Dimensionsformeln für Räume von hermiteschen Spitzenformen vom Grad 2

  • Dimension formulas for spaces of Hermitian cusp forms of rank two

Freh, Andreas; Krieg, Aloys (Thesis advisor); Raum, Martin (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aaachen University, 2017

Abstract

Braun introduced the concepts of Hermitian modular forms and Hermitian cusp forms as a generalization of elliptic and Siegel modular forms. In this thesis we use the Selberg trace formula to determine exact dimension formulas for the spaces of Hermitian cusp forms of degree $k$ with respect to a main congruence subgroup $\Gamma_2(q)$ of rank $2$.We have to restrict ourselves to the cases $q\in\mathbb{N}$ where $q\geq2$. In addition, we only get results in the case, where the class number of the imaginary quadratic field is $1$.We shall first derive the Selberg trace formula for Hermitian cusp forms. We will also consider possible transformations of the trace formula, which allow us to evaluate the trace formula in the later chapters.We will also determine convergence-generating factors that allow us to exchange summation and integration. In particular, we derive a form of the Selberg trace formula, which allows us to calculate the dimension of the cusp forms with respect to a main congruence subgroup $\Gamma_2(q)$, but to determine representatives in the Hermitian modular group $\Gamma_2$.We develop methods to determine representatives of a conjugacy class in $ \Gamma_2 $ and the unitary group $\mathcal{U}(2,\mathbb{C})$. Then we will determine which conjugacy classes do not contribute to thedimension formula. For these, we do not need to know any representatives in $\Gamma_2 $. For the remaining conjugacy classes, we will specify a representative system and determine the contributions of these conjugacy classes.In addition, we derive a formula for the index of a main congruence subgroup in the modular group and use a formula from Shimura to calculate the volume of a fundamental domain. Using these results we can evaluate the trace formula.The calculation contains limit values that contain so-called Shintani Zeta functions. We will determine these and use them to give exact numerical values for the dimension formula.Finally, we give a brief overview of possible generalizations for the results of this thesis and indicate the greatest difficulties that come along with these generalizations.