Local analysis of the Kuznetsov formula and the density conjecture

TL;DR

This paper proves Sarnak's spherical density conjecture for principal congruence subgroups of SL(n, Z), providing new bounds and estimates that advance understanding of automorphic forms and their distributions.

Contribution

It establishes the conjecture for arbitrary levels and extends the density theorem to co-compact cases, introducing new bounds for Whittaker functions and Kloosterman sets.

Findings

Proved Sarnak's spherical density conjecture for all levels of SL(n, Z)

Extended the density theorem to co-compact subgroups

Developed new lower bounds for Whittaker functions and estimates for Kloosterman sets

Abstract

We prove Sarnak's spherical density conjecture for the principal congruence subgroup of SL(n, Z) of arbitrary level. Applications include a complete version of Sarnak's optimal lifting conjecture for principal congruence subgroups of SL(n, Z), as well as a transfer of the density theorem to certain co-compact situations. The main ingredients are new lower bounds for Whittaker functions and strong estimates for the cardinality of ramified Kloosterman sets.