Sup-norm Bounds for Automorphic Forms and Eisenstein Series

ALM 19, pp.407–444 and International Press Beijing–Boston

Sup-norm Bounds for Automorphic Forms and Eisenstein Series

Dedicated to Stephen Kudla’s 60th Birthday

Jay Jorgenson

and J¨urg Kramer

Abstract

For any finite volume hyperbolic Riemann surface M of positive genus, letµcan

denote the (1,1)-form associated to the canonical metric andµshypthe (1,1)-form associated to the hyperbolic metric, both forms scaled such that the volume of M is one. LetdM be the sup-norm of the quotientµcan/µshyp. In [13] optimal bounds fordM through covers were obtained. In the present paper, we revisit this problem and give an alternative proof for the optimal bounds established in [13].

Our approach here takes as its starting point the key relation from [14] involving as the main term an integral over time of a heat kernel onM. By suitably decomposing this integral, one obtains bounds in terms of classical parabolic Eisenstein series, the hyperbolic Eisenstein series defined in [21], and elliptic Eisentein series defined in [24].

2000 Mathematics Subject Classification: 11M36, 11F11, 11F72, 58G25, 30F35.

Keywords and Phrases: Eisenstein series, automorphic forms, heat kernels, Green’s functions.

1 Introduction

1.1. Associated to any finite volume hyperbolic Riemann surface M of positive genus, there are a number of natural metrics to consider. The hyperbolic metric onM comes from the point of view of the uniformization theorem, which allows us to representM as the quotient space Γ\H, where Hdenotes the upper half-plane and Γ is a Fuchsian subgroup of the first kind of PSL2(R). The hyperbolic metric

∗Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, NY 10031, U.S.A., jjorgenson@mindspring.com. The first author acknowledges support from grants from the NSF and PSC-CUNY.

†Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany, kramer@math.hu-berlin.de. The second author acknowledges support from the DFG Graduate SchoolBerlin Mathematical Schooland from the DFG Research Training Group Arithmetic and Geometry.

is the unique metric which is compatible with the complex structure ofM and is complete with constant curvature equal to negative one. The canonical metric on M comes from the point of view of complex geometry, where one uses the Abel- Jacobi map to embed the compactificationM ofM into its Jacobian Jac(M), and then a metric on M is obtained through the pull-back of the natural flat K¨ahler metric on Jac(M), when the Jacobian is viewed as a complex torus. From the definition of the Abel-Jacobi map, one can easily show that the canonical metric equals (up to scaling) the average of the squared absolute values of an orthonormal basis of holomorphic weight two forms onM. Let us use the notationµshyp and µcanto denote the (1,1)-forms associated to the hyperbolic and canonical metrics, respectively, where each metric has been scaled so thatM has volume one.

A natural topic to investigate is to compare the hyperbolic and canonical met- rics. Aside from intrinsic interest, being a question that could have been posed nearly 150 years ago, the problem possesses additional significance due to modern developments in mathematics. Specifically, Proposition 3.1 of [27] states a relation involving the degree of a modular parameterization of an elliptic curve and the Petersson norm of certain holomorphic modular forms. Since the canonical metric is an average value of such norms, one can relate the Petersson norm of such forms to the quotientµcan/µshyp. The literature contains many articles which document the relation between degrees of modular parameterizations and fundamental ques- tions in number theory, such as the abc-conjecture. Therefore, we can take the question of understanding bounds for holomorphic modular forms as having both classical significance as well as modern importance.

1.2. With the above discussion, let us focus on the following problem. For any finite volume hyperbolic Riemann surfaceM, let

dM := sup

z∈M

µcan(z) µshyp(z).

The main result in [1] is the boundd_Γ0(N)\H =O(N^2+ε); in [23] the bound was improved to d_Γ0(N)\H =O(N^ε). The methods of proof in [1] and [23] employ a significant amount of arithmetic associated to the congruence subgroup Γ0(N). In [13], the authors use geometric considerations, namely heat kernel analysis and the maximum principle for holomorphic functions, and prove the following result.

If M is a finite degree cover of M0, then dM = OM0(1); in particular, for the modular curves, the main theorem in [13] asserts thatdΓ0(N)\H=O(1), which can be shown to be optimal.

1.3. In the present article, we revisit the problem of proving the optimal bound fordM derived in [13] by studying one of the key identities from our previous work.

Specifically, in [14], an identity is proved which relatesµcanto µshyp and an inte- gral involving the hyperbolic heat kernel onM. The analysis in the present paper involves a detailed study of the hyperbolic heat kernel onM, sufficient to reprove the main result from [13] by examining the aforementioned relation betweenµcan

and µshyp. The hyperbolic heat kernel on M is written as a sum over the uni- formizing group Γ, and this series is decomposed into terms involving parabolic, hyperbolic, and elliptic terms. In each case, the series of terms can be bounded

involving special values of non-holomorphic Eisenstein series ats= 2, wheresis the spectral parameter. By now, non-holomorphic Eisenstein series attached to parabolic subgroups are classical objects in mathematics. Kudla and Millson de- fined non-holomorphic Eisenstein series attached to hyperbolic subgroups; see [21].

With [21] as motivation, we defined non-holomorphic Eisenstein series attached to elliptic subgroups in [17]. The results in the unpublished article [17] have been absorbed into the present paper.

