Selberg zeta function and hyperbolic eisenstein series

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Selberg zeta function and hyperbolic eisenstein series

Thérèse Falliero

To cite this version:

Thérèse Falliero. Selberg zeta function and hyperbolic eisenstein series. 2019. �hal-01551102v2�

EISENSTEIN SERIES.

TH ´ER `ESEFALLIERO

1. Introduction

The relation between the geometric and the spectral properties of a Riemann surface (e.g., volume, periodic geodesics, etc., among the geometric objects, and eigenvalues, resonances, au- tomorphic functions, etc., among the spectral entities) is an important subject with a long, rich history.

Here we consider a family of degenerating geometrically finite hyperbolic surfaces (S_l) with one or more non separating geodesics or geodesics of a funnel being pinched.

The aim of the paper is to study the behavior of the hyperbolic Eisenstein series of (S_l) through degeneration (for results on degenerating Eisenstein series, see, for example [18], [19], [7], [8]), on the left of the critical axis {s ∈ C,Res = 1/2}. For purposes of illustration, it is desirable to restrict ourselves to the weight 0 case. We plan to devote one later paper to a treatment of more general weights.

Because of the behavior of the eigenvalues through degeneration (see [14]), it is not possible to define a good notion of convergence for hyperbolic Eisenstein series in any domain that intersects the critical axis (at least if the surfacesS_lare compact). To overthrow this problem we consider the product of the Selberg Zeta function and hyperbolic Eisenstein series which gives a holomorphic function on Res= 1/2 (except eventually ats= 1/2).

More precisely, letS_l= Γ_l\Hbe a family of geometrically finite hyperbolic surfaces degenerating to the surfaceSwith only one geodesic c_lbeing pinched. The group Γ_l contains the transformation σ_l(z) = e^lz corresponding to c_l and the left half-collar for c_l is in S_l\F₁. Let S₀ = Γ\H be the component ofS

(1) containing, in the non separating case, p, the cusp arising from the right half-collar forcl; q will denote the cusp arising from the left half-collar.

(2) containing, in the funnel’s geodesic boundary case, the compact core, the cusp will be denoted by ∞.

Let Zl(s) be the Selberg Zeta function of Sl, zl(s) these of the hyperbolic cylinder hσ_li\H and E_l(z, s) the hyperbolic Eisenstein series associated to c_l.

Let denote by

E_l^∗(z, s) = Z_l(s) z_l(s)

1

l^sEl(z, s). Consider two cases

(1) c_l is not separating,

(2) c_l is the geodesic boundary of a funnel.

We know by ([21], [4] and [5]) that the family (E_l^∗(., s))_l converges uniformly on every compact subsets of Res >1/2 and S0 to,

(1) Z₀(s)(E_p(., s) +E_q(., s)) in the separating case, (2) Z0(s)E∞(., s) in the funnel geodesic boundary case, whereZ0(s) corresponds to the Selberg Zeta function ofS0.

We summarize the main results:

(1) In the finite volume case:

2010Mathematics Subject Classification. Primary 30F30, 32N10, 47A10 ; Secondary 53C20, 11M36, 11F12.

Key words and phrases. Eisenstein series, Selberg Zeta function, Degenerating surfaces.

1

• the meromorphic family of weighted hyperbolic Eisenstein series (_l¹sE_l(., s))_l converges uniformly on every compact subsets of S0 and on 0<Res <1/2, to 0;

• the family (E_l^∗(., s))_l doesn’t converge on Res >0;

• in the non compact case, the scattering matrix and the (classical) Eisenstein series converge for Res <1/2;

• the weighted Selberg Zeta function Zl(s)

z_l(s), doesn’t converge on 0<Res <1/2 even if the S_l are non compact.

(2) In the infinite volume case, withc_l the geodesic boundary of a funnel:

• for Res > 1/2, 2(1−s) 6= 0,−1, ...,in particular for 1/2 < Res < 3/2, the relative scattering determinant, τ_l(s), tends to 1

2 sin^2π₂s asltends to 0;

• the weighted Selberg Zeta function doesn’t converge on 0<Res <1/2;

• the meromorphic family of weighted hyperbolic Eisenstein series (_l¹sE_l(., s))_l converges uniformly on every compact subsets of S0 and on 0<Res <1/2, to 0;

• the family (E_l^∗(., s))_lconverges uniformly on every compact subsets ofS₀ and Res >0, toZ₀(s)E∞(., s).

Remark 1.1. (1) In particular we answer a remark of Schulze ([21], p.122).

(2) We can also note the parallel with his result ([21], Theorem 40).

(3) The domains in the last two points of the preceding main results, are not the best, but in this introduction, we let them because our aim is just to show, we can pass the critical axis.

An application expected would be, in particular, to obtain some characterization of embedded eigenvalues. More precisely, an idea was to exploit the fact that (classical) Eisenstein series don’t have any pole on Res= 1/2, while hyperbolic Eisenstein series have simple poles on Res= 1/2.

In view of the summarized results, such application doesn’t exist in the finite volume case. In the infinite volume case, to achieve such a characterization we need, in particular, to control the multiplicity of a resonance. If we had such a result as Corollary 4.2 in [25], then an answer could be given.

2. Notations We recall here, briefly, some notations.

A family of degenerating geometrically finite hyperbolic surfaces consists of a surface M and a smooth family (g_l)_l>0 of Riemannian metrics that meet the following assumptions:

(1) The Riemannian manifoldM_l= (M, g_l) is a geometrically finite hyperbolic surface for each l.

