On the spectral expansion of hyperbolic Eisenstein series

Transcription

1 On the spectral expansion of hyperbolic Eisenstein series J. Jorgenson, J. Kramer, and A.-M. v. Pippich Abstract In this article we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in [7], the scalar-valued hyperbolic Eisenstein series is defined for each primitive, hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going beyond the results in [7] and [11], we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface by first proving that the hyperbolic Eisenstein series is in L. Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion. 1 Introduction 1.1. Summary. Let Γ PSL R be a Fuchsian group of the first kind acting by fractional linear transformations on the upper half-plane H := {z C z = x + iy, y > }. Let X := Γ\H, which is a Riemann surface of finite volume with respect to the natural hyperbolic metric induced from H. Associated to any hyperbolic element γ Γ, we define a scalar-valued hyperbolic Eisenstein series E hyp,γ z, s, which is analogous to the form-valued hyperbolic Eisenstein series defined in [7]; see also [11], section 3. We first prove that the hyperbolic Eisenstein series is in L X. The main result of this article is the determination of the full spectral expansion of E hyp,γ z, s based on an explicit computation of the inner product of E hyp,γ z, s with any eigenfunction of the hyperbolic Laplacian see Theorem 4.1. The knowledge of the spectral expansion of E hyp,γ z, s enables us to also determine its meromorphic continuation see Theorem Comparison with known results. As stated, hyperbolic Eisenstein series have been considered elsewhere, most notably in [], [7], and [11]. In [7], the authors define a form-valued hyperbolic Eisenstein series; their main result, which is an analogue of the classical Kronecker limit formula, is that the constant term in a Laurent expansion at the first pole of their hyperbolic Eisenstein series is the harmonic form dual to the cycle determined by the hyperbolic element γ Γ from which they define their series. In addition, the authors prove the meromorphic continuation and establish the location of singularities when X is compact, though they do not explicitly evaluate the spectral expansion of their form-valued hyperbolic Eisenstein series, nor do they study the case when X is non-compact. In [11], the author proves the meromorphic continuation of the scalar-valued hyperbolic Eisenstein series using perturbation theory, but, again, does not discuss the full spectral expansion. More significantly, the consideration in [11] restricts attention to the case when X is compact, whereas the computations here simply require X to have finite hyperbolic volume. In this article, we obtain the main results analogous to theorems of [7] by first explicitly computing the spectral expansion of our hyperbolic Eisenstein series, then we extract all other results as corollaries. In the case when X is non-compact, we establish the asymptotic behavior of E hyp,γ z, s as z tends to a cusp of X, which has not been established elsewhere, so then we can consider both compact and non-compact finite volume Riemann surfaces simultaneously. We are confident that the techniques developed here will apply to other types of hyperbolic Eisenstein series. For example, in [11], the author studies hyperbolic Eisenstein series which are twisted 1

2 by modular symbols. In order to apply the ideas from the present paper, we need at our disposal an inner product for functions on H, whose functional equation when acted upon by Γ, agrees with that of this more general Eisenstein series. In [6], the authors interprete the higher-order parabolic Eisenstein series as components of eigensections of certain unipotent bundles on X. We are confident that such an interpretation can be made for hyperbolic Eisenstein series twisted by modular symbols, at which time the techniques of the present article will apply. We will leave this problem for future study. In a different direction, the article [] studies the asymptotic behavior of hyperbolic Eisenstein series when considering a degenerating sequence of finite volume hyperbolic Riemann surfaces. In brief, the main result in [] is that the limit of the properly scaled hyperbolic Eisenstein series associated to the pinching geodesic from a degenerating sequence of Riemann surfaces is equal to the parabolic Eisenstein series associated to the newly formed cusp on the limit surface. The method of proof in [] involves a detailed analysis of the differential equation satisfied by the hyperbolic Eisenstein series. The main results in [] are reproved in [3] using counting function arguments and Stieltjes integral representations of various Eisenstein series. Background material and notation.1. Basic notation. As mentioned in the introduction, we let Γ PSL R denote a Fuchsian group of the first kind acting by fractional linear transformations on the upper half-plane H := {z C z = x + iy, y > }. We let X := Γ\H, which is a Riemann surface, and denote by p : H X the natural projection. The hyperbolic line element ds hyp, resp. the hyperbolic Laplacian hyp, are given as ds hyp := dx + dy y, resp. hyp := y x + y. Under the change of coordinates x := e ρ cosθ and y := e ρ sinθ, the hyperbolic line element, resp. the hyperbolic Laplacian, are rewritten as ds hyp = dρ + dθ sin θ, resp. hyp = sin θ ρ + θ. In a slight abuse of notation, we will at times identify X with a fundamental domain in H say, a Ford domain, bounded by geodesic paths and identify points on X with their preimages in such a fundamental domain. Given any measurable functions f and g on X, their inner product is defined by f, g := fzgzµ hyp z, where µ hyp z := X dx dy dθ dρ y, or, in other coordinates, µ hyp z = sin θ. Throughout this paper we will assume that f and g have sufficiently many derivatives and moderate growth when X is non-compact, so then we have, by Green s theorem, the identity We refer to [1] and [4] for precise details as to when 1 is valid. hyp f, g = f, hyp g. 1.. The Γ-function. The classical Γ-function will play an important role in our computations, so we will summarize here the relevant properties of the Γ-function which we need. Recall that Γs is defined for Res > by the integral Γs := e t t s dt t.

