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# On Vector-Valued Automorphic Forms Open Access

## Descriptions

Other title
Subject/Keyword

Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Bajpai, Jitendra K
Supervisor and department
Prof. Terry Gannon, Department of Mathematical and Statistical sciences, University of Alberta
Examining committee member and department
Prof. Terry Gannon, Department of Mathematical and Statistical sciences, University of Alberta
Prof. Arturo Pianzola, Department of Mathematical and Statistical sciences, University of Alberta
Prof. Chris Cummins, Department of Mathematical and Statistical sciences, Concordia University
Prof. Thomas Hillen, Department of Mathematical and Statistical sciences, University of Alberta
Prof. Manish Patnaik, Department of Mathematical and Statistical sciences, University of Alberta
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2015-01-30T11:49:09Z
2015-06
Degree
Doctor of Philosophy
Degree level
Doctoral
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.
Language
English
DOI
doi:10.7939/R3FN10X98
Rights
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
J. Bajpai, S. Kimport, J. Liang, D. Ma, and J. Ricci. Bilateral series and Ramanujan’s radial limits. Accepted for publication in Proceedings of the American Mathematical Society, 2013.

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