Related papers: On the $L^{2}$-restriction norm problem for closed…

Let $\pi$ be a Hecke-Maass cusp form for $\rm SL_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbf{Z})$. Let $\chi$ be a primitive Dirichlet character of modulus $M=M_1M_2$ with $M_i$ prime, $i=1,2$.…

Let $f \in C^2(\mathbb{T}^2)$ have mean value 0 and consider $$ \sup_{\gamma~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|\gamma|} \left| \int_{\gamma}{ f ~~d\mathcal{H}^1}\right| },$$ where $\gamma$ ranges over all closed geodesics…

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $\mu$ in generic position i.e. away from Weyl chamber walls and…

We define and analyse certain $L^2$ norm on moduli space of Higgs bundles over $\mathbb{CP}^1$ with certain singularity. We prove that certain limit of our metrics here is the regulated $L^2$ norm proposed on the central fiber, first…

Let $(M,g)$ be an $n$-dimensional compact boudaryless Riemannian manifold with nonpositive sectional curvature, then our conclusion is that we can give improved estimates for the $L^p$ norms of the restrictions of eigenfunctions to smooth…

Let $f$ and $g$ be normalized Hecke-Maass cusp forms for the full modular group having spectral parameters $t_f$ and $t_g$ respectively with $t_f,t_g\asymp T\rightarrow \infty $. In this paper we show that the Rankin Selberg $L$-function…

We consider an overdetermined problem for Laplace equation on a disk with partial boundary data where additional pointwise data inside the disk have to be taken into account. After reformulation, this ill-posed problem reduces to a bounded…

In this paper, we prove uniform bounds for $\rm GL (3)\times GL(2)$ $L$-functions in the $\rm GL(2)$ spectral aspect and the $t$ aspect by a delta method. More precisely, let $\phi$ be a Hecke--Maass cusp form for $\rm SL(3,\mathbb{Z})$ and…

Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as $g\to \infty$, for a generic surface in $\mathcal{M}_g$, the error term…

We establish "higher depth" analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic L-functions of GL_d over a general number field. This is a generalization of the result of Deninger about the regularized…

Let $f$ be a holomorphic cusp form of weight $k$ with respect to full modular group $SL_2(\mathbb{Z})$ satisfying a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. Good gave the approximate functional…

Let $\psi$ be an $L^2$-normalized Hecke-Maass form with a large spectral parameter $\lambda>0$ on a compact arithmetic congruence hyperbolic 3-manifold $X=\Gamma\backslash\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$, and let $Y$ be a totally…

Given a maximal even-integral lattice $\cL$ of signature $(m+, 2-)$ with an odd $m\geq 3$, we consider the holomorphic cusp forms $F$ of weight $l$ on the bounded symmetric domain of type IV of dimension $m$ with respect to the discriminant…

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex…

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances…

We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.…

Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.

Let $D\geq 1$ and $q\geq 3$ be two integers. Let $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. Let ${\mathcal T}(D)$ denote the Terwilliger algebra of $H(D)$. Let $V(D)$ denote the standard ${\mathcal…

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$…

Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic…