Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms A. Knightly and C. Li December 12, 2012 Abstract We give an adelic treatment of the Kuznetsov trace formula as a rel- ative trace formula on GL(2) over Q. The result is a variant which in- corporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. We include a proof of a Weil bound for the general- ized twisted Kloosterman sums which arise on the geometric side. As an application, we show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidis- tributed relative to the Sato-Tate measure in the limit as the level goes to infinity. Contents 1 Introduction 3 1.1 Somehistory ............................. 3 1.2 Overviewofthecontents. 6 1.3 Acknowledgements .......................... 9 2 Preliminaries 10 2.1 NotationandHaarmeasure . 10 2.2 Characters and Dirichlet L-functions ................ 12 3 Bi-K -invariant functions on GL2(R) 15 3.1 Severalguises∞ ............................. 15 3.2 TheHarish-Chandratransform . 18 3.3 TheMellintransform. 19 3.4 TheSelbergtransform . 20 3.5 The principal series of G(R)..................... 22 4 Maass cusp forms 26 4.1 Cuspformsofweight0........................ 26 4.2 Heckeoperators............................ 28 4.3 AdelicMaassforms.......................... 31 1 5 Eisenstein series 36 5.1 Induced representations of G(A) .................. 36 5.2 DefinitionofEisensteinseries . 39 5.3 The finite part of φ .......................... 40 K K1(N) 5.4 An orthogonal basis for H(χ1,χ2) ∞× ............ 43 5.5 Evaluationofthebasiselements . 46 5.6 FourierexpansionofEisensteinseries. 48 5.7 Meromorphiccontinuation. 51 5.8 Charactersums............................ 53 6 The kernel of R(f) 55 6.1 Thespectraldecomposition . 55 6.2 Kernelfunctions ........................... 59 6.3 A spectral lower bound for Kh h∗ (x,x)............... 62 6.4 The spectral form of the kernel∗ of R(f) .............. 65 7 A Fourier trace formula for GL(2) 70 7.1 Convergenceofthespectralside . 70 7.2 Cuspidalcontribution . 70 7.3 Residualcontribution . 74 7.4 Continuouscontribution . 74 7.5 Geometricside ............................ 77 7.5.1 Firstcellterm ........................ 78 7.5.2 Secondcellterms. 80 7.6 Finalformulas ............................ 84 7.7 Classicalderivation. 87 8 Validity of the KTF for a broader class of h 90 8.1 Preliminaries ............................. 91 8.2 Smoothtruncation . 100 8.3 Comparing the KTF for h and hT .................104 8.4 R0(f) for f not smooth or compactly supported. 106 8.5 ProofofProposition8.29 . 112 9 Kloosterman sums 119 9.1 AboundfortwistedKloostermansums . 120 9.2 Factorization ............................. 128 9.3 ProofofTheorem9.2. 130 10 Equidistribution of Hecke eigenvalues 132 Notation index 142 Subject index 145 2 1 Introduction 1.1 Some history A Fourier trace formula for GL(2) is an identity between a product of two Fourier coefficients, averaged over a family of automorphic forms on GL(2), and a series involving Kloosterman sums and the Bessel J-function. The first example, arising from Petersson’s computation of the Fourier coefficients of Poincar´eseries in 1932 [P1] and his introduction of the inner product in 1939 [P2], has the form Γ(k 1) am(f)an(f) k S(m, n; c) 4π√mn − k 1 2 = δm,n+2πi Jk 1( ), (4π√mn) − f c − c f (N) c NZ+ ∈FXk k k ∈X where (N) is an orthogonal basis for the space of cusp forms S (Γ (N)), and Fk k 0 S(m, n; c)= e2πi(mx+nx)/c xx 1 mod c ≡X is a Kloosterman sum. Because of the existence of the Weil bound S(m, n; c) τ(c)(m,n,c)1/2c1/2 (1.1) | |≤ where τ is the divisor function, and the bound k 1 1/2 Jk 1(x) min(x − ,x− ) − ≪ for the Bessel function, the Petersson formula is useful for approximating ex- pressions involving Fourier coefficients of cusp forms. For example, Selberg used it in 1964 ([Sel3]) to obtain the nontrivial bound (k 1)/2+1/4+ε an(f)= O(n − ) (1.2) (k 1)/2+ε in the direction of the Ramanujan-Petersson conjecture an(f)= O(n − ) subsequently proven by Deligne. In his paper, Selberg mentioned the problem of extending his method to the case of Maass forms. This was begun in the late 1970’s independently by Bruggeman and Kuznetsov ([Brug], [Ku]). The left-hand side of the above Petersson formula is now replaced by a sum of the form am(uj)an(uj ) h(tj ) 2 , (1.3) uj cosh(πtj ) u Xj ∈F k k where m,n > 0, is an (orthogonal) basis of Maass cusp forms of weight k = 0 F 1 2 and level N = 1, tj is the spectral parameter defined by ∆uj = ( 4 + tj )uj for the Laplacian ∆, and h(t) is an even holomorphic function with sufficient decay. 3 There is a companion term coming from the weight 0 part of the continuous spectrum, describable in terms of the Eisenstein series 1 y1/2+s E(s,z)= (Re(s) > 1 ,y > 0,z = x + iy). 2 cz + d 1+2s 2 c,d Z (c,dX∈)=1 | | More accurately, it involves the analytic continuation to s on the imaginary line. This analytic continuation is provided by the Fourier expansion 1/2+s 1/2 s √π Γ(s)ζ(2s) E(s,z)= y + y − (1.4) Γ(1/2+ s)ζ(1+2s) 2y1/2π1/2+s + σ (m) m sK (2π m y)e2πimx. Γ(1/2+ s)ζ(1+2s) 2s | | s | | m=0 X6 2s Here σ2s(m)= 0<d m d is the divisor sum, and Ks is the K-Bessel function. The continuous contribution| to the Kuznetsov/Bruggeman formula is the fol- P lowing integral of the product of two Fourier coefficients of E(it,z) against the function h(t): it 1 ∞ (m/n) σ (m)σ (n) 2it 2it h(t)dt. (1.5) π ζ(1+2it) 2 Z−∞ | | The Fourier trace formula is then the equality between the sum of (1.3) and (1.5) on the so-called spectral side, with the geometric side given by δ ∞ 2i S(m, n; c) ∞ 4π√mn h(t) t m,n h(t) tanh(πt) tdt + J ( ) dt. π2 π c 2it c cosh(πt) Z−∞ c Z+ Z−∞ X∈ (1.6) Using this together with the Weil bound (9.2), Kuznetsov proved a mean-square estimate for the Fourier coefficients an(uj ) ([Ku], Theorem 6), which immedi- ately implies the bound a (u ) n1/4+ε n j ≪j,ε ε in the direction of the (still open) Ramanujan conjecture an(uj )= O(n ). (See also [Brug], 4.) This extended Selberg’s result (1.2) to the case of Maass forms. Kuznetsov§ also “inverted” the formula to give a variant in which a gen- eral test function appears on the geometric side in place of the Bessel integral. (Motohashi has given an interesting conceptual explanation of this, showing that the procedure is reversible, [Mo2].) This allows for important applications to bounding sums of Kloosterman sums. Namely, Kuznetsov proved that the estimate S(m, n; c) Xθ+ε (1.7) c ≪m,n,ε c X X≤ 1 holds with θ = 6 ([Ku], Theorem 3). The Weil bound alone yields only θ = 1 2 , showing that Kuznetsov’s method detects considerable cancellation among the Kloosterman sums due to the oscillations in their arguments. Linnik had 4 conjectured in 1962 that (1.7) holds with θ = 0, and Selberg remarked that this would imply the Ramanujan-Petersson conjecture for holomorphic cusp forms of level 1, ([Sel3]; see also 4 of [Mu]). By studying the Dirichlet series § S(m, n; c) Z(s,m,n)= , c2s c X Selberg also codified a relationship between sums of Kloosterman sums and the 1 smallest eigenvalue λ1 of the Laplacian, leading him to conjecture that λ1 4 3 ≥ for congruence subgroups. He obtained the inequality λ1 16 using the Weil bound (9.2). This inequality is also a consequence of the gen≥eralized Kuznetsov formula given in 1982 by Deshouillers and Iwaniec ([DI]). Fourier trace formulas have since become a staple tool in analytic number theory. We mention here a sampling of notable results in which they have played a role. Deshouillers and Iwaniec used the Kuznetsov formula to deduce bounds for very general weighted averages of Kloostermans sums, showing in particular that Linnik’s conjecture holds on average ([DI], 1.4). They list some interesting consequences in 1.5 of their paper. For example,§ there are infinitely many primes p for which p+1§ has a prime factor greater than p21/32. They also give applications to the Brun-Titchmarsh theorem and to mean-value theorems for primes in arithmetic progressions (see also [Iw1], 12-13). Suppose f(x) Z[x] is a quadratic polynomial with§ negative discriminant. ∈ ν If p is prime and ν is a root of f in Z/pZ, then the fractional part p [0, 1) is independent of the choice of representative for ν in Z. Duke, Friedlander,{ }∈ and Iwaniec proved that for (p,ν) ranging over all such pairs, the set of these fractional parts is uniformly distributed in [0, 1], i.e. for any 0 α<β 1, ≤ ≤ # (p,ν) p x,f(ν) 0 mod p,α ν <β { | ≤ ≡ ≤ { p } } (β α) # p x p prime ∼ − { ≤ | } as x ([DFI]). Their proof uses the Kuznetsov formula to bound a certain related→ Poincar´eseries∞ via its spectral expansion. See also Chapter 21 of [IK]. Applications of Fourier trace formulas to the theory of L-functions abound. Using the results of [DI], Conrey showed in 1989 that more than 40% of the zeros of the Riemann zeta function are on the critical line ([Con]).1 Motohashi’s book [Mo1] discusses other applications to ζ(s), including the asymptotic formula for its fourth moment.