3 3 Integration by parts shows that Γs satisfies the recursion formula Γs + 1 = sγs, which also provides the meromorphic continuation of Γs to all s C; its continuation has singularities at the non-positive integers, and each singularity is a simple pole with residue at s = n equal to 1 n /n! n N. From the recursion formula, one can show that the function gx, s := defined for x R and s C, satisfies the relation s Γs + 1 Γ s + ix/ + 1 Γs ix/ + 1, gx, s = ss 1 s gx, s. + x Furthermore, gx, s is bounded in the vertical strip a < Res < b for any a, b R satisfying 1 < a < b. Similarly, and in fact equivalently, the function satisfies hs := Γs 1/ + ir j/γs 1/ ir j / Γ s/ hs + = ss 1 + λ j s hs, where λ j = 1/4 + rj ; it is bounded in the vertical strip a < Res < b for any a, b R satisfying < a < b. Among the many known identities for the Γ-function, we shall make use of the following fundamental relation Γs + 1/Γs/ = π 1 s Γs. In addition to the above identities, we shall make use of Stirling s asymptotic formula for the Γ-function, which states that log Γs = 1 logπ 1 logs + s logs s + o1, 3 which holds when s provided s remains in a sector of the form args < π ε for some ε >. In particular, we have for fixed σ R and t the asymptotics log Γσ + it = 1 logπ + σ 1 logt πt + it logt it + o1 4 with an implied constant depending on σ. For both formulas we refer to [4], p Hyperbolic Eisenstein series. Let γ be a primitive hyperbolic element of Γ. Hence there is an element σ PSL R such that σ 1 e l γ/ γσ = e lγ/, 5 where l γ denotes the hyperbolic length of the closed geodesic on X in the homotopy class determined by γ. We note that := p 1 = σ L, where L := {z H x = Rez = } is the positive y-axis, and that Γ γ := Stab Γ Lγ = γ.

4 4 Using the coordinates ρ = ρz and θ = θz introduced in subsection.1, the hyperbolic Eisenstein series E hyp,γ z, s associated to γ Γ is defined by E hyp,γ z, s := sinθσ 1 ηz s. 6 η Γ γ\γ Recalling that the hyperbolic distance d hyp z, L from z to the geodesic line L is characterized by the formula sinθz coshd hyp z, L = 1, we can rewrite the hyperbolic Eisenstein series 6 as E hyp,γ z, s = coshdhyp ηz, s. η Γ γ\γ Referring to [], [3], [1], or [11], where detailed proofs are provided, we recall that the series 6 converges absolutely and locally uniformly for any z H and s C with Res > 1, and that it is invariant with respect to Γ. A straightforward computation shows that the series 6 satisfies the differential equation hyp s1 s E hyp,γ z, s = s E hyp,γ z, s Spectral expansions. Under the hypotheses made in subsection.1, there is a spectral expansion in terms of the eigenfunctions ψ j associated to the discrete eigenvalues λ j of the hyperbolic Laplacian hyp and the parabolic Eisenstein series E par,p associated to the cusps P of X. For any function f on X, for which f and hyp f are bounded, the spectral expansion is given by the identity fz = f, ψ j ψ j z + 1 4π j= P cusp f, E par,p E par,p z, 1/ + ir dr. 8 We refer to [4] for all aspects of these results, in particular to Theorem 7.3 on p Counting functions. Using the notations of subsection.3, we define the hyperbolic counting function N hyp,γ T ; z as N hyp,γ T ; z := card { η Γ γ \Γ d hyp ηz, < T } Equivalently, the function N hyp,γ T ; z counts the number of geodesic paths from z X to the closed geodesic on X of length less than T. Using the counting function N hyp,γ T ; z we can express the hyperbolic Eisenstein series 6 as a Stieltjes integral, namely we have E hyp,γ z, s = This representation of E hyp,γ z, s plays an important role in [3]. coshu s dnhyp,γ u; z. 9 3 Preliminary inner product computations 3.1. Lemma. For any x R and any s C with Res > 1, we have π sinu se xu du = πe πx/ s Γs + 1 Γs + ix/ + 1Γs ix/ + 1.

