Shimura lift of real analytic Poincaré series and Hilbert modular Eisenstein series
The author starts from the Siegel theta function for some indefinite integral ternary quadratic form \(Q_p\) of signature \((1,2)\) where \(p=1\) or \(p\) is an odd prime number. He then studies the Shimura lift of real analytic automorphic forms with respect to this theta kernel. Applying this lift to Selberg's real analytic Poincaré series \(P_\nu (z,s)\) he obtains a new class of Poincaré series associated with closed geodesics on \(\Gamma_0(p) \setminus \mathbb{H}\). (The latter series may be regarded as the real analytic counterparts of some series introduced by Zagier in the holomorphic case [cf. \textit{D. Zagier}, Invent. Math. 30, 1-46 (1975; Zbl 0308.10014)].) The first main result of the work under review roughly reads as follows: If \(\nu> 0\) is a fundamental discriminant then the theta lift of \(P_\nu (z,s)\) is related to a (real analytic) Hilbert modular Eisenstein series for \(\mathbb{Q}(\sqrt{\nu})\) on the diagonal via an operator-valued zeta-function defined by means of the Hecke operators. The second main result states: If \(\psi\) is an even normalized Hecke-Maaß form for \(\text{SL}_2 (\mathbb{Z})\) and \(\nu> 0\) a fundamental discriminant then the sum of the various integrals of \(\psi\) against the \(h(\nu)\) (narrow class-number) Hilbert modular Eisenstein series for \(\mathbb{Q}(\sqrt{\nu})\) on the diagonal is a multiple of the \(L\)-series for \(\psi\). The author extended his work on the theta lift of the series \(P_\nu(z,s)\) to the \(n\)-dimensional case in his work [Forum Math. 11, 483-502 (1999; Zbl 0926.11028)]. The work under review is also closely related with the following paper [\textit{Wolfgang Bauer}, Der Thetalift reell-analytischer Eisenstein- und Poincaréreihen. Mathematica Gottingensis 4 (1996; Zbl 0972.11035)].
scientific article; zbMATH DE number 1336839