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The second moment of \(\mathrm{GL} (4) \times \mathrm{GL} (2)\) \(L\)-functions at special points. (English) Zbl 1458.11076

Let \(\phi\) be a \(\operatorname{GL}(n)\) Hecke-Maass form, and let \(\{u_1(z), u_2(z), \dots\}\) be an orthonomal basis of Hecke-Maass forms over \(\operatorname{SL}(2, \mathbb{Z})\) with eigenvalues \(1/4 + t_1^2, 1/4 + t_2^2, \dots\) respectively. The study of the Rankin-Selberg \(L\)-functions \(L(s, u_j \times \phi)\) at the special points \(1/2 + i t_j\) has its unique arithmetic and analytic interests. On the one hand, in the case \(n = 2\), these special points are zeros of the Selberg zeta function, so the behavior of the special values \(L(1/2 + i t_j, u_j \times \phi)\) controls the error term in the prime geodesic theorem [H. Iwaniec, J. Reine Angew. Math. 349, 136–159 (1984; Zbl 0527.10021)]. Further, the nonvanishing of these special values is also closely related to the Phillips-Sarnak theory [R. S. Phillips and P. Sarnak, Invent. Math. 80, 339–364 (1985; Zbl 0558.10017)] of deformation of cusp forms. On the other hand, at these special points the \(L\)-functions \(L(s, u_j \times \phi)\) are of analytic conductor \(T^n\) rather than \(T^{2n}\), due to a conductor-dropping phenomenon, which poses both new challenges and new opportunities for the study of their subconvexity bounds and high moments; in particular, an estimate of the second moment \[ \sum_{t_j \leq T} \left| L\left(u_j \times \phi, \frac{1}{2} + i t_j\right) \right|^2 \ll T^{2 + \varepsilon} \] often leads to parallel estimates about the \(2n\)-th moment of the \(\operatorname{GL}(2)\) \(L\)-functions \[ \sum_{t_j \leq T} \left| L\left(u_j, \frac{1}{2} + i t_j\right) \right|^{2n} \ll T^{2 + \varepsilon} \] We note that the bound \(T^{2 + \varepsilon}\) is optimal in view of the Lindelöf Hypothesis.
The \(n = 3\) case was studied by M. P. Young [Math. Ann. 356, No. 3, 1005–1028 (2013; Zbl 1312.11038)]. In this paper, the authors establish the result in the \(n = 4\) case. The proof employs various large sieve estimates, as well as the \(\operatorname{GL}(4)\) Voronoï summation formula and the functoriality of H. H. Kim [J. Am. Math. Soc. 16, No. 1, 139–183 (2003; Zbl 1018.11024)] for the exterior square \(L\)-function \(L(\phi, \wedge^2, s)\).
We should note that there are some typos in this paper. The definition of the Godement-Jacquet \(L\)-function \(L(\phi, s)\) of a \(\operatorname{GL}(4)\) Hecke Maass form \(\phi\), as given at the beginning of §2, does not agree with Definition 9.4.3 in [D. Goldfeld, Automorphic forms and \(L\)-functions for the group \(\operatorname{GL}(n, \mathbb{R})\). Cambridge: Cambridge University Press (2006; Zbl 1108.11039)]; rather, the formula given in the paper is the \(L\)-function \(L(\tilde{\phi}, s)\) for the dual Maass form \(\tilde{\phi}\) of \(\phi\). This does not affect the validity of the result, but may change the dependence of the \(\ll\)-constant in the above estimate upon the form \(\phi\). Another typo is in the reference item [19] in the paper for the work of Young [loc. cit., Zbl 1312.11038], where the given bibliographic information actually refers to another paper of M. P. Young [Adv. Math. 226, No. 4, 3550–3578 (2011; Zbl 1231.11112)] on the integral second moment of the Rankin-Selberg \(L\)-function.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F72 Spectral theory; trace formulas (e.g., that of Selberg)

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References:

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