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Second moments of twisted Koecher-Maass series. (English) Zbl 1345.11033

Summary: K. Imai [Am. J. Math. 102, 903–936 (1980; Zbl 0447.10028)] considered the twisted Koecher-Maass series for Siegel cusp forms of degree 2, twisted by Maass cusp forms and Eisenstein series, and used them to prove the converse theorem for Siegel modular forms. They do not have Euler products, and it is not even known whether they converge absolutely for \(\operatorname{Re}(s)>1\). Hence the standard convexity arguments do not apply to give bounds. In this paper, we obtain the average version of the second moments of the twisted Koecher-Maass series, using Titchmarsh’s method of Mellin inversion. When the Siegel modular form is a Saito Kurokawa lift of some half integral weight modular form, a theorem of W. Duke and Ö. Imamoḡlu [Int. Math. Res. Not. 1996, No. 7, 347–355 (1996; Zbl 0849.11039)] says that the twisted Koecher Maass series is the Rankin-Selberg \(L\)-function of the half-integral weight form and Maass form of weight 1/2. Hence as a corollary, we obtain the average version of the second moment result for the Rankin-Selberg \(L\)-functions attached to half integral weight forms.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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