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 Author Bajpai, Jitendra K Title On Vector-Valued Automorphic Forms URL https://era.library.ualberta.ca/files/chh63sv954 Publication Date 2015 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 2017-02-26 Date Indexed 2017-06-06 Grantor University of Alberta

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…theory (RCFT) form a vvmf of weight 0. The Borcherds lift associates vvmf for a Weil representation to automorphic forms on orthogonal groups with infinite product expansions, which can arise as denominator identities in Borcherds-Kac-Moody…

…This chapter is ended with an example of vector-valued modular forms . Chapter 3 discusses the lift of an automorphic form of any Fuchsian group G of the first kind . This chapter also shows how the induction of any representation (multiplier)…

…C× be a 1-dimensional representation . Then , a scalar-valued meromorphic function f : H → C is a weight w meromorphic scalar-valued automorphic form of G with respect to σ , if (i) f (γτ ) = σ(γ)(cτ + d)w f…

…equations on the sphere instead on the upper half plane . In some sense, this thesis strengthens the relation between differential equations and vectorvalued automorphic forms (vvaf ) by showing that for any triangle group G, defining a Riemann’s…

representation ρ : G → GLd (C) (c) Nw (ρ) the space of nearly holomorphic vvaf of G of weight w ∈ 2Z with respect to cusp c ∈ CG and admissible multiplier ρ M!w (ρ) the space of all weakly holomorphic vvaf of G of weight w…

automorphic forms X[n] the nth vector-valued Fourier coefficients of vvaf X(τ ) Z Zm Q R C C[x] 4 We end this section by fixing the notation for the modular group PSL2 (Z) by Γ(1) as well as for any number N…

…developing the theory of vector-valued automorphic forms (vvaf) of Fuchsian groups. In order to be explicit about the use of the terms vvmf and vvaf, we will make the following distinction between them : vvaf for a group commensurable with Γ(1…

representation, then classify all the vvaf of G with respect to the multiplier ρ . In an attempt to answer the above , this is answered with certain restrictions and qualifications . More precisely, • first restricted only to the world of genus-0 Fuchsian groups…