Seminar: On a singular character deformation of the Selberg zeta function for $\Gamma_0(4)$

Speaker: Dieter Mayer (TU Clausthal)

2012.10.04 | Jane Jamshidi

 Date Wed 10 Oct Time 11:00 — 12:00 Location G2

Abstract:

The numerical approach via the transfer operator to the behaviour of the zeros of Selberg's zeta function under a singular character deformation has shown several unexpected phenomena as the character tends to the trivial character. Besides the accumulation of zeros on the critical line , corresponding to resonances of the corresponding twisted Laplacian and well known already to Selberg, a phenomenon of avoided crossing of the zeros on the critical line or the approach of zeros to the critical line in an infinite sequence of loops have been found. These different behaviours obviously depend on a symmetry of the transfer operator reflecting also an even-odd symmetry of the eigenfunctions of the corresponding twisted Laplacian. By using the automorphic spectral theory and especially the relation of the zeros of the Selberg function to the singularities of the scattering matrix and the embedding of this matrix into a meromorphic family of so called extended scattering matrices, introduced by Bruggeman, allows us to prove at least some of these asymptotic phenomena.

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