On the local $L^2$-Bound of the Eisenstein Series

Subhajit Jana, Amitay Kamber

Published 2022-10-28Version 1

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. In this work, we show an upper bound of the squared $L^2$-norms of the unitary Eisenstein series restricted to a fixed compact domain, of \emph{poly-logarithmic} strength on an average, for a large class of reductive groups. Moreover, as an application of our method, we prove the optimal lifting property for $\mathrm{SL}_n(\mathbb{Z}/q\mathbb{Z})$ for square-free $q$, as well as the Sarnak--Xue counting property for the principal congruence subgroup of $\mathrm{SL}_n(\mathbb{Z})$ of square-free level. This makes the recent results of Assing--Blomer unconditional.