Related papers: Sup-norm bounds for Jacobi cusp forms

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the…

Let $\pi$ be a fixed Hecke--Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be a prime. Let $L(s,\pi\otimes \chi)$ be the $L$-function associated to $\pi\otimes…

In this note we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $GL_2(\mathbb{Z}_p)$-type $V$ in $\pi$ and proof…

In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/\mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series…

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations.…

In this paper we derive an explicit formula for the Hirzebruch-Mumford volume of an indefinite lattice L of rank at least 3. If \Gamma is an arithmetic subgroup of the group O(L) of isometries of L and L has signature (2,n), then an…

For positive integers $k$ and $N$, we describe how to compute the natural action of $SL_2(\mathbb{Z})$ on the space of cusp forms $S_k(\Gamma(N))$, where a cusp form is given by sufficiently many terms of its $q$-expansion. This will reduce…

Let $\pi$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form and $\chi$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$…

Using a theorem of Jackiw and Pi expressing the delicate balance of the spin and the orbital momentum, we systematically classify the flat-space massless Lagrangian quantum field theories that are invariant under the global conformal group…

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarn\'{\i}k limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This…

Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level…

Let F be a square integrable Maass form on the Siegel upper half space of rank 2 for the Siegel modular group Sp(4, Z) with Laplace eigenvalue lambda. If, in addition, F is a joint eigenfunction of the Hecke algebra, we show a power-saving…

We complete the program indicated by the Ansatz of D'Hoker and Phong in genus ~4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Anstaz. We prove $\dim [\Gamma_4(1,2),8]_0=2$ and…

We consider a geometrically finite discrete group of conformal transformations of the sphere. Further we consider distributions which are supported on the limit set and are invariant with conformal weight. We estimate their regularity in…

For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $\Gamma$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the…

We find the average value of the $L^4$-norm of holomorphic Hecke cusp forms of weight within a dyadic interval, up to an error which saves a power of the weight.

Let $\lambda(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean…

In this paper, we study quadratic forms in spaces of holomorphic cusp forms. We show, conditionally, that when two quadratic forms in Hecke eigenforms share no common diagonal terms, their inner product is expected to converge to the sum of…

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…

Let $G$ be a simple non-compact linear Lie group. Let $\pi$ be any irreducible unitary representation of $G$ with infinitesimal character $\Lambda$ whose continuous part is $\nu$. The beautiful Helgason-Jonson bound in 1969 says that the…