(2) There are finitely many disjoint open subsets that are diffeomorphic to cylinders R\Z×Ji

where J_i ⊂R is a connected neighborhood of 0, of cuspidal and funnel types. A cuspidal type contains a cusp Ci, 1≤i≤nc, which is isometric to the half parabolic cylinderC∞= ([0,∞[)_r×(R/Z)x with the metricds²=dr²+e^−2rdx². A funnel type contains a funnelFj, 1 ≤j ≤n_f, which is isometric to a half hyperbolic cylinder Γ_lj\H, F_lj = (R⁺)_r×(R/Z)_x with the metric ds² =dr²+l_j²cosh²(r)dx² and l_j is the lenght of the geodesic boundary.

(3) There are finitely many disjoint open subsets C_l,i⊂M that are diffeomorphic to cylinders R\Z ×J_l,i where J_l,i ⊂ R is a connected neighborhood of 0 with the metric (x, a) 7→

(l_i(l)²+a²)dx²+ ((l_i(l)²+a²)⁻¹da² and l_i(l)→0 as l→0. The curvec_i =R\Z× {0}is a closed geodesic of length li(l).

(4) The complement of (C₁∪...∪C_nc)∪(F₁∪...∪F_nf)∪_iC_l,i, where we may have someF_j ⊂ C_l,i is relatively compact.

(5) On M₀ := M\ ∪_ic_i, the metrics g_l converge smoothly to a hyperbolic metric g₀ as l→ 0.

M0 is a possibly non connected hyperbolic surface that contains a pair of cusps for each i.

We set M =K∪C∪F ,whereC =C1∪...∪Cnc and F =F1∪...∪Fnf.

The standard funnel of diameter l > 0, F_l, is the half hyperbolic cylinder hz 7→ e^lzi\H, F_l = (R⁺)_r ×(R/Z)_x with the metric ds² = dr² +l²cosh²(r)dx², with hz 7→ e^lzi the cyclic group generated by the transformationz7→e^lz.

The standard cusp C∞ is the half parabolic cylinder Γ∞\H,C∞= ([0,∞[)_r×(R/Z)_x with the metric ds² =dr²+e^−2rdx².

A standard collar for a geodesic of lengthlis a cylinder isometric tohz7→e^lzi\C withC ={z= re^iθ,1 ≤ r ≤ e^l, l < θ < π−l} ⊂ H with the restriction of the hyperbolic metric. There is a constantk₀ (the short geodesic constant) such that each closed geodesic on M_l of length at most k0has a neighbourhood isometric to the standard collar and each cusp for M_l has a neighbourhood isometric to the standard cusp; furthermore, the collars for short geodesics and the cusp regions are all mutually disjoint.

We define the function r as the distance to the compact coreK and the functionρ by

(1) ρ(r) =

2e^−r inF e^−r inC ,

withρextended to a smooth non vanishing function insideK in some arbitrary way. We will adopt (ρ, t)∈(0,2]×R/l_jZas the standard coordinates for the funnelF_j, wheretis the arc length around the central geodesic at ρ= 2.

For the cusp, our standard coordinates (ρ, t)∈(0,1]×R/Zare based on the model defined by the cyclic group Γ∞. The cusp boundary isy= 1, so thaty=e^r and ρ= 1/y. We sett≡x (modZ).

Here we consider a family of surfaces S_l = Γ_l\H degenerating to the surface S with only one geodesic c_l being pinched to form a pair of cusps on S. As is common, we realize H as {z ∈ C,Imz >0}. Writing z=x+iy, then the hyperbolic metricds² and hyperbolic Laplacian acting on function can be expresses as

(2) ds²= dx²+dy²

y² and ∆_l=y² ∂²

∂x² + ∂²

∂y²

,

Γ_l containing the transformation σ_l(z) = e^lz corresponding to c_l. We denote by p and q the two cusps of S 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.Many mathematicians investigated, by various parameterizations, degeneration of hyperbolic surfaces and the asymptotic behavior of several functions. Basically those various parameterizations turn out to be almost the same powerful tools (see Remark??). We will start with [25]. LetK_lbeS_l minusC_lthe standard collar forc_l. There exist homeomorphisms fl fromSl\c_l toS, withfl tending to isometriesC²-uniformly on the compact coreKl⊂Sl; define πl = f_l⁻¹. Let S0 = Γ\H be the component of S containing p and conjugate Γ to represent the cusp by the translation w 7→ w+ 1. In the following, p = ∞. In a similar way, without loss of generality we can representq by 0.

Here, we will opt, as definition of the hyperbolic Eisenstein series, that which one can find in [15].

The exponent of convergenceδof a Fuchsian group Γ is defined to be the abscissa of convergence of the Dirichlet series

δ= inf{s >0,X

T∈Γ

e^{−sd(z,T w)}<∞}

for somez, w∈H, whered(z, w) denotes the hyperbolic distance fromz∈H tow∈H.

Letδl be the exponent of convergence for Γl, then for Res > δl:

(3) E_l(z, s) = X

hσ_li\Γ_l

sin^s(θγz).

A very quickly way to verify the convergence for Res > δl, of the right hand series is to use Proposition 13 of [16], which remains true in the infinite volume case.

Recall the definition of the (classical) Eisenstein series. We will assimilate a parabolic point with its cusp. The stabilizer of a cusp a is an infinite cyclic group generated by a parabolic motion,

Γa={γ∈Γ :γa=a}=hγ_ai,

say. There exists a σ_a ∈ SL₂(R), called a scaling matrix of the cusp a, such that σ_a∞ = a, σ⁻¹_a γ_aσ_a=

1 1 0 1

;σ_ais determined up to composition with a translation from the right. For Y ≥1, the semi-strip C∞(Y) = {z =x+iy,0 < x <1, y ≥Y} is mapped into the cuspidal zone C_p(Y) =σ_pC∞(Y) with length 1/Y horocycle.

