29 problems

Let $\phi$ be a Hecke–Maass form for the full modular group, let $K$ be a compact subset of the modular surface $\mathbb{X}$, let $2<p\le\infty$, and let $\epsilon>0$. Then Iwaniec…

Let $\Gamma=SL_2(\mathbb{Z})$ be the full modular group, let $\mathbb{X}=\Gamma\backslash\mathbb{H}$ be the modular surface, and let $\phi$ be an even Hecke–Maass cusp form with ei…

Let $q$ be a parameter and let $\mathbf{E}_{q}(y)$ denote the function introduced through the Farey transfer operators. For the previously considered cases $q=1$ and $q=\frac12$, o…

Consider the Laplace spectrum of Maass forms on $SL(2,\mathbb Z)\backslash\mathbb H$. Cartier's simplicity conjecture. The spectrum should be simple: there should not be two linear…

Let $\phi$ be a Maass form with Laplace eigenvalue $\frac14$. Cassels's conjecture. Must $\phi$ arise from a Galois representation? The source presents this as an open conjectural…

Let $\{f_i\}_{i=1}^I$ be an orthogonal family of Hecke–Maass cusp forms and let $\{\widetilde{E_{T_j}}\}_{j=1}^J$ be an orthogonal family of normalized real Eisenstein series with…

Let $\{f_i\}_{i=1}^I$ be an orthogonal family of Hecke–Maass cusp forms, with $\langle f_i,f_{i'}\rangle=0$ for $i\ne i'$, and let $C_n$ denote the $n$th moment of a standard real…

Let $\mathbb{X}=\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, let $\{f_i\}$ be Hecke–Maass cusp forms normalized by … and define the Gaussian moments by…

Hua–Huang–Li conjecture. The powers $\{f_j^{a_j}\}_{j=1}^J$ are statistically independent: for every $\psi\in\mathcal{C}_c^\infty(\mathbb{X})$,

Let $N$ be the level, and let $\{\mu_j^1\}_j$ be an orthonormal basis of Maass forms of weight $1$ on $\Gamma_0(N)$, with spectral parameters $t_j$ and Fourier coefficients…

Fix an integer $N\geq 2$. Let $\ell_1,\ell_2,\ldots,\ell_N$ be integers greater than one, and let $\eta_1,\eta_2,\ldots,\eta_N$ be distinct Maass forms, with $\eta_j$ a Maass form…

Let $(\phi_j)_{j=1}^r$ be a finite orthogonal family of Hecke–Maaß cusp forms for $\operatorname{SL}(2,\mathbb{Z})$. Let $\mathcal{V}(x)$ be the associated vector of additive-twist…

Let $\Gamma$ be a Fuchsian group of the first kind with cusps, and let $\phi$ be a Maaß cusp form for $\Gamma$ of integer weight. For $\Gamma=\operatorname{SL}(2,\mathbb{Z})$, the…

Fix $n\geq 2$. Let $\{\phi_j\}_{j=1,2,\ldots}$ be an orthogonal basis of Maass cusp forms for $\operatorname{SL}(n,\mathbb Z)$, with Langlands parameters $\alpha^{(j)}$ and normali…

Let $q$ and $\chi$ be fixed, and let $g\in\mathcal{B}_0^{\ast}(q,\chi)$ be a real-valued newform with spectral parameter $t_g$. Equip $\Gamma_0(q)\backslash\mathbb{H}$ with hyperbo…

Let $g\to\frac12+it_g$ be a sequence of newforms in $\mathcal{B}_0^{\ast}(q,\chi)$, and let $R$ denote the radius of a hyperbolic ball. The volume-level shrinking-ball equidistribu…

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…

Let $n_0(q,T)$ be the least integer such that two nonmonomial Maaß newforms of weight $0$, level $q$, and principal character, with Laplacian eigenvalues in $(1/4,T]$ and identical…

Let $q$ be an odd positive integer, and let $s(q)$ denote the number of distinct odd primes $p$ for which $p^2$ divides $q$. Consider the new part of the discrete spectrum of the L…

Let $\mathcal{B}_N$ be the family of Maass cusp forms of level $N$, let $h_T$ be the weight function defined in the source, let $\phi$ be an even Schwartz function with compactly s…

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

Let $u_j$ be an even Hecke–Maass cusp form with spectral parameter $t_j$, so its Laplace eigenvalue is $1/4+t_j^2$. Young's logarithmic geodesic $L^2$ conjecture. … as…

The weak orthogonality relation. For positive integers $m_1,\ldots,m_{N-1}$,

Let $\phi$ be a Maass-Hecke cusp form with spectral parameter $t_\phi$, Fourier coefficients $\rho_\phi(n)$, and let $\delta_{0,m}$ denote the Kronecker delta. For…