The density conjecture for principal congruence subgroups (Q6566368)

Let \(q>1\) be a square free positive integer and \(\Gamma(q)\) be the principal congruence group \(\Gamma(q)=\text{ker}(SL_n({\mathbb{Z}})\mapsto SL_n({\mathbb{Z}}/q{\mathbb{Z}}))\). Let \(\mathcal{F}_{\Gamma(q)}(M)\) be the set of cusp forms \(\varpi\) whose archimedean Langlands parameters \(\mu_{\varpi,\infty}\) have norm bounded by \(M\). Let \(v\) be an unramified place of \({\mathbb{Q}}\). For \(\sigma>0\), denote by \(N_v(\sigma,{\mathcal{F}})\) the number of cusp forms in \(\mathcal{F}_{\Gamma(q)}(M)\) that violates Ramanujan conjecture at place \(v\) by amount \(\sigma\). Sarnak's density conjecture states that \N\[\NN_v(\sigma,{\mathcal{F}})\ll_{v,\epsilon,M,n}[SL_n({\mathbb{Z}}):\Gamma(q)]^{1-\frac{2\sigma}{n-1}+\epsilon}.\]\NThis paper proves the conjecture and gives a polynomial bound in \(M\): for some constant \(K\) depending only on \(n\) \N\[\NN_v(\sigma,{\mathcal{F}})\ll_{v,\epsilon,n}M^K[SL_n({\mathbb{Z}}):\Gamma(q)]^{1-\frac{2\sigma}{n-1}+\epsilon}.\N\]\NThe proof is based on Kuznietsov trace formula. The key original ingredients are a sharp bound of Kloosterman sum, and a lower bound of the average of Fourier coefficients.\N\NThe paper gives two applications of the density conjecture on counting problems. One is an upper bound on matrices in \(\Gamma(q)\) whose norm is bounded by a positive number \(T\). The other is an upper bound on number of elements in \(SL_n({\mathbb{Z}}/q{\mathbb{Z}})\) whose lifts to \(SL_n({\mathbb{Z}})\) have norms all bigger than \(q^{1+\frac{1}{n}+\epsilon}\). For both applications, the density theorem needs to be extended to include noncuspidal spectrum. Both results are stated as conditional on a technical hypothesis on the size of truncated Eisenstein series. These results are now established unconditionally by a work of Jana-Kamber (without proving the technical hypothesis) by establishing some bounds on \(L^2\) norm of Eisenstein series.