View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by MPG.PuRe Asymptotic trace formula for the Hecke operators Junehyuk Jung Naser T. Sardari (With an appendix by Simon Marshall) Abstract Given integers m, n and k, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the mth and nth Fourier coefficients of an orthonormal basis of Sk (N)∗ (the weight k newforms 1 with fixed square-free level N) provided that 4π√mn k = o k 3 . Moreover, | − | we establish an explicit formula with a power saving error term for the trace of the Hecke operator on S (N)∗ averaged over k in a short interval. By bounding the Tn∗ k second moment of the trace of n over a larger interval, we show that the trace of n T 1 T is unusually large in the range 4π√n k = o n 6 . As an application, for any fixed | − | prime p coprime to N, we show that there exists a sequence kn of weights such { } that the error term of Weyl’s law for is unusually large and violates the prediction Tp of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd, Jakobson and Sarnak [GJS99, Theorem 1.4] with an improved exponent. Acknowledgments. J.J. thanks S.M. and Department of Mathematics of UW-Madison for invitation and sup- port. J.J. also thanks Sug Woo Shin, Peter Jaehyun Cho, and Matthew Young for many helpful comments. J.J. was supported by NSF grant DMS-1900993, and by Sloan Research Fellowship. S.M. was supported arXiv:1808.04015v3 [math.NT] 2 Aug 2020 by NSF grant DMS-1902173. N.T.S. was supported by NSF grant DMS-2015305 and is grateful to Max Planck Institute for Mathematics in Bonn and Institute For Advanced Study for their hospitalities and finan- cial supports. N.T.S. thanks his Ph.D. advisor Peter Sarnak for several insightful and inspiring conversations regarding the error term of the Weyl law while he was a graduate student at Princeton University. 1 Introduction In this paper, we give bounds for the error term of Weyl’s law for the Hecke eigenvalues of the family of classical holomorphic modular forms with a fixed level. We briefly describe J. Jung: Department of Mathematics, Brown University, Providence, RI 02912 USA; e-mail: june- [email protected] N. T. Sardari: The Institute For Advanced Study, Princeton, NJ 08540 USA; e-mail: [email protected] S. Marshall: Department of Mathematics, UW-Madison, Madison, WI 53706 USA; e-mail: mar- [email protected] ) Mathematics Subject Classification (2010): Primary 11F25; Secondary 11F72 1 2 Junehyuk Jung, Naser T. Sardari, (With an appendix by Simon Marshall) this family, its Weyl’s law, and known bounds and predictions on its error term. Next, we explain our results and compare them with the previous results and predictions. Let a b Γ (N) := SL (Z): c 0 (mod N) 0 c d ∈ 2 ≡ be the Hecke congruence subgroup of level N. Let Sk (N) be the space of even weight k Z modular forms of level N. It is the space of the holomorphic functions f such that ∈ az + b f =(cz + d)k f (z) cz + d a b for every Γ (N), and f converges to zero as it approaches each cusp (we have c d ∈ 0 finitely many cusps for Γ0 (N) that are associated to the orbits of Γ0 (N) acting by Möbius 1 transformations on P (Q)) [Sar90]. It is well-known that Sk (N) is a finite dimensional vector space over C, and is equipped with the Petersson inner product k dxdy f,g := f (z) g (z)y 2 . h i Γ0(N) H y Z \ Assume that n is fixed and is coprime to N. Then the nth (normalized) Hecke operator acting on S (N) is given by Tn k k 1 − k az + b (f)(z) := n 2 d− f . (1.1) Tn d adX=n b (modX d) The Hecke operators form a commuting family of self-adjoint operators with respect to the Petersson inner product, and therefore Sk (N) admits an orthonormal basis Bk,N con- sisting only of joint eigenfunctions of the Hecke operators. Any form f belonging to Bk,N is referred as a (Petersson normalized) holomorphic Hecke cusp form. Let ∞ k 1 − f (z)= ρf (n)n 2 e (nz) n=1 X be the Fourier expansion of f B at the cusp . For gcd(n, N)=1, we denote by ∈ k,N ∞ λf (n) the nth (normalized) Hecke eigenvalue of f, i.e., f = λ (n)f, Tn f and we have ρf (n)= ρf (1)λf (n) [Iwa97, p. 107]. By the celebrated result due to Deligne [Del74], we have λ (n) σ(n), (1.2) | f | ≤ for all n, where σ is the divisor function. 