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Local analysis of the Kuznetsov formula and the density conjecture

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arxiv 2404.05561 v1 pith:GNI3QGNT submitted 2024-04-08 math.NT

Edgar Assing , Valentin Blomer , Paul D. Nelson This is my paper · ORCID
classification math.NT
keywords conjecturedensitycongruenceprincipalsarnakanalysisapplicationsarbitrary
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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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  1. On the spectral aspect density hypothesis and application

    math.NT 2025-04 accept novelty 8.0 of 10

    For n >= 4, the paper establishes Sarnak's density hypothesis in the spectral aspect for GL_n(Z) cuspidal representations and uses it to prove the Diophantine exponent of the SL_n(Z[1/p])-action is optimal (kappa = 1).

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