48 problems

Let $\varphi\in\mathcal{B}(q)$ be a Hecke–Maaß cusp form, and let $\lambda_{\varphi}(m)$ denote its Hecke eigenvalue for $(m,q)=1$. Ramanujan–Petersson conjecture. For every…

Let $\pi$ be an unramified cuspidal representation of $\operatorname{GL}_N$ and let $\alpha_1,\ldots,\alpha_N$ be its Satake parameters at a prime $\mathfrak{p}$. Generalized Raman…

Let $piinmathfrak{F}m$ be an automorphic representation, and let $alpha{r,pi}(p)$ denote its Satake parameters at a prime $p$. Generalized Ramanujan–Petersson conjecture. The Sat…

Ramanujan–Selberg conjecture. The Langlands parameters satisfy

Let $F$ be a Hecke–Maass cusp form for $\operatorname{SL}_{N}(\mathbb{Z})$, and let … be its Satake parameter at a prime $p$, where $\mathsf{T}_0$ is the standard maximal torus of…

Consider cusp forms with suitably normalized Hecke eigenvalues at primes. Ramanujan--Petersson conjecture. The suitably normalized Hecke eigenvalues of cusp forms are bounded on pr…

The generalized Selberg conjecture and generalized Ramanujan conjecture. One may take $\theta_n=0$ in these bounds.

Let $\pi$ be an automorphic representation and let $\pi_\nu$ be its local component at a place $\nu$. When $\pi_\nu$ is unramified, write $\alpha_\pi(j,\nu)$ for its local paramete…

Ramanujan conjecture. The eigenvalues of $A_v$ have absolute value $1$ for cuspidal automorphic representations. The source states that the meromorphic continuation and functional…

Let $\pi=\mathop{\otimes'}_{p\in\mathbb{P}\cup\{\infty\}}\pi_p$ be an automorphic cuspidal form on $GL_2(\mathbb{A}_{\mathbb{Q}})$, with local Hecke parameters…

Let $\pi$ be an unramified automorphic representation of $GL(N)$ at the archimedean places, with archimedean parameters $\kappa_j(\pi)$. Ramanujan conjecture at the archimedean pla…

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 $c_E(|d|)$ denote the Fourier coefficients of the half-integral weight modular form associated with the elliptic curve $E$ through the Waldspurger–Shimura–Kohnen–Zagier formula…

Among automorphic representations of $\operatorname{GSp}(4)$, Saito–Kurokawa lifts provide genuine non-tempered examples, but they are non-generic. Generalized Ramanujan conjecture…

Let $F$ be a number field and let $\pi=\otimes_v\pi_v$ be a cuspidal representation of $\operatorname{PGL}_n(\mathbb{A})$ such that every $\pi_v$ is unramified. For each archimedea…

Let $A=\mathbb{F}_q[T]$, let $mathfrak p\trianglelefteq A$ be a maximal ideal of degree $d$, and let $n\geq 1$. For integers $k,l$, write $\mathbf{T}_{mathfrak p}$ for the correspo…

Weak generalized Ramanujan conjecture. There exists $\eta>0$ such that, for every $n\geqslant 1$,

Let $k$ be a number field, let $\mathrm{G}$ be a connected reductive group over $k$, and let $\pi=\bigotimes_v\pi_v$ be an automorphic representation of $\mathrm{G}(\mathbb{A}_k)$…

Let $k$ be a number field, let $\mathrm{G}$ be a reductive group over $k$, and let $\pi=\bigotimes_v\pi_v$ be an automorphic representation in the discrete spectrum of…

Let $G$ be a quasi-split reductive group, and let an Arthur packet mean the packet attached to an Arthur parameter for $G$. A member is locally generic with respect to a Whittaker…

Ramanujan conjecture. The Langlands parameters satisfy $\operatorname{Re}(\mu_{i})=0$ for $i=1,2,3$. This is the expected temperedness of the associated automorphic representation…

Let $\psi_v$ be a local $A$-parameter, let $\Pi_{\psi_v}$ be its local $A$-packet, and let $\Pi_{\phi_{\psi_v}}\subset\Pi_{\psi_v}$ be the corresponding $L$-packet. Let $r(\pi)$ de…

Let $L$ be a degree-$3$ conductor-$1$ L-function, and for each prime $p$ let $a_p$ denote its coefficient at $p$. The compact group $\operatorname{SU}_3$ is the special unitary gro…

Let $\{F_j\}$ be the orthonormal basis of the space of Maass cusp forms over $SL(3,\mathbb{Z})$, and let $\{g\}$ be the orthonormal basis of the space of Maass cusp forms over…

Ramanujan–Petersson conjecture. The exponent $\vartheta=0$ is admissible. Unconditionally, the source states that the currently known bound is $\vartheta\leq 5/14$, so the conjectu…