5 5 Proof. We set fx, s := π sinu se xu du. Using integration by parts, we arrive at the relation, as long as Res > 1, fx, s = As discussed in subsection., the function satisfies the relation gx, s = ss 1 s fx, s. + x s Γs + 1 Γs + ix/ + 1Γs ix/ + 1 gx, s = ss 1 s gx, s. + x Obviously, both fx, s and gx, s are bounded and holomorphic, and gx, s in the vertical strip 1 < Res < 4. Therefore, the function hx, s = fx, s/gx, s satisfies hx, s = hx, s and is bounded and holomorphic for all s C, and hence is constant in s, meaning hx, s = Cx. To evaluate Cx, let us take s =. For this, we have Also, fx, = gx, = π e xu du = 1 x 1 e πx. 1 Γix/ + 1Γ ix/ + 1. Taking w = ix/ and using the well-known identity ΓwΓ1 w = π sinπw, we find Γix/ + 1Γ ix/ + 1 = ix Γix/Γ1 ix/ = ix π sinπix/. Writing sinπix/ = i sinhπx/, we then have Therefore, gx, = Cx = 1 Γix/ + 1Γ ix/ + 1 = sinhπx/. πx fx, gx, = 1 e πx /x sinhπx//πx = πe πx/. Recalling that fx, s = Cxgx, s, the stated assertion now follows. 3.. Lemma. For any s C with Res > 1, the hyperbolic Eisenstein series E hyp,γ z, s is bounded as a function of z X. If X is non-compact and P X is a cusp satisfying P = τi for suitable τ PSL R, we have the estimate Ehyp,γ z, s = O Imτ 1 z Res as z P.

6 6 Proof. If X is compact, the boundedness of the hyperbolic Eisenstein series 6 follows from the discussion in subsection.3, or the analysis given in [3], section 4.1. It remains to determine the asymptotic behavior of E hyp,γ z, s, when X is non-compact and z approaches a cusp of X. Without loss of generality, we may assume that the cusp P of X corresponds to the cusp i of a fundamental domain F H of X, which we identify with X and fix for this proof. We then choose y sufficiently large such that every point on the geodesic H has imaginary part less than y. Let now z = x + iy X be such that y > y, and let L denote the horocycle about z at height y ; the hyperbolic distance d := d hyp z, L from z to L equals d = logy/y. We consider the counting function N hyp,γt ; L := card{η Γ γ \Γ d hyp η L, < T }. 1 Now, every element of the set { η Γγ \Γ d hyp ηz, < T } corresponds to a geodesic path L on X from z to of length less than T, which necessarily intersects the horocycle L on X. Let d 1 be the length of the portion of L from z to L, and let d be the length of the portion of L from L to. Trivially, we have that d 1 + d is the length of L and that d 1 d. Therefore, we find d 1 + d < T, and hence d < T d. This analysis proves the inclusion of sets { η Γγ \Γ d hyp ηz, < T } {η Γ γ \Γ d hyp η L, < T d} which implies the inequality N hyp,γ T ; z N hyp,γt d; L 11 for T > d. Trivially, we also have that N hyp,γ T ; z = for T < d. Recalling the representation 9 of the hyperbolic Eisenstein series, we have Ehyp,γ z, s Res coshu dnhyp,γ u; z coshu Res dn hyp,γ u d; L. d d Using the elementary bound coshu e u / and letting v = u d, we get the estimate Ehyp,γ z, s Res y e vres dn y hyp,γv; L. 1 The result now follows from elementary counting arguments which imply that the integral in 1 converges for Res > 1 see [5] and [8] Lemma. For any smooth, bounded, real-valued function φ on X, we have for sufficiently small ε > the estimate E hyp,γ, φ = s πγs + 1 Γ s/ + 1 φz ds hyp z + O ε/ s as s, where the implied constant depends on φ and ε. Proof. Without loss of generality it suffices to prove the lemma in the case when σ in 5 is the identity matrix. Then, using the series expansion for the hyperbolic Eisenstein series 6, we can unfold the integral in question, resulting in the expression E hyp,γ, φ = l γ π φ e ρ e iθ sinθ s dθ dρ sin θ.

