12 problems

Let $\Gamma\subseteq \operatorname{SL}_2(\operatorname{\mathbb{Z}})$ be a Fuchsian group of the first kind, let $f\in S_k(\Gamma)$ be a normalized Hecke cusp form, and let $\mu_f$…

Let $\psi\colon \mathbb{R}^+ \to \mathbb{R}$ be a smooth compactly supported function, and let $f(z)$ run over $L^2$-normalized holomorphic Hecke cusp forms of weight $k$. Young's…

Holomorphic QUE conjecture. Fix a bounded continuous function $\phi$ on $\operatorname{Sp}_{2n}(\mathbb Z)\backslash\mathbb H_n$. If $F\in S_k(\operatorname{Sp}_{2n}(\mathbb Z))$ t…

Let $\mathfrak{X}:=\mathcal{H}/SL(2,\mathbb{Z})$ be the modular surface, let $\Delta$ be its Laplacian, and let $\phi_i$ be a sequence of $L^2$-normalized eigenfunctions of…

Holomorphic mass equidistribution conjecture. For every such $\Psi$,

Let $u_j$ be a Hecke–Maass cusp form with Laplace eigenvalue $1/4+T_j^2$, Hecke eigenvalues $\lambda_{u_j}(m)$, parity $u_j(-x+iy)=\delta_j u_j(x+iy)$ with $\delta_j\in\{-1,1\}$, a…

Fix $n\in\mathbb{Z}_{\geqslant 0}$ and let $N\geqslant n$ tend to infinity. Let $\mathbf{Y}_{-N..N}$ be the space of paths used above, let $\mathcal{F}_{-N..N}$ be the specified fa…

Let $f$ be a classical holomorphic newform of fixed weight $k$ and level $q$, with mass $d\nu_f=|f(z)|^2y^{k-2}\,dx\,dy$ on $Y_0(q)=\Gamma_0(q)\backslash\mathbb H$. Let…

Let $X$ be the relevant hyperbolic three-manifold, let $\rho_m$ be the representation defining the homogeneous bundle $X\times_K\rho_m$, and let $\{s_n\}$ be a sequence of $L^2$-no…

Let $(M,g)$ be a compact smooth Riemannian manifold with ergodic geodesic flow, and let $(\psi_j)_{j\geq1}$ be any orthonormal basis of eigenfunctions. For an eigenfunction, let…

Let $\Phi^t_H$ be an ergodic Hamiltonian flow on an energy shell $\mathcal{E}_E$, with quantum Hamiltonian $\hat H_\hbar$ and eigenstates $\psi_{\hbar,j}$ of energies…

Let $\mathbbm{H}$ be the upper half plane with hyperbolic measure $d\mu z=y^{-2}dxdy$, let $\Gamma=\operatorname{SL}_2(\mathbbm{Z})$, and set $X=\Gamma\backslash\mathbbm{H}$. Let…