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

TL;DR

This paper demonstrates that for certain number fields with small regulators relative to their discriminant, a significant proportion of class group characters have non-vanishing Hecke L-functions at the central point, using advanced equidistribution and orbit analysis.

Contribution

It establishes a quantitative non-vanishing result for class group L-functions in number fields with small regulators, linking regulator size to L-function behavior.

Findings

Fraction of non-vanishing class group characters is at least proportional to | ext{Disc}|^{-1/4- ext{epsilon}}.

The proof combines equidistribution of Eisenstein periods with orbit escape of mass analysis.

Results apply to number fields with regulators smaller than a constant times the discriminant to the 1/4 power.

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 $\gg_{n,\varepsilon} |\operatorname{Disc}|^{-1/4-\varepsilon}$. 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.