Bounds for dM in terms of various Eisenstein series at s = 2 stem from the group theoretic decomposition of a sum over Γ of a function, a method that is well-known from the development of the Selberg trace formula. The series obtained by studying the parabolic and elliptic terms are bounded by finite sums of special values of Eisenstein series, in effect because there are a finite number of non-conjugate (in Γ), primitive classes of parabolic and elliptic elements in Γ. By contrast, there are an infinite number of non-conjugate (in Γ), primitive, hyperbolic elements, so the resulting bound for dM in terms of special values of hyperbolic Eisenstein series needs to be studied in depth. Beyond the formal bound fordM, we utilize two additional results. First, we employ the spectral expansion of scalar-valued hyperbolic Eisenstein series, which differ slightly from the form- valued hyperbolic Eisenstein series defined in [21]. The spectral expansion we need is proved in [18]. Second, we use results from [31] which studies the asymptotic behavior of periods of functions when integrated over primitive, closed geodesics of increasing length. To be honest, we require a version of the main theorem of [31] which asserts the dependence on the error term in the asymptotic formula proven in terms of the eigenvalue of the function whose period is under study; see Subsection 2.7.

As stated, the analysis of the present paper yields another proof of the main theorem of [13] using hyperbolic geometry, Eisenstein series of various types (parabolic, hyperbolic, and elliptic), asymptotics for periods of eigenfunctions of the Laplacian, as proved in [31], and the main identity from [14] relatingµcan to hyperbolic geometric quantities.

1.4. We find it fascinating that the special value of the Eisenstein series which appears is at the values= 2 of the spectral parameter. The same special value of parabolic Eisenstein series has appeared elsewhere. In [29], the author is studying various problems in the theory of infinite energy harmonic maps, and numerous bounds involve the parabolic Eisenstein series ats= 2. The results from [29] are used in [30] when studying the disappearance of cusp forms through deformations, which itself is related to the outstanding problem of determining the existence of L²-eigenfunctions onM, i.e., the Phillips-Sarnak conjecture. In [28], the authors study curvature forms of metrics on the moduli space of Riemann surfaces. If the surfaces are non-compact, the curvature forms include a term involving the parabolic Eisenstein series ats= 2.

1.5. This article is organized as follows. In Section 2, we establish notation and recall necessary background results from throughout the literature. In Section 3, we define the three types of non-holomorphic Eisenstein series associated to M; parabolic, hyperbolic, and elliptic. In Section 4, we extend the key identity from

[14] to the general setting of finite volume, hyperbolic Riemann surfaces which may have cusps as well as elliptic fixed points. As stated above, the identity involves the heat kernel onM, which can be written as a sum of three sub-series corresponding to the parabolic, hyperbolic, and elliptic elements in Γ. These three sub-series are studied separately in Sections 5, 6, and 7. Finally, in Section 8, we put together the main computations from Sections 5, 6, and 7 in order to establish the optimal bound fordM.

1.6. We are very happy to dedicate the present paper to Stephen S. Kudla in honor of his 60th birthday, and to call attention to his article [21]. The study of the arithmetic and geometry of hyperbolic and elliptic Eisenstein series seems to be a fruitful area of investigation, and we first learned of hyperbolic Eisenstein series when James Cogdell suggested to the first-named author to study [21]. Many important theorems in mathematics have been obtained from non-holomorphic, parabolic Eisenstein series, and each result can be posed as a question to be studied for either hyperbolic or elliptic Eisenstein series. We refer the reader to the articles [5], [6], [19], and [24] for recent results.

We thank the editors of the present volume for the opportunity to contribute our article in honor of Stephen Kudla. We thank Anna von Pippich for many suggestions which helped with the exposition of the article, as well as D. Hejhal and S. Zelditch for informative mathematical discussions relating to this article.

Also, the first-named author thanks James Cogdell for the suggestion to study [21], which begin the mathematics in this article. We anticipate that the mathematical influence of [21] will continue from here.

2 Background material

2.1. Hyperbolic metric. Let Γ⊆PSL2(R) be any Fuchsian group of the first kind acting by fractional linear transformations on the upper half-planeH:={z∈ C|z=x+iy , y >0}. Let M be the quotient space Γ\Handg the genus ofM. Denote by T the set of torsion points (i.e., elliptic fixed points) of M and by C the set of cusps ofM; we putt:=|T |andc:=|C|. Ifp∈ T, we letmp denote the order of the torsion pointp; we setmp = 1, ifpis a regular point ofM. Locally, away from the torsion points, we identifyM with its universal coverH, and hence, denote the points onM \ T by the same letter as the points onH.

We denote by ds²_hyp the line element, resp. byµhyp the volume form corre- sponding to the hyperbolic metric on M, which is compatible with the complex structure ofM and has constant curvature equal to minus one. Locally onM\ T, we have

ds²_hyp(z) :=dx²+ dy²

y² , resp. µhyp(z) := dx∧dy y² .

We denote the hyperbolic distance between z, w ∈ M by disthyp(z, w) and we

recall that the hyperbolic volume volhyp(M) ofM is given by the formula vol_hyp(M) = 4π

µ

g−1 + c 2+1

2 X

p∈T

³ 1− 1

mp

´¶

.

We define the (1,1)-form corresponding to the scaled hyperbolic metric, which measures the volume ofM to be one, by

µshyp:= 1

volhyp(M)µhyp.

We note thatµhyp andµshyp are smooth differential forms onM \ T.

In addition to the cartesian coordinates x, y, we will also make use of eu- clidean and hyperbolic polar coordinates. The euclidean polar coordinates ρ = ρ(z),θ=θ(z) of the pointzcentered at the origin are related tox, yin the way

x:=e^ρcos(θ), y:=e^ρsin(θ). (1) The line element ds²_hyp, resp. volume formµhyp are expressed as follows in these coordinates

ds²_hyp(z) = dρ²+ dθ²

sin²(θ) , resp. µhyp(z) = dρ∧dθ sin²(θ) .

