Skip to main content

Weighted Distribution of Low-lying Zeros of GL(2)  $L$ -functions

  • Andrew Knightly (a1) and Caroline Reno (a1)

We show that if the zeros of an automorphic $L$ -function are weighted by the central value of the $L$ -function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$ -value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$ -values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

Hide All

This work was partially supported by grant #317659 from the Simons Foundation to author A. K. Section 3 is based in part on the University of Maine MA thesis of author C. R.

Hide All
[BBDDM] Barrett, O., Burkhardt, P., DeWitt, J., Dorward, R., and Miller, S. J., One-level density for holomorphic cusp forms of arbitrary level . Res. Number Theory 3(2017), Art. 25.
[BBR] Blomer, V., Buttcane, J., and Raulf, N., A Sato–Tate law for GL(3) . Comment Math. Helv. 89(2014), no. 4, 895919.
[BLGHT] Barnet-Lamb, T., Geraghty, D., Harris, M., and Taylor, R., A family of Calabi-Yau varieties and potential automorphy II . Publ. Res. Inst. Math. Sci. 47(2011), no. 1, 2998.
[Br] Bruggeman, R. W., Fourier coefficients of cusp forms . Invent. Math. 45(1978), no. 1, 118.
[BrM] Bruggeman, R. and Miatello, R., Eigenvalues of Hecke operators on Hilbert modular groups . Asian J. Math. 17(2013), no. 4, 729757.
[CDF] Conrey, J. B., Duke, W., and Farmer, D. W., The distribution of the eigenvalues of Hecke operators . Acta Arith. 78(1997), no. 4, 405409.
[FW] Feigon, B. and Whitehouse, D., Averages of central L-values of Hilbert modular forms with an application to subconvexity . Duke Math. J. 149(2009), no. 2, 347410.
[FMP] File, D., Martin, K., and Pitale, A., Test vectors and central L-values for GL(2) . Algebra Number Theory 11(2017), no. 2, 253318.
[GMR] Gun, S., Murty, M. R., and Rath, P., Summation methods and distribution of eigenvalues of Hecke operators . Funct. Approx. Comment. Math. 39(2008), part 2, 191204.
[GR] Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products. 7th ed., Elsevier/Academic Press, San Diego, 2007.
[Gu] Guo, J., On the positivity of the central critical values of automorphic L-functions for GL(2) . Duke Math. J. 83(1996), no. 1, 157190.
[ILS] Iwaniec, H., Luo, W., and Sarnak, P., Low lying zeros of families of L-functions . Inst. Hautes Études Sci. Publ. Math. 91(2000), 55131.
[JK] Jackson, J. and Knightly, A., Averages of twisted L-functions . J. Aust. Math. Soc. 99(2015), no. 2, 207236.
[KL1] Knightly, A. and Li, C., Traces of Hecke operators. Mathematical Surveys and Monographs, 133, American Mathematical Society, Providence, RI, 2006.
[KL2] Knightly, A. and Li, C., Petersson’s trace formula and the Hecke eigenvalues of Hilbert modular forms. In: Modular forms on Schiermonnikoog, Cambridge University Press, Cambridge, 2008.
[KL3] Knightly, A. and Li, C., Weighted averages of modular L-values . Trans. Amer. Math. Soc. 362(2010), no. 3, 14231443.
[KL4] Knightly, A. and Li, C., Kuznetsov’s formula and the Hecke eigenvalues of Maass forms . Mem. Amer. Math. Soc. 224(2013), no. 1055.
[Ko] Kowalski, E., Families of cusp forms. In: Actes de la Conférence “Théorie des Nombres et Applications”, Publ. Math. Besançon Algèbre Théorie Nr., Presses Univ. Franche-Comté, Besançon, 2013, pp. 5–40.
[KS1] Katz, N. and Sarnak, P., Random mtrices, Frobenius eigenvalues and monodromy. American Mathematical Society Colloquium Publications, 45, American Mathematical Society, Providence, RI, 1999.
[KS2] Katz, N. and Sarnak, P., Zeros of zeta functions and symmetries . Bull. Amer. Math. Soc. 36(1999), 126.
[KST] Kowalski, E., Saha, A., and Tsimerman, J., Local spectral equidistribution for Siegel modular forms and applications . Compos. Math. 148(2012), no. 2, 335384.
[Li1] Li, C., Kuznietsov trace formula and weighted distribution of Hecke eigenvalues . J. Number Theory 104(2004), no. 1, 177192.
[Li2] Li, C., On the distribution of Satake parameters of GL 2 holomorphic cuspidal representations . Israel J. Math. 169(2009), 341373.
[MR] Michel, P. and Ramakrishnan, D., Consequences of the Gross-Zagier formulae: stability of average L-values, subconvexity, and non-vanishing mod p . Number theory, analysis and geometry, Springer, New York, 2012, pp. 437–459.
[Mi] Miller, S. J., An orthogonal test of the L-functions ratios conjecture . Proc. Lond. Math. Soc. (3) 99(2009), no. 2, 484520.
[MS] Murty, R. and Sinha, K., Effective equidistribution of eigenvalues of Hecke operators . J. Number Theory 129(2009), no. 3, 681714.
[MT] Matz, J. and Templier, N., Sato–Tate equidistribution for families of Hecke-Maass forms on . arxiv:1505.07285.
[Na] Nagoshi, H., Distribution of Hecke eigenvalues . Proc. Amer. Math. Soc. 134(2006), no. 11, 30973106.
[Ne] Newman, M., Integral matrices. Pure and Applied Mathematics, 45, Academic Press, New York-London, 1972.
[RR1] Ramakrishnan, D. and Rogawski, J., Average values of modular L-series via the relative trace formula . Pure Appl. Math. Q. 1(2005), no. 4, 701735.
[RR2] Ramakrishnan, D. and Rogawski, J., Erratum: Average values of modular L-series via the relative trace formula . Pure Appl. Math. Q. 5(2009), no. 4, 1469.
[Sa] Sarnak, P., Statistical properties of eigenvalues of the Hecke operators. In: Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Progr. Math., 70, Birkhäuser, Boston, MA, 1987, pp. 321–331.
[Se] Serre, J.-P., Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p . J. Amer. Math. Soc. 10(1997), no. 1, 75102.
[ST] Shin, S. W. and Templier, N., Sato–Tate theorem for families and low-lying zeros of automorphic L-functions . Invent. Math. 203(2016), no. 1, 1177.
[Su] Sugiyama, S., Asymptotic behaviors of means of central values of automorphic L-functions for GL(2) . J. Number Theory 156(2015), 195246.
[SuT] Sugiyama, S. and Tsuzuki, M., Relative trace formulas and subconvexity estimates of L-functions for Hilbert modular forms . Acta Arith. 176(2016), no. 1, 163.
[T] Tsuzuki, M., Spectral means of central values of automorphic L-functions for GL(2) . Mem. Amer. Math. Soc. 235(2015), no. 1110.
[W] Wang, Y., The quantitative distribution of Hecke eigenvalues . Bull. Aust. Math. Soc. 90(2014), 2836.
[Z] Zhou, F., Weighted Sato–Tate vertical distribution of the Satake parameter of Maass forms on PGL(N) . Ramanujan J. 35(2014), no. 3, 405425.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed