Fontaine–Mazur conjecture

In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by about when p-adic representations of Galois groups of number fields can be constructed from representations on étale cohomology groups of a varieties. Some cases of this conjecture in dimension 2 have been proved by.

The first conjecture stated by Fontaine and Mazur assumes that $$\rho \colon \mathrm{Gal}(\overline{\mathbb{Q}}|\mathbb{Q}) \to \mathrm{GL}(\overline{\mathbb{