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

Let $f$ be a holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k>1$. In these notes, we follow Munshi to prove the Burgess bound $$ L(1/2+it,f)\ll_{f,\varepsilon} (1+|t|)^{1/2-1/8+\varepsilon}. $$

Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L…

In present article, we consider a $L^2$-orthogonal decomposition of the second fundamental form of a closed spacelike hypersurface in a Lorentzian spacetime and its applications to the study of some algebraic-differential properties of the…

Let $f$ be a normalized holomorphic cusp form with a square-free level $N$ and weight $k$. Using a pre-trace formula, we establish a sup-norm bound of $f$ such that $\|y^kf(z)\|_{\infty} \ll N^{-1/6+\epsilon}$ where the trivial bound is…

We prove a Burgess-like subconvex bound for twisted L-functions of a fixed irreducible cuspidal automorphic representation of GL(2) over a totally real number field. The proof is based on a spectral decomposition of shifted convolution sums…

The purpose of this paper is to prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs…

Given a compact segment, $\beta$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $\beta$. We prove unconditionally a sharp lower bound for Eisenstein series along…

Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error…

This paper uses algebro-topological techniques such as characteristic classes and obstruction theory, together with the $h$-principles for $\widetilde{\mathrm{G}}_2$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms recently established by the author…

We establish a correspondence between information geometry and gauge theory. First, we define an important class of statistical manifolds, that is normalized and satisfies a conservation field equation. Second, we prove that for a…

If $f\in L^2(R^d)$ and if the function $f(x)f(y)$ is close in $L^2(R^{2d})$ norm to a radially symmetric function of $(x,y)$ then $f$ is close in $L^2$ norm to a centered Gaussian function. This is proved in a quantitative form with the…

We prove a sub-convex estimate for the sup-norm of $L^2$-normalized holomorphic modular forms of weight $k$ on the upper half plane, with respect to the unit group of a quaternion division algebra over $\mf Q$. More precisely we show that…

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight $k$ with respect to a finite index subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$. We also prove corresponding bounds for the supremum over a compact…

The Katz-Sarnak density conjecture states that, as the analytic conductor $R \to \infty$, the distribution of the normalized low-lying zeros (those near the central point $s = 1/2$) converges to the scaling limits of eigenvalues clustered…

We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the…

This paper establishes an arithmetic intersection formula for central L-derivatives in higher weights.We prove that for a general cusp form (extending the previous result for newforms), the derivative is represented by the global height…

If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for…

Suppose $0 < \alpha \leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable function, and $A_\alpha(H)$ is the infimum of all numbers $C$ for which the inequality $\int_B H(x) dx \leq C R^\alpha$ holds for all balls $B \subset \Bbb R^n$…

The winding of a closed oriented geodesic around the cusp of the modular orbifold is computed by the Rademacher symbol, a classical function from the theory of modular forms. In this article, we introduce a new construction of winding…

Recently Wolff obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a…