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


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.

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