Master Degree Project Presentation: The geometric theory of classical automorphic forms and a rigid direct-limit construction of Cech cohomology

Zoom link: https://uu-se.zoom.us/j/66673942848

Abstract: This thesis is concerned with the geometric foundations of the theory of automorphic forms. The principal objects are the upper half-plane, its hyperbolic metric, and quotients by Fuchsian groups. The thesis develops the preliminary geometry needed for a later cohomological treatment. We determine the holomorphic automorphism group of the upper half-plane, study its invariant metric and volume form, classify linear fractional transformations, and analyse discrete subgroups of SL(2,R). We then construct the compactified quotient. These constructions explain why the quotient is naturally a Riemann surface and provide the geometric setting in which automorphic forms may subsequently be interpreted as sections of holomorphic line bundles. Later chapters will develop the proposed rigid direct-limit model for Čech cohomology and develop Riemann–Roch and Serre duality, and then apply to dimension formulas for automorphic forms.