12 problems

Ramanujan conjecture. As $|D|\mapsto\infty$, for every $\alpha>0$,

Let $F$ be a number field, let $N\in\mathbb{N}$, and let $\pi\in\mathcal{AR}_{cusp}(GL_n)$. For every place $v$ of $F$, let $\pi_v$ denote the local component of $\pi$, and call it…

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 and let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_k)$. A local component $\pi_v$ is tempered when its local repres…

Let $F\in S_k(\Gamma^{(2)}_0(N))$ be a Hecke eigenform at good primes that is non-CAP, meaning that it is of general type or Yoshida type. For each prime $p$ with $\gcd(p,N)=1$, le…

Let $G=SO(n,1)$, and let $\widehat{G}^{\operatorname{sph}}_{\operatorname{Aut}}$ denote the automorphic dual of the spherical unitary dual of $G$. Generalized Ramanujan conjecture.…

Let $G$ be a real, semisimple, almost-simple and simply connected Lie group, let $\Gamma_1$ be a cocompact arithmetic lattice of $G$, and let $(\Gamma_N)$ be a sequence of finite-i…

Let $k$ be a global field and let $\Pi$ be a cuspidal automorphic representation of ${\rm GL}_n(\mathbb{A}_k)$. Symmetric-power automorphy conjecture. For every $n$, the symmetric-…

Generalized Ramanujan conjecture. For every prime $\mathfrak{p}\notin R_\pi$ and every $1\leq j\leq d$, $|\alpha_\pi(j,\mathfrak{p})|=1$, while for every prime…

Let $f(z) = \sum^{\infty}_{n=1} a_f(n) n^{\frac{k-1}{2}} q^n$ be a half-integral weight modular form of weight $k=\ell+\frac{1}{2}$ on $\Gamma_0(4N)$, where $k \in \mathbb{N}$,…

Let $E$ be a number field, let $p$ be a rational prime that is unramified and does not split completely in $E$, and let $\nu\mid p$. Let $f_p$ denote the relevant residue-degree qu…