 1)} ∑_{γ∈Γ} γ ∗ [(\frac{y}{|z|})^{s−1} Im(z^{−1}dz)] = Im(θ^{s}(z)) ,

where

θ^{s}(z) = \frac{1}{k(s − 1)} ∑_{γ∈Γ} γ ∗ [(\frac{y}{|z|})^{s−1} \frac{dz}{z} ] ,

and k(s−1) = \frac{Γ(1/2)Γ(s/2)}{Γ(1/2 + s/2)}. The first part of the question has been yet answered (see

[F]). Now we want to prove the analytic continuation of η̂^{s} at s = 1. For this, first of all, we are going to show that θ^{s}(z) − i(E_{∞}(1, z) − E_{0}(1, z)) is square integrable. What we have to do is to investigate the Fourier expansion of θ^{s}, at each inequivalent cusp : 0 and ∞, and to show that y|θ^{s}(z)| is bounded. We have the following proposition:

Proposition 3.2. At ∞

θ^{s}(z) = (\frac{1}{i} + O(1/y)) dz ,

and at 0

θ^{s}(z) = (−\frac{1}{iez^{2}} + O(1/y)) dz .

Proof. We begin by estimating the constant term of the Fourier expansion of θ^{s} at the cusp ∞ . For this, we write for σ = Re s > 1 :

θ^{s}(z) = \frac{1}{k(s − 1)} ∑_{γ∈Γ_{∞}\Γ} \frac{y^{s−1}}{|cz + d|^{2(s−1)}(cz + d)^{2}} ζ(γz, s) dz ,

where

ζ(Z, s) = ∑_{n∈ℤ} \frac{1}{|n + Z|^{s−1}(n + Z)} .

We have, where we note I_{∞} the orbit of the identity modulo Γ_{∞}:

θ^{s}(z) = \frac{1}{k(s − 1)} [y^{s−1}ζ(z, s) + ∑_{γ∈Γ_{∞}\Γ−I_{∞}} \frac{y^{s−1}}{|cz + d|^{2(s−1)}(cz + d)^{2}} ζ(γz, s) ] dz

= \frac{1}{k(s − 1)} [y^{s−1}ζ(z, s) + ∑_{S_{z}−I_{∞}} \frac{y^{s−1}}{|cz + d|^{2(s−1)}(cz + d)^{2}} ζ^{∗}(γz, s)

+ ∑_{S_{z}−I_{∞}} \frac{y^{s−1}}{|cz + d|^{2(s−1)}(cz + d)^{2}} \frac{1}{|γz|^{s−1}(γz)}] dz

Here we introduced, S_{z} a set of orbit representatives modulo Γ_{∞} such that

| Re γz| ≤ 1/2 and ζ^{∗}(Z, s) = ∑_{n∈ℤ^{∗}} \frac{1}{|n + Z|^{s−1}(n + Z)} .

Now look at the Fourier expansion of ζ(z, s) at ∞. Poisson’s summation formula is here applicable with the result:

ζ(z, s) = ∑_{n=−∞}^{+∞} A_{n}(z, s), Im z > 0 ,

with

A_{n}(z, s) = ∫_{−∞}^{+∞} \frac{e^{−2πinv}}{(v + z)|v + z|^{s−1}} dv .

So we have

A_{0}(z, s) = ∫_{−∞}^{+∞} \frac{1}{(v + z)|v + z|^{s−1}} dv

= \frac{1}{y^{s−1}} ∫_{−∞}^{+∞} \frac{1}{(i + t)|i + t|^{s−1}} dt = \frac{1}{i y^{s−1}} ∫_{−π/2}^{+π/2} cos^{s−1}θ dθ
= \frac{k(s − 1)}{i y^{s−1}} .

Moreover, utilizing [R] p.127, there exists a constant C, independent of s and n, such that:

|A_{n}(z, t)| ≤ C \frac{e^{−π|n|y}}{y^{σ}} ,

Therefore

Lemma 3.1. Near ∞, ζ has the following behavior:

ζ(z, s) = \frac{k(s − 1)}{iy^{s−1}} + O(\frac{e^{−πy}}{y^{s}}) .

Return to the first term of θ^{s} in the sum: \frac{1}{k(s−1)}y^{s−1}ζ(z, s) dz. It can be written as:

\frac{1}{k(s − 1)}y^{s−1}ζ(z, s) dz = [\frac{1}{i} + O(\frac{e^{−πy}}{y^{s}})] dz .

