Hostname: page-component-77f85d65b8-t6st2 Total loading time: 0 Render date: 2026-04-14T01:57:09.551Z Has data issue: false hasContentIssue false
  • English
  • Français

Conditional lower bounds on the distribution of central values: The case of modular forms

Part of: Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  01 December 2025

Didier Lesesvre Open the ORCID record for Didier Lesesvre [Opens in a new window]  and
Ade Irma Suriajaya Open the ORCID record for Ade Irma Suriajaya [Opens in a new window]
Didier Lesesvre*
Affiliation:
University of Lille , France
Ade Irma Suriajaya
Affiliation:
Kyushu University, Japan e-mail: adeirmasuriajaya@math.kyushu-u.ac.jp
*
e-mail: didier.lesesvre@univ-lille.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Radziwiłł and Soundararajan unveiled a connection between low-lying zeros and central values of L-functions, which they instantiated in the case of quadratic twists of an elliptic curve. This article addresses the case of the family of modular forms in the level aspect, and proves that the logarithms of central values of associated L-functions approximately distribute along a normal law with mean $-\tfrac 12 \log \log c(f)$ and variance $\log \log {c(f)}$, where $c(f)$ is the analytic conductor of f, as predicted by the Keating–Snaith conjecture.

Keywords

Automorphic formPetersson trace formulaLow-lying zeroL-functions

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11M41: Other Dirichlet series and zeta functions

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 30
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The authors acknowledge support from the R-CDP-24-004-C2EMPI project, the CNRS (PEPS), and JSPS KAKENHI Grant Number 22K13895.

References

Baluyot, S., Chandee, V., and Li, X., Low-lying zeros of a large orthogonal family of automorphic $L$ -functions. Preprint, 2023. arXiv:2310.07606.Google Scholar
Bombieri, E. and Hejhal, D. A., On the distribution of zeros of linear combinations of Euler products . Duke Math J. 80(1995), 821–862.CrossRefGoogle Scholar
Cheek, T., Gilman, P., Jaber, K., Miller, S. J., and Tomé, M.-H., On the density of low-lying zeros of a large family of automorphic $L$ -functions. Preprint, 2024. arXiv:2408.09050.Google Scholar
Cogdell, J. and Michel, P., On the complex moments of symmetric power $L$ -functions at s = 1 . Int. Math Res. Not. 31(2004), 1561–1617.CrossRefGoogle Scholar
Devin, L., Fiorilli, D., and Södergren, A., Families of holomorphic cusp forms: The weight aspect . Quart. J. Math. 73(2022), 1402–1426.CrossRefGoogle Scholar
Dusart, P., Estimates of the k-th prime under the Riemann hypothesis . Ramanujan J. 47(2018), 141–154.CrossRefGoogle Scholar
Granville, A. and Soundararajan, K., Large character sums . J. Amer. Math. Soc. 14(2000), 365–397.CrossRefGoogle Scholar
Hough, B., The distribution of the logarithm in an orthogonal and a symplectic family of L-functions . Forum Math 26(2014), 523–546.10.1515/forum-2011-0105CrossRefGoogle Scholar
Iwaniec, H., Topics in classical automorphic forms, American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004.Google Scholar
Iwaniec, H., Luo, W., and Sarnak, P., Low-lying zeros of families of $L$ -functions . Pub. Math. IHÉS 91(2000), 55–131.CrossRefGoogle Scholar
Iwaniec, H. and Sarnak, P., Perspectives on the analytic theory of $L$ -functions. In: Alon, N. (ed.), et al., GAFA 2000. Visions in mathematics–Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25-September 3, 1999. Part II. Basel: Birkhäuser, 2000, pp. 705–741.Google Scholar
Montgomery, H. and Vaughan, R. C., Multiplicative number theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University press, Providence, RI, 2010.Google Scholar
Murty, R., Problems in analytic number theory, Graduate Texts in Mathematics, Springer, Providence, RI, 2008.Google Scholar
Radziwiłł, M. and Soundararajan, K., Moments and distribution of central $L$ -values of quadratic twists of elliptic curves . Invent. Math. 202(2015), 1029–1068.CrossRefGoogle Scholar
Radziwiłł, M. and Soundararajan, K., Conditional lower bounds on the distribution of central values in families of $L$ -functions. Acta Arith. 214(2024), 481–497.Google Scholar
Rudnick, Z. and Sarnak, P., Zeroes of principal $L$ -functions and random matrix theory . Duke J. Math. 81(1996), 269–322.CrossRefGoogle Scholar
Sarnak, P., Definition of families and their symmetries. Letter, 2008.Google Scholar
Sarnak, P., Shin, S. W., and Templier, N., Families of $L$ -functions and their symmetries. In: Families of automorphic forms and the trace formula, Simons Symposia, Springer, 2016, pp. 531–578.10.1007/978-3-319-41424-9_13CrossRefGoogle Scholar
Steuding, J., Value-distribution of L-functions, Lecture Notes in Mathematics, 1877, Springer, Providence, RI, 2007.Google Scholar