Abstract:
Let $f$ be a Hecke--Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda$. It is shown that $\pnorm{f}_\infty \ll_{\lambda,\epsilon} N^{-1/6}+\epsilon} \pnorm{f}_2$ for any $\epsilon>0$.

Abstract:
In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma_0(N)^+$, where N>1$ is a square-free integer. After we prove that $\Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to N=5 and N=6 have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $\Gamma_0(5)^+$ than for $\Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first 3557 of $\Gamma_0(5)^+$ and the first 12474 eigenvalues of $\Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.

Abstract:
For integers $k\geq 2$, we study two differential operators on harmonic weak Maass forms of weight $2-k$. The operator $\xi_{2-k}$ (resp. $D^{k-1}$) defines a map to the space of weight $k$ cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are "dual" under $\xi_{2-k}$ to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of $D^{k-1}$.

Abstract:
In this paper we study two quantum mechanical systems on punctured surfaces modeled by hyperbolic spaces, namely the cases of the singly punctured two-torus and triply punctured two-sphere. We study the systems using their Maass waveforms in connection with the eigenfunctions of the Laplacian. The energy eigenfunctions on such surfaces are precisely the eigenfunctions of the hyperbolic Laplacian satisfying $\Gamma $($2)$-automorphicity for the triply punctured sphere and $\Gamma ^{\prime}$-automorphicity for singly punctured torus. We introduce the algorithm of numerically computing the Maass cusp forms on these two surfaces and report on the (preliminary) computational results of the lower-lying eigenvalues for each odd and even Maass cusp forms on both surfaces.

Abstract:
Let $\phi$ denote a primitive Hecke-Maass cusp form for $\Gamma_o(N)$ with the Laplacian eigenvalue $\lambda_\phi=1/4+t_{\phi}^2$. In this work we show that there exists a prime $p$ such that $p\nmid N$, $|\alpha_{p}|=|\beta_{p}| = 1$, and $p\ll(N(1+|t_{\phi}|))^c$, where $\alpha _{p},\;\beta _{p}$ are the Satake parameters of $\phi$ at $p$, and $c$ is an absolute constant with $0

Abstract:
For nonuniform cofinite Fuchsian groups $\Gamma$ which satisfy a certain additional geometric condition, we show that the Maass cusp forms for $\Gamma$ are isomorphic to 1-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.

Abstract:
In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight 0 on the full modular group introduced by Lewis and Zagier.

Abstract:
We prove a quantitative statement of the quantum ergodicity for Maass-Hecke cusp forms on $SL(2,\mathbb{Z})\backslash \mathbb{H}$. As an application of our result, we obtain a sharp lower bound for the $L^2$-norm of the restriction of even Maass-Hecke cusp form $f$'s to any fixed compact geodesic segment in $\{iy~|~y>0\} \subset \mathbb{H}$, with a possible exceptional set which is polynomially smaller in the size than the set of all $f$. We also improve $L^\infty$ estimate for Maass-Hecke cusp forms given by Iwaniec and Sarnak, for almost all Maass-Hecke cusp forms. We then deduce that the number of nodal domains of $f$ which intersect a fixed geodesic segment increases with the eigenvalue, with a small number of exceptional $f$'s. In the recent work of Ghosh, Reznikov, and Sarnak, they prove the same statement for all $f$ without exception, assuming the Lindelof Hypothesis and that the geodesic segment is long enough. For almost all Maass-Hecke cusp forms, we give better lower bound of number of nodal domains.

Abstract:
We establish lower bounds on the sup norm of Hecke-Maass cusp forms on congruence quotients of ${\rm GL}_n(\mathbb{R})$. The argument relies crucially on uniform estimates for Jacquet-Whittaker functions. These purely local results are of independent interest, and are valid in the more general context of split semi-simple Lie groups. Furthermore, we undertake a fine study of self-dual Jacquet-Whittaker functions on ${\rm GL}_3(\mathbb{R})$, showing that their large values are governed by the Pearcey function.