The hyperbolic polar coordinates%=%(z),ϑ=ϑ(z) of the pointz centered ati are given by

%(z) := disthyp(i, z), ϑ(z) :=∠(L, Te z), (2) where Le denotes the positive y-axis and T_z is the tangent line to the geodesic passing throughiandzat the pointi. The line element ds²_hyp, resp. volume form µhyp are expressed as follows in these coordinates

ds²_hyp(z) = d%²+ sinh²(%) dϑ², resp. µhyp(z) = sinh(ϑ) d%∧dϑ.

The hyperbolic Laplacian ∆hyp is locally, onM \ T, given by

∆hyp:=−y² µ ∂²

∂x² + ∂²

∂y²

¶ .

From [22], p. 10, we have for any smooth functionf onM dzd^c_zf =− 1

4π(∆hypf)µhyp.

2.2. Hyperbolic heat kernels. Fort >0 andρ≥0, we define KH(t;ρ) :=

√2e^−t/4 (4πt)^3/2

Z∞

ρ

r e^−r2/t

pcosh(r)−cosh(ρ)dr.

The hyperbolic heat kernel onHis defined by KH(t;z, w) :=KH

¡t; disthyp(z, w)¢

(z, w∈H).

The hyperbolic heat kernel onM is now defined by Khyp(t;z, w) :=X

γ∈Γ

KH(t;z, γw) (z, w∈M).

We note that the hyperbolic heat kernel satisfies the heat equation, i.e., µ

∆hyp+ ∂

∂t

¶

Khyp(t;z, w) = 0.

We end this subsection by recalling the following heat kernel estimates:

Khyp(t;z, w)− 1

volhyp(M)=OM

¡e^−cM;z,wt¢

(z, w∈M;t→ ∞), (3) Khyp(t;z, w) =OM

¡e^−cM;z,w/t¢

(z, w∈M;z6=w;t→0), (4) Khyp(t;z, z)−KH(t; 0) =OM

¡e^−cM;z/t¢

(z∈M;t→0); (5) here cM;z,w refers to a positive constant depending on M and z, w; furthermore we letc_M;z:=c_M;z,z. The constant c_M;z,w can be uniformly bounded away from zero whenz andware restricted to any compact subsets ofM, which is the only situation we will consider when employing the above bound. The bound (3) can be derived from [11], p. 103, Theorem 7.3, formula (7.15), applied toKhyp(t;z, w) for fixedw∈M; the bounds (4) and (5) follow from [4], p. 154, formula (45).

2.3. Hyperbolic Green’s functions. Forz, w∈Hands∈Csatisfying Re(s)>

1/2, the free-space Green’s function onHis defined by (see [9], p. 31, taking into account that our normalization differs from Hejhal’s by a factor of−4π)

gH,s(z, w) := Γ(s)² Γ(2s)

à 1−

¯¯

¯¯z−w z−w¯

¯¯

¯¯

2!s

F Ã

s, s,2s; 1−

¯¯

¯¯z−w z−w¯

¯¯

¯¯

2! ,

where Γ(s) is the Gamma function andF(a, b, c;w) is Gauss’ hypergeometric func- tion. The hyperbolic Green’s function onHis now defined by

gH(z, w) :=gH,1(z, w).

The well-known formula forF(1,1,2;w) yields the useful simple formula gH(z, w) =−log

ï¯

¯¯z−w z−w¯

¯¯

¯¯

2!

. (6)

The hyperbolic Green’s function onHis related to the hyperbolic heat kernel on Hthrough the formula

gH(z, w) = 4π Z∞

0

KH(t;z, w) dt. (7)

Forz, w∈M and s∈Csatisfying Re(s)>1, the free-space Green’s function on M is now defined by the series (see [9], p. 31)

ghyp,s(z, w) :=X

γ∈Γ

gH,s(z, γw).

Fors∈Cwith Re(s)>1, this series is absolutely and locally uniformly convergent onH×Has long as (the lifts of)z andwdo not lie in the same Γ-orbit (see [9], Proposition 6.2). For fixedz, w∈M, the series defines a holomorphic function in the half-plane{s ∈C|Re(s)>1}, which admits a meromorphic continuation to alls∈Cwith a pole at s= 1 (see [9], p. 250). The hyperbolic Green’s function ghyp(z, w) onM (z, w∈M;z6=w) is defined as the constant term in the Laurant expansion ofghyp,s(z, w) ats= 1. The hyperbolic Green’s function onM is related to the hyperbolic heat kernel onM through the formula

ghyp(z, w) = 4π Z∞

0

µ

Khyp(t;z, w)− 1 vol_hyp(M)

¶ dt.

The heat kernel estimates (3) and (4) imply that the above integrand is integrable wheneverz6=w.

We conclude this subsection by recalling the characteristic properties of the hyperbolic Green’s function onM. Forz, w∈M, z6=w,z /∈ T, ands∈Cwith Re(s)>1, the free-space Green’s function onM satisfies the differential equation (see [9], p. 250)

dzd^c_zghyp,s(z, w) =s(s−1)ghyp,s(z, w)µhyp(z), which gives rise to the differential equation

dzd^c_zghyp(z, w) +δw(z) =µshyp(z)

for the hyperbolic Green’s function onM; here z, w∈M with z /∈ T. Next, the hyperbolic Green’s function onM satisfies the normalization condition

Z

M

g_hyp(z, w)µ_hyp(z) = 0 (w∈M), which follows from

Z

M

Khyp(t;z, w)µhyp(z) = 1 (w∈M).

