TY - JOUR

T1 - Asymptotic trace formula for the Hecke operators

AU - Jung, Junehyuk

AU - Talebizadeh Sardari, Naser

N1 - Funding Information:
J.J. thanks S.M. and Department of Mathematics of UW-Madison for invitation and support. 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 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 financial 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. N.T.S. was supported Grant no. 1902185 and S.M. was supported Grant no. 1501230
Funding Information:
J.J. thanks S.M. and Department of Mathematics of UW-Madison for invitation and support. 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 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 financial 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. N.T.S. was supported Grant no. 1902185 and S.M. was supported Grant no. 1501230
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/10/1

Y1 - 2020/10/1

N2 - 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 with fixed square-free level N) provided that |4πmn-k|=o(k13). 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 second moment of the trace of Tn over a larger interval, we show that the trace of Tn is unusually large in the range |4πn-k|=o(n16). 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 Tp 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.

AB - 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 with fixed square-free level N) provided that |4πmn-k|=o(k13). 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 second moment of the trace of Tn over a larger interval, we show that the trace of Tn is unusually large in the range |4πn-k|=o(n16). 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 Tp 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.

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U2 - 10.1007/s00208-020-02054-w

DO - 10.1007/s00208-020-02054-w

M3 - Article

AN - SCOPUS:85089179775

SN - 0025-5831

VL - 378

SP - 513

EP - 557

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

ER -