Edgar Assing, Valentin Blomer, Paul D. Nelson
Published 2024-04-08Version 1
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.
The density conjecture for principal congruence subgroups
On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$
A relative trace formula approach to the Kuznetsov formula on $GSp_4$
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