THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES IGOR PROKHORENKOV 1. Introduction to the Selberg Trace Formula This is a talk about the paper H. P. McKean: Selberg’s Trace Formula as applied to a compact Riemann surface (1972). For simplicity, assume that M is a compact Riemannian manifold. Consider classical mechanics on M , where a free particle on M moves along geodesics. If M has infinite fundamental group, then in each free homotopy class of a curve on M , there is a unique closed geodesic. If M is a Riemann surface of genus ≥ 1, we can look at the length of the (unique) closed shortest geodesic in each equivalence class from π1 (M ). Next, consider quantum mechanics. Eigenvalues of the Laplacian on the manifold 0 = γ1 < γ2 ≤ γ3 ≤ ... ↑ +∞ γj is the energy of the j th “pure” state. We expect that there is a relation between the classical data (lengths of closed geodesics) and quantum data (eigenvalues). The Selberg trace formula provides this link. So there should be a “Selberg trace formula” on any manifold. There are many examples of this. When the manifold has a lot of symmetry (eg hyperbolic space mod a subgroup of P SL (2, R)), there is an example. Starting with the 19th century: the Poisson summation formula. Let M = CL be the 2-torus, where L is an integral lattice. We let L be the integral span of 1 and a + ib. Consider the Laplacian on this surface. Then the Laplacian will be 2 ∂ ∂2 + ∆=− ∂x21 ∂x22 acting on L-periodic functions. The eigenfunctions are f (x) = exp (2πiω 0 · x) , where ω 0 is an element of the dual lattice: that is ω 0 ·ω ∈ Z for all ω ∈ L. The corresponding eigenvalues are 4π 2 |ω 0 |2 . The Poisson summation 1 2 IGOR PROKHORENKOV formula relates the theta functions of L and its dual lattice L0 , as follows: ! area (M ) X X |ω|2 2 0 2 exp −4π |ω | t = exp − , 4πt ω∈L 4t ω 0 ∈L0 or X λj | ! X area (M ) |ω|2 exp (−λj t) = exp − 4πt 4t |ω| length of closed geodesics {z } | {z } quantum side classical side The theta function in physics language is the partition function. Also, this is the trace of the heat kernel: X exp (−λj t) , tr (exp (−t∆)) = λj where exp (−t∆) is the heat operator. The heat equation is ∂ K = −∆x K; K (0, x, y) = δ (x − y) , ∂t Z tr (exp (−t∆)) = K (t, x, x) dx M In the case of the torus, K T (t, x, x) = X 2 K R (t, γ (x) , x) . γ∈L On the circle, 1 K S (t, x, x) = X K R (t, x + n, x) . n∈Z To prove the Poisson summation formula, one expands the left side in terms of the Fourier series 2and uses the known heat kernel for R : |x−y| 1 R √ K (t, x, y) = 4πt exp − 4t . The knowledge of the spectrum of the Laplacian determines |a| and |b|, as follows. For our torus with the lattice above, area(M ) = |b|. After this, subtract terms from both sides the parts corresponding to q geodesics with length 1, 2, 3, ... The next geodesic will be |a|2 + |b|2 , so we can find |a|. So we can determine the torus up to reflection. Next, we generalize to a hyperbolic Riemann surface M with genus g ≥ 2. Then the spectrum σ (M ) of the Laplacian is 0 = γ1 < γ2 ≤ γ3 ≤ ... → +∞. THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES 3 The Selberg trace formula relates the trace ∞ X tr (exp (−t∆)) = exp (−tλj ) j=1 of the heat kernel to the kind of dual theta function. The role of the lattice L is taken over by the conjugacy classes Q of G = π1 (M ), identified with a subgroup of SL (2, R). Then M = SL (2, R) G. The numbers |w| are replaced by l (Q) = 2 cosh −1 1 tr (Q) . 2 Here, Q is a free deformation class of closed paths on M , and l (Q) is the length of the shortest path in this class. There is a famous (noncompact) cases that we will not cover M = SL (2, R) SL (2, Z) or M = SL (2, R) Γ where Γ is an algebraic subgroup of SL (2, Z). Audrey Terras and Serge Lang have good books on the subject. Also there is a survey paper by Werner M¨ uller. 