 Main
The Algebra And Arithmetic Of VectorValued Modular Forms On $\Gamma_{0}(2)$
 Author(s): Gottesman, Richard Benjamin
 Advisor(s): Mason, Geoff
 et al.
Abstract
In this thesis, we investigate the module structure and the arithmetic of vectorvalued modular forms. We show that for certain subgroups $H$ of the modular group, the module $M(\rho)$ of vectorvalued 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 twodimensional 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 vectorvalued 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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