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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 ith p-adic étale cohomology group $$V_{X,i}:=H^i_{\mathrm{\acute{e}t}}(X\times\overline{K},\mathbf{Q}_p)$$ is a finite-dimensional Qp-vector space with a continuous action by the absolute Galois group GK of K. It satisfies several important properties of which the following are relevant to the Fontaine–Mazur conjecture: These properties are then both true for an GK-subquotient of VX,i.
 * 1) Let v be a finite place of K not dividing p at which X has good reduction, then VX,i is unramified at v. Since such an X is unramified at all but finitely many places, this is true of VX,i.
 * 2) Let v be a finite place of K above p, then VX,i is de Rham at v.

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


 * Definition: Let &rho;: GK → GL(n, Qp) be a continuous, irreducible representation. Then &rho; is called geometric if
 * &rho; is unramified at all but finitely many places of K;
 * &rho; 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 &rho;: GK → GL(n, Qp) is a continuous, irreducible, geometric representation, then &rho; comes from geometry.

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