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Traces of Hecke operators acting on three-dimensional hyperbolic space. (English) Zbl 1076.11034

Reine Mathematik. Münster: Univ. Münster, Fachbereich Mathematik und Informatik (Dissertation). vi, 154 p. (2004).
This doctoral thesis is based on the monograph: J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Springer (1998; Zbl 0888.11001). The hyperbolic upper half space \(\mathbb{H}\) is the set of all Hamiltonian quaternions of the form \(P= z+ rj= x+ yi+ rj\) with \(x,y,r\in\mathbb{R}\), \(r> 0\). The group \(\text{GL}_2(\mathbb{C})\) acts on \(\mathbb{H}\) by \(MP= q^{-1}(aP+ b)(cP+ d)^{-1}q\) for \(M=\left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix}\right)\in \text{GL}_2(\mathbb{C})\), \(q= \sqrt{\text{det\,}M}\); be aware that products in \(\mathbb{H}\) do not commute. Let \({\mathcal O}\) be the ring of integers in an imaginary-quadratic number field \(K\). Then in particular the discrete group \(\Gamma= \text{GL}_2({\mathcal O})\) acts on \(\mathbb{H}\). Let \(L^2(\Gamma\setminus\mathbb{H})\) be the Hilbert space of measurable functions on \(\mathbb{H}\) which are \(\Gamma\)-invariant and square-integrable with respect to the hyperbolic metric. The Laplace-Beltrami operator \(\Delta\) acts on twice differentiable functions in \(L^2(\Gamma\setminus\mathbb{H})\). The eigenvalues of \(-\Delta\) are real and \(\geq 0\). For \(v\in{\mathcal O}\), \(v\neq 0\), let \({\mathcal M}_v\) be the set of \(2\times 2\)-matrices with entries in \({\mathcal O}\) and determinant \(v\), and let \({\mathcal T}_v\) be a system of representatives for the right cosets \(\Gamma\cdot S\), \(S\in{\mathcal M}_v\). Then the Hecke operator \(T_v\) is defined by its action \[ (T_v f)(P)= \sum_{M\in{\mathcal T}_v} f(MP) \] on \(\Gamma\)-invariant functions \(f\). All Hecke operators commute with \(-\Delta\). The result of this thesis is a formula for the trace \(\text{tr}_\lambda T_v\) of \(T_v\) acting on the Hilbert space \({\mathcal E}_\lambda\) of eigenfunctions of \(-\Delta\) for a fixed eigenvalue \(\lambda\). This is achieved under the restriction that the class number of \(K\) is 1, which implies that there is only 1 orbit of cusps of \(\Gamma\). The author expects that future research will overcome the technical difficulties which arise from more cusp orbits in the case of class number \(>1\). In an intermediate result, the trace \(\text{tr}_\lambda T_v\) is represented as an integral, involving Eisenstein series and an automorphic kernel, with contributions from all \[ \gamma\in\Gamma^*_v= \left\{{1\over \sqrt{v}} \begin{pmatrix} a & b\\ c & d\end{pmatrix}\,\biggl|\, a,b,c,d\in{\mathcal O},\, ad- bc= v\right\}. \] According to Selberg, \(\Gamma^*_v\) is decomposed into \(\Gamma\)-conjugacy classes, and their contributions are treated separately. Difficulties arise for non-parabolic elements in \(\Gamma^*_v\) which have fixed points in \(K\cup\{\infty\}\); then an integral diverges, and an asymptotic formula has to be established for integrals over truncated fundamental domains of \(\Gamma\).
The final result exhibits a wonderful connection with quadratic forms and solutions of Pell equations over the ring \({\mathcal O}\). For \(a,b,c\in{\mathcal O}\) with \(d= b^2- 4ac\neq 0\), consider the quadratic form \(q(u, v)= au^2+ buv+ cv^2\) with discriminant \(d\). Denote by \(\Omega\) the set of all discriminants \(d\neq 0\) in \({\mathcal O}\). For \(d\in\Omega\) the solutions of the Pell equation \(t^2- du^2= 4\) form a group which is a direct product of an infinite cyclic group, generated by a fundamental solution \(\varepsilon_d= {1\over 2}(t_0+ u_0\sqrt{d})\), and a finite cyclic group, generated by a root of unity \(\zeta_d= {1\over 2}(t_1+ u_1\sqrt{d})\). Now the trace is identified with a residue of a certain Dirichlet series, \[ \text{tr}_\lambda T_v= 2|v|^\tau\cdot \text{res}_{s=\tau} L^\times_v(s). \] Here \(\tau= 1+\kappa i\), \(\kappa= \sqrt{\lambda-1}\) if \(\lambda> 1\), \(\kappa=- i\sqrt{1-\lambda}\) if \(0<\lambda< 1\), and the Dirichlet series \(L^\times_v\) is a sum of two components \(L_v\) and \(\widetilde L_v\), one of which is \[ L_v(s)= \sum_{d\in\Omega}\, \sum_u {h_d\cdot\log|\varepsilon_d|\over \text{ord}(\zeta_d)}\cdot| du^2|^{-s}, \] where \(h_d\) is the number of classes of quadratic forms with discriminant \(d\), and \(u\) runs over a system of representatives of \({\mathcal O}\) modulo units for which a solution \(t\) of \(t^2- du^2= 4v\) exists.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F25 Hecke-Petersson operators, differential operators (one variable)

Citations:

Zbl 0888.11001
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