The Eisenstein series for the cusp ais then defined by Ea(z, s) = X

Γa\Γ

y(σ⁻¹_a γz)^s ,

wheresis a complex variable with Res > δ_l. E_a(z, s) has Fourier expansion at the cuspb (see for example, [13] Theorem 3.4, [10] Proposition 8.6, [17] Section 2.2, [23] Section 3.1, [6])

(4) Ea(σbz, s) =δa by^s+ϕa by^1−s+X

n6=0

ϕa b(n, s)Ws(nz), W_s(z) is expressed in terms of the Whittaker function,W_0,s−¹

2 by: W_s(z) =e^2iπnxW_0,s−¹

2

(4π|n|y), z=x+iy.

Whittaker functionsW_0,s−¹

2,M_0,s−¹

2 are defined in terms of the modified Bessel functions I and K, in the following way (see [1], Sections 9.6-9.7; [6], p.172):

W_0,s−¹

2(t) = t

π ¹₂

K_s−¹

2

t 2

, M_0,s−¹

2(t) = Γ

s+1 2

4^s−12t¹²I_s−¹

2

t 2

.

K_s−1/2(t) the MacDonald-Bessel function, has the following asymptotics, independent of the parameters,

K_s−¹

2(t)∼ rπ

2te^−t, ast→+∞, I_s−¹

2(t)∼ e^t

√

2πt, ast→+∞. Moreover we have then the asymptotic behaviors:

W_0,s−¹

2(t) ∼ e^−t/2 t→ ∞;

(5)

M_0,s−¹

2(t) ∼ Γ(s+ 1 2)4^s−12

√π e^t/2 t→ ∞. (6)

The Eisenstein series initially defined as a serie for Res > 1 can be continued meromorphically to the alls-plane. The poles of Ea(z, s) in Res >1/2 are among the poles ofϕa a(s) and they are simple.

The Selberg Zeta function is defined for Res > δl by an absolutely convergent product Z_l(s) = Πcz(lc, s), wherez(lc, s) = Π^∞_k=0(1−e^−(s+k)lc)².

The first product ranges over the set of all unoriented, primitive, closed geodesicscof the surface, wherelcdenotes its length. Zl(s) extends meromorphically tos∈C(see for example, [2], Chapter 10). Note that z(l_c, s) is an entire function with zeros {n∈Z, n ≤0}. When c=c_l, the pinched geodesic, we will denote byz_l(s) =z(l_cl, s).

We review basic material on the expansion of eigenfunctions. Our essentials references are [6], [10].

A fundamental domain inH for the cyclic sungroup Γl generated by σl is the semi annulus:

(7) z=re^iθ 1≤r > e^l,0< θ < π

which is homeomorphic to the annuluse^−2π2/l<|t|<1 under the conformal projection t=z^2πil :H−→C− {0}.

We consider the Fourier expansion of an eigenfunction f, ∆_lf +s(1−s)f = 0, for the hyperbolic Laplacian ∆l,f invariant for σl.

f(z) =X

n

fn(θ)r^¯n, ¯n= 2iπn/l in the coordinates (7), then

(8) d²fn

dθ² +

s(1−s) sin²θ + ¯n²

fn= 0. Settingfn(θ) =gn(u), u=icotθ, then

d²g_n

du² + 2u u²−1

dg_n du +

s(1−s)

u²−1 −n¯² 1 (u²−1)²

g_n= 0.

Fay observes that solutions can be given in terms of the Gauss hypergeometric function F. We follow him ([6], Formula (87)) with the

Definition 2.1. We let

(9) N_sⁿ(θ) = (sinθ)^sexp [iθ(s+ ¯n)]F

s+ ¯n, s,2s;−2isinθe^iθ be the unique solution to(8) for which

(10) lim

θ→0θ^−sN_sⁿ(θ) = lim

θ→π(π−θ)^−sN_sⁿ(π−θ) = 1. N_sⁿ(θ) is analytic for all 2s6=−1, ....

To keep in mind:

Notation 2.1. We will denote R_l(s) the resolvent operator defined for Res >1/2, s6∈[1/2,1] by R_l(s) = (∆_l+s(1−s))⁻¹.

Its kernel will be denoted by G^l_s(z, w).

In the following, we will need the

Definition 2.2. A family (f_n) of meromorphic functions on a domain O inC is called bounded in O if

(1) ∃(z_m)_m a discrete subset in O,

(2) ∀K compact set ofO\{z_m}, ∃n(K),∀n≥n(K), fn has no pole in K, (3) M_K = sup_n≥n(K)(sup_z∈K|f_n(z)|)<+∞.

Definition 2.3. A family (fn) of meromorphic functions on a domain O inC is called bounded in O if

(1) ∃(z_m)m a discrete subset in O,

(2) ∀K compact set of O\{z_m}, ∃n(K), ∀n ≥ n(K), f_n has no pole in K, and the family (f_n)_n≥n(K) converges uniformly on K.

For degenerating surfaces the convergence of small eigenvalues, that is eigenvalues in the range [0,1/4) is well understood ([3], [12], [14], [25], [21]). This result, throughout this paper, is implicitly used.

3. The compact case

To be more precise, we return first to the case of a family of degenerating compact Riemann surfaces (S_l) and we show, with the notations of Section 2.1

Proposition 3.1. The weighted hyperbolic Eisenstein series (_l¹sE_l(z, s))_l converges uniformly on every compact subsets of S0 and 0<Res <1/2, to zero, as a meromorphic family.

Before proving this proposition, we need to recall some results on hyperbolic Eisenstein series.

In the compact caseδl= 1 and the hyperbolic Eisenstein series is first given as a convergent series for Res >1 (Formula (3)). The analytic continuation of E_l has been carried on in [?], [20], [15].

Succently speaking it follows from the relation that links the hyperbolic Eisensein series to the resolvent:

∆_lE_l(z, s) +s(1−s)E_l(z, s) =s²E_l(z, s+ 2).