7 7 For sufficiently small ε >, let us write l γ π... = l γ π/ ε... + l γ π/+ε π/ ε... + l γ π π/+ε Given ε >, there is a constant a ε with < a ε < 1 and sinθ a ε, whenever θ [, π/ ε] [π/ + ε, π]. Then, we have the bound... l γ π/ ε φ e ρ e iθ sinθ lγ s dθ dρ + π π/+ε φ e ρ e iθ sinθ s dθ dρ = O a s ε 13 as s. Hence we can write E hyp,γ, φ = l γ π/+ε π/ ε φ e ρ e iθ sinθ s dθ dρ + O a s ε 14 as s. Now, using a Taylor series expansion for the function φ with respect to the variable θ about the point θ = π/, we have that l γ for θ π/ < ε. Observe that φ e ρ e iθ dρ = φz ds hyp z + O θ π/ 15 π/+ε π/ ε π/ θ sinθ s dθ = ε ε u cosu s du, which we use for simplicity of exposition. Now, on the interval [ ε, ε], we consider the estimate where u cosu s gu, gu := u 1 b ε u s with b ε := 1 ε 4. Since the function gu assumes its extrema on [ ε, ε] for u = ±b 1/ ε s 3 1/, we get the bound u cosu s s = O 1/ s, bε s 3 s 3 from which we derive ε ε u cosu s du = O ε/ s 16 as s. By combining 14, 15, and 16, we arrive at E hyp,γ, φ = φz ds hyp z π/+ε π/ ε sinθ s dθ + O ε/ s 17

8 8 as s. Noting that π/ ε π s dθ sinθ + π/+ε as s, combined with 17, we arrive at the estimate E hyp,γ, φ = φz ds hyp z as s. To finish, we use Lemma 3.1 with x = to give E hyp,γ, φ = s πγs + 1 Γ s/ + 1 s dθ sinθ = O a s ε π sinθ s dθ + O ε/ s φz ds hyp z + O ε/ s as s. This completes the proof of the lemma. 4 Spectral expansion and meromorphic continuation We are now in position to state and prove the main result of this paper Theorem. For any s C with Res > 1, the hyperbolic Eisenstein series E hyp,γ z, s associated to γ Γ admits the spectral expansion E hyp,γ z, s = j= a j,γ s ψ j z + 1 4π The coefficient a j,γ s is given by the formula P cusp a j,γ s = π Γs 1/ + ir j/γs 1/ ir j / Γ s/ a 1/+ir,γ,P s E par,p z, 1/ + ir dr. 18 ψ j z ds hyp z; 19 here we have written the eigenvalue λ j of the eigenfunction ψ j in the form λ j = 1/4 + r j. An analogous formula holds for the coefficient a 1/+ir,γ,P s; it is given at the end of the proof below. Proof. The hyperbolic Eisenstein series E hyp,γ z, s is a smooth function on X, which is bounded by Lemma 3.. The differential equation 7 allows us to conclude that hyp E hyp,γ z, s is also smooth and bounded on X. The existence of the spectral expansion 18 now follows from [4], Theorem 7.3. The coefficient a j,γ s is given by the inner product E hyp,γ, ψ j, which converges by the asymptotic bound proved in Lemma 3., and known asymptotic bounds for eigenfunctions of the hyperbolic Laplacian. Using the differential equation 7 and integration by parts, which is justified again using Lemma 3. and [4], Theorem 3.1, we have the relation which implies λ j a j,γ s = λ j E hyp,γ, ψ j = E hyp,γ, hyp ψ j = hyp E hyp,γ, ψ j = s1 sa j,γ s + s a j,γ s +, a j,γ s + = ss 1 + λ j s a j,γ s.

