Local analysis of the Kuznetsov formula and the density conjecture

2024/04/08 by Assing, Edgar, Blomer, Valentin, Nelson, Paul D.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.05561

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.

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