3 3 t We use the divisor function parameterized by t, defined by σt (n)= d|n d . When t =0, we drop 0, and use σ instead of σ0. P Asymptotic trace formula 3 Under Langlands’ philosophy, the Hecke operator is the p-adic analogue of the Tp Laplace operator, in the following sense. The eigenvalues of determine the Satake pa- Tp rameters of the associated local representation πp of GL2 (Qp) just as the Laplace eigen- value of the Maass form determines the associated local representation π of GL2 (R). ∞ Now fix a rational prime p that is coprime to N, and let 1 µk,N := δλf (p) dim (Sk (N)) f B ∈Xk,N be the spectral probability measure associated to acting on S (N). Using the Eichler– Tp k Selberg trace formula, Serre [Ser97] proved that µ converges weakly to µ as k +N k,N p → with gcd(N,p)=1, where µ is the Plancherel measure of GL (Q ) given by ∞ p 2 p 1 2 2 1 x p +1 − 4 µp (x) := dx. 1 1 2 π 2 p 2 + p− 2 x − Moreover, if we let Γ(k 1) ν := − ρ (1) 2δ , k,N (4π)k 1 | f | λf (p) − f B ∈Xk,N then it follows from the Petersson trace formula (see Section 2) that νk,N converges weakly to the semi-circle law 1 x2 µ (x) := 1 dx, ∞ π r − 4 as k + N with gcd(N,p)=1. →∞ 1.1 Quantitative rate of convergence Given two probability measures µ1 and µ2 on R, we denote the discrepancy between them by D (µ1,µ2) , where D (µ ,µ ) := sup µ (I) µ (I) : I = [a, b] R . 1 2 {| 1 − 2 | ⊂ } In [GJS99], Gamburd, Jakobson and Sarnak studied the spectrum of the elements in the group ring of SU (2), and proved 1 D (µ ,µ )= O . (1.3) k,2 p log k Moreover, [GJS99, Theorem 1.4] is equivalent to the existence of a sequence of integers kn such that →∞ 1 D (µkn,2,µp) 1 . (1.4) ≫ 2 2 kn (log kn) This is a corollary of their lower bound for the variance of the trace of the Hecke operators by varying the weight k (Theorem 1.6). 4 Junehyuk Jung, Naser T. Sardari, (With an appendix by Simon Marshall) 1.2 Bounds for the error term of Weyl’s law We now present some details regarding the philosophical analogy between Weyl’s law and µ µ . To this end, we first review Weyl’s law. Let X Rd be a bounded domain k,N → p ⊂ with smooth boundary. Let T be a positive real number, and let N (T ) be the number of Dirichlet Laplacian eigenvalues of X less than T 2 (counted with multiplicity). It was conjectured independently by Sommerfeld and Lorentz, based on the work of Rayleigh on the theory of sound, and proved by Weyl [Wey11] shortly after, that N (T )= c vol (X) T d (1 + o (1)) as λ , d →∞ d where cd is a constant depending only on d and vol (X) is the volume of X in R . This gives the distribution of the eigenvalues of the Laplace–Beltrami operator as T . As →∞ in Langlands’ philosophy, this is analogous to the convergence of µ µ giving the k,N → p distribution of the Hecke eigenvalues as k . →∞ More generally, let M d,g be a compact smooth Riemannian manifold of dimension d. Let N(T ) be the number of eigenvalues of the Laplace–Beltrami operator ∆ less − g than T 2, counted with multiplicity. Then Hörmander [Hör68] proved that d N (T )= cdvol (M) T + RM (T ) , d 1 where RM (T ) = O T − . In fact, this general estimate is sharp for the round sphere d. However, given a manifold the question of finding the optimal bound for the M = S M error term R (T ) is a very difficult problem. An analogue of R (T ) for µ µ is M m k,N → p the discrepancy D(µk,N ,µp). Remark. If M is a symmetric space, then Weyl’s law is formulated and expected to hold in great generality for families of automorphic forms [SST16, Conjecture 1]. We now restrict to the case d =2, and discuss the relation between the size of RM (T ) and the geodesic flow on the unit cotangent bundle S∗M, predicted by the correspondence principle. The two extreme behaviors that the geodesic flow can have are being chaotic or completely integrable, and in these two cases the correspondence principle predicts the distribution of eigenvalues to be modeled by a large random matrix, and a Poisson process, respectively [Ber85, Ber86].