9 9 From subsection., we recall the function which satisfies the recursion formula hs = Γs 1/ + ir j/γs 1/ ir j / Γ s/ hs + = ss 1 + λ j s hs. 1 From and 1, we conclude that the quotient a j,γ s/hs is invariant under s s + ; furthermore, it is bounded in a vertical strip, say < Res < 5. Therefore, the quotient a j,γ s/hs is constant. In other words, we have a j,γ s = b j,γ Γs 1/ + ir j/γs 1/ ir j / Γ s/ for some constant b j,γ, which is independent of s, but possibly depends on j and γ. We are left to determine the constant b j,γ, which we will do now. Using Stirling s formula 3 for real s tending to infinity, we get Γs 1/ + irj /Γs 1/ ir j / log Γ = o1 3 s 1// as s. Using Stirling s formula 3 a second time for real s tending to infinity, we find Γs 1// log = 1 logs 1// + o1 4 Γs/ 4 as s. Combining the asymptotics 3, 4 with, we obtain, log E hyp,γ, ψ j = logb j,γ 1 logs 1/ + 1 log + o1 5 as s. Now, recall Lemma 3.3 with φ = ψ j, namely the formula E hyp,γ, ψ j = s πγs + 1 Γ s/ + 1 ψ j z ds hyp z + O c s ε as s. Using, we can rewrite the Γ-factor as s πγs + 1 Γ s/ + 1 = Γs + 1/ π Γs/ + 1. Using Stirling s formula 3 a third time for real s tending to infinity, we get the asymptotics Γs + 1/ log = 1 Γs/ + 1 logs 1/ + 1 log + o1 7 as s. Combining 7 with 6, yields the formula log E hyp,γ, ψ j = log ψ j z ds hyp z + log π 1 logs 1/ + 1 log + o1 8 as s. Finally, by comparing 5 with 8, we find b j,γ = π ψ j z ds hyp z, as claimed. Proceeding as in the discrete case, we obtain for the coefficient a 1/+ir,γ,P s the formula a 1/+ir,γ,P s = Γs 1/ + ir/γs 1/ ir/ π Γ E par,p z, 1/ + ir ds hyp z. s/ This completes the proof of the theorem. 6

10 1 4.. Theorem. The hyperbolic Eisenstein series E hyp,γ z, s admits a meromorphic continuation to all s C. The singularities of the function Γ s/ E hyp,γ z, s are located at the points a s = 1/ ± ir j n, where n N and λ j = 1/4 + r j is the eigenvalue of the L -eigenfunction ψ j on X, with residues Res s=1/±irj n[ Γ s/ E hyp,γ z, s ] = 1 n π Γ±ir j n n! ψ j z ψ j z ds hyp z. b s = 1 ρ n with n N >, or s = ρ n with n N, where w = ρ is a pole of the Eisenstein series E par,p z, w satisfying < Reρ < 1/, with residues [ Res s=1 ρ n Γ s/ E hyp,γ z, s ] = 1n π Γ1/ ρ n n! CT w=ρ E par,p z, w Res w=ρ E par,p z, w ds hyp z+ P cusp +Res w=ρ E par,p z, w CT w=ρ E par,p z, w ds hyp z. In case s = ρ n, the Γ-factor in the above formula has to be replaced by Γ 1/ + ρ n. Proof. In order to derive the meromorphic continuation of E hyp,γ z, s we use the spectral expansion 18. We start by giving the meromorphic continuation for the series in 18 arising from the discrete spectrum. The explicit formula 19 in terms of Γ-functions proves the meromorphic continuation for the coefficients a j,γ s to all s C. Now, using the well-known sup-norm bound sup ψ j z = O rj z X for the eigenfunctions together with Stirling s formula 4, we find a j,γ s ψ j z = O r Res j e πrj/, which proves that the series in 18 arising from the discrete spectrum is locally absolutely and uniformly convergent as a function of s C away from the poles. The location of the poles calculation and the determination of the residues arising from this part is straightforward referring to the corresponding facts for the Γ-function recalled in subsection.. We now turn to give the meromorphic continuation of the integral in 18 arising from the continuous spectrum. Assuming 1/ < Res < 5/ and using the residue theorem we can rewrite the integral in question as π Γs 1 + w/γs w/ E par,p z, w E par,p z, w ds hyp z dw = 4πi Rew=1/ π Γs 1 + w/γs w/ E par,p z, w E par,p z, w ds hyp z dw+ 4πi Rew= 1/ π + ρ pole of E par,p z,w <Reρ<1/ Γs 1 + ρ/γs ρ/

