WEYL S LAW FOR HYPERBOLIC RIEMANN SURFACES

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1 WEYL S LAW OR HYPERBOLIC RIEMANN SURACES MATTHEW STEVENSON Abstract. These are notes for a talk given in Dima Jakobson s class on automorphic forms at McGill University. This is a brief survey of the results from Chapters and of [3]; specifically, we sketch a proof of the Selberg trace formula and of Weyl s law for hyperbolic Riemann surfaces. The exposition is inspired by [4]. If Ω R n is some open domain, the flat Laplacian has a discrete real spectrum which tends to +. It is a classical result that the number of Dirichlet eigenvalues N(λ) which are less than or equal to some λ is asymptotically given by N(λ) vol(ω)ω n (π) n λn/ as λ +, where ω n is the volume of the unit ball in R n. Given a hyperbolic Riemann surface, it is then natural to consider the number of eigenvalues of the hyperbolic Laplacian belonging to the discrete spectrum less than some fixed number. This question is much more difficult in this context, and herein we explore some classical results in this direction.. Preliminaries Let Γ SL(, R) be a uchsian group of the first kind whose fundamental domain = Γ\H has cusps a,..., a m R { }. Let σ i ( SL(, ) R) be such that σ i ( ) = a i and σ i Γ ai σ i = Γ, where Γ ai denotes the n stabilizer of a i in Γ and Γ = { : n Z} is the stabilizer of in SL(, Z). The Eisenstein series associated to the cusp a i is the function of two complex variables given by E i (z, s) = Im(σ i γz) s. (.) γ Γ ai \Γ This series converges locally uniformly for z, s H with Re(s) >, and is smooth in z. Remark that by reordering the sum, we find that E i (γz, s) = E i (z, s) for any γ Γ i.e. the Eisenstein series is automorphic with respect to Γ in the first argument. In addition, z E i (z, s) = s( s)e i (z, s) because Im(z) s is an eigenfunction with eigenvalue s( s) and the composition of an eigenfunction with an isometry is still an eigenfunction (with the same eigenvalue even!), so the Eisenstein series is in fact an automorphic form. Recall that the zeroth ourier coefficient of the Eisenstein series E k (z, s) in the cusp a l is given by E k (σ l (x + iy), s)dx = δ k,l y s + ϕ k,l (s)y s, (.) where ϕ k,l (s) is a meromorphic function of s. The scattering matrix is given by Φ(s) = (ϕ k,l ) m k,l=, however it is often its determinant ϕ(s) = det(φ(s)) in which we are interested, or a variant thereof such as ϕ (s) := tr(φ (s)φ (s)). (.3) These byproducts of the scattering matrix will appear in, and be integral to, our discussion of the Selberg trace formula. Date: April 8, 4.

2 MATTHEW STEVENSON. Selberg Trace ormula Herein, we provide an intuitive introduction to the Selberg trace formula. Given a point-pair invariant kernel k(z, w) = k(u(z, w)), the associated automorphic kernel is the function K(z, w) := γ Γ k(z, γw). The integral operator with kernel K is said to be of trace class if K(z, z) dµ(z) <, i.e. the kernel K is absolutely integrable on the diagonal of. In this case, the trace of K is defined to be Tr(K) := K(z, z)dµ(z) = k(z, γz)dµ(z), γ Γ where we are obviously ignoring, for the moment, issues of convergence. Consider the spectral decomposition K(z, w) = j= h(t j)u j (z)u j (w), where the u j s denote the eigenfunctions corresponding to the discrete spectrum {λ j = 4 + t j } j= of the hyperbolic Laplacian on Γ\H. The trace Tr(K) is defined to be the integral over the diagonal, so Tr(K) = K(z, z)dµ(z) = h(t j ) u j (z) dµ(z) = h(t j ). (.) j= Equating these two expressions for the trace, we get the pre-trace formula for the automorphic kernel: Tr(K) = k(z, γz)dµ(z) = h(t j ). (.) γ Γ j= Remark that on the left-hand side we have geometric data and on the right-hand side we have spectral data; this will be a recurring theme. or each conjugacy class C Γ, we have a partial kernel K C (z, w) = γ C k(z, γw), whose trace is given by Tr(K C ) = k(z, γz)dµ(z). γ C ix some γ C, then note that Tr(K C ) = τ Z(γ)\Γ k(z, τ γτz)dµ(z) = Z(γ)\H j= k(z, γz)dµ(z), This integral is distinctly easier to analyze, especially if we understand the action of γ (that is, is it parabolic, hyperbolic, or elliptic). Now, this expression depends only on the conjugacy class of γ in SL(, R), and since each conjugacy class of SL(, R) has a representative that is parabolic, hyperbolic, or elliptic, we can take γ (or more precisely, take gγg for some g SL(, R)) to be parabolic, hyperbolic, or elliptic. In this case, it remains to consider the following equality, Tr(K) = Tr(K {} ) + Tr(K C ) + Tr(K C ) + Tr(K C ), (.3) C : parabolic C : hyperbolic C : elliptic where K {} denotes the partial kernel associated to the conjugacy class of the identity. By analyzing these sums on a case-by-case basis, we can get an amazing equality involving both spectral and geometric data of the Riemann surface Γ\H, namely the Selberg trace formula. More generally, let k be the kernel of an invariant integral operator (and hence k(z, w) = k(u(z, w)) i.e. it is a point-pair invariant). Define h(t) = /+it (u)k(u)du, (.4) Recall the following classification of isometries of H (which follows from the Iwasawa decomposition of SL(, R)): a parabolic element acts by translations and has fixed point ; a hyperbolic element acts by dilation and has fixed points, ; an elliptic element acts by rotations and has fixed point i, the axis of rotation.

