Related papers: The sup-norm problem beyond the newform

Let $\pi$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $\phi$ be an $L^2$-normalized automorphic form in the space of $\pi$. The sup-norm…

The sup-norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function-field version of this problem can be reduced to the geometric problem of finding the largest dimension…

We give a lower bound for the sup-norm of an $L^2$-normalized newform in an irreducible, unitary, cuspidal representation $\pi$ of $GL_2$ over a number field. When the central character of $\pi$ is sufficiently ramified, this bound improves…

In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on $\text{PGL}_n$, whose associated automorphic representation has finite conductor $C(\pi)=p^c$ with $c\rightarrow \infty$, and its local…

Let $D$ be an indefinite quaternion division algebra over $\mathbb{Q}$. We approach the problem of bounding the sup-norms of automorphic forms $\phi$ on $D^\times(\mathbb{A})$ that belong to irreducible automorphic representations and…

Let $N$ be a prime and $\phi$ be a Hecke-Maass cuspidal newform for the Hecke congruence subgroup $\Gamma_0(N)$ in $\operatorname{SL}_n(\mathbb{R})$. Let $\Omega$ be an adelic compactum and let $\Omega_N$ be its projection to $\Gamma_0(N)…

Let $\pi$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$-function of $\pi$ in the $t$-aspect. More generally, we address the spectral aspect…

Let $\phi$ be an $L^2$-normalized Hecke--Maa{\ss} cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $\Omega$ is a compact…

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…

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank $L$-functions. The goal of this note is to explain how the method can be applied to the…

We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the…

Recently, the problem of bounding the sup norms of $L^2$-normalized cuspidal automorphic newforms $\phi$ on $\text{GL}_2$ in the level aspect has received much attention. However at the moment strong upper bounds are only available if the…

For reductive groups $G$ over a number field we discuss automorphic liftings from cuspidal irreducible automorphic representations $\pi$ of $G(\mathbb{A})$ to cuspidal irreducible automorphic representations on $H(\mathbb{A})$ for the…

Let $\pi$ be a cuspidal automorphic representation for $\mathrm{GL}(n)$ over a number field. We establish a conditional upper bound on the number of cuspidal isobaric summands in the symmetric $k$-th power lift of $\pi$, assuming that the…

In this paper we prove a new subconvexity result for the standard L-function of a unitary cuspidal automorphic representation $\pi$ of $\text{GL}_n$, where the finite set of places $S$ with large conductors is allowed to vary, provided that…

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f…

We prove the subconvexity conjecture for sup-norms of automorphic forms for congruence subgroups of SL(n, Z) that satisfy the Ramanujan conjecture at infinity.

Let $\Gamma\subseteq\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d$-dimensional space of cusp forms $\mathcal{S}_{k}^{\Gamma}$ of weight $2k$ for $\Gamma$,…

We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by…

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of…