Non-vanishing of Class Group L-functions for Number Fields with a Small Regulator

Abstract

Let $E/\mathbb{Q}$ be a number field of degree $n$. We show that if $\mathrm{Reg}(E){\ll }_n|\mathrm{Disc}(E){|}^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is ${\gg }_{n,\epsilon }|\mathrm{Disc}{|}^{-1/4-\epsilon }$. 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.

to appear in Compos. Math., updated following referee comments

References in corpus (1)

• Distribution of periodic torus orbits and Duke's theorem for cubic fields