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

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

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
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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
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Copyright
© The Author(s) 2020

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