23 problems

Level-aspect sup-norm conjecture. With $F$ and $\Gamma$ as above, as $N\to\infty$,

Weight-aspect sup-norm conjecture. If $F$ is an eigenfunction of all Hecke operators, then as $k\to\infty$,

Weight-aspect Bergman-kernel size conjecture. As $k\to\infty$,

Let $\mathrm{SK}_k(N)$ denote the space of Saito-Kurokawa lifts of weight $k$ and level $N$, and let $F\in\mathrm{SK}_k(N)$ be an $L^2$-normalized newform. Write…

Let $\mathrm{SK}_k(N)$ denote the space of Saito-Kurokawa lifts of weight $k$ and level $N$, let $B_k^{SK}(N)$ be an orthonormal basis, and define … with…

Let $J^{cusp}_{k,1}(N)$ denote the space of Jacobi cusp forms of index $1$ and level $N$, and let $\phi\in J^{cusp}_{k,1}(N)$ be an $L^2$-normalized newform. Write…

Let $J^{cusp}_{k,1}(N)$ denote the space of Jacobi cusp forms of index $1$ and level $N$, and let $\sup(J^{cusp}_{k,1}(N))$ denote its associated supremum quantity. Jacobi index-on…

Let $g\in \operatorname{SL}_{2}(\mathbb{R})$. For parameters $\ell$, $\delta$, $L$, and $\heartsuit$, consider pairs $\gamma_1,\gamma_2\in R(\ell;g)$ lying in…

Let $Y$ be a compact negatively curved locally symmetric space of dimension $d$ and rank $r$, and of congruence type. Consider well-balanced Hecke--Maass forms on $Y$ of high frequ…

Let $\phi$ be a Hecke–Maass cusp form on an arithmetic hyperbolic surface, normalized in $L^2$, and let $\lambda$ be its Laplace–Beltrami eigenvalue. Arithmetic hyperbolic surface…

Let $F\in\mathrm{SK}_k$ be a Hecke eigenform, let $F^\circ$ denote its pullback to the diagonal, and suppose that $F^\circ\ne0$ and $\lVert F^\circ\rVert_2=1$. The Saito–Kurokawa p…

Let $n$ be positive, let $m$ be a positive scalar index, and let $J^{cusp}_{k,m;(n,1)}$ denote the space of Jacobi cusp forms of weight $k$ and index $m$. Let…

Let $g,n$ be positive integers, let $S\in\Lambda_g^+$ be a symmetric positive-definite half-integral matrix, and let $J^{cusp}_{k,S;(n,g)}$ be the space of Jacobi cusp forms of wei…

Let $S_k^n$ denote the space of Siegel cusp forms of degree $n$ and weight $k$, and let $F\in S_k^n$ be an $L^2$-normalised Hecke eigenform. Write $\|F\|_\infty$ for its sup-norm.…

Optimal level-aspect sup-norm bound. It is conjectured that, for every $\epsilon>0$,

Let $\Gamma=\operatorname{SL}_2(\mathbb{Z})$, let $\mathbb{H}$ be the upper half-plane, and let $g\in\mathcal{B}_0(\Gamma)$ be a Hecke–Maaß eigenform with spectral parameter $t_g$.…

Let $F$ be a number field, let $G=\operatorname{GL}_2$, and let $\pi$ be an automorphic representation with finite conductor $N$ and local conductor exponents $n_v$ and central-cha…

Let $f$ be a newform on $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ with level $M$ and conductor $N$, and let $N_1$ be the smallest integer such that $N\mid N_1^2$. Assume that…

Let $f$ be an $L^2$-normalized Hecke--Maass newform of level $N$, with spectral parameter bounded in terms of $T$, and let the square-ful part of $N$ be the product of the prime-po…

Let $U(z)$ be a Hecke–Maass cusp form with Laplace eigenvalue $1/4+T^2$, Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$,…

Let cusp forms vary in the level aspect, with their modulus measured by the sup norm. Folklore conjecture. All cusp forms should be uniformly small in the level aspect. Templier pr…

Let $N\geq 1$, let $\Gamma_0(N)\subset SL_2(\mathbb{Z})$ be the Hecke congruence subgroup of level $N$, let $\chi$ be a Dirichlet character modulo $N$, and let $f\in S_2^*(N,\chi)$…

Let $f$ be a cusp form with archimedean parameter $t_f$, and let $\\|f\\|_2$ and $\\|f\\|_{\infty}$ denote its $L^2$- and sup-norms, respectively. Sup-norm conjecture. One might co…