A framework for some distinguished series of orthogonal type modular forms

• Konstruktionen für einige ausgezeichnete Serien von Modulformen zu orthogonalen Gruppen

Woitalla, Martin; Krieg, Aloys (Thesis advisor); Gritsenko, Valery (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen, 2016

Abstract

In this thesis we consider modular forms of orthogonal type. We consider lattices which can be represented as an orthogonal sum of two integral hyperbolic planes anda negative definite even lattice. The modular groups are subgroups of finite index ofthe group of all lattice automorphisms intersected with the real spinor kernel. The associated modular forms are defined on a homogeneous domain of type IV in Cartan’s classification. Due to some low-dimensional exceptional isogenies this setting covers degree two modular forms of Siegel, Hermitian and quaternionic type which have been studied intensively by Igusa, Freitag, Krieg and others. In 2010 Gritsenko found three towers of orthogonal type modular forms with simple divisors. We resume Gritsenko’s construction and develop a framework to construct modular forms for several seriesof lattices. In contrast to earlier approaches our methods are very direct and reflect the configuration of the underlying definite lattice. This allows us to construct all generators of the associated graded rings of modular forms. Moreover we provide very natural coordinates in this case. We perform this process completely for one of the forementioned series. We indicate how our methods can be adapted to other series.