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The Prime Geodesic Theorem

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The Prime Geodesic Theorem The Prime Geodesic Theorem Matt Tyler Throughout these notes, we write Γ for SL2(Z). Recall that a quadratic form Q(x; y) = [a; b; c] = ax2 +bxy+cy2 is primitive if gcd(a; b; c) = 1. The discriminant of [a; b; c] is d = b2 −4ac. We will be concerned here with the indefinite quadratic forms, which are those forms with non-square discriminant d > 0. The group Γ acts on the set of quadratic forms via the action α β Q (x; y) = Q(αx + βy; γx + δy): (0.1) γ δ This action preserves both the discriminant and the property of being primitive. We say that two quadratic forms in the same orbit are equivalent. The stabilizer of the quadratic form [a; b; c] in Γ is the group x−by 2 −cy 2 2 ∼ Γ[a;b;c] = ± x+by j x − dy = Z × Z=2Z: (0.2) ay 2 2 2 If x0; y0 > 0 are the fundamental solutions to the Pell equation x − dy , then we have x0−by0 2 −cy0 Γ[a;b;c] = ±M[a;b;c] where M[a;b;c] = x0+by0 : (0.3) ay0 2 p x0+ dy0 The two important quantities depending on the discriminant d are d = 2 and h = jfequivalence classes of primitivegj, which are analogous to the fundamental unit and narrow d forms of discriminantp d class number for Q( d), respectively. Let D be the set of positive discriminants fd > 0 j d ≡ 0; 1 (mod 4); d non-squareg, and let D(x) = fd 2 D j d ≤ xg. The purpose of these notes is to prove the following asymptotic expression due to Sarnak for the average value of hd ordered by d. Theorem 0.1. 1 X 16 li(x2) h = + O x2=3+ jD(x)j d 35 x d2D(x) R u dt where li(u) is the logarithmic integral 2 log t , as in the ordinary prime number theorem. 1 Summing by parts, we immediately obtain the following corollary. Corollary 0.2. 1 X 8 h log = x + O x2=3+ : jD(x)j d d 35 d2D(x) By way of comparison, the following asymptotic expression for the average value of hd log d ordered by discriminant was noticed by Gauss and confirmed by Siegel. Theorem 0.3. 1 X π2 p h log = x + O(x log x): jfd 2 D j d ≤ xgj d d 9ζ(3) d2D d≤x In Section 1, we will prove Theorem 0.1 (assuming the prime geodesic theorem for Γ and an asymptotic expression for jD(x)j) by exploiting a correspondence between equivalence classes of primitive binary quadratic forms, primitive hyperbolic conjugacy classes in Γ, and closed geodesics on ΓnH. The main ingredient in the proof of the prime geodesic theorem is the Selberg trace formula, which we will explain in section 2. We will then prove the prime geodesic theorem in section 3. In appendix A, we prove the asymptotic formula for jD(x)j. The material in section 1 and appendix A comes from Sarnak [3], and the material in sections 2 and 3 comes primarily from the books of Bergeron [1] and Iwaniec [2]. Contents 1 Closed Geodesics on SL2(Z)nH 3 2 The Selberg Trace Formula 4 2.1 The spectral trace . .7 2.2 Parabolic classes . .8 2.3 The identity class . .8 2.4 Hyperbolic classes . .8 2.5 Elliptic classes . .8 2.6 The full trace formula . .9 3 The Prime Geodesic Theorem 9 A Asymptotics for jD(x)j 12 2 1 Closed Geodesics on SL2(Z)nH Let H be the Poincar´emodel of the hyperbolic plane, which is the upper half-plane fx+iy j y > 0g with the Riemannian metric. The group SL2(R) acts on H via fractional linear a b az+b transformations c d z = cz+d , which are isometries of H. Our goal in this section is to explain the correspondence between equivalence classes of primitive binary quadratic forms, primitive hyperbolic conjugacy classes in Γ, and closed geodesics on ΓnH. We begin with the correspondence between the latter two col- lections (which actually holds for any discrete subgroup Γ0 of SL2(R)). We say that g 2 SL2(R) is hyperbolic if its fixed points as a linear fractional transformation are in R^ = R [ f1g and distinct. If i g is hyperbolic, then it is conjugate (in SL2(R)) to a homothety 1=2 p 0 for some p > 1 (with p determined by the formula 0 p−1=2 tr g = p1=2 + p−1=2), and we