Let us work on the investigation of the second term. For γ ∈ S_{z} − I_{∞} we have

| ∑_{n∈ℤ^{∗}} \frac{1}{|n + γz|^{s−1}(n + γz)}| ≤ ∑_{n∈ℤ^{∗}} \frac{1}{||n| − 1/2|^{σ}} = 2 ∑_{n≥1} \frac{1}{(n − 1/2)^{σ}} ,

and so

| ∑_{S_{z}−I_{∞}} \frac{y^{s−1}}{|cz + d|^{2(s−1)}(cz + d)^{2}} ζ^{∗}(γz, s)| ≤ ( ∑_{S_{z}−I_{∞}} \frac{y^{σ−1}}{|cz + d|^{2σ}} ) 2(2^{σ} − 1)ζ(σ) ,

with ∑_{n≥1}(n − 1/2)^{−s} = (2^{s} − 1)ζ(s). To estimate this sum we use the first paragraph 3.1:

∑_{Γ_{∞}\Γ−I_{∞}} \frac{y^{σ}}{|cz + d|^{2σ}} = O(y^{1−σ}) . (3)

Rewrite the last term in θ^{s} as follow and denote S_{z}^{∗} = S_{z} − I_{∞}:

∑_{S_{z}−I_{∞}} \frac{y^{s−1}}{|cz + d|^{2(s−1)}(cz + d)^{2}} \frac{1}{|γz|^{s−1}(γz)} = ∑_{S_{z}−I_{∞}} \frac{y^{s−1}}{|cz + d|^{(s−1)}(cz + d)|az + b|^{s−1}(az + b)} .

We have the inequalities:

| ∑_{S_{z}−I_{∞}} \frac{y^{s−1}}{|cz + d|^{(s−1)}(cz + d)|az + b|^{s−1}(az + b)}| ≤ ∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{s−1}}{|cz + d|^{2σ}} \frac{1}{|γz|^{σ}} ∑_{n∈ℤ} \frac{1}{|neγz + 1|^{σ}}

≤ ∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|cz + d|^{2σ}} \frac{1}{|γz|^{σ}} + ∑_{Γ_{0}\Γ} \frac{y^{σ−1}}{|cz + d|^{2σ}} \frac{1}{|γz|^{σ}} ∑_{n∈ℤ^{∗}} \frac{1}{|neγz + 1|^{σ}} .

First it follows that:

∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|cz + d|^{2σ}} \frac{1}{|γz|^{σ}} = ∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|cz + d|^{σ}|az + b|^{σ}}
≤ ( ∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|cz + d|^{2σ}} )^{1/2} ( ∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|az + b|^{2σ}} )^{1/2}

Now we have an estimation of (see (3)):

∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|cz + d|^{2σ}} ≤ ∑_{Γ_{∞}\Γ−I_{∞}} \frac{y^{σ−1}}{|cz + d|^{2σ}} ,

moreover,

∑_{Γ_{0}\S_{z}^{∗}} \frac{y^{σ−1}}{|az + b|^{2σ}} ≤ ∑_{Γ_{0}\Γ} \frac{y^{σ−1}}{|az + b|^{2σ}} ,

and we know it’s a O(1/y) near ∞.

The second sum to estimate is:

∑_{Γ_{0}\Γ} \frac{y^{σ−1}}{|cz + d|^{2σ}} \frac{1}{|γz|^{σ}} ∑_{n∈ℤ^{∗}} \frac{1}{|neγz + 1|^{σ}} = ∑_{Γ_{0}\Γ} \frac{y^{σ−1}}{|az + b|^{2σ}e^{σ}} ∑_{n∈ℤ^{∗}} \frac{1}{|n − σ_{0}γz|^{σ}} ,

with σ_{0}(Z) = −\frac{1}{eZ}. We can find a set of orbit representatives modulo Γ_{0} such that | Re(−σ_{0}γz)| ≤ 1/2, so the second term is majored by

∑_{Γ_{0}\Γ} \frac{y^{σ−1}}{|az + b|^{2σ}} e^{σ} 2(2^{σ} − 1)ζ(σ) .

Then we investigate the Fourier expansion at the cusp 0 . The method is similar and we don’t give all the details. We write

θ^{s}(z) = \frac{1}{k(s − 1)} ∑_{Γ_{0}\Γ} \frac{y^{s−1}}{|az + b|^{2(s−1)}(az + b)^{2} e^{s}} ζ(−σ_{0}γz, s) dz
= \frac{1}{k(s − 1)} (\frac{y^{s−1}}{|z|^{2(s−1)}z^{2} e^{s}} ζ(−σ_{0}z, s) dz
+ ∑_{Γ_{0}\Γ−I_{0}} \frac{y^{s−1}}{|az + b|^{2(s−1)}(az + b)^{2} e^{s}} ζ(−σ_{0}γz, s) dz) .

