Skip to main content
Open Access Publications from the University of California

UC Santa Cruz

UC Santa Cruz Electronic Theses and Dissertations bannerUC Santa Cruz

The Algebra And Arithmetic Of Vector-Valued Modular Forms On $\Gamma_{0}(2)$

  • Author(s): Gottesman, Richard Benjamin
  • Advisor(s): Mason, Geoff
  • et al.

In this thesis, we investigate the module structure and the arithmetic of vector-valued modular forms. We show that for certain subgroups $H$ of the modular group, the module $M(\rho)$ of vector-valued modular forms for a representation $\rho$ of $H$ is a free module of dimension $\textrm{dim } \rho.$ In the case when $\rho$ is an irreducible two-dimensional representation of $\Gamma_{0}(2)$, we compute a basis for $M(\rho)$ using the modular derivative. We then express the component functions of an element $F$ of $M(\rho)$ of minimal weight in terms of the Gaussian hypergeometric series, a Hauptmodul of $\Gamma_{0}(2)$, and the Dedekind $\eta$-function. This allows us to obtain explicit formulas for the Fourier coefficients of $F$. We say that a function $f$ whose Fourier coefficients are algebraic numbers has unbounded denominators if the sequence of the denominators of the Fourier coefficients of $f$ is unbounded. We show that if $\rho$ has certain properties then the Fourier coefficients of a normalization of each of the component functions of $F$ are algebraic numbers. Moreover, we show that both component functions of this normalization have unbounded denominators. We then prove that if $X$ is any vector-valued modular form for $\rho$ whose component functions have Fourier coefficients that are algebraic numbers then both of the component functions of $X$ have unbounded denominators.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View