# Quaternionic modular forms of degree two over Q(-3,-1)

• Quaternionische Modulformen zweiten Grades über Q(-3,-1)

Gehre, Dominic Steffen; Krieg, Aloys (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2012, 2013)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2012

Abstract

This thesis mainly deals with the analysis of quaternionic modular forms of degree two over the maximal order $O=+(1+i_1sqrt{3})/2+ i_2+(i_2+i_1i_2sqrt{3})/2$, where $H=R+i_1R+i_2R+i_1i_2R$ is the skew field of quaternions. These are defined as follows: Let $Sp_2(O)leqGL_4(O)$ be the quaternionic modular group over $O$ and $mathcal{H}(H)subsetH^{2imes 2}otimes_{H}C$ the quaternionic half-space of degree two, $GammaleqSp_2(O)$ a subgroup and $u$ a multiplier system of weight $k$ -- note that all these terms are defined in the thesis in detail. Then a holomorphic function $f:mathcal{H}(H)ightarrowC$ is called a quaternionic modular form of degree two with respect to $Gamma$ and $u$ of weight $k$ if [f((AZ+B)(CZ+D)^{-1})=u(M)(det(check{C}check{Z}+check{D}))^{k/2} f(Z)] holds for all $M=igl(egin{smallmatrix} A & B \ C & D end{smallmatrix}igr)inGamma$, with $~^{vee};$'' denoting the standard embedding of $H$ in $C^{2imes2}$. One of the main issues is to determine the graded ring of all quaternionic modular forms. In order to be able to do so, the thesis pursues several essential issues concerning quaternionic modular forms. Essentially, certain important classes of quaternionic modular forms are analyzed in detail. Moreover, an approach to answer the question concerning the graded ring is worked out, making use of a certain reduction process. The first chapter serves as an introduction. Quaternionic modular forms are defined and some first fundamental results are developed. This includes the determination of all multiplier systems of the quaternionic modular group and first dimensional bounds. In the second chapter quaternionic theta-series are examined in general -- and theta-constants in particular. Regarding similar topics concerning modular forms, theta-constants often sufficed to describe the occuring graded rings of modular forms. Therefore, in the thesis quaternionic theta-constants are analyzed in great detail, answering questions concerning their transformation behavior, Fourier-expansions and restrictions to half-spaces of lower dimension. The third chapter deals with so-called quaternionic Maaß lifts. Mainly, the theory of quaternionic Maaß lifts of odd weight with respect to non-trivial multiplier systems is established. Here one should note that the resulting existence of non-trivial quaternionic modular forms of odd weight with respect to the whole quaternionic modular group is a significant difference to the theory of quaternionic modular forms over the Hurwitz order. To analyze quaternionic Maaß lifts of odd weight the theories off elliptic Hecke-operators and elliptic newforms are presented. Some new results are obtained in order to be able to describe the spaces of elliptic modular forms serving as input for the Maaß lifts. In the fourth chapter quaternionic Eisenstein-series are studied. The main goal here is to determine their Fourier-expansions. Actually, quaternionic Eisenstein-series turn out to be Maaß lifts for the trivial character. To prove this the term of quaternionic Hecke-operators is needed. In the thesis the action of the special Hecke-operators $mathcal{T}_2(p)$ on quaternionic Maaß lifts for the trivial character is analyzed in full detail. Among other results, it is shown that these Hecke-operators preserve the spaces of quaternionic Maaß lifts, which in turn yields answers concerning the Fourier-expansions of the Eisenstein-series. In chapter five the connection between quaternionic modular forms and special orthogonal modular forms is illustrated. Orthogonal modular forms are introduced in general, while special lattices and their attached spaces of orthogonal modular forms are analyzed in detail. This information is needed for the subsequent reduction process. In the final chapter said reduction process is presented, which reduces the problem of determining the graded ring of quaternionic modular forms to analyzing modular forms on half-spaces of lower dimension. For this purpose, the important concept of Borcherds products is required. Ultimately, the original issue is reduced to the examination of paramodular forms of degree two and level 7. Unfortunately, it has not been possible to completely describe the graded ring of these paramodular forms of level 7, but nonetheless all manageable results are collected. Moreover, the worked out reduction process shows how to obtain structural conclusions concerning quaternionic modular forms once the problems regarding the paramodular forms are solved. Finally, the thesis ends by constructing a maximal system of algebraically independent quaternionic modular forms, which builds up on large parts of the developed results.