Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T08:54:11.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-vanishing of class group L-functions for number fields with a small regulator

Part of: Forms and linear algebraic groups Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  17 December 2020

Ilya Khayutin
Ilya Khayutin*
Affiliation:
Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL60208, USAkhayutin@northwestern.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.

Keywords

L-functionclass groupnon-vanishingescape of massequidistributionperiodic torus orbit

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions 11E45: Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 11 , November 2020 , pp. 2423 - 2436
Check for updates
Copyright
© The Author(s) 2020

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.)

References

Balasubramanian, R. and Kumar Murty, V., Zeros of Dirichlet $L$-functions, Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 567–615.CrossRefGoogle Scholar
Blanksby, P. E. and Montgomery, H. L., Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369.CrossRefGoogle Scholar
Blomer, V., Non-vanishing of class group $L$-functions at the central point, Ann. Inst. Fourier (Grenoble) 54 (2004), 831–847.CrossRefGoogle Scholar
Blomer, V., Harcos, G. and Michel, P., Bounds for modular $L$-functions in the level aspect, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), 697–740.CrossRefGoogle Scholar
Bombieri, E. and Gubler, W., Heights in diophantine geometry, New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006).Google Scholar
Burgess, D. A., On character sums and $L$-series. II, Proc. Lond. Math. Soc. (3) 13 (1963), 524–536.CrossRefGoogle Scholar
Dimitrov, V., A proof of the Schinzel–Zassenhaus conjecture on polynomials, Preprint (2019), arXiv:1912.12545.Google Scholar
Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391–401.CrossRefGoogle Scholar
Duke, W., Number fields with large class group, in Number theory, CRM Proceedings and Lecture Notes, vol. 36 (American Mathematical Society, Providence, RI, 2004), 117–126.CrossRefGoogle Scholar
Duke, W., Friedlander, J. B. and Iwaniec, H., The subconvexity problem for Artin $L$-functions, Invent. Math. 149 (2002), 489–577.CrossRefGoogle Scholar
Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A., Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J. 148 (2009), 119–174.CrossRefGoogle Scholar
Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A., Distribution of periodic torus orbits and Duke's theorem for cubic fields, Ann. of Math. (2) 173 (2011), 815–885.CrossRefGoogle Scholar
Epstein, P., Zur Theorie allgemeiner Zetafunktionen. II, Math. Ann. 63 (1906), 205–216.CrossRefGoogle Scholar
Fröhlich, A., Artin root numbers and normal integral bases for quaternion fields, Invent. Math. 17 (1972), 143–166.CrossRefGoogle Scholar
Hecke, E., Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z. 6 (1920), 11–51.CrossRefGoogle Scholar
Iwaniec, H. and Sarnak, P., The non-vanishing of central values of automorphic $L$-functions and Landau–Siegel zeros, Israel J. Math. 120 (2000), 155–177.CrossRefGoogle Scholar
Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Luo, W., Rudnick, Z. and Sarnak, P., On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), 387–401.CrossRefGoogle Scholar
Luo, W., Rudnick, Z. and Sarnak, P., On the generalized Ramanujan conjecture for ${\rm GL}(n)$, in Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proceedings of Symposia in Pure Mathematics, vol. 66 (American Mathematical Society, Providence, RI, 1999), 301–310.CrossRefGoogle Scholar
Michel, P. and Venkatesh, A., Heegner points and non-vanishing of Rankin/Selberg $L$-functions, in Analytic number theory, Clay Mathematics Proceedings, vol. 7 (American Mathematical Society, Providence, RI, 2007), 169–183.Google Scholar
Rademacher, H., On the Phragmén–Lindelöf theorem and some applications, Math. Z. 72 (1959–1960), 192–204.CrossRefGoogle Scholar
Sarnak, P., Shin, S. W. and Templier, N., Families of $L$-functions and their symmetry, in Families of automorphic forms and the trace formula, Simons Symposium (Springer, Cham, 2016), 531–578.CrossRefGoogle Scholar
Shankar, A., Södergren, A. and Templier, N., Sato–Tate equidistribution of certain families of Artin $L$-functions, Forum Math. Sigma 7 (2019), e23.CrossRefGoogle Scholar
Soundararajan, K., Nonvanishing of quadratic Dirichlet $L$-functions at $s=\frac {1}{2}$, Ann. of Math. (2) 152 (2000), 447–488.CrossRefGoogle Scholar
Soundararajan, K., Weak subconvexity for central values of $L$-functions, Ann. of Math. (2) 172 (2010), 1469–1498.CrossRefGoogle Scholar
Stewart, C. L., Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. Math. France 106 (1978), 169–176.CrossRefGoogle Scholar
Terras, A. A., Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Amer. Math. Soc. 183 (1973), 477–486.CrossRefGoogle Scholar
Terras, A., The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions, J. Number Theory 12 (1980), 258–272.CrossRefGoogle Scholar
Wielonsky, F., Séries d'Eisenstein, intégrales toroïdales et une formule de Hecke, Enseign. Math. 31 (1985), 93–135.Google Scholar