The third property to be mentioned states that the hyperbolic Green’s function on M is bounded for z 6= w and that the function ghyp(z, w) +mplog|z−w|² is bounded as z approaches the point w = p. The last property to be recalled states that the hyperbolic Green’s function onM inverts the dzd^c_z-operator; more precisely, iff is a bounded function onM with

Z

M

f(z)µhyp(z) = 0,

then

Z

M

ghyp(z, w)dzd^c_zf(z) =f(w) (w∈M).

The analysis given in [9] readily extends to prove that d_zd^c_zg_hyp,s(z, z) = X

γ∈Γ γ6=id

d_zd^c_zg_H,s(z, γz)

forz∈M ands∈Csatisfying Re(s)>1. This shows in particular that the series on the right-hand side is absolutely and locally uniformly convergent in the range under consideration. Ultimately, our work will extend this equality tos= 1 in a conditionally converging sense (see Section 8).

2.4. Canonical metric. Assumingg >0, we denote byS2(Γ) theg-dimensional C-vector space of cusp forms of weight 2 with respect to Γ equipped with the Petersson inner product

hf, hi:=

Z

M

f(z)h(z) dx∧dy;

heref, h∈S2(Γ). Choosing an orthonormal basis{f1, . . . , fg}ofS2(Γ), we define the (1,1)-formµcan corresponding to the canonical metric ofM by

µcan(z) := 1 g· i

2 Xg

j=1

|fj(z)|²dz∧d¯z.

Elementary linear algebra shows thatµcanis independent of the choice of the cho- sen orthonormal basis ofS2(Γ). Further computations using the classical Riemann- Roch theorem show that µcan is non-vanishing, and hence defines a metric on M\ T.

2.5. Canonical Green’s functions. The canonical Green’s function gcan(z, w) on M (z, w ∈ M; z 6= w) is a smooth function on (M ×M)\∆(M) with a logarithmic singularity along the diagonal ∆(M). It is uniquely characterized by the two properties

dzd^c_zgcan(z, w) +δw=µcan(z) (z, w∈M;z /∈ T), (8) Z

M

g_can(z, w)µ_can(z) = 0 (w∈M). (9)

The existence and uniqueness of gcan(z, w) is obtained as follows: Consider the compactificationM ofM by adding the cusps. After reuniformization one obtains a compact Riemann surface M⁰ of genus g with the same underlying complex structure asM. The existence and uniqueness of the canonical Green’s function

onM⁰satisfying the analogues of properties (8) and (9) is now well-known. Trans- ferring the resulting function back toM and restricting it toM, yields the desired functiong_can(z, w).

Canonical Green’s functions on quotient spaces of genus zero having elliptic fixed points can be explicitly written down using affine coordinates; see [22], p. 26, in the compact and smooth setting. For the sake of brevity, we omit the general formulas from the present discussion.

2.6. The prime geodesic theorem. LetH(Γ) denote a complete set of rep- resentatives of the Γ-conjugacy classes of the maximal hyperbolic subgroups of Γ. If H ∈ H(Γ), then H =hγ_Hi with generator γ_H uniquely determined up to inversion. The real number `H denotes the length of the closed geodesic LH on M determined byγH.

The prime geodesic counting function π(u) is defined for u ∈ R>1 by the formula

π(u) := X

H∈H(Γ) e^`H<u

1.

Equivalently, the quantity π(u) counts the number of non-conjugate, primitive, hyperbolic elements of Γ such that the corresponding geodesics have length less than log(u). From [8], p. 46, Proposition 2.11, we recall that the prime geodesic counting function can be “trivially” bounded asπ(u) =O(u).

For any eigenvalueλj of the hyperbolic Laplacian ∆hyp satisfying 0≤λj <

1/4, we set

s_j :=1 2 +

r1 4−λ_j;

we note that 1/2< sj≤1. In addition, we define for any 0< ε≤1/4 analogously sε:=1

2 + r1

4−ε . Recalling the logarithmic integral

li¡ u^sj¢

:=

u^sj

Z

2

dξ log(ξ) , the prime geodesic theorem states

π(u) = X

0≤λj<1/4

li¡ u^sj¢

+O³

u^3/4log(u)^−1/2´

foru >2, where the implied constant depends solely onM (see [10], p. 474).

2.7. Periods of eigenfunctions. Letψbe any smooth, bounded function onM and letCdenote any bounded, continuous path onM. The period ofψalongCis

defined to be the integral

Z

C

ψ(z) dshyp(z).

A generalization of the prime geodesic theorem can be considered by studying the counting function

π(u;ψ) := X

H∈H(Γ) e^`H<u

Z

LH

ψ(z) dshyp(z).

The asymptotic behavior ofπ(u;ψ) asutends to infinity is studied in detail in [31].

Specifically, the following statement is proven. Letψbe either a non-constant,L²- normalized eigenfunction of the hyperbolic Laplacian with eigenvalueλ=s(1−s), or the parabolic Eisenstein series with spectral parameter s = 1/2 +ir, hence λ= 1/4 +r²(see Subsection 3.1 for more details). Then, there existsε=ε(Γ)>0 such that the asymptotic formula

π(u;ψ) =Oλ

¡u^1−ε¢

(10) holds asu→ ∞.

