10 problems

Conjecture (strong, or binary, Goldbach conjecture). Every even integer $n>2$ is the sum of two primes: …

Conjecture (Fontaine–Mazur). Every irreducible geometric representation $\rho\colon G_K\to\mathrm{GL}_n(\overline{\mathbb Q}_p)$ comes from algebraic geometry.

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. There are infinitely many primes $p$ such that $p+2$ is also prime: …

Conjecture. For every admissible family, there are infinitely many positive integers $n$ such that $f_1(n),\dots,f_k(n)$ are all prime.

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

Conjecture. For every integer $n\ge1$ there is a prime strictly between $n^2$ and $(n+1)^2$: …

Conjecture (Elliott–Halberstam). The primes have level of distribution $\theta$ for every $\theta<1$. That is, for every fixed $\theta<1$ and every $A>0$ there is a constant…