The paper is concerned about the following question about automorphic representations of \(\mathrm{GL}_n\) over \(\mathbb{Q}\) in families. Fix a place \(v\) of \(\mathbb{Q}\). For every automorphic representation \(\pi\) of \(\mathrm{GL}_n\), denote by \(\mu_\pi(v) = (\mu_\pi(v, 1), \ldots, \mu_\pi(v, n))\) its local spectral parameter at \(v\), and write \(\sigma_\pi(v) = \min_j |\operatorname{Re}\mu_\pi(v, j)|\), which measures the distance of \(\pi_v\) to the tempered spectrum. For \(\sigma \geq 0\) and a finite family \(\mathcal{F}\) of automorphic representations of \(\mathrm{GL}(n)\), define \(N_v(\sigma, \mathcal{F})\) be the number of \(\pi\)’s with \(\sigma_\pi(v) \geq \sigma\).
Let \(q\) be a prime, and let \(\Gamma_0(q)\) be the subgroup of \(\mathrm{SL}_n(\mathbb{Z})\) consisting of matrices whose last row is \(\equiv (0, \ldots, 0, *) \pmod{q}\). Now consider the family \(\mathcal{F}_I(q)\) of cuspidal representations \(\pi\) generated by Maass forms of level \(\Gamma_0(q)\) with Laplace eigenvalue in a finite interval \(I\). Theorem 1 is the following assertion: Let \(n \geq 3\) and \(v \neq q\); let \(\varepsilon, \sigma > 0\). We have \(N_v(\sigma, \mathcal{F}_I(q)) \ll_{I, v, n, \varepsilon} q^{n-1-4\sigma + \varepsilon}\).
This reduces to known results if \(n=2\) or if \((n, v) = (3, \infty)\). For higher \(n\) the result is completely new.
The proof of Theorem 1 is based on an in-depth analysis of the arithmetic side of the Kuznetsov trace formula, with a test function that blows up for exceptional Langlands parameters at \(v\). This turns out to be easier to use than the Arthur-Selberg trace formula. Specifically, one needs an explicit analysis of the \(\mathrm{GL}_n\) Kloosterman sums \(S_{q, w}(M, N, (q, \ldots, q))\), recorded in Theorem 3.
As by-products, one obtains a large sieve inequality (Theorem 4), as well as a best-possible bound for the second moment of \(L\)-functions on the critical line (Corollary 5), namely: \(\sum_{\pi \in \mathcal{F}_I(q)} |L(\frac{1}{2} + it, \pi)|^2 \ll_{I, t, n, \varepsilon} q^{n-1+\varepsilon}\) under the same assumptions.

Reviewer: Wen-Wei Li (Beijing)

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