11 11 CT w=ρ E par,p z, w Res w=ρ E par,p z, w ds hyp z+ Res w=ρ E par,p z, w CT w=ρ E par,p z, w ds hyp z. 9 While the left-hand side integral in 9 is holomorphic for 1/ < Res < 5/, the integral on the right-hand side is holomorphic for 1/ < Res < 3/. Since the sum in 9 is meromorphic for all s C, formula 9 establishes the meromorphic continuation of the term 1 4π P cusp a 1/+ir,γ,P s E par,p z, 1/ + ir dr 3 in the spectral expansion 18 to the half-plane Res > 1/. Now, moving the integral along Rew = 1/ in 9 to the vertical line Rew = 3/ using Cauchy s theorem, we obtain the meromorphic continuation of 3 to the half-plane Res > 3/; note that this time no further residues occur, since E par,p z, w has no poles in the strip 3/ < Rew < 1/. Continuing in this way, we obtain the meromorphic continuation of 3 to all s C. The location of the poles and their residues can finally be easily read off from the sum over the poles of the Eisenstein series in 9 along the same lines as it was done for the discrete part Remark. As in [7], one can consider an analogue of the Kronecker limit formula, which amounts to understanding the second order term in the Laurent expansion of E hyp,γ z, s at a pole. From the spectral expansion given in Theorem 4.1, one easily obtains the spectral expansion of the function which appears in the next order term of the Laurent expansion at a pole Remark. As stated in the introduction, hyperbolic Eisenstein series twisted by modular symbols were defined in [11], and their meromorphic continuation was determined using perturbation theory. In the language of [9], one can refer to such series as higher order, non-holomorphic, hyperbolic Eisenstein series. In [6], the authors study higher order, non-holomorphic, parabolic Eisenstein series using the framework of unipotent vector bundles. Beginning with the linear algebra associated to vector bundles, one can define an inner product for smooth sections, which would point toward an inner product for the higher order, non-holomorphic, hyperbolic Eisenstein series defined in [11]. With this, the methods of the present paper can be applied after establishing a spectral theorem associated to unipotent vector bundles. References [1] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, [] T. Falliero, Dégénérescence de séries d Eisenstein hyperboliques, Math. Ann , [3] D. Garbin, J. Jorgenson, M. Munn, On the appearance of Eisenstein series through degeneration, to appear in Comment. Math. Helv.. [4] H. Iwaniec, Spectral methods of automorphic forms, Graduate Studies in Mathematics 53, Amer. Math. Soc., Providence,. [5] J. Jorgenson, R. Lundelius, Convergence of the normalized spectral counting function on degenerating hyperbolic Riemann surfaces of finite volume, J. Funct. Anal , 5 57.

12 1 [6] J. Jorgenson, C. O Sullivan, Unipotent vector bundles and higher-order non-holomorphic Eisenstein series, preprint, 6. [7] S. Kudla, J. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math , [8] R. Lundelius, Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume, Duke Math. J , 1 4. [9] C. O Sullivan, Properties of Eisenstein series formed with modular symbols, J. Reine Angew. Math. 518, [1] A.-M. v. Pippich, Elliptische Eisensteinreihen, Diplomarbeit, Humboldt-Universität zu Berlin, 5. [11] M. Risager, On the distribution of modular symbols for compact surfaces, Int. Math. Res. Not. 41 4, Jay Jorgenson Department of Mathematics The City College of New York Convent Avenue at 138th Street New York, NY 131 U.S.A. jjorgenson@mindspring.com Jürg Kramer Institut für Mathematik Humboldt-Universität zu Berlin Unter den Linden 6 D-199 Berlin Germany kramer@math.hu-berlin.de Anna-Maria von Pippich Institut für Mathematik Humboldt-Universität zu Berlin Unter den Linden 6 D-199 Berlin Germany apippich@math.hu-berlin.de