In fact, a stronger statement than (10) is given on p. 5 of [31]: Specifi- cally, an asymptotic expansion of the counting function π(u;ψ) is proven out to Oλ(u^19/20), with lead terms expressed in terms of certain integrals involvingψand eigenfunctions as well as constants which depend on small eigenvalues. Elementary considerations, as discussed on p. 85 of [31], bound these constants in the form O(r^ke^πr/2), and the main results from [25] bound the integrals involving ψ and the eigenfunctions associated to the small eigenvalues byO(e^−πr/2). Combining these results, the finite series which appears in the expansion of (10) on p. 5 of [31] has a bound of the formO(λ^k) for somek.

With the above discussion, we can assume in the sequel that the dependence of the estimate (10) with respect toλcan be bounded in the formO(λ^k) for some k. These considerations, which were provided to us by S. Zelditch, constitute the crucial step in providing all details in proving this statement. We will leave a complete proof analysis of the method of proof in [31] which utilizes the bounds from [25] to the interested reader.

3 Eisenstein series

3.1. Parabolic Eisenstein series. LetP(Γ) denote a complete set of represen- tatives of the Γ-conjugacy classes of the maximal parabolic subgroups of Γ; we note that the setP(Γ) is finite. If P ∈ P(Γ), then P =hγPi with generator γP

uniquely determined up to inversion. There existsσP ∈PSL2(R) such that σ⁻¹_P γPσP =

µ1 1 0 1

¶

=:γ∞ ⇐⇒ γP =σPγ∞σ_P⁻¹. (11)

Fors ∈Cwith Re(s) >1, theparabolic Eisenstein series Epar,P(z, s) associated toP is defined by the series

Epar,P(z, s) := X

η∈P\Γ

Im¡

σ_P⁻¹ηz¢_s

. (12)

The parabolic Eisenstein series (12) is absolutely and locally uniformly convergent for z ∈ H and s ∈ Cwith Re(s) > 1; it is invariant under the action of Γ and satisfies the differential equation

¡∆hyp−s(1−s)¢

Epar,P(z, s) = 0.

For proofs of theses facts and further details, we refer to [9] or [20].

3.2. Hyperbolic Eisenstein series. Let H(Γ) denote a complete set of rep- resentatives of the Γ-conjugacy classes of the maximal hyperbolic subgroups of Γ. If H ∈ H(Γ), then H =hγ_Hi with generator γ_H uniquely determined up to inversion. There existsσH ∈PSL2(R) such that

σ⁻¹_H γHσH=

µe^{`H/2</sup> 0 0 e<sup>−`H/2}

¶

=:γ0,H ⇐⇒ γH =σHγ0,Hσ_H⁻¹; (13) the real number`Hdenotes the length of the closed geodesicLHonM determined byγH.

Fors∈Cwith Re(s)>1, the hyperbolic Eisenstein series Ehyp,H(z, s) asso- ciated toH is defined by the series

Ehyp,H(z, s) := X

η∈H\Γ

sin¡

θ(σ⁻¹_H ηz)¢_s

(14)

using the polar coordinates (1). The hyperbolic Eisenstein series (14) is absolutely and locally uniformly convergent forz∈Hands∈Cwith Re(s)>1; it is invariant under the action of Γ and satisfies the differential equation

¡∆hyp−s(1−s)¢

Ehyp,H(z, s) =s²Ehyp,H(z, s+ 2). (15) Denoting byLeH the lift ofLH to the universal coverH, we have the relation

sin¡

θ(σ_H⁻¹z)¢

·cosh¡

disthyp(z,LeH)¢

= 1, and the hyperbolic Eisenstein series (14) can be rewritten as

Ehyp,H(z, s) = X

η∈H\Γ

cosh¡

disthyp(ηz,LeH)¢_−s . For proofs of theses facts and further details, we refer to [21].

In the sequel we will have to make use of the spectral expansion of the hyperbolic Eisenstein series which was established in [18]. Theorem 4.1 therein

states

Ehyp,H(z, s)

= X∞

j=0

aj,H(s)ψj(z) + 1 4π

X

P∈P(Γ)

Z∞

−∞

a_1/2+ir,H,P(s)Epar,P(z,1/2 +ir) dr . (16)

The coefficientaj,H(s) is given by the formula aj,H(s) =√

π Γ((s−1/2 +irj)/2)Γ((s−1/2−irj)/2) Γ(s/2)²

Z

LH

ψj(z) dshyp(z);

here we have written the eigenvalueλj of the eigenfunctionψj in the form λj = 1/4 +r_j². An analogous formula holds for the coefficienta_1/2+ir,H,P(s); it is given at the end of the proof of Theorem 4.1 in [18].

3.3. Elliptic Eisenstein series. LetE(Γ) denote a complete set of representa- tives of the Γ-conjugacy classes of the maximal elliptic subgroups of Γ; we note that the setE(Γ) is finite. IfE∈ E(Γ), thenE=hγEiwith generatorγE uniquely determined up to inversion. There existsσE ∈PSL2(R) such that

σ_E⁻¹γEσE=

µ cos¡ π/mE

¢ sin¡ π/mE

¢

−sin¡ π/mE

¢cos¡ π/mE

¢

¶

=:γi,E ⇐⇒ γE=σEγi,Eσ_E⁻¹; (17) the natural numbermEdenotes the order of the torsion pointtEonM determined by the maximal elliptic subgroupE, which also equals the order|E| of the finite groupE.