We easily compute the constant term of the Fourier expansion at 0 of ζ(−σ_{0}z, s):

ζ(−σ_{0}z, s) = − (\frac{k(s − 1)}{i(\frac{y}{e|z|^{2}} ^{s−1}} ) + O(e^{−π \frac{y}{e|z|^{2}}} ) .

Now we have the following estimations for the terms in the sum of θ^{s}:

| ∑_{Γ_{0}\Γ−I_{0}} \frac{y^{s−1}}{|az + b|^{2(s−1)}(az + b)^{2} e^{s}} ζ^{∗}(−σ_{0}γz, s)| ≤ ∑_{Γ_{0}\Γ−I_{0}} \frac{y^{σ−1}}{|az + b|^{2σ}} 2 ∑_{n≥1} \frac{1}{(n − 1/2)^{σ}} .

| ∑_{Γ_{0}\Γ−I_{0}} \frac{y^{s−1}}{|az + b|^{2(s−1)}(az + b)^{2} e^{s}} \frac{1}{(−σ_{0}γz)|σ_{0}γz|^{s−1}} | ≤ ∑_{Γ_{∞}\(Γ_{0}\Γ−I_{0})} \frac{y^{σ−1}}{|cz + d|^{σ}|az + b|^{σ}}

+ ∑_{Γ_{∞}\(Γ_{0}\Γ−I_{0})} \frac{y^{σ−1}}{|cz + d|^{2σ}} ∑_{n∈ℤ^{∗}} \frac{1}{|n + γz|^{σ}} .

The second term of the right hand side is majored by

∑_{Γ_{∞}\Γ} \frac{y^{σ−1}}{|cz + d|^{2σ}} 2 ∑_{n≥1} \frac{1}{(n − 1/2)^{σ}} ,

and the first term by

( ∑_{Γ_{∞}\Γ} \frac{y^{σ−1}}{|cz + d|^{2σ}} )^{1/2} ( ∑_{Γ_{0}\Γ−I_{0}} \frac{y^{σ−1}}{|az + b|^{2σ}} )^{1/2} .

By propositions (3.1) and (3.2), we conclude:

Proposition 3.3. The 1-forms θ^{s}(z) − 1/i(E_{∞}(1, z) − E_{0}(1, z)) and η̂^{s}(z) + Re(E_{∞}(1, z) − E_{0}(1, z)) are square integrables.

Now we recall some properties of η̂^{s} (see [F]):

Theorem 3.1. For Re s > 1, the Eisenstein series associated to the infinite geodesic η =]p, q[ converges uniformly on compact subsets, it represents a closed 1-form whose cohomology class is dual to η. Moreover in Re s > 1, it is an analytic function of s and satisfies the differential functional equation:

Δ η̂^{s} = s(1 − s)[η̂^{s} − η̂^{s+2}] .

Denote Φ^{s}(z) = η̂^{s}(z) + Re(E_{∞}(1, z) − E_{0}(1, z)), by the results (3.3) and (3.1), Φ^{s} is square integrable on M and satisfies:

Δ Φ^{s} = s(1 − s)[Φ^{s} − Φ^{s+2}] . (4)

Since Φ^{s} is square integrable on M, we can use the method develop in [K-M]. The 1-form Φ^{s} may be expanded in eigen-forms of Δ:

Φ^{s}(z) = ∑_{n=1}^{∞} a_{n}(s)ν_{n}(z) + ∑_{𝔞∈\{±0,±∞\}} \frac{1}{4πi} ∫_{−∞}^{+∞} h_{𝔞}(s, t)E_{𝔞}(1/2 + it, z) dt ,

where a_{n}(s) = ⟨Φ^{s}, ν_{n}⟩ , h_{𝔞}(s, t) = ⟨Φ^{s}, E_{𝔞}(1/2 + it, .)⟩. Remark that h_{− 𝔞}(s, t) = \overline{h_{𝔞}(s, t)}. We assume Δν_{n} = λ_{n}ν_{n}. By (4), ΔΦ^{s} is square integrable and we have:

⟨ΔΦ^{s}, ν_{n}⟩ = ⟨Φ^{s}, Δν_{n}⟩ = λ_{n}⟨Φ^{s}, ν_{n}⟩ ,
⟨ΔΦ^{s}, E_{𝔞}(1/2 + it, .)⟩ = ⟨Φ^{s}, ΔE_{𝔞}(1/2 + it, .)⟩ = (1/4 + t^{2})⟨Φ^{s}, E_{𝔞}(1/2 + it, .)⟩ .

