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 ...
Let K be a number field. Given a smooth proper n-dimensional variety[1] X over K, its ith p-adic étale cohomology group 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 .
Abstracting the properties of Galois representations that come from geometry Fontaine and Mazur introduced the following definition:
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:
In the case of two-dimensional representations: Fontaine–Mazur–Langlands.[3]