The global Langlands conjecture

Let $F$ be a finite extension of $\mathbb Q$, let $\mathbb A_F$ be its ring of adeles, and fix a prime $p$ together with an isomorphism $\overline{\mathbb Q}_p\cong\mathbb C$. Consider certain irreducible continuous, almost everywhere unramified and de Rham at places dividing $p$, $n$-dimensional representations of $\operatorname{Gal}(\overline F/F)$ over $\overline{\mathbb Q}_p$, and certain $L$-algebraic, cuspidal automorphic representations of $\operatorname{GL}_n(\mathbb A_F)$ over $\mathbb C$. The global Langlands conjecture. There is a bijection between their isomorphism classes, preserving invariants by matching eigenvalues of Frobenius elements with Satake parameters. This is a central organizing conjecture relating Galois representations and automorphic representations; the stated general correspondence remains open.