So, by (4) we find the recurrence relations:

1. a_{n}(s) = \frac{s(s − 1)}{λ_{n} + s(s − 1)}a_{n}(s + 2)

2. h_{𝔞}(s, t) = \frac{s(s − 1)}{1/4 + t^{2} + s(s − 1)}h_{𝔞}(s + 2, t).

From this we deduce in particular, that Φ^{s} has a meromorphic continuation to Re s > 1/2 and s = 1 is a regular value, so

Theorem 3.2. The 1-form η̂^{s} has a meromorphic continuation to Re s > 1/2, with s = 1 a regular point and η̂ is the harmonic dual form to η.

Remark 3.1. In the case, the Hecke congruence group of level N, Γ_{0}(N) has no elliptic points, we obtain an automorphic realization of the modular symbol of Mazur ([Ma]). For example, for N = 4, let \overline{H} = H ∪ ℚ ∪ \{i∞\}, Y = Γ_{0}(4)\\overline{H} = Γ(2)\\overline{H} (noting

h_{2} = ( \begin{matrix}1 & 0 \\ 0 & 1/2 \end{matrix} ) we have h_{2} Γ_{0}(4) h_{2}^{−1} = Γ(2) and, if ∀γ ∈ Γ(2), f(γz) = f(z), then

setting g(Z) = f(h_{2} Z), the fonction g verifies ∀γ ∈ Γ_{0}(4), g(γZ) = g(Z)). Consider the linear functional on differential forms:

{i∞} = ∫_{]0,i∞[} : H^{0}(Y, Ω_{Y/ℂ}^{1}) → ℂ .

The modular symbol associated to i∞: {i∞}, is represented by

− Re(E_{∞}(1, z) − E_{0}(1, z)) = − Re(θ(0, z)^{4} dz)
= − Re lim_{s→1}(θ^{s}(z))

where the 1-form: θ^{s}(z)

= ( ∑_{{(m,n,p,q)∈ℤ^{4} \atop (1+2m)(1+2q)−4np=1}} \frac{1}{|(2m + 1)z + 2n|^{s−1}|2pz + (2q + 1)|^{s−1}} \frac{dz}{((2m + 1)z + 2n)(2pz + (2q + 1))}) .

References

[F] Falliero,T., Séries d’Eisenstein Hyperboliques et Capacité d’unions d’intervalles, Thèse, Université Bordeaux I (1998).

[Fay] Fay,J., Fourier coefficients of the resolvent for a Fuchsian group , Journal für die reine und angewandte Mathematik 293-294 (1977), 144-203.

[G] Gérardin,P., Formes automorphes associées aux cycles géodésiques des surfaces de Riemann hyperboliques, Séminaire Bourbaki 562 (1980).

[I] Iwaniec,H., Introduction to the Spectral Theory of Automorphic Forms, Biblioteca de la revista matematica iberoamericana, 1995.

[K-M] Kudla,S.and Millson,J., Harmonic differentials and closed geodesics on a Riemann Surface,

Invent.Math. 54 (1979), 193-211.

[KM1] Kudla,S.and Millson,J., The Poincaré dual of a geodesic algebraic curve in a quotient of the 2-ball, Can. J. Math. 33, vol.2, (1979), 485-499

[KM2] Kudla,S.and Millson,J., Geodesic cycles and the Weil representation I; quotients of hyperbolic space and Siegel modular forms, Compo. Math. 45, vol.2, (1982), 207-271.

[Ma] Mazur,B., Courbes elliptiques et symboles modulaires, Séminaire Bourbaki 414 (1972).

[M] Mumford,D., Tata lectures on Theta I, Progress in math. 28 (1983).

[R] Rademacher,H., Topics in Analytic Number Theory , Springer-Verlag Berlin Heidelberg 169 (1973).

[S] Siegel,C.L., Topics in Complex function theory,, Tracts in Mathematics Number 25, vol.2, Wiley-Interscience, 1971.

U.F.R. Mathématiques et Informatique

Université Bordeaux I

351, cours de la Libération , 33405 Talence Cedex

France

falliero@math.u-bordeaux.fr