2. Riemann surface formula Let M be a compact Riemann surface of genus g ≥ 2. By the Riemann uniformization theorem, the universal cover is the upper half plane H. H = {(x1 + ix2 ) : x2 > 0} This is also called the Poincar´e hyperbolic plane, with metric ds2 = dx21 + dx22 x22 (You can also realize this as the Poincare disk.) The fundamental group π1 (M ) acts by deck on H that are isometries transformations a b z 7→ az+b , such that det = 1 with a, b, c, d ∈ R. So SL (2, R) cz+d c d is the group of isometries of H. Thus, we can identify π1 (M ) with a subgroup G of SL (2, R). The G has a fundamental region in H that is a hyperbolic polygon with 4g sides. 4 IGOR PROKHORENKOV What must be true about G in order that HG is a compact Riemann that SL (2, R) = KAN , where K = SO (2) = surface? Note cos (θ) − sin θ is the stabilizer of i. The group A is the group sin θ cos θ a 0 : a > 0 . The group N is the group of of magnifications 0 a−1 1 b horizontal translations, N = :b∈R . 0 1 Proposition 1. For any g ∈ SL (2, R), g is conjugate to • rotation iff tr (g) < 2 (elliptic) • magnification iff tr (g) > 2 (hyperbolic) • translation iff tr (g) = 2 (parabolic) The hyperbolic distance d (x, y) satisfies d (x, gx) = d (kx, kgx) = d kx, kgk −1 kx , for all k, g in SL (2, R). So inf d (x, gx) = inf d kx, kgk −1 kx x x = inf d x, kgk −1 x x Think of elements of G as homotopy classes of closed paths on M with fixed base point. Free homotopy classes of M are identified with the conjugacy class Q = {kgk −1 : k ∈ G}. Every nontrivial element of G is conjugate to a hyperbolic element. (Proof: if an element g of G is not the identity, then there is a geodesic of minimum length connecting some x to gx, but if g is parabolic or elliptic, this length can go to zero.) Thus it is conjugate to a magnification, and thus ` (g n ) = |n| ` (g) is true, where ` (g) = inf x d (x, gx) is the length of the shortest path. For every g ∈ G that is nontrivial, it can be expressed in a unique way as the positive power of a primitive element p ∈ G (primitive: it is not the power of any other element of G). Proposition 2. As p runs through the inconjugate primitive elements in G and n through the positive integers, the conjugacy class Q = kpn k −1 : k ∈ GGp runs through the conjugacy classes of G. Here, Gp is the centralizer of p. Moreover, for fixed p, n, elements kpk −1 run once through Q as k runs through GGp . THE SELBERG TRACE FORMULA OF COMPACT RIEMANN SURFACES Note that d (x, y) = cosh−1 1 + m 0 p∼ . 0 m−1 kx−yk 2x2 y2 5 , ` (pn ) = n |log m2 | where Theorem 3. (Selberg trace formula) Start with a function K : R → R that decays sufficiently rapidly as x → ∞. Then KH (x, y) = K (cosh d (x, y)) is a function on H × H. It induces a symmetric kernel on M × M via X KM (x, y) = KH (x, gy) . g∈G Then KM (x, hy) = KM (x, y) for all h ∈ G. Then, with dx = Z KM (x, x) dx trKM : = dx1 dx2 x22 = hyperbolic volume element M = area (M ) K (1) + ∞ X ` (p) X p cosh ` (pn ) − 1 n=1 inconjugate Z ∞ cosh `(pn ) K (b) db p . b − cosh ` (pn ) primitive p Proof. Let F be a fundamental domain of M . We have Z KM (x, x) dx trKM = F XZ K (cosh d (x, gx)) dx = g∈G M = area (M ) K (1) + = area (M ) K (1) + = area (M ) K (1) + ∞ X X X Z n=1 inconjugate k∈GGp primitive p ∞ X X X n=1 inconjugate k∈GGp primitive p ∞ X X Z K cosh d x, kpn k −1 x F Z K (cosh d (x, pn x)) dx F K (cosh d (x, pn x)) dx n=1 inconjugate primitive p Fp where Fp is a fundamental domain for Gp . Then p is conjugate to some magnification x 7→ m2 x. Then Fp = {x1 ∈ R : 1 ≤ x2 ≤ m2 } . The formula follows from a direct calculation. dx 6 IGOR PROKHORENKOV In the particular case where KH (t) is the fundamental solution of the heat equation on H. Then KH (t) = exp (−t∆). Then ∞ X tr (exp (−t∆)) = exp (−tγn ) n=0 = area (M ) e−t/4 Z ∞ 2 be−b /4t db sinh 21 b (4πt)3/2 0 ∞ 1X X e−t/4 −|`(pn )|2 /4t ` (p) e . + 2 n=1 inconjugate sinh 21 ` (pn ) (4πt)1/2 primitive p

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