Fors∈Cwith Re(s)>1, theelliptic Eisenstein series Eell,E(z, s) associated toE is defined by the series

Eell,E(z, s) := X

η∈E\Γ

sinh¡

%(σ⁻¹_E ηz)¢_−s

(18) using the hyperbolic polar coordinates (2). The elliptic Eisenstein series (18) is absolutely and locally uniformly convergent forz∈Hwithz6=ησEifor anyη∈Γ ands ∈Cwith Re(s)>1; it is invariant under the action of Γ and satisfies the differential equation

¡∆hyp−s(1−s)¢

Eell,E(z, s) =−s²Eell,E(z, s+ 2).

For proofs of theses facts and further details, we refer to [24] or [19].

4 A fundamental identity

In this section, we discuss a fundamental identity relating the hyperbolic and the canonical metric. This relation has already been presented in special contexts (see [15] and [16]).

4.1. Proposition. With the above notations, the following relation holds true for z∈M\ T

g µcan(z) = µ 1

4π + 1

vol_hyp(M)

¶

µhyp(z) +1 2



 Z∞

0

∆hypKhyp(t;z, z)dt



µhyp(z).

(19) Proof. In case Γ is cocompact and torisonfree, the proof is given in [15]. If Γ is cocompact with torison, the proof given in [15] can easily be adapted as long as z∈M\ T. If Γ is no longer cocompact, but cofinite, the proof given in [16] applies (see also [3]).

4.2. Remark. Formally, we have Z∞

0

∆hypKhyp(t;z, z) dt= Z∞

0

∆hyp

X

γ∈Γ γ6=id

KH(t;z, γz) dt

= X

γ∈Γ γ6=id

∆hyp

Z∞

0

KH(t;z, γz) dt= 1 4π

X

γ∈Γ γ6=id

∆hypgH(z, γz).

It will turn out that this formal computation can be justified if the latter sum is taken in a conditionally convergent sense, i.e., by individually summing over the parabolic, the elliptic, and (suitably) the hyperbolic elements of Γ; the details will be specified in the subsequent Sections 5, 6, 7, and 8.

4.3. Lemma. For anyγ=¡_{a b}

c d

¢∈PSL2(R), we have

∆_hypg_H(z, γz) =−4y²

µ 1

(cz¯z+dz−a¯z−b)² + 1

(czz¯+d¯z−az−b)²

¶ . Proof. Using the explicit formula (6) forgH(z, w), the proof is a straightforward computation, namely we have

∆hypgH(z, γz) =−4y² ∂

∂z

∂

∂z¯ µ

−log

¯¯

¯z−γz z−γz¯

¯¯

¯²

¶

= 4y² ∂

∂z

∂

∂z¯

¡log(z−γz) + log(¯z−γ¯z)−log(z−γ¯z)−log(¯z−γz)¢

= 4y² ∂

∂z

³ 1

z−γ¯z ·∂γz¯

∂¯z − 1

¯ z−γz

´

=−4y² µ 1

(z−γz)¯² ·∂γz¯

∂¯z + 1

(¯z−γz)² ·∂γz

∂z

¶ . Employing

∂γz

∂z = 1

(cz+d)², the claim follows.

5 Estimates in the parabolic case

5.1. Lemma. (i)For a parabolic elementγ=¡_1u

0 1

¢∈PSL2(R), we have

∆hypgH(z, γz) =−8y² u²−4y² (u²+ 4y²)².

(ii)Let P =hγPibe a parabolic subgroup of PSL2(R)generated by γP =¡_1u

0 1P

¢. Then, we have

X

γ∈P\{id}

∆hypgH(z, γz) = 2

µ 2π/uP·y sinh(2π/uP·y)

¶₂

−2.

Proof. The first assertion follows immediately from Lemma 4.3. As for the second claim, we recall the identity

X∞

n=−∞

n²−w² (n²+w²)² =−

µ π sinh(πw)

¶2

,

which can be proven using elementary arguments from complex analysis (see, e.g., [7], p. 36, formula 1.421.5).

5.2. Lemma. The series

X

γ∈Γ\{id}

γparabolic

∆hypgH(z, γz)

is absolutely and locally uniformly convergent forz∈H.

Proof. Using the finite setP(Γ) of representatives of Γ-conjugacy classes of max- imal parabolic subgroups of Γ (see subsection 3.1), we have the disjoint union decompositions

©γ∈Γ\ {id} |γparabolicª

= [

P∈P(Γ)

[

Qconj.toP

¡Q\ {id}¢

= [

P∈P(Γ)

[

η∈P\Γ

¡η⁻¹P η\ {id}¢

= [

P=hγPi∈P(Γ)

[

η∈P\Γ

[∞

n=−∞n6=0

{η⁻¹γ_Pⁿη}.

This gives X

γ∈Γ\{id}

γparabolic

∆hypgH(z, γz) = X

P=hγPi∈P(Γ)

X

η∈P\Γ

X∞

n=−∞n6=0

∆hypgH(z, η⁻¹γ_Pⁿηz).

ForP=hγPi ∈ P(Γ), we choose σP andγ∞ as in (11), which leads to X

γ∈Γ\{id}

γparabolic

∆_hypg_H(z, γz) = X

P=hγPi∈P(Γ)

X

η∈P\Γ

X∞

n=−∞n6=0

∆_hypg_H¡

z, η⁻¹σ_Pγⁿ_∞σ⁻¹_P ηz¢

= X

P=hγPi∈P(Γ)

X

η∈P\Γ

X∞

n=−∞n6=0

∆hypgH

¡σ⁻¹_P ηz, γ_∞ⁿσ_P⁻¹ηz¢ .

