Asymptotic trace formula for the Hecke operators

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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.

Original languageEnglish (US)
Pages (from-to)513-557
Number of pages45
JournalMathematische Annalen
Issue number1-2
StatePublished - Oct 1 2020

All Science Journal Classification (ASJC) codes

  • General Mathematics


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