Applications of the Kuznetsov formula on GL(3) II: the level aspect

Abstract

Abstract. We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelöf on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maaß forms violating the Ramanujan conjecture at finite places.

Applications of analytic newvectors for $$\mathrm {GL}(n)$$

“…The families usually are defined in terms of some intrinsic attributes of the automorphic representations π , such as, their levels (non-archimedean), weights, spectral parameters, Laplace eigenvalues (archimedean), or analytic conductors. A plethora of works have been done in these aspects on various higher rank and higher dimensional arithmetic locally symmetric spaces among which we refer to [2][3][4][5][6]8].…”

“…An application of this kind in the question of weighted counting of automorphic forms for PGL r (Z), for r ≥ 2, with bounded analytic conductors is provided in [27,Theorem 9]. In this article we will give a few other averaging applications of this kind which (including their proofs) are mostly influenced by the recent works [2,3,5,6,18,19,33,39].…”

A Density Theorem for $\operatorname{Sp}(4)$

“…Proof. The case Z ≪ 1 is proved in [BBM17,Blo19]. For each µ ∈ Ω, choose an open set S µ ⊆ R 2 + such that ℜW µ (y) = 0 for all y ∈ S µ or ℑW µ (y) = 0 for all y ∈ S µ .…”

“…Density theorems have attracted much attention in the history, and many strong density results are known for various automorphic families on GL(2) with different settings [Hux86,Sar87,Iwa90,BM98,BM03,BBR14]. Via Kuznetsov-type trace formulae on GL(3), strong density results on GL(3) were obtained in [Blo13,BBR14,BBM17]. Blomer [Blo19] further generalised the technique to obtain results in GL(n) beyond Sarnak's density hypothesis.…”

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

“…Potentially, the Kuznetsov trace formula can handle much more complicated situations. For instance, Blomer-Buttcane-Maga [9] obtained a sixth moment formula for the L-values of automorphic forms on GL 3 averaged over a family of size N 2+o (1) , so that the logarithmic ratio of conductor versus family is 6/2 = 3 > 1. An advantage of our method is that one can deal with arbitrary GL n uniformly by a relatively soft method except that the exact evaluation of local zeta-integrals is required to compute the main term.…”