The large sieve for self-dual Eisenstein series of varying levels (Q6595576)

Let \(k\) be a positive integer, and let \(\theta\) run over all Dirichlet characters modulo \(k\). Let \(Q \geq 1\) be a real number, and for each \(Q/2 < q \leq Q\) with \((q, k) = 1\), let \(\chi\) run over primitive Dirichlet characters modulo \(q\). Finally, let \(T \geq 1\) be a real number, and let \(|t| \leq T\). Then, define \(\mathcal{F}\) to consist of the characters \(\chi \theta\), with corresponding data \N\[\N\lambda_{\chi\theta,t}(a,b)=\chi\theta(a)\overline{\chi\theta}(b)(a/b)^{it},\N\]\Nwith \( N/2 < ab \leq N\) and \((a, b) = 1\). Let denote by \N\[\N\Delta(Q, k, T, N)=\max_{|\alpha|=1}\int_{T/2\leq t\leq T}\sum_{\substack{Q/2<q\leq Q\\ (q,k)=1}}\mathop{{\sum}^*}_{\chi\, (\mathrm{mod}\, q)}\sum_{\theta\, (\mathrm{mod}\, k)} \left |\sum_{\substack{N/2<ab\leq N\\ (a,b)=1}} \alpha_{a,b}\lambda_{\chi\theta,t}(a,b)\right|^2 dt.\N\]\NThe main result of the paper is the following estimate \N\[\N\Delta(Q, k, T, N)\ll_{\varepsilon} (QkTN)^\varepsilon(Q^2kT+N).\N\]\NThis result appears to be the first sharp large sieve inequality for a \(\mathrm{GL}_2\) family with varying levels. Furthermore, it is optimal up to the \(\varepsilon\)-aspect.