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Title On Vector-Valued Automorphic Forms
Publication Date
Degree PhD
Discipline/Department Department of Mathematical and Statistical Sciences
Degree Level doctoral
University/Publisher University of Alberta
Abstract Let $\rG$ be a genus 0 Fuchsian group of the first kind\,, $w \in 2\Z$ and $\rho : \rG \longrightarrow \mr{GL}_{d}(\C)$ be any admissible representation of $\rG$ of rank $d$\,. Then this dissertation deduces that the space $\mc{M}^{!}_{w}(\rho)$ of rank $d$ weakly holomorphic vector-valued automorphic forms of weight $w$ with respect to $\rho$ is a free module of rank $d$ over the ring $R_{_{\rG}}$ of weakly holomorphic scalar-valued automorphic functions\,. Note that almost every $\rho$ is admissible\,. Let $\mr{H}$ be any finite index subgroup of $\rG$ and $\rho$ be any rank $d$ admissible multiplier of H then this thesis establishes that the lift of any vector-valued automorphic form of $\mr{H}$ with respect to $\rho$ is a rank $d\times [\rG:\mr{H}]$ vector-valued automorphic form of $\rG$ with respect to the induced admissible multiplier $\mr{Ind}_{_{\mr{H}}}^{^{\rG}}(\rho)$\,. In case $\rG$ is a triangle group of type $(\ell, m, n)$ we show that to classify the rank 2 vector-valued automorphic forms is equivalent to classify the solutions of Riemann's differential equation of order 2\,. When $\rG$ is a modular triangle group then we also classified the primes for which the denominator of Fourier coefficients of at least one of the components of any rank $2$ vector-valued modular form with respect to some rank 2 admissible multiplier $\rho$ will be divisible by $p$ \ie the Fourier coefficients will have unbounded denominators\,. Such components are noncongruence scalar-valued automorphic forms of $\ker(\rho)$\,. In addition this thesis also proves the modularity of the bilateral series associated to various mock theta functions and provide the closed formula of the associated Ramanujan's radial limit for all of Ramanujan's 5th order mock theta functions as well as few other mock theta functions of various order.
Subjects/Keywords Automorphic Forms, Representation Theory; Fuchsian groups, Triangle groups; Vector-valued automorphic forms;
Language en
Rights Unrestricted: open access
Country of Publication ca
Record ID oai:alberta:chh63sv954
Repository alberta
Date Retrieved
Date Indexed 2017-06-06
Grantor University of Alberta

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