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In number theory, the Fontaine–Mazur conjecture provides a conjectural characterization of those p-adic Galois representations of number fields which "come from geometry". It is named after ...

Representations coming from geometry

Let K be a number field. Given a smooth proper n-dimensional variety X over K, its i^th p-adic étale cohomology group $V_{X,i}:=H_{\mathrm {{\acute {e}}t} }^{i}(X\times {\overline {K}},\mathbf {Q} _{p})$ is a finite-dimensional Q_p-vector space with a continuous action by the absolute Galois group G_K of K. It satisfies several important properties of which the following are relevant to the Fontaine–Mazur conjecture:

1. Let v be a finite place of K not dividing p at which X has good reduction, then V_X,i is unramified at v. Since such an X is unramified at all but finitely many places, this is true of V_X,i .
2. Let v be a finite place of K above p, then V_X,i is de Rham at v.

These properties are then both true for an G_K-subquotient of V_X,i .

Geometric Galois representations

Abstracting the properties of Galois representations that come from geometry Fontaine and Mazur introduced the following definition:

Definition: Let ρ: G_K → GL(n, Q_p) be a continuous, irreducible representation. Then ρ is called geometric if

1. ρ is unramified at all but finitely many places of K;
2. ρ is de Rham at places above p.

The conjecture and partial results

Fontaine and Mazur were then lead to conjecture that the two conditions imposed in the definition of a geometric Galois representation in fact characterize the collection of Galois representations coming from geometry. Specifically:

Fontaine–Mazur conjecture: If ρ: G_K → GL(n, Q_p) is a continuous, irreducible, geometric representation, then ρ comes from geometry.

In the case of two-dimensional representations: Fontaine–Mazur–Langlands.