53 problems

Let $\mathbb{H}:=\{x+iy:x\in\mathbb{R},y>0\}$, let $\Gamma=\operatorname{SL}(2,\mathbb{Z})$, and let $\mathbb{X}=\Gamma\backslash\mathbb{H}$ with measure…

Ivić's moment conjecture. In 2002, Ivić conjectured that

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…

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 $\phi$ be a Maass cusp form on $\mathbb{X}$, let $\lambda_\phi$ be its Laplace eigenvalue, and let $N(\phi)$ denote the number of connected components of…

Divisor-bound conjecture. The coefficients satisfy

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 $\phi$ be a hypothetical Maass form with integral coefficients $a_n$, and let $a_p$ denote its coefficients at primes. Sarnak's integral-coefficient consequence of the Ramanuja…

Let $r$ be the spectral parameter of a Maass cusp form, with eigenvalue … Let $F$ be any compact regular subregion of the fundamental domain $\mathcal F$, and let $\sigma$ denote t…

Let $q\in\mathbb{N}$, let $z_0\in\Gamma_0(q)\backslash\mathbb{H}$, and let $\mathcal{B}(q)$ be an orthonormal Hecke eigenbasis of cuspidal Laplace eigenfunctions, with spectral par…

Let $q\in \mathbb{N}$ be fixed, let $\Gamma_0(q)\subseteq \operatorname{SL}_2(\mathbb{Z})$ be the Hecke congruence subgroup, and let $\mathcal{B}(q)$ be an orthonormal Hecke eigenb…

Let $b_p^{(n)}$ be the Hecke eigenvalue of the Maass cusp form $\nu_n$ at a prime $p$, let $b_1^{(n)}$ be its first coefficient, and let $R_n$ denote the Laplace-eigenvalue paramet…

Let $\nu_n$ be Maass cusp forms with Laplace eigenvalues parametrized by $R_n$, and let $b_p^{(n)}$ denote their Hecke eigenvalues at a prime $p$. Horizontal Sato–Tate conjecture.…

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…

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

Bernstein–Reznikov conjecture. One has

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…

Let $\phi$ be a Hecke–Maaß cusp form for $\Gamma_0(N)$. For each positive integer $q$, define … where $L_\phi(x)=L(\phi,1/2,x)$ is the central value of the additive twist. Additive…

Let $F$ be a Siegel cusp form of degree $2$ and level $N=1$ lying in the Maass subspace, and write $a(T)$ for its Fourier coefficients. Das–Kohnen conjecture. For every…

Let $N$ be squarefree, and consider the Laplacian spectrum of Maass cusp forms of level $N$ with trivial character. Simplicity and Poissonian spacing conjecture. The Laplacian spec…

Reality conjecture. The spectral parameter $t_g$ is real for every $g\in B(N,\psi)$.