Related papers: Equidistribution of holomorphic cusp forms on thin…

Let $Y_1$ be a compact arithmetic hyperbolic surface associated to a maximal quaternion order, let $Y_q$ be a cover associated to an Eichler suborder of prime level $q$, and let $\iota_q$ be embedding of $Y_q$ as the Hecke correspondence…

We consider the question of Quantum Unique Ergodicity for quasimodes on surfaces of constant negative curvature, and conjecture the order of quasimodes that should satisfy QUE. We then show that this conjecture holds for Eisenstein series…

We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres in the Cayley graph converges to Haar measure.

We investigate analogues of some of the classical results in homogeneous dynamics in non-linear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma<G$ a discrete subgroup. For a large class of…

Let $\Gamma$ be a Schottky semigroup in $\mathrm{SL}_2(\mathbf{Z})$, and for $q\in \mathbf N$, let $\Gamma(q):=\{\gamma\in \Gamma: \gamma= e \text{ (mod $q$)}\}$ be its congruence subsemigroup of level $q$. We prove the following uniform…

A phenomenological analysis based on the published branching fractions and $CP$ asymmetry observables of the $B\to K\pi$, $B\to\pi\pi$ and $B\to KK$ dataset is performed. The amplitude decomposition by the topological diagrams and the…

We prove the convergence of the normalized Fubini-Study measures and the logarithms of the Bergman kernels of various Bergman spaces of holomorphic and weakly holomorphic sections associated to a singular Hermitian holomorphic line bundle…

We derive a set of light-cone sum rules relating the hadronic form factors relevant for $B\to K\pi\ell^+\ell^-$ decays to the $B$-meson light-cone distribution amplitudes (LCDAs). We obtain the sum rule relations for all $B\to K\pi$ form…

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for…

The approximation properties of the trigonometric sums U_{n,p}^\psi of a special type are investigated on the classes C^\psi_{\beta, \infty} of (\psi,\beta)-differentiable (in the sense of Stepanets) periodical functions. The solution of…

We prove a harmonically weighted equidistribution result for the $p$-th Satake parameters of the family of automorphic cuspidal representations of $\operatorname{PGSp}(2n)$ of fixed weight $\mathtt{k}$ and prime-to-$p$ level $N\to \infty$.…

Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb…

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$…

We show that the image of repeated differentiation on weak cusp forms is precisely the subspace which is orthogonal to the space of weakly holomorphic modular forms. This gives a new interpretation of the weakly holomorphic Hecke…

For a complete hyperbolic surface whose fundamental group is finitely generated and has critical exponent bigger than 1/2, we obtain an effective equidistribution of closed horocycles in its unit tangent bundle. This extends a result of…

This thesis contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a…

The $B\to\pi K$ decays with significant electroweak penguin contributions show a puzzling pattern. We explore this "$B\to\pi K$ puzzle" through a systematic strategy. The starting point, which is essentially unaffected by electroweak…

The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by…

Let $\Gamma$ be a lattice in $G=\mathrm{SL}(2,\mathbb{C})$. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in $\Gamma\backslash G$. Our method of proof relies…