The sixth moment of automorphic \(L\)-functions (Q5918583)

Let \(S_k(\Gamma_0(q), \chi)\) denote the space of cusp forms of integral weight \(k\geq 2\) for the congruence subgroup \(\Gamma_0(q)\) of the modular group and the nebentypus character \(\chi \pmod q\). Denote by \(\mathcal{H}_{\chi}\) an orthogonal basis (with respect to the Petersson inner product) for \(S_k(\Gamma_0(q), \chi)\) consisting of Hecke cusp forms, normalized so that the first Fourier coefficient is \(1\). For each \(f\in\mathcal{H}_{\chi}\), let \(L(f,s)\) be the \(L\)-function associated to \(f\), and let \[ \Lambda\left(f, \frac{1}{2}+s\right) = \left(\frac{q}{4\pi ^2}\right)^{s/2} \Gamma \left(s+\frac{k}{2}\right) L\left(f, \frac{1}{2}+s\right) \] be the completed \(L\)-function associated to \(f\). The main result of the paper under review is an asymptotic formula for the sixth moment of the family of those \(L\)-functions. More precisely, the main theorem can be stated as follows: For a prime \(q\) and odd integer \(k\geq 3\), we have \[ \frac{2}{\phi(q)} \sum_{\stackrel{\chi \pmod q}{\chi(-1)=(-1)^k}} \sum_{f\in\mathcal{H}_{\chi}}^{h} \left| L\left(f, \frac{1}{2}\right)\right|^6 \ll (\log q)^9 \] as \(q \to \infty\).