Asymptotic trace formula for the Hecke operators
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Mathematische Annalen
Asymptotic trace formula for the Hecke operators Junehyuk Jung1 · Naser Talebizadeh Sardari2 With an appendix by Simon Marshall. Received: 1 November 2019 / Revised: 21 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 1 k newforms √ 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 Tn∗ on Sk (N )∗ averaged over k in a short interval. By bounding the we show that the trace of Tn second moment of the trace of Tn over a larger interval, √ 1 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 T p is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd et al. (J Eur Math Soc 1(1):51–85, 1999) [Theorem 1.4] with an improved exponent. Mathematics Subject Classification Primary 11F25 · Secondary 11F72
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 this family, its Weyl’s law, and known bounds and predictions on its error
Communicated by Kannan Soundararajan.
B
Naser Talebizadeh Sardari [email protected] Junehyuk Jung [email protected]
1
Department of Mathematics, Brown University, Providence, RI 02912, USA
2
The Institute For Advanced Study, Princeton, NJ 08540, USA
123
J. Jung et al.
term. Next, we explain our results and compare them with the previous results and predictions. Let a b 0 (N ) := ∈ S L 2 (Z) : c ≡ 0 (mod N ) c d 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 = (cz + d)k f (z) f cz + d a b ∈ 0 (N ), and f converges to zero as it approaches each cusp (we c d have finitely many cusps for 0 (N ) that are associated to the orbits of 0 (N ) acting by Möbius transformations on P1 (Q)) [31]. It is well-known that Sk (N ) is a finite dimensional vector space over C, and is equipped with the Petersson inner product for every
f , g :=
0 (N )\H
f (z) g (z)y k
d xd y . y2
Assume that n is fixed and is coprime to N . Then the nth (normalized) Hecke operator Tn acting on Sk (N ) is given by Tn ( f ) (z) := n
k−1 2
d −k
ad=n
b
f
(mod d)
az + b . d
(1.1)
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 consisting o
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