SettingyP,η:= Im(σ⁻¹_P ηz), we estimate using Lemma 5.1 (i) withu=n∈Z

¯¯

¯∆hypgH

¡σ⁻¹_P ηz, γ_∞ⁿσ_P⁻¹ηz¢¯¯

¯= 8y_P,η² ·

¯¯

¯¯

n²−4y_P,η² (n²+ 4y²_P,η)²

¯¯

¯¯≤ 8y_P,η² n²+ 4y_P,η² . This gives

X∞

n=−∞n6=0

¯¯

¯∆_hypg_H(σ_P⁻¹ηz, γⁿ_∞σ⁻¹_P ηz)

¯¯

¯≤ X∞

n=1

16y_P,η² n²+ 4y_P,η² .

To ease notation, set w := 2yP,η for the moment. Observing the elementary estimate

X∞

n=1

w²

n²+w² ≤ w² 1 +w² +

Z∞

1

w² t²+w²dt

≤w²+ Z∞

1

w²

t²+w²dt=w²+w µπ

2 −arctan

³1 w

´¶

≤w²+w·w= 2w², we get

X

γ∈Γ\{id}

γparabolic

¯¯∆hypgH(z, γz)¯

¯≤ X

P∈P(Γ)

X

η∈P\Γ

32y²_P,η

= 32 X

P∈P(Γ)

X

η∈P\Γ

Im¡

σ_P⁻¹ηz¢₂

= 32 X

P∈P(Γ)

Epar,P(z,2),

with the parabolic Eisenstein seriesEpar,P(z, s) associated to P evaluated ats= 2. The absolute and locally uniform convergence of the series in question now follows.

5.3. Proposition. Let Γ be a subgroup of finite index in the Fuchsian subgroup Γ0 of the first kind. Then, we have

X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz)≤ X

γ∈Γ\{id}

γparabolic

∆hypgH(z, γz) ≤ 0.

Proof. We denote by Mpar(Γ) the set of all maximal parabolic subgroups of Γ.

We observe that there is a bijection ϕ : M_par(Γ)−→ M^' _par(Γ₀), which is given as follows: ForP ∈ Mpar(Γ), there exists a maximal parabolic subgroupP0⊂Γ0

containing P, and we set ϕ(P) := P0; the inverse map is given by ϕ⁻¹(P0) :=

P0∩Γ.

We note that the set of parabolic elements of Γ different from the identity can be written as the disjoint union of the setsP\ {id}with P running through Mpar(Γ). Choosing forP=hγPi ∈ Mpar(Γ), the quantitiesσP andγ∞as in (11), lettingwP :=σ_P⁻¹z, and applying Lemma 5.1 (ii) withuP = 1, we compute

X

γ∈Γ\{id}

γparabolic

∆hypgH(z, γz) = X

P∈Mpar(Γ)

X∞

n=−∞n6=0

∆hypgH

¡z, σPγ_∞ⁿσ⁻¹_P z¢

= X

P∈Mpar(Γ)

X∞

n=−∞n6=0

∆hypgH

¡wP, γ_∞ⁿwP

¢

= 2 X

P∈Mpar(Γ)

õ 2πIm(wP) sinh(2πIm(wP))

¶₂

−1

!

. (20)

Elementary calculus shows that the function h(x) :=

µ x sinh(x)

¶₂

−1

is negative, bigger than−1, and strictly monotone decreasing forx >0. Using the negativity of the function h(x), we derive the claimed upper bound immediately from formula (20).

We are left to prove the claimed lower bound. Replacing Γ by Γ0, we obtain as above

X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz) = 2 X

P0∈Mpar(Γ0)

õ 2πIm(wP0) sinh(2πIm(wP0))

¶₂

−1

!

; (21)

here wP₀ := σ⁻¹_P

0z with the scaling matrix σP₀ for a generator γP₀ of P0 ∈ Mpar(Γ0). Letting p := [P0 : P], the relation between σP0 and σP is given by the formula

σP0 =σP

µ1/√ p 0

0 √

p

¶ .

This together with the bijection between Mpar(Γ0) and Mpar(Γ) allows us to rewrite (21) in the form

X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz) = 2 X

P∈Mpar(Γ)

õ 2πpIm(wP) sinh(2πpIm(wP))

¶₂

−1

! .

The claimed lower bound now immediately follows from the fact that the function h(x) is monotone decreasing.

5.4. Proposition. Let Γbe a subgroup of finite index inΓ₀. Then, we have sup

z∈H

¯¯

¯¯

¯ X

γ∈Γ\{id}

γparabolic

∆hypgH(z, γz)

¯¯

¯¯

¯=OΓ0(1).

Proof. By Proposition 5.3, it suffices to prove sup

z∈H

à X

γ∈Γ₀\{id}

γparabolic

∆hypgH(z, γz)

!

=OΓ0(1).

By the Γ0-invariance of the series under consideration, it suffices to bound it onM0:= Γ0\H. On the compact set given by M0 minus the union of sufficiently small neigborhoods around the cusps ofM0, the series in question can be uniformly bounded. It remains to find uniform bounds in the neighborhoods of the (finitely many) cusps ofM0. For this we proceed as follows.

