Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T07:53:09.751Z Has data issue: false hasContentIssue false
  • English
  • Français

One level density of low-lying zeros of families of L-functions

Part of: Discontinuous groups and automorphic forms Exponential sums and character sums Algebraic number theory: global fields Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  07 September 2010

Peng Gao  and
Liangyi Zhao
Peng Gao
Affiliation:
Division of Mathematical Science, School of Physics & Mathematical Science, Nanyang Technological University, Singapore 637371 (email: penggao@ntu.edu.sg)
Liangyi Zhao
Affiliation:
Division of Mathematical Science, School of Physics & Mathematical Science, Nanyang Technological University, Singapore 637371 (email: lzhao@pmail.ntu.edu.sg)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

Keywords

one level densitylow-lying zerosHecke eigenformsquadratic Dirichlet characterscubic Dirichlet charactersquartic Dirichlet characters

MSC classification

Secondary: 11F11: Holomorphic modular forms of integral weight 11L40: Estimates on character sums 11M06: $zeta (s)$ and $L(s, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M50: Relations with random matrices 11R16: Cubic and quartic extensions
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 1 - 18
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Baier, S. and Young, M. P., Mean values with cubic characters, J. Number Theory 130 (2010), 879–903.CrossRefGoogle Scholar
[2]Baier, S. and Zhao, L., On the low-lying zeros of Hasse–Weil L-functions for elliptic curves, Adv. Math. 219 (2008), 952–985.CrossRefGoogle Scholar
[3]Brumer, A., The average rank of elliptic curves, I, Invent. Math. 109 (1992), 445–472.CrossRefGoogle Scholar
[4]Davenport, H., Multiplicative number theory, Graduate Texts in Mathematics, vol. 74, third edition (Springer, New York, 2000).Google Scholar
[5]David, C., Fearnley, J. and Kisilevsky, H., On the vanishing of twisted L-functions of elliptic curves, Experiment. Math. 13 (2004), 185–198.CrossRefGoogle Scholar
[6]Deligne, P., La conjecture de Weil. I., Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.CrossRefGoogle Scholar
[7]Dueñez, E. and Miller, S. J., The low lying zeros of a GL(4) and a GL(6) family of L-functions, Compositio Math. 142 (2006), 1403–1425.CrossRefGoogle Scholar
[8]Dueñez, E. and Miller, S. J., The effect of convolving families of L-functions on the underlying group symmetries, Proc. London Math. Soc. (3) 99 (2009), 787–820.CrossRefGoogle Scholar
[9]Fouvry, E. and Iwaniec, H., Low-lying zeros of dihedral L-functions, Duke Math. J. 116 (2003), 189–217.CrossRefGoogle Scholar
[10]Gao, P., n-level density of the low-lying zeros of quadratic Dirichlet L-functions, arXiv:0806.4830.Google Scholar
[11]Güloğlu, A. M., On low lying zeros of automorphic L-functions, PhD thesis, The Ohio State University (2005).Google Scholar
[12]Güloğlu, A. M., Low-lying zeros of symmetric power L-functions, Int. Math. Res. Not. 2005 (2005), 517–550.CrossRefGoogle Scholar
[13]Heath-Brown, D. R., The average rank of elliptic curves, Duke Math. J. 122 (2004), 225–320.CrossRefGoogle Scholar
[14]Hecke, E., Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), 11–51.CrossRefGoogle Scholar
[15]Hughes, C. and Miller, S. J., Low-lying zeros of L-functions with orthogonal symmetry, Duke Math. J. 136 (2007), 115–172.CrossRefGoogle Scholar
[16]Hughes, C. P. and Rudnick, Z., Linear statistics of low-lying zeros of L-functions, Q. J. Math. 54 (2003), 309–333.CrossRefGoogle Scholar
[17]Ireland, K. and Rosen, M., A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, second edition (Springer, New York, 1990).CrossRefGoogle Scholar
[18]Iwaniec, H., Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
[19]Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[20]Iwaniec, H., Luo, W. and Sarnak, P., Low lying zeros of families of L-functions, Publ. Math. Inst. Hautes Études Sci. 91 (2000), 55–131.CrossRefGoogle Scholar
[21]Jutila, M., On the mean value of L(1/2,χ) for real characters, Analysis 1 (1981), 149–161.CrossRefGoogle Scholar
[22]Landau, E., Verallgemeinerung eines Polyaschen Satzes auf algebrasche Zahlkörper, Göttinger Nachr. (1918), 478–488.Google Scholar
[23]Luo, W., On Hecke L-series associated with cubic characters, Compositio Math. 140 (2004), 1191–1196.CrossRefGoogle Scholar
[24]Miller, S. J., One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries, Compositio Math. 140 (2004), 952–992.CrossRefGoogle Scholar
[25]Miller, S. J., A symplectic test of the L-functions ratios conjecture, Int. Math. Res. Not. IMRN (2008), 36pp., Art. ID rnm146.Google Scholar
[26]Özlük, A. E. and Snyder, C., On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis, Acta Arith. 91 (1999), 209–228.CrossRefGoogle Scholar
[27]Ricotta, G. and Royer, E., Statistics for low-lying zeros of symmetric power L-functions in the level aspect, Forum Math., to appear, arXiv:math.NT/0703760.Google Scholar
[28]Royer, E., Petits zéros de fonctions L de formes modulaires, Acta Arith. 99 (2001), 147–172.CrossRefGoogle Scholar
[29]Rubinstein, M. O., Low-lying zeros of L-functions and random matrix theory, Duke Math. J. 209 (2001), 147–181.CrossRefGoogle Scholar
[30]Rudnick, Z. and Sarnak, P., Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996), 269–322.CrossRefGoogle Scholar
[31]Soundararajan, K., Nonvanishing of quadratic Dirichlet L-functions at , Ann. of Math. (2) 152 (2000), 447–488.CrossRefGoogle Scholar
[32]Sunley, J. E., On the class numbers of totally imaginary quadratic extensions of totally real fields, Bull. Amer. Math. Soc. 78 (1972), 74–76.CrossRefGoogle Scholar
[33]Young, M. P., Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2005), 205–250.CrossRefGoogle Scholar