Universal minoration of certain quadratic integrals of automorphic \(L\) functions (Q2080949)

Let \(\pi\) be an automorphic representation of \(\mathrm{GL}(n, \mathbb{A}_{\mathbb{Q}})\) that is a parabolic induction from unitary cuspidal representations. Suppose that (i) \(m \leq 2\), (ii) \(\pi_\infty\) is self-dual, and (iii) \(\pi_v\) is tempered for every place \(v\). The author announces four main theorems, sketches their proofs and discusses possible ramifications. Let \(s = \frac{1}{2} + i\tau\) be a zero of \(L(s, \pi)\) on the critical line. One has the following universal lower bounds. Theorem A. When \(L(s, \pi)\) has no \(\zeta\) factor, \(\int_{\mathbb{R}} \left| \frac{L(\frac{1}{2} + it, \pi)}{\frac{1}{2} + it -s} \right|^2 dt > 2\pi \log 2\). Theorem B. When \(\pi\) may contain \(\zeta\) factors, the same integral is \(> 2\pi \left(\log - \frac{2|\kappa|}{1-s}\right)\) and \(> \pi \log 2\), where \(\kappa = \mathrm{Res}_{s=1} L(s, \pi)\). When the role of \(s\) is replaced by \(0\), one has the following results. Theorem C. When \(L(s, \pi)\) has no \(\zeta\) factor, \(\int_{\mathbb{R}} \left| \frac{L(\frac{1}{2} + it, \pi)}{\frac{1}{2} + it} \right|^2 dt > \pi\). Theorem D. In the general case, the integral above is \(> \frac{\pi}{2}\). The hypothesis \(m \leq 2\) implies that all integrals above are finite, by the subconvexity results of \textit{P. Michel} and \textit{A. Venkatesh} [Publ. Math., Inst. Hautes Étud. Sci. 111, 171--271 (2010; Zbl 1376.11040)]. In general, they would follow from some approximation to the generalized Lindelöf hypothesis. The sketched arguments are based on studying Mellin transforms, using tools such as estimates of \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Can. J. Math. 57, No. 3, 494--505 (2005; Zbl 1160.11340)]. As an application, one has the Theorem 3. For \(\pi\) cuspidal and non-trivial subject the hypotheses above, the Dirichlet series for \(L(s, \pi)\) converges for \(\mathrm{Re}(s) > 1 -\frac{2}{m+1}\).