13 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 $F\in\mathrm{SK}_k$ 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 $S_k^n$ 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.…

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