# Jacobiformen über den Cayley-Zahlen

• Jacobi forms over the Cayley numbers

Dieckmann, Cornelia; Krieg, Aloys (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2014)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2014

Abstract

The classical theory of Jacobi forms on (mathcal{H}_1 imes C), was systematically described by Eichler and Zagier. Higher-dimensional generalizations of these Jacobi forms were considered by Ziegler and Gritsenko. Jacobi forms of this type naturally appear in the Fourier-Jacobi expansion of Siegel modular forms. Eie and Krieg introduced Jacobi forms on (mathcal{H}_1 imes mathcal{C}_{C}) in 1991. Analogously to the classical case, these Jacobi forms also occur as Fourier-Jacobi coefficients of modular forms on the half space of the Cayley numbers of degree 2. Consequently, the Fourier-Jacobi expansion of modular forms on the Cayley half space of degree 3 yields Jacobi forms on (mathcal{H}_1 imes mathcal{C}_{C}^{2}) and on (mathcal{H}(2,mathcal{C}) imes mathcal{C}_{C}^{1 imes 2}) respectively, where (mathcal{H}(2,mathcal{C})) denotes the Cayley half space of degree 2. The precise investigation of both types of Jacobi forms is one of the essential aims of this thesis, which is structured as follows: In the first chapter we will introduce the Cayley numbers and quote their most important properties. For this, we mainly refer to the works of Conway and Smith as well as Rehm. Beside the integral Cayley numbers (mathcal{O}_{mathcal{C}}), we focus on the algorithm to determine all right and left divisors respectively of an integral Cayley number. In collaboration with Derek Smith we managed to characterise the set of right and left divisors of odd norm with respect to their congruence properties modulo (2 mathcal{O}_{mathcal{C}}). In chapter 2 we investigate matrices over Cayley numbers. Especially hermitian and positive (semi)definite matrices are of interest. We will explain how these terms have to be interpretated in the octonionic case. In addition to that, we establish some characterizations for positive (semi)definite matrices which will be useful in the course of this thesis. In the third chapter we introduce the concepts of the Cayley half space, the modular group over the Cayley numbers as well as the octonionic modular forms. We observe that Baily, Krieg and Walcher use a definition for the modular group which is different from the one Eie uses, for instance. We will show that these different definitions still describe the same group. Functions which naturally occur when we consider Jacobi forms are the so called theta series. Since it is important to know their properties exactly in order to establish connections between Jacobi forms and special modular forms, we will discuss theta series in detail. Particularly one interesting question is how theta constants behave under the transpose map (Z mapsto Z^{tr}). In some cases we can use the new algorithm to determine all right and left divisors of odd norm of chapter 1. In all remaining cases an easy behaviour can be excluded. This again leads to the question how theta constants behave when we restrict them to the quaternionic half space. We will give an answer to this question in the last part of chapter 4 This part, written in collaboration with A. Krieg, has already been published. In chapter 5 we will introduce the first kind of Jacobi forms on (mathcal{H}_1 imes mathcal{C}^2), which are called Jacobi forms of weight (k in ) and index R, where R denotes an hermitian 2x2 matrix over the integral Cayley numbers. We will see that the Fourier-Jacobi coefficients of modular forms on the Cayley half space of degree 3 are examples of such Jacobi forms. In addition to that, we will show that every Jacobi form of weight k and index R can be written as a linear combination of special, linear independent theta series, where the occuring coefficients are classical modular forms with respect to a main congruence group of (extnormal{SL}(2,)). This circumstance provides an estimation of the dimension of the space of Jacobi forms. At the end of this chapter we indicate further examples of Jacobi forms of weight k and index R, the so called Jacobi-Eisenstein series. In the last chapter of this thesis we will consider a second kind of Jacobi form, which is defined on (mathcal{H}(2,mathcal{C}) imes mathcal{C}_{C}^{1 imes 2}). They are called Jacobi forms of weight (k in ) and index (m in N_0), and have already been studied by Eie. However, we will show that we can weaken Eie's definition of an Jacobi form without losing the claimed transformation properties. Especially, we will prove that in this case the so called Koecher effect holds, which provides a Fourier expansion that runs only over positive semidefinite matrices. Even for this kind of Jacobi forms we obtain a representation as a linear combination of special theta series. But in this case, the corresponding coefficients form a vector valued modular form over the Cayley half space of degree 2. Again we establish a connection between vector valued modular forms and Jacobi forms from this fact.