Fontaine–Mazur conjecture

Let $K$ be a number field with absolute Galois group $G_K=\operatorname{Gal}(\bar K/K)$, and let $p$ be a prime. Let

$$\rho\colon G_K\longrightarrow \mathrm{GL}_n(\overline{\mathbb Q}_p)$$

be a continuous representation. For a place $v$ of $K$, let $G_{K_v}\subseteq G_K$ be a decomposition group at $v$. Call $\rho$ geometric if

1. $\rho$ is unramified at all but finitely many places of $K$, and
2. for every place $v\mid p$, the restriction $\rho|_{G_{K_v}}$ is de Rham in the sense of Fontaine's $p$-adic Hodge theory. Equivalently, by the $p$-adic monodromy theorem, it is potentially semistable.

Say that $\rho$ comes from algebraic geometry if there are a smooth projective variety $X$ over $K$, integers $i\ge0$ and $r\in\mathbb Z$, such that $\rho$ is isomorphic to a subquotient of the $G_K$-representation

$$H^i_{\mathrm{\acute et}}\bigl(X_{\bar K},\overline{\mathbb Q}_p\bigr)(r).$$

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

The converse is a theorem. Étale cohomology of a smooth projective variety is unramified outside the primes of bad reduction and $p$, and it is de Rham at $p$ by the comparison theorems of Faltings and Tsuji. A solution must prove the conjecture for every number field $K$, prime $p$ and dimension $n$. Fontaine and Mazur also state consequences for specific cases. For example, an irreducible representation unramified at all places above $p$ and at almost all other places should have finite image [1].

Background

Fontaine and Mazur formulated the conjecture in 1993 [1]. It gives a purely local criterion, at the primes above $p$, for a global $p$-adic Galois representation to be motivic. The theory of Fontaine provides the local condition: being de Rham is the $p$-adic analogue of the Hodge decomposition.

The conjecture connects with the Langlands program. For $K=\mathbb Q$ and $n=2$, Fontaine and Mazur conjectured that an irreducible odd geometric representation with distinct Hodge–Tate weights is, up to a twist, the representation attached to a classical cuspidal eigenform. Such representations come from algebraic geometry by work of Eichler, Shimura, Deligne and Scholl. In general, combined with the Langlands correspondence, the conjecture predicts that geometric representations correspond to algebraic automorphic representations. The case $n=1$ is known by class field theory and the theory of locally algebraic abelian representations of Serre.