5 problems

Problem (Hilbert's twelfth problem). For every number field $K$, give an explicit construction of $K^{\mathrm{ab}}$ of the same kind. That is, specify analytic functions attached t…

Conjecture (generalized Ramanujan conjecture for $\mathrm{GL}_n$). For every number field $F$, every $n\ge2$, every such $\pi$, and every place $v$ of $F$, archimedean or not,…

Conjecture (global functoriality). For every $F$, $H$, $G$ and $\rho$ as above, and every automorphic representation $\pi'$ of $H(\mathbb A_F)$, there is an automorphic representat…

Conjecture. For every number field $K$ and every elliptic curve $E/K$ with $\operatorname{End}_K(E)=\mathbb Z$, $E$ is modular.

Conjecture (Deligne). For every such $K$, $\ell$, $X$ and $w$, and every integer $k$, each eigenvalue $\alpha$ of $\Phi$ on $\operatorname{Gr}^M_kV$ is an algebraic number such tha…