Related papers: Effective sup-norm bounds on average for cusp form…

If $\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group…

Let $f$ be a primitive Maass cusp form for a congruence subgroup $\Gamma_0(D) \subset $ SL($2,\mathbb{Z}$) and $\lambda_f(n)$ its $n$-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many $\lambda_f(n)$…

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to…

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} +…

Using the amplification technique, we prove that `mass' of the pullback of the Saito-Kurokawa lift of a Hecke eigen form $g\in S_{2k}$ is bounded by $k^{1-\frac{1}{210}+\epsilon}$. This improves the previously known bound $k$ for this…

We show that noncongruence subgroups of SL_2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Gamma(N) of level…

The vacuum stability lower bound on the mass of the Higgs boson is numerically investigated in an $SU(2)_L \otimes SU(2)_R$ symmetric Yukawa model, which describes two heavy degenerate fermion doublets in the limit of vanishing gauge…

We study the pointwise supremum of convex integral functionals $\mathcal{I}_{f,\gamma}(\xi)= \sup_{Q} \left( \int_\Omega f(\omega,\xi(\omega))Q(d\omega)-\gamma(Q)\right)$ on $L^\infty(\Omega,\mathcal{F},\mathbb{P})$ where…

Let $\Gamma$ be a non-elementary, non-convex-cocompact Kleinian group acting on $\mathbb{H}^{d}$. We show that the Hausdorff dimension of the sublinearly conical Myrberg limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$.…

Let $\pi$ be a cuspidal representation on $\GL(2,\mathbb{A}_{\mathbb{Q}}).$ We give nontrivial lower and upper bounds for average of absolute values of Dirichlet coefficients associated to $\pi;$ and nontrivial upper bound in the case of…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…

Let $\rho:\Gamma\rightarrow PSL_d(\mathbb{K})$ be a Zariski dense Borel-Anosov representation, for $\mathbb{K}$ equal to $\mathbb{R}$ or $\mathbb{C}$. Let $o$ be a form of signature $(p,d-p)$ on $\mathbb{K}^d$ (where $0<p<d)$. Let…

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\mathrm{SL}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\mathrm{Sp}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Our…

First we give a review of the spurion formalism and the exact renormalization group equations for soft supersymmetry breaking parameters in general gauge theories. Next we discuss the minimal supersymmetric standard model coupled to…

A cusp form $f(z)$ of weight $k$ for $\SL_{2}(\Z)$ is determined uniquely by its first $\ell := \dim S_{k}$ Fourier coefficients. We derive an explicit bound on the $n$th coefficient of $f$ in terms of its first $\ell$ coefficients. We use…

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq 1}$ denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums…

Let $\Gamma\subseteq PSL_2({\bf R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and…

Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j…

A new solution to the requirement of two-loop finiteness of the soft supersymmetry breaking terms (SSB) parameters is found in Finite-Gauge-Yukawa unified theories. The new solution has the form of a sum rule for the relevant scalar masses,…

Recently, Bo'az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag's quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies…