Density theorems for GL(n)

Abstract

Strong bounds - going beyond Sarnak's density hypothesis - are obtained for the number of automorphic forms for the congruence subgroup Gamma_0(q) of SL_n(Z) violating the Ramanujan conjecture at any given unramified place. The proof is based on a relative trace formula of Kuznetsov type and best-possible bounds for certain Kloosterman sums for GL(n). Further applications are given.

Some updates and corrections. To appear in Invent. Math

Cited by in corpus (4)

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• An unconditional $\mathrm{GL}(n)$ large sieve
• The density hypothesis for horizontal families of lattices
• Analytic newvectors for ${\mathrm{GL}}_n(\mathbb{R})$