Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to a congruence subgroup. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.

[1]  Tejasi Bhatnagar.  MODULAR FORMS AND MODULAR CURVES , 2018 .

[2]  Jean-Pierre Serre,et al.  Répartition asymptotique des valeurs propres de l’opérateur de Hecke _ , 1997 .

[3]  Analytic isolation of newforms of given level , 2016, 1601.02530.

[4]  Djordje Milićević,et al.  The Second Moment of Twisted Modular L-Functions , 2014, 1404.7845.

[5]  M. Murty,et al.  Effective equidistribution of eigenvalues of Hecke operators , 2009 .

[6]  Ian Petrow,et al.  A generalized cubic moment and the Petersson formula for newforms , 2016, Mathematische Annalen.

[7]  Proceedings of the London Mathematical Society , 1877, Nature.

[8]  J. Hoffstein,et al.  Appendix: An Effective Zero-Free Region , 1994 .

[9]  Jun-Ichi Igusa.  KRONECKERIAN MODEL OF FIELDS OF ELLIPTIC MODULAR FUNCTIONS.* 1 , 1959 .

[10]  B. M. Fulk.  MATH , 1992 .

[11]  W. Li.  L-series of Rankin type and their functional equations , 1979 .

[12]  The Fourth Power Mean of Dirichlet's L-Functions , 1981 .

[13]  Peter Humphries.  Density theorems for exceptional eigenvalues for congruence subgroups , 2016, Algebra & Number Theory.

[14]  E. Kowalski,et al.  The analytic rank of $J_0 ( q )$ and zeros of automorphic $L$-functions , 1999 .

[15]  S. Gelbart,et al.  A relation between automorphic representations of GL(2) and GL(3) , 2003 .

[16]  Wen-Ching Winnie Li,et al.  Newforms and functional equations , 1975 .

[17]  Ian Petrow,et al.  Elliptic curves over a finite field and the trace formula , 2015, 1510.03980.

[18]  Fred Diamond,et al.  A First Course in Modular Forms , 2008 .

[19]  W. Banks.  Twisted symmetric-square $L$-functions and the nonexistence of Siegel zeros on $GL(3)$ , 1997 .

[20]  Akshay Venkatesh.  Large sieve inequalities for GL(n)-forms in the conductor aspect , 2006 .

[21]  R. Schulze-Pillot,et al.  Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients , 2016, International Journal of Number Theory.

[22]  Charles Li,et al.  Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms , 2012, 1202.0189.

[23]  S. Gelbart,et al.  A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$ , 1978 .

[24]  Steven J. Miller,et al.  One-level density for holomorphic cusp forms of arbitrary level , 2016, 1604.03224.

[25]  M. Ng,et al.  The basis for space of cusp forms and Petersson trace formula , 2012 .

[26]  D. Rouymi.  Formules de trace et non-annulation de fonctions $L$ automorphesau niveau $\mathfrak{p}^{\nu}$ , 2011 .

[27]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[28]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[29]  Peter Sarnak,et al.  Low lying zeros of families of L-functions , 1999, math/9901141.

[30]  Carlos J. Moreno,et al.  Algebraic curves over finite fields: Frontmatter , 1991 .

[31]  P. Kam,et al.  : 4 , 1898, You Can Cross the Massacre on Foot.

[32]  M. Ziegler.  Volume 152 of Graduate Texts in Mathematics , 1995 .

[33]  Barbara Schneider,et al.  Basel , 2000 .

[34]  A simplified trace formula for Hecke operators for Γ , 1992 .

[35]  W. Duke,et al.  A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations , 2000 .

[36]  M. Tsfasman,et al.  Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound , 1982 .