say the norm Ng of g is p. 1 1 − 2 2 We say that a hyperbolic element γ 2 Γ is primitive if it is not a non-trivial power in Γ, so that every hyperbolic element of Γ ΓnH is a power of a unique primitive hyperbolic in Γ. Note that con- jugates of hyperbolic elements are hyperbolic, and similarly for primitive hyperbolics, so we may speak of primitive hyperbolic conjugacy classes in Γ. The geodesics on H are the semicircles perpendicular to the real line and the vertical lines, so there is a unique geodesic connecting any two points in R^. Each hyperbolic element g 2 SL2(R) maps the geodesic ag (of length log Ng) connecting the two fixed points of g back to itself. For γ 2 Γ hyperbolic, aγ is a closed geodesic on ΓnH, and any Γ-conjugate of γ gives rise to the same geodesic. This gives a map n o n o primitive hyperbolic −!∼ closed geodesics on ; (1.1) conjugacy classes [γ] ΓnH of length log Nγ and it is easy to see that this map is bijective. Returning to binary quadratic forms, recall the matrix M[a;b;c] for a quadratic form [a; b; c] −1 (defined in (0.3)), which is a primitive hyperbolic with trace x0 = d + d and hence norm 2 d. The map [a; b; c] 7! M[a;b;c] is Γ-invariant and in fact gives a bijection nequivalence classes of primitiveo ∼ nprimitive hyperbolico binary quadratic forms −! conjugacy classes (1.2) 2 which sends a form of discriminant d to conjugacy class of norm d. 3 Combining the correspondences (1.1) and (1.2), we arrive at the following proposition. Proposition 1.1. The lengths of the closed geodesics on ΓnH are the numbers 2 log d with multiplicity hd. In light of this reformulation, Theorem 0.3 follows from the asymptotic expression 35 jD(x)j = x + O x2=3+ (1.3) 16 proved in the appendix, as well as the following result. Theorem 1.2 (Prime geodesic theorem). ! x3=4 π(x) = li(x) + O log x where π(x) is the number of closed geodesics on ΓnH of length at most log x. Proof of Theorem 0.1. By the prime geodesic theorem and the proposition above, ! X x3=4 h = π(x2) = li(x2) + O : (1.4) d log x d2D(x) Dividing by (1.3), we find 1 X 16 li(x2) h = + O x2=3+ ; (1.5) jD(x)j d 35 x d2D(x) as claimed. 2 The Selberg Trace Formula Given a function k : H ! H ! C, consider the integral operator L with kernel k defined by Z (Lf)(z) = k(z; w)f(w)dµw: (2.1) H This operator is SL2(R)-invariant (i.e. it commutes with precomposition with g for all g 2 SL2(R)) if and only if k(gz; gw) = k(z; w) for all g 2 SL2(R); (2.2) 4 in which case k depends only on the hyperbolic distance ρ(z; w) between z and w. This hyperbolic distance satisfies jz − wj2 cosh ρ(z; w) = 1 + 2u(z; w) where u(z; w) = ; (2.3) 4 Im z Im w so we may write k(z; w) as k(u(z; w)) where k(u) is a function in one variable u ≥ 0. These invariant integral operators have the following important property. Whenever f : 2 @2 @2 H ! C is an eigenfunction of the Laplacian ∆ = y @x2 + @y2 , f is also an eigenfunction of L. In particular, if (∆ + λ)f = 0, then Z k(z; w)f(w)dµw = h(t)f(z) (2.4) H 1 2 where λ = 4 + t and h(t) is the Selberg/Harish-Chandra transformation of k defined by Z 1 k(u) q(v) = p du; v u − v g(r) = 2q((sinh r=2)2); (2.5) Z 1 h(t) = eirtg(r)dr: −∞ If we restrict the domain of L to automorphic functions (i.e. functions invariant under Γ), then we may write Z (Lf)(z) = K(z; w)f(w)dµw (2.6) ΓnH P where K(z; w) = γ2Γ k(z; γw). The Selberg trace formula comes from evaluating the trace Tr K = R K(z; z)dµz of K in two different ways, one using the spectrum of the ΓnH Laplacian and one using the geometry of the Riemann surface ΓnH. It will be more convenient to write the formula in terms of the function h and its Fourier transform h^(r) = 1 R 1 irt 2π −∞ e h(t)dt. In order for all of our sums and integrals to converge, we impose the following conditions on h: h(t) is even; 1 h(t) is holomorphic in the strip j Im tj ≤ + , (2.7) 2 h(t) (jtj + 1)−(2+) in this strip: To begin, we may write Tr K = P R k(z; γz)dµz.
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