In the compact case there exists a complete orthonormal basis {ψ_j} for L²(S_l) such that ∆_lψ_j = λjψj, with 0 =λ0 < λ1≤λ2 ≤...→ ∞.

The resolvent R_l(s) : L²(S_l) → L²(S_l) has a meromorphic continuation to the whole complex plane, more precisely it’s analytic for s(1−s) 6= λ_j where λ_j is an eigenvalue of the Laplacian, for λ 6= λ_j, G^l_s(z, w) = P∞

n=0

ψn(z)ψn(w)

λ−λ_n . The hyperbolic Eisenstein series admits the spectral expansion

E_l(z, s) =

∞

X

j=0

a_j,l(s)ψ_j(z). The coefficienta_j,l is given by the formula

a_j,l(s) =√

πΓ((s−¹₂ +ir_j)/2)Γ((s−¹₂ −ir_j)/2) Γ(s/2)²

Z

cl

ψ_j(z)ds(z),

here we have written the eigenvalueλ_j of the eigenfunctionψ_j in the formλ_j = ¹₄+r²_j, (ψ_j)j≥0 is a complete orthonormal system of smooth eigenfunctions. In conclusion ([20]), the function E_l(z, s) has meromorphic continuation to the whole s plane. At a regular point s₀, E_l(z, s) is square integrable on Γl\H. The eventual poles are located at −2n+sj where sj(1−sj) is an eigenvalue of the automorphic Laplacian and n∈ N. They are simple, except in the eventual case s_j = 1/2, of order 2. The pole ats= 1 is simple with residue

2l vol(Γ_l\H).

What can leave us pressing Proposition 3.1, are the following facts:

Lemma 3.1. (1) The family (_l¹sE_l(π_l(.), s))_l of meromorphic functions, is bounded in 0 <

Res <1/2.

(2) The residus at the s∈]0,1/2[ corresponding to small eigenvalues tend to zero.

(3) The expression of the scalar product h_l1−s¹ E_l(z,1−s),_l¹s¯E_l(z,¯s)i.

Proof. To show point (1), the method used for Res >1/2, the method in [4], p. 367-370, applies directly.

For point (2), it is necessary to emphasize on the dependence on l of the eigenvalues. Let’s write s^±_j(l) = ¹₂ ±r_j(l) whith 0< r_j(l)≤1/2.

Let λ_j 0 ≤ λ_j < 1/4 the small eigenvalues on S₀. We know (see for example [3]) that for l sufficiently small, for 0≤j ≤M,λj(l)→λj, equivalently rj(l)→rj.

We use [15] Theorem 2.

Res_s⁺

j(l)(1

l^sΓ(s/2)²Γ(s−1/2)⁻¹E_l(z, s)) = 1 l^s+j(l)

Γ(s⁺_j/2(l))²Γ(rj(l))⁻¹Res_s⁺

j(l)E_l(z, s)

= 2√ πψ_j(z)

Z

cl

ψ_j(z)ds(z), which can represent, depending on the multiplicity of λ_j(l), a finite sum.

We have a similar expression for the residu at s⁻_j (l). We deduce the relation Res_s⁻

j(l)(1

l^sEl(z, s)) =l^s+j(l)−s

− j(l)

Γ(s⁻_j(l)/2)⁻²Γ(s⁺_j (l)/2)²Γ(rj(l))⁻¹Γ(−r_j(l)) Res_s⁺

j(l)(1

l^sEl(z, s)) ;

but we have for 0≤j≤M, that Res_s⁺

j(l)(_l¹sE_l(z, s))→Res_s⁺

j(E∞+E₀), then the result.

For point (3), let’s write s= ¹₂ +r with Rer >1/2. We compute h 1

l^1−sE_l(z,1−s), 1

l^¯sE_l(z,¯s)i= 1 l

X

j

a_j,l 1

2 −r

a_j,l 1

2 + ¯r

= π

lΓ²(^1/2+r₂ )Γ²(^1/2−r₂ )

×

X

j

Z

c_l

ψj(z)ds(z)

2

Γ

−r+ir_j 2

Γ

−r−ir_j 2

Γ

r−ir_j 2

Γ

r+ir_j 2

. We use know the relation−wΓ(w)Γ(−w) = _sin(πw)^π to find

h_l1−s¹ E_l(z,1−s),_l¹¯sE_l(z,s)i¯ = ^π3

lΓ²(^1/2+r₂ )Γ²(^1/2−r₂ )

P

j

R

clψj(z)ds(z)

2 2

r²+r²_j

1

cos(πr)+cos(πirj); and we recognize

Z

cl

G^l_s(z, w)ds(z)

2

L²

=X

j

|R

c_lψj(z)ds(z)|² r²+r_j² .

We also use [6], Formula (99) that connects in particular for any fuchsian group the hyperbolic Eisenstein series to the following integral of the kernel of the resolventIl=Re^l

1 G^l_s(z, iy⁰)dlny⁰. Because this result will be important for the following sections, we write

Lemma 3.2. For any Fuchsian group Γl, for s, Res > −1/2, not a pole of G^l_s and z not in the standard collar, we have

Z e^l 1

G^l_s(z, iy⁰)dlny⁰=−4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)El(z, s) +H(z, s)

withH(z, s) =O(1)El(z, s+ 2),O(1)bounded independentely of Γl and ofz in πl(C), C a compact of S0.

Proof. More precisely, [6], Formula (99) gives, forz not in the standard collar Z e^l

1

G^l_s(z, iy⁰)dlny⁰ = −4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)e^πi2s X

hσ_li\Γ_l

(sin argγz)^se^{∓is(argγz−π2)} .F(s, s,2s; 2

1∓icot argγz).

where the± is chosen as ∓(argγz−^π₂)>0. We can then verify that e^{∓is(argγz−π2)}=

e^−isπ/2(cos argγz+isin argγz)^s if argγz < π/2 e^−isπ/2(−cos argγz+isin argγz)^s if argγz ≥π/2 LetC be a compact ofS0,π_l(C)⊂S_l\C_l.

