Title Introduction Mass forms D-functions Period functions Summary Title.pdfVIP

Title Introduction Mass forms D-functions Period functions Summary Title Maass cusp forms and “two-sided L-functions” with simple functional equation Tobias Mu?hlenbruch Institute of Theoretical Physics TU Clausthal tobias.muehlenbruch@tu-clausthal.de http://home.tu-clausthal.de/?tmu/ Advances in Number Theory and Random Matrix Theory Rochester, 5–9 June 2006 Title Introduction Mass forms D-functions Period functions Summary Outline of the presentation 1 Introduction Motivation Notation 2 Mass forms Multiplier systems Maass cusp forms with real weight Special functions Ψk,ν; Fourier expansion of Maass cusp forms 3 D-functions Two sided L-series D0 and D1 with simple functional equation Example: Powers of Dedekind η-function Other L-series’ results: Maass, “simplified Maass” and Bykovskii Sketch of Proof: Maass cusp forms ? D-functions Summary of “different L-functions” 4 Period functions What are period functions Period functions and D-functions 5 Summary Summary References Title Introduction Mass forms D-functions Period functions Summary Motivation The situation: Modular cusp forms and Maass cusp forms are similar objects. The L-series attached to modular and Maass cusp forms satisfy different functional equations. The usual L-series for real weights do not work very well in my application. The questions: Can we find a “uniform” statement of L-series which covers both cases? Do we need 2 functional equations for L-series attached to Maass cusp forms? My application: How can we generalize a result of J.Lewis and D.Zagier in [LZ01] using L-series for real weights? Title Introduction Mass forms D-functions Period functions Summary Notation Z denotes the set of integers R, C denote the real line and the complex plane H upper half-plane {z ∈ C; Im(z) 0} with hyperbolic metric ds2 = dx∧dx+dy∧dy y2 C′ = Cr (?∞, 0] cut-plane SL2(Z) full modular group generated by S = ( 0 1 ?1 0 ) and T = ( 1 0 1 1 ) , i.e. SL2(Z) = 〈S ,T | S4 = (ST )6 = 1〉. γz = aγz+bγ cγz+dγ for γ = ( a