Certain Dirichlet series attached to automorphic forms over imaginary quadratic fields
In the paper under review (the author's Ph.D. thesis) the subseries \[ \mathcal R(s,f) = \sum_{0 \neq \xi \in L \cap \mathbb{Q}} \theta (\xi) | \xi |^{-s} \] of the usual Dirichlet series \(\mathcal D(s,f)\) associated to certain eigenfunctions \(f\) of the Laplace-Beltrami operator on the hyperbolic 3-space \(\mathrm{SL}_2(\mathbb{C})/\mathrm{SU}_2\) are studied. These functions \(f\) have to be invariant under a subgroup of \(\mathrm{GL}_2(\mathbb{F})\) where \(\mathbb{F}\) is an imaginary quadratic number field and where in \(\mathcal D(s,f)\) the summation is over a whole 2-dimensional lattice \(L\) in \(\mathbb{F}\) instead and where the coefficients \(\theta(\xi)\) are induced by the Fourier expansion of \(f\). By Langlands theory it is well known that the series \(\mathcal D(s,f)\) has an analytic continuation to the whole complex plane and satisfies a functional equation. Now, the subseries \(\mathcal R(s,f)\) inherits these properties provided \(f\) is a cusp form. For the proof, real analytic functions on the upper- half-plane invariant under a certain subgroup of \(\mathrm{GL}^+_2 (\mathbb{Q})\) as well as Eisenstein series play an important role. Furthermore, adopting the adelic viewpoint a one-to-one correspondence between automorphic forms \(\Phi\) of the adelization of \(\mathrm{GL}_2(\mathbb{F})\) and \(h\)-tuples \((f_1, \dots, f_n)\) of automorphic forms on hyperbolic 3-space is established, \(h\) being the class number of \(\mathbb{F}\). Following \textit{G. Shimura} [Duke Math. J. 45, 637--679 (1978; Zbl 0394.10015)] the analogous subseries \(\mathcal R(s,\Phi) = \sum_{n \ge 1} c(n,\Phi) n^{-s}\) has also an analytic continuation and satisfies a functional equation if \(\Phi\) is a cusp form. Moreover it has an Euler product of degree 4 if \(\Phi\) is a common eigenfunction of all Hecke operators and satisfies \(\Phi(t.x) = \psi(t) \Phi(x)\) for a Hecke character \(\Psi\) and all ideles \(t\). Similar results have been achieved for the Hilbert modular group of a real quadratic number field [\textit{T. Asai}, J. Math. Soc. Japan 28, 48--61 (1976; Zbl 0313.10026)]; for nontrivial representations of \(\mathrm{SU}_2\) analogous results may be expected.
scientific article; zbMATH DE number 559008
scientific article