Langlands program

Statement

The following is the global reciprocity conjecture for $\mathrm{GL}_n$, one central part of the program. Let $F$ be an algebraic number field, let $G_F$ be its absolute Galois group, let $n \ge 1$, and let $\ell$ be a prime. Fix a field isomorphism $\iota \colon \overline{\Q}_\ell \to \C$. Let $\rho \colon G_F \to \mathrm{GL}_n(\overline{\Q}_\ell)$ be an irreducible Galois representation. Assume that $\rho$ is unramified at all but finitely many places of $F$ and de Rham at the places above $\ell$. Then there is a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\mathbb{A}_F$ is the adele ring of $F$, with the following property. For all but finitely many finite places $v$ of $F$, both $\rho$ and $\pi_v$ are unramified at $v$. Let $\mathrm{Frob}_v$ be a Frobenius element at $v$. Then the Satake parameters of $\pi_v$ are, up to the standard normalising twist, the images under $\iota$ of the eigenvalues of $\rho(\mathrm{Frob}_v)$.