∀γ ∈ hσ_li\Γ_l, if argγz < π/2, then 0<argγz ≤l; if argγz≥π/2, 0< π−argγz≤l. In every case 0<sin argγz≤sinl.

Moreover

cos argγz+isin argγz = 1 + 2 sinargγz

2 (−sinargγz

2 +icosargγz 2 ) cos(π−argγz) +isin(π−argγz) = 1 + 2 sinθ

2(−sinθ

2+icosθ

2) withθ=π−argγz . Now, if we have 0 < α≤ l0, cos(l0/2)≤ cos(α/2)<1 and sin^α₂ =O(sinα). We deduce that, in the case argγz < π/2, cos argγz+isin argγz= 1 +O(sin argγz/2) = 1 +O(sin argγz) and, in the case argγz≥π/2, with alwaysθ=π−argγz, cos(π−argγz) +isin(π−argγz) = 1 +O(sinθ/2) = 1 +O(sinθ) = 1 +O(sin argγz).

Then e^πi2se^{∓is(argγz−π2)} = (1 +O(1) sin argγz)^s = 1 +O(1) sin argγz, O(1), s−holomorphic, bounded independently of Γ_l and zinπ_l(C),C compact ofS0, for alls. More precisely

e^πi2se^{∓is(argγz−π2)}= 1 + 2ssinθ

2(−sinθ

2 +icosθ

2) +O((sin argγz)²), (11) withθ=π−argγz if argγz≥π/2, θ= argγz if not.

It is well-known (see for example, [10], p. 603) that F(a, b;c;Z)/Γ(c) defines a holomorphic function for (a, b, c, z)∈C×C×C×(C−[1,∞[). ThenF(s, s,2s;Z)/Γ(2s) is s−holomorphic and the eventual poles of F(s, s,2s;Z) are in {n/2, n ∈Z, n≤0}. Moreover we easily verify that the set of poles is{n/2, n∈Z, n <0}.

In the same way _{1∓icot arg}² _γz = 2_{sin arg}^{sin arg}_{γz∓icos arg}^γz _γz approaches 0 and F(s, s,2s; 2

1∓icotargγz) = 1 +s 2

2 sin argγz sin argγz∓icos argγz

+O((sin argγz)²)

with a uniform O-term for sbounded, Res >−1/2, bounded independently of Γ_l and z inπ_l(C), C compact ofS₀.

Calculating thescoefficient in the producte^πi2se^{∓is(argγz−π2)}F(s, s,2s;_1∓icot²_argγz), we obtain, with the same definition of θ as in (11): 2(sin^θ₂)²e^iθ. Then e^πi2se^{∓is(argγz−π2)}F(s, s,2s;_1∓icot²_argγz) = 1+O(1)(sin argγz)²with a uniformO(1)-term forsbounded, Res >−1/2, bounded independently of Γl and z inπl(C),C compact ofS0.

Then, for Res >1

(12) I_l+4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)E_l(z, s) =−4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)X

Γl\Γ

(sin argγz)^sO(1)(sin argγz)². The right member of the last equality is a convergent and analytic series for Res >−1. Then by analytic continuation we can deduce that for Res >−1/2,s not a pole ofG^l_s

I_l = −4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)E_l(z, s)−4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)X

Γl\Γ

(sin argγz)^sO(1)(sin argγz)²

= −4^s−1 π

Γ⁴(s) Γ²(2s)N_s⁰(π

2)El(z, s) +H(z, s).

whereH(z, s) = 0(E_l(z, s+ 2)).

Remark 3.1. This Lemma 3.2 will be improved in the infinite volume case.

We are now able to prove Proposition 3.1.

Proof. In the compact case we have Gs =G1−s, so we obtain, from Lemma 3.2, in particular for 0<Res <1/2,

(13) E_l(z, s) =K(s)E_l(z,1−s) +O(E_l(z,3−s)) +O(E_l(z, s+ 2)),

withK(s) depending only ons. Then dividing by 1/l^s and using the known results on the conver- gence of hyperbolic Eisensteins series (see [4]) we have

E_l(z, s)

l^s =O(l^1−2s), and the result.

Remark 3.2. In view of Lemma 3.2, we can also notice the formula in [16], Proposition 11.

When S_l is compact, you have to give a sense to the convergence of (_l¹sE_l(z, s))_l) on a domain that intersect Res = 1/2 because the eigenvalues of Sl cluster at every point of the continuous spectrum [¹₄,+∞[ ofS₀ as l →0 (see for example [14]). To remove this problem we multiply the hyperbolic Eisenstein series by the Selberg Zeta function to obtain holomorphic functions. More precisely, we recall (see for example [9], p.72) that in the compact case

(1) Zl(s) is actually an entire function;

(2) Z_l(s) has ”trivial” zeross=−k,k≥1, with multiplicity (2g−2)(2k+ 1);

(3) s= 0 is a zero of multiplicity 2g−1;

(4) s= 1 is a zero of multiplicity 1;

(5) the nontrivial zeros ofZl(s) are located at ¹₂±irj of order forrj 6= 0, equal to the dimension of theλ_j-eigenspace except fors= 1/2 whereZ_l(s) has a zero of multiplicity equal to twice the dimension of the (1/4)-eigenspace.

It follows that for Res >0, poles of the hyperbolic Eisenstein series are zeros of the Selberg Zeta function and the multiplicity of a pole ofE_l(z, s) is less or equal the multiplicity of the corresponding zero ofZl(s), then

Lemma 3.3. The product Zl(s)El(z, s) is a holomorphic function in Res >0.

