Journal ArticleNon-vanishing of class group <i>L</i> -functions for number fields with a small regulatorKhayutin, IlyanullLet $E/\mathbb {Q}$ 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.2020-11-0110223509Compositio Mathematica156112423 to 24360010-437Xhttps://doi.org/10.1112/S0010437X200074721946333National Science Foundation