3 WEYL S LAW OR HYPERBOLIC RIEMANN SURACES 3 where /+it is the Gauss hypergeometric function (there is nothing really special about using s (u), we just want to test the kernel k against some eigenfunction of the hyperbolic Laplacian). Assume that in addition that h satisfies the following conditions: () h(z) is an even holomorphic function defined in the strip {z C: Im(z) + ɛ}. () We have the growth estimate h(z) there. Denote by g(w) = π j= ( z +) +ɛ eitw h(t)dt its inverse ourier transform. Theorem. (Selberg Trace ormula) Let h and g be as above, then h(t j ) h(r) ϕ ϕ ( + ir)dr = parabolic + identity + hyperbolic + elliptic, (.5) where parabolic = h() 4 trace(i Φ( )) α g() log α π where here α is the number of inequivalent cusps of Γ\H, and identity = area(γ\h) h(r) tanh(πr)rdr, h(r) Γ i( + ir)dr, Γ and hyperbolic = g(l log p) log p p l/ p, l/ P l= where the sum is over primitive hyperbolic conjugacy classes P of order p (in the sense that the primitive element dilates by the factor p), and elliptic = cosh(π( l/m)r) h(r) dr, m sin(πl/m) R cosh(πr) <l<m where the sum is over primitive elliptic conjugacy classes R of order m (in the sense that the primitive element is a rotation of order m). Remark that for a general kernel, we get a second term on the left-hand side, which serves to measure or account for the continuous spectrum. This term is necessary because the partial kernels corresponding to parabolic conjugacy classes are not absolutely integrable in the cusps, and the Eisenstein series E a (z, + it) is not square-integrable (see page 38 of [3] for further details). Proof. More as an example, we will compute the identity motion (the rest of the details are found in Chapter of [3]). Since k(z, z) = k(u(z, z)) is a point-pair invariant, we have that Tr(K {} ) = k(z, z)dµ(z) = k() dµ(z) = k()area(γ\h). However, from the start of the class, recall that k(u) = s (u)h(r) tanh(πr)rdr = k() = π h(r) tanh(πr)rdr, where s = / + it and s (u) is the Gauss hypergeometric function 3 The result follows. Recall that γ Γ is primitive if it generates the pointwise stabilizer of the set ix(γ ) of its fixed points, i.e. γ = Stab Γ (ix(γ )). Given γ Γ\{±}, it determines a primitive conjugacy class C (i.e. one which does not contain the identity), and every other class C which has the same fixed points as γ (modulo the action of Γ, of course) arises as a unique power of C, say C = C l for some l Z\{}. If C is elliptic, we have in addition that l < m. See page 37 of [3] for further details. 3 or s = / + it, the Gauss hypergeometric function is s(u) = π π (u + + u(u + ) cos θ) s dθ. In particular, s() =.