Corollary 3.1. The familyE_l^∗(z, s) = ^Z_z^l(s)

l(s) 1

l^sEl(z, s) cannot converge for0<Res <1/2as l→0.

Proof. To see this, from Proposition 3.1 and the preceding study (13) , _l¹sE_l(z, s) = O(l^1−2s) on 0<Res <1/2. Moreover we know (see [24]) that on 0<Res <1/2,

(14) Z_l(s)

z_l(s) = Z_l(1−s)

z_l(1−s)O(l^4s−2). Then ^Z_z^l(s)

l(s) 1

l^sEl(z, s) =O(l^2s−1). Simply observe that for 0<Res <1/2 andl→0 the factorl^2s−1

is divergent.

4. The non-compact finite volume case

Here again we show that the meromorphic family, (_l¹sE_l(z, s)) tends to zero withlon 0<Res <

1/2. For this purpose we will follow the same method as the preceding case.

Doing this we will find again the convergence of the scattering matrix in the finite volume (non compact) case on every compact of Res >1/2\{s₀, ..., s_M}, where theses_j correspond to the small eigenvalues of the limit surface and this will enable us to lift a condition of Schulze [21], p.122.

In this case S_l, being a noncompact finite-volume hyperbolic surface, has finitely many cusps and Eisenstein series associated to them. The Laplacian on ∆l has absolutely continuous spectrum [1/4,∞) with multiplicity the number of cusps ofS_l. The discrete spectrum consists of finitely many eigenvalues in [0,1/4): λn=sn(1−sn), wheresn= ¹₂+irn, ˜sn= ¹₂−irn,sn∈[¹₂,1]∪[¹₂,¹₂+i∞).

There are examples with infinitely many embedded eigenvalues in [1/4,∞). The theory of Eisenstein series and their analytic continuation furnishes the continuous spectrum. Without loss of generality we can assume that there is only one cusp p with E_p^l(z, s) the associated Eisentein series. The constant term in the Fourier expansion of the Eisenstein series, ϕ_l(s) is meromorphic in C. Its only poles in Res ≥ 1/2 are in (¹₂,1]; these poles {ρ₀, ρ₁, ..., ρ^l_Me} are finite and simple. They coincide with the poles ofE_p^l(z, s) in Res≥1/2. The residues at these poles furnish solutions to the eigenvalue problem, called the residual spectrum ofS_l. The poles of ϕ_l(s) in Res <1/2 yield resonances .

L²(S_l) = H_res+H_cusp+H_cont

= X

0≤n≤M_e^l

hψ_ni+ X

n>M_e^l

hϕ_ni+H_cont

∆_lψ_n+ρ_n(1−ρ_n)ψ_n= 0, ∆_lϕ_n+c_n(1−c_n)ϕ_n= 0.

The orthogonal complement in L²(S_l) of the continuous and residual spectrum is the cuspidal space. It is invariant under ∆_l, and the resolvent is compact when restricted to the cuspidal space.

The resolvent R_l(s) : L²(S_l) → L²(S_l) has a meromorphic continuation to the whole complex plane. More precisely ([10], p.250), the singularities of the kernel of the resolvent correspond to {s_n,s˜n} ∪ {1/2} ∪ {the poles of ϕ_l}. {s_n,s˜n} are simple poles with the possible exception of the pole ats= 1/2.

The kernel of the resolvent is given for Res >1/2 byG^l_s(z, w) =P∞ n=0

ψn(z)ψn(w)

λ−λ_n +_2π¹ R+∞

0

Ep(z,¹₂+ir)Ep(w,¹₂−ir) s(1−s)−(¹

4+r²) dr.

We return now to the study of the hyperbolic Eisenstein series. The hyperbolic Eisenstein series admits the spectral expansion

El(z, s) =

∞

X

j=0

aj,l(s)ψj(z) + 1 4π

Z +∞

−∞

a_1/2+ir,l(s)E_p^l(z,1/2 +ir)dr . The coefficienta_j,l is given by the formula

a_j,l(s) = √

πΓ((s−¹₂ +ir_j)/2)Γ((s−¹₂ −ir_j)/2) Γ(s/2)²

Z

cl

ψ_j(z)ds(z) a_1/2+ir,l(s) = √

πΓ((s−¹₂ +ir)/2)Γ((s−¹₂ −ir)/2) Γ(s/2)²

Z

cl

E_p^l(z,1/2 +ir)ds(z) The eventual poles are located at the points

(1) s= 1/2±ir_j−2n, n∈Nand they are simple except in the eventual cases_j = 1/2, of order 2.

(2) s=ρ−2n, wheren∈Nandt=ρis a pole of the Eisenstein seriesE_p^l(z, t) with Re(ρ)<1/2.

We will prove that

Proposition 4.1. _l¹sE_l(z, s) = O(l^1−2s) on 0 < Res < 1/2, such that s(1−s) belongs to the resolvent set of the Laplacian onS0.

Before we need

Lemma 4.1. On Res <1/2, as a family of meromorphic function, ϕ_l(s) tends to ϕ₀(s)

Proof. We know the convergence of the family ofs-meromorphic functions (E_p^l(π_l(w), s))_ltoE_p⁰(w, s) uniformly on any compact subset of S0 (see [19], [7] for Res >1, [21] and [11] for Res >1/2).

AsR1

0 E_p^l(z, s)dx=y^s+ϕ_l(s)y^1−swe deduce the convergence of (ϕ_l(s)) toϕ₀(s) on every compact of Res > 1/2\{s₀, ..., s_Me} (see also [10], Proposition 12.5). By Hurwitz’s theorem, the zeros of ϕl converge to those ofϕ0. Then, for Res >1/2

1 ϕ_l(s)

is a meromorphic family converging to 1

ϕ0(s).