Recall thatP(Γ0) denotes a complete set of representatives of the Γ0-conjugacy classes of the maximal parabolic subgroups of Γ0. Then, arguing as in the proof of Lemma 5.2, we have

©γ∈Γ0\ {id} |γparabolicª

= [

P∈P(Γ0)

[

Qconj.toP

¡Q\ {id}¢

= [

P∈P(Γ0)

[

η∈P\Γ0

¡η⁻¹P η\ {id}¢

= [

P=hγPi∈P(Γ0)

[

η∈P\Γ0

[∞

n=−∞n6=0

{η⁻¹γⁿ_Pη},

which leads to X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz) = X

P=hγPi∈P(Γ0)

X

η∈P\Γ0

X∞

n=−∞n6=0

∆hypgH

¡z, η⁻¹σPγ_∞ⁿσ⁻¹_P ηz¢

= X

P=hγPi∈P(Γ0)

X

η∈P\Γ0

X∞

n=−∞n6=0

∆hypgH

¡σ_P⁻¹ηz, γ_∞ⁿσ_P⁻¹ηz¢ .

Using Lemma 5.1 (ii) withuP = 1, we get X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz) = 2 X

P=hγPi∈P(Γ0)

X

η∈P\Γ0

õ 2πIm(σ⁻¹_P ηz) sinh(2πIm(σ_P⁻¹ηz))

¶2

−1

! .

We are now able to show that the series in question is uniformly bounded as z tends to a cusp ofM0. By suitably conjugating Γ0inside of PSL2(R), if necessary, we may assume that the cusp in question isi∞with cusp width one. To simplify notation, we denote the resulting Fuchsian subgroup again by Γ0. Let thenP0 be the stabilizer ofi∞in Γ0. We can viewP0 as an element ofP(Γ0), and we may chooseσP0 = id. We have

X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz) = 2 X

P∈P(Γ0) P6=P0

X

η∈P\Γ0

õ 2πIm(σ⁻¹_P ηz) sinh(2πIm(σ_P⁻¹ηz))

¶2

−1

!

+ 2 X

η∈P0\Γ0

η6=id

õ 2πIm(ηz) sinh(2πIm(ηz))

¶₂

−1

! + 2

õ 2πIm(z) sinh(2πIm(z))

¶₂

−1

! .

Applying the bound

0≥ µ x

sinh(x)

¶₂

−1≥ −x² forx >0, we get

0≥ X

γ∈Γ0\{id}

γparabolic

∆hypgH(z, γz)≥ −8π² X

P∈P(Γ0) P6=P0

X

η∈P\Γ0

Im(σ⁻¹_P ηz)²

−8π² X

η∈P0\Γ0

η6=id

Im(ηz)²+ 2

õ 2πIm(z) sinh(2πIm(z))

¶₂

−1

! .

With the parabolic Eisenstein seriesEpar,P(z, s), we get

0≥ X

γ∈Γ0\{id}

γparabolic

∆_hypg_H(z, γz)≥ −8π² X

P∈P(Γ0) P6=P0

E_par,P(z,2)

−8π²¡

Epar,P0(z,2)−Im(z)²¢ + 2

õ 2πIm(z) sinh(2πIm(z))

¶₂

−1

! .

If Im(z) tends to ∞, all terms involving the parabolic Eisenstein series are uni- formly bounded (see [20], p. 12, Theorem 2.1.2). The last term is trivially bounded by−2 from below. This completes the proof of the proposition.

6 Estimates in the elliptic case

6.1. Hyperbolic polar coordinates. From Subsection 2.1, formula (2), we recall the hyperbolic polar coordinates%(z) and ϑ(z) for z=x+iy ∈H. These

coordinates satisfy the set of equations x²+¡

y−cosh¡

%(z)¢¢2

= sinh²¡

%(z)¢

¡ ,

x+ tan¡ ϑ(z)¢¢₂

+y²= tan²¡ ϑ(z)¢

+ 1, which are equivalent to

x²+y²+ 1 = 2ycosh¡

%(z)¢ , x²+y²−1 =−2xtan¡

ϑ(z)¢ .

6.2. Lemma. (i) For an elliptic element γ =

³ cos(α) sin(α)

−sin(α) cos(α)

´

∈ PSL2(R), we have

∆hypgH(z, γz) =−8y² (x²+y²+ 1) sin²(α)−4y²cos²(α)

¡(x²+y²+ 1) sin²(α) + 4y²cos²(α)¢₂

=−2 cosh²¡

%(z)¢

sin²(α)−cos²(α)

¡cosh²¡

%(z)¢

sin²(α) + cos²(α)¢₂.

(ii)Let E=hγi,Ei be an elliptic subgroup ofPSL2(R) of ordermE>1 generated by the elementγi,E=

³ cos(π/mE) sin(π/mE)

−sin(π/mE) cos(π/mE)

´

. Then, we have X

γ∈E\{id}

∆hypgH(z, γz) = 2 Ã

4m²_E sinh²¡

%(z)¢· tanh^2mE¡

%(z)/2¢

¡1−tanh^2mE¡

%(z)/2¢¢₂ −1

! .

Proof. The first assertion follows immediately from Lemma 4.3 and the use of the hyperbolic polar coordinates given in 6.1. In order to prove the second claim, we first note the formula

X

γ∈E\{id}

∆hypgH(z, γz) =−2

mXE−1

n=1

cosh²¡

%(z)¢ sin²¡

πn/mE

¢−cos²¡ πn/mE

¢

¡cosh²¡

%(z)¢ sin²¡

πn/mE

¢+ cos²¡ πn/mE

¢¢₂.

PutX := cosh²¡

%(z)¢

for the moment; noteX ≥1. For realα, we then consider the smooth real function

F(α) := Xsin²(α)−cos²(α)

¡Xsin²(α) + cos²(α)¢₂.

Since F(α) is an even function with period π, it has a Fourier expansion of the form

F(α) = X∞

m=1

am(X)·cos(2mα).