4 4 MATTHEW STEVENSON 3. Weyl s Law Let {λ j } j= denote the discrete spectrum of the hyperbolic Laplacian on Γ\H, then we can write λ j = 4 + t j. Our goal is to understand the asymptotics of the counting function, which is N Γ (λ) := {j : t j λ}. (3.) To account for the continuous spectrum, we consider the winding number M Γ (λ), which is defined as M Γ (λ) := λ λ ϕ ( + ir)dr. (3.) or a generic group Γ, the two quantities N Γ (λ) and M Γ (λ) are coupled, and cannot really be estimated separately (or more precisely, it is not known how to accurately estimate them separately). To estimate their sum, we will apply the Selberg Trace ormula. Let δ > be small, and define h(t) = e δt, so its inverse ourier transform is the heat kernel g(x) = e x /4δ δ. The identity motion contributes area(γ\h) e δt tanh(πt)tdt = area(γ\h) e δu tanh(π u)du = area(γ\h) + O(), where the last integral is computed by recognizing it as the Laplace transform of tanh(π u). One can show that the contributions of the hyperbolic and elliptic motions are bounded, so we just subsume them into O(). inally, the parabolic motion contributes terms of the following form: a log δ δ + b δ + O(), for some constants a, b >, which depend only on Γ. These can be computed in greater detail, but require precise asymptotic estimates of the psi-function ψ := Γ Γ, which can be found in Appendix B of [3]. Substituting these computations into the Selberg Trace ormula, we have that for any δ >, e δt j ϕ ( + dt it)e δt = area(γ\h) δ + blog δ + a + O(). (3.3) δ δ j= The strategy now is the following: show that the winding number is monotonically increasing in λ, so (by cleverly choosing a measure) we can use Karamata s theorem to deduce Weyl s law. Define a Borel measure µ on [, ) by λ λ µ([, λ]) := ϕ ( + it)dt = ϕ ( + i s) ds s then its Laplace transform, in the sense of Eq. (5.), is given by µ(δ) = ϕ ( + i s)e δs ds s = ϕ ( + it)e δt dt Remark that µ([, λ]) and µ(δ) are monotone increasing as λ and δ + respectively, so by Karamata s theorem it follows that for λ > sufficiently large and δ > sufficiently small, λ ϕ ( + it)dt = µ([, λ]) µ(δ) = ϕ ( + it)e δt dt The above considerations imply the following result, known as Weyl s law: N Γ (λ) + M Γ (λ) area(γ\h) λ as λ +. (3.4)

5 WEYL S LAW OR HYPERBOLIC RIEMANN SURACES 5 4. Weyl s Law for Congruence Groups More specific asymptotics can be obtained if the uchsian group is a principal congruence group, that is, it is of the form Γ(N) = ker(sl(, Z) SL(, Z/NZ)) for some N Z (or variants thereof). Denote the quotient of the hyperbolic plane by X(N) = Γ(N)\H, the modular surface of level N. Specifically, we can decouple the estimates for M Γ(N) (λ) and N Γ(N) (λ) by using a classical result of Huxley from [], which computes the determinant of the scattering matrix to be ϕ(s) := ( ) l A s ( Γ( s) Γ(s) ) k χ L( s, χ), (4.) L(s, χ) where the product runs over Dirichlet characters χ (there is one for each m Z >, as each one is determined by a group homomorphism (Z/mZ) C ) and for some k, l Z and A >. Consequently, we have the estimate ϕ ( + ir) = O(log(4 + r )) as r. (4.) Note that both of these results are highly nontrivial. It follows that for large λ, M Γ(N) (λ) λ λ log(4 + r )dr Cλ log λ for some constant C >. Thus, the maximum growth term in Weyl s law (i.e. λ ) has to come from N Γ(N) (λ), and so we have that N Γ(N) (λ) area(x(n)) λ as λ. (4.3) This result is due to Selberg along with the stronger fact that the remainder is O(λ log λ). In addition, these results holds for other congruence groups, such as the Hecke group {( ) } a b Γ (N) = SL(, Z): c mod N. c d Corollary. There exists infinitely-many linearly independent cusp forms for congruence groups. 5. Appendix: A Tauberian Theorem Let µ be a Borel measure on [, ), then its Laplace transform is the function µ: (, ) R given by Theorem 3. (Karamata s Theorem) or any r and a R, µ(t) := lim tr µ(t) = a t + e tx dµ(x). (5.) lim x x r µ([, x]) = urther details and related results are given in Section 6. of []. References a Γ(r + ). [] W. Arendt, Heat Kernels. ISEM (5/6). [] M. Huxley, Scattering matrices for congruence subgroups. Modular orms, R. Rankin Ed, (984), [3] H. Iwaniec, Spectral Methods of Automorphic orms. AMS Volume 53, (). [4] W. Müller, Weyl s Law in the Theory of Automorphic orms. Groups and Analysis, The Legacy of Hermann Weyl, Cambridge Univ. Press., (8), pp