Now for Res <1/2, Re(1−s)>1/2 and for l≥0,ϕ_l(s) = 1/ϕ_l(1−s), so the conclusion.

Corollary 4.1. E_p^l(z, s) for Res <1/2, tends to E_p⁰(z, s).

Proof. of Proposition 4.1. In the finite volume non compact case, the resolvent is no more symetric and we have the relation

Gs(z, iy⁰) =G1−s(z, iy⁰)− 1

2s−1E_p^l(z, s)E_p^l(iy⁰,1−s). Hence we have for 0<Res <1/2

(15) Z e^l

1

G_s(z, iy⁰, l)dlny⁰= Z e^l

1

G1−s(z, iy⁰, l)dlny⁰− 1

2s−1E_p^l(z, s) Z e^l

1

E_p^l(iy⁰,1−s)dlny⁰. For 0 < Res <1/2,

Z e^l 1

E_p^l(iy⁰,1−s)dlny⁰ = lO(1), see [21] Definition 19, 20 pp. 138-139 and Theorem 30 p. 147.

Dividing (15) byl^sand using, like in the compact case, Lemma (3.2), the conclusion follows from

the preceding study.

As in the compact case we multiply the hyperbolic Eisenstein series by the Selberg Zeta function to obtain meromorphic functions, holomorphic on Res > 0\{1 −ρk,0 ≤ k ≤ M_e^l,1/2}. More precisely, we recall (see for example [10], p.498) that in the finite volume non compact case

(1) Z_l(s) is meromorphic on the s-plane;

(2) Z_l(s) has ”trivial” zeross=−k,k≥1.

(3) the nontrivial zeros ofZ_l(s) are located at pointssj, on the real line Res= 1/2,sj = ¹₂±ir_j of multiplicity, for r_j 6= 0, equal to the dimension of theλ_j-eigenspace.

(4) for s = 1/2 Z_l(s) has a zero or pole of multiplicity equal to twice the dimension of the (1/4)-eigenspace minus the number of inequivalent cusps ¹₂Tr[I−ϕl(¹₂)];

(5) the nontrivial zeros ofZ_l(s) are located for Res >1/2 at pointssj and ρ_k, 0≤k≤M_e^l in ]¹₂,1] of multiplicity equal to the dimension of the corresponding eigenvalue;

(6) it has additional non-trivial zeros in Res <1/2 at the poles of ϕl(s) with the same multi- plicity.

(7) Zl(s) has poles at s=−l+ 1/2,l≥1, of order 1.

In order to write the functional equation, we need the entire function G∞ which is defined in terms of the Barnes G-function ([2],p. 44)

G(s+ 1) = (2π)^s/2e−s/2−(γ+1)s²/2Π^∞_k=1 1 + s

k k

e^−s+s2/(2k), (16)

G∞(s) = (2π)^−sΓ(s)G(s)² (17)

We then have a relation between Zl(s) and Zl(1−s) in the finite volume and non compact case (see for example [2], p. 45, also [10], formula (5.7) p.499)

Z_l(s) Zl(1−s) =

G∞(s) G∞(1−s)

−χ(S_l)

.

It allows us to answer the remark in the same manner as in the compact case, because we have again ^Z_z^l(s)

l(s) = ^Z_z^l(1−s)

l(1−s)O(l^4s−2).

Lemma 4.2. The quotient Z_l(s)

z_l(s) cannot converge forRes < ¹₂ as l→0.

Then, from Proposition (4.1), ^Z_z^l(s)

l(s) 1

l^sE_l(z, s) =O(l^2s−1) and Corollary 4.2. The meromorphic family,E_l^∗(z, s) = ^Z_z^l(s)

l(s) 1

l^sE_l(z, s)cannot converge for0<Res <

1/2 as l→0.

Remark 4.1. In the finite volume case, looking at the particular point s = 1/2, we could already conclude that the convergence of the family ^Z_z^l(s)

l(s) 1

l^sE_l(z, s)to the desired series is not true in general:

in the case Z0 has a pole at s= 1/2 which occurs when twice the dimension of the 1/4 eigenvalue space minus 1/2 Tr[I−Φ(1/2)]<0.

5. The infinite volume case

For the remainder of this section, assume, for simplicity, that Γ is a Fuchsian group of the second kind whose fundamental domainDhas no cusps and only one free sideD ∩R= [1, e^l] corresponding to the cyclic subgroup generated by σl(z) =e^lz. The corresponding funnel is denoted byFl. A collarC forσ_l is the cylinder

C ={z=re^iθ,1≤r ≤e^l, cl < θ < π−cl}/hz→e^lzi

for a choice of constantc. Forlsmall the standard collar,c= 1, embeds inSlto give a neighborhood of σ_l, and has: area ≈2, width≈2 log¹_l, and each boundary of length ≈1.

Here also, we adapt the notations in Borthwick [2].

The Poisson operator is the map

ES_l(s) :C^∞(∂S_l)−→C^∞(S_l)

which maps f ∈C^∞(∂F_l) to

EF(s)f(ρ, t) = Z l

0

E^fl(s;ρ, t, t⁰)f(t⁰)dt⁰; where

E^fl(s, z, t⁰) = (1−2s) lim

ρ⁰→0ρ^0−sGs(z, z⁰) ;

and S_l(s) is the corresponding scattering matrix. The funnel scattering operator SF_l is defined in the same way.

The relative scattering determinantτSl(s) is a Fredholm determinant. S_rel^l (s)−I is a smoothing operator, with S_rel^l (s) =S_Fl(1−s)S_l(s).

τ_Sl(s) = detS_rel^l (s) = Π^∞_k=1

1 +λ_k(S_rel^l (s)−I) whereλ_k(S_rel^l (s)−I) are the eigenvalues of A_l:=S_rel^l (s)−I.

Theorem 5.1. ForRes >1/2 we have

τSl(s)→ 1

2(sinπ/2s)², l→0, .

We recall the following eigenfunction expansion (see [6] formula (98)’)

(18) G^l_s(z, z⁰) = 1

1−2s X

n∈Z

F_n^l(z, s)N_sⁿ(ϕ⁰)|z⁰|^n¯ when argz >argz⁰=ϕ⁰ and z⁰ in the collar; here

(19) F_n^l(z, s) = 4^s−1 π

Γ(s)²Γ(s+ ¯n)Γ(s−n)¯ Γ(2s)Γ(2s−1)

1 l

X

Γ_l\Γ

N_s⁻ⁿ(π−argγz)|γz|^−¯n, with Res >1 to ensure convergence of the series.

We recall know the standard Fourier expansion for the resolvent of the limit surface S0= Γ\H, Γ containing a parabolic subgroup Γ∞ generated by the transformationz7→z+ 1 (see [6] formula (73), [10], p. 42) . For y⁰ > y, the resolvent has an eigenfunction expansion

(20) Gs(z, z⁰) = 1

1−2s X

n

Fn(z, s)W_s−¹

2

(4π|n|y⁰)e^2πinx0 where then^th Fourier coefficient is given by

(21) F_n(z, s) = 1 4π

Γ(s)

|n|Γ(2s−1) X

γ∈Γ∞\Γ

M_s−1

2(4π|n|Imγz)e^{−2πinReγz}, with the convention that the constant term in (20) isy^01−sE∞(z, s).

We begin with estimating the Fourier coefficients of the resolvent of the laplacian onS_l. We are now going to establish the convergence of the Fourier coefficients for Res >1. Following straight fully the method to study the degeneration of hyperbolic Eisenstein series like in [4], or [?], we find that for Res >1

Proposition 5.1. (1) The family (l^1−sF₀^l(., s))_l converges uniformly on compact subsets ofS₀ and on compact subsets of Res >1 to ^4s−1_π Γ(2s)Γ(2s−1)^Γ(s)4 1

l^s+1E∞(., s).

(2) For n6= 0, the family (l^se^2|n|π

2

l F_n^l(., s))_l converges uniformly on compact subsets of S₀ and on compact subsets of Res >1 to

Proof. To study the left side of the collar forσ_l, use the change of variables lζ =−log(−z), with the principal branch; then

(22) argz=π−lImζ, |z|= exp(−lReζ).

We briefly recall the method and refer to [4], [?], for more details.

Conjugate Γ_l by the mapζ to obtain ˜Γ_l acting on S_l={ζ,0<Imζ < π/l}.

Given > 0, denote by G the set of cosets and representatives for hz 7→ z+ 1i\Γ such that sup ImA(F) < for [A]6∈ G and letR_l be the corresponding cosets of hz 7→ z+ 1i\Γ˜_l with the corresponding representatives. The setG is finite and so isR_l. We have

X

hσ_li\Γ_l

N_s⁰(π−argγz) = X

hz→z+1i\Γ˜l−R_l

N_s⁰(lImδζ) +X

Rl

N_s⁰(lImδζ). By formula (9), as l → 0, N_s⁰(lImδζ) ∼ l^sIm^sδζ; thus for _l¹s

P

hσ_li\Γ_lN_s⁰(π −argγz), in the last equality the second sum is uniformly close to the corresponding term for E∞(., s).

For the first sum recall that

N_s⁰(θ) = (sinθ)^sexp(iθs)F

s, s,2s;−2isinθe^iθ .

Withθ=π−argγz =lImδζ, and that, forδ ∈ hz→z+ 1i\Γ˜l−R_l, Imδζ ≤2, thenlImδζ→0 uniformly in δ∈ hz→z+ 1i\Γ˜_l−R_l, we obtain that

1 l^s

X

hz→z+1i\Γ˜l−R_l

N_s⁰(lImδζ) =O(^s−1) see also [7]. Now

F₀^l(z, s) = 4^s−1 π

Γ(s)⁴ Γ(2s)Γ(2s−1)

1 l

X

Γl\Γ

N_s⁰(π−argγz) and by the preceding change of variables (l^s−1F₀^l(., s))l tend to

4^s−1 π

Γ(s)⁴ Γ(2s)Γ(2s−1)

X

δ

Im(δζ)^s

uniformly on compact subsets ofS0 and on compact subsets of Res >1.

In a similar way

N_sⁿ(θ) = (sinθ)^sexp [iθ(s+ ¯n)]F

s+ ¯n, s,2s;−2isinθe^iθ

We have the formula

1

1−2sF_n^l(z, s)N_sⁿ(ϕ⁰) = 1 l

Z e^l 1

G^l_s(z, z⁰)|z⁰|^−¯ndln|z⁰| with the change of variable t⁰= ln|z⁰|we obtain

1

1−2sF_n^l(z, s)N_sⁿ(ϕ⁰) = 1 l

Z l 0

G^l_s(z, z⁰)e^−2iπnt0/ldt⁰. Using the change of variables in Schulzet⁰ =−lx⁰ we write

1 l

Z l 0

G^l_s(z, z⁰)e^−2iπnt0/ldt⁰ = Z 0

−1

G^l_s(z, z⁰)e^2iπnx0dx⁰

With the convergence of the resolvent (see Schulze prop 6, p.126), we have the convergence of R1

0 G^l_s(z, z⁰)e^−2iπnRew0dRew⁰ to (1) for n6= 0,F_n(w, s)W_0,s−1 2

(4π|n|y⁰) (2) for n= 0, _1−2s¹ P

γ(Imγw)^s= _1−2s¹ E∞(w, s)