Maass-sche Spitzenformen, die Vermutung von Phillips und Sarnak und die Transfer-Operatoren kofiniter Fuchs-scher Gruppen
Final Report Abstract
In GAFA 4 (1994), 93-118, Phillips and Sarnak studied the spectrum of the Laplace-Beltrami operator for the principal congruence subgroup Gamma(2) under the deformation of a family of characters introduced originally by Selberg for the conjugate Hecke congruence subgroup Gamma0(4). They showed that Weyl's law holds only for an at most countable set of characters in this family and that the even new Maass cusp forms for the trivial character cannot be limits of Maass cusp forms for the nontrivial characters. Selberg on the other hand showed in Collected papers, vol 2 (1989), 45-46, for the corresponding character family for the conjugate group Gamma0(4), that the limit to the trivial character is rather singular where the resonances accumulate in this limit at the critical line Re s =1/2 and some of them tend to - infinity for the character approaching another arithemtic one. The reason for the singular behaviour is the change of the multiplicity of the continuous spectrum from 3 for the trivial character to 1 for the nontrivial characters. The transfer operator approach allowed us to confirm these results numerically by tracing the low lying zero's of the corresponding family of Selberg's zeta functions Z(s) which can be expressed by the Fredholm determinants of this family of transfer operators. An important ingredient in the numerical calculation of these zeros is the existence of a symmetry operator P commuting with the transfer operators, which allows to express the zeta function similar to the modular group as the product of two determinants of a reduced transfer operator In Algebra and Number Theory 6 (2012) 587-610, we showed that this symmetry operator P corresponds just to the only automorphism of the Maass wave forms for the group Gamma0(4) with nontrivial character. We also determined there the number of symmetries for the general group Gamma0(N). The numerical results we got are the following: all the zeros of the Selberg function Z(S) on the critical line belonging to the eigenvalue -1 of the reduced transfer operator and corresponding to odd Maass cusp forms, stay on this line without ever crossing each other when approaching the trivial character. Thereby the phenomenon of avoided crossing takes place seemingly on analytic curves in the plane spanned by the critical line and the parameter of the characters. These curves cut the critical line exactly at 'odd' zeros of the Selberg function. Some of the zeros corresponding to resonances of the Laplacian belonging to the eigenvalue 1 of the reduced transfer operator and corresponding to the even Eisenstein series, converge to the critical line and approach together with one of the afore mentioned odd zeros the point s=1/2 on the real line. These are just the Selberg resonances accumulating at the critical line. That every of these even zero's is joining an odd zero was however not mentioned by Selberg. Another set of smooth curves of even zero's of Z(s) in the complex s-plane shows a rather curious behaviour not known up to now in the literature: they touch the critical line in a sequence of points which converge, when approaching the trivial character, to even zeros of the Selberg function, and which moreover are closely related to the locations where the avoided crossings of the odd zeros take place. Also the Selberg resonances tending to -infinity could be confirmed by our numerical calculations. Applying the automorphic spectral theory of R. Bruggeman, which allows automorphic forms with exponential growth properties in the cusps, we could prove in a common paper with him in Experimental Mathematics 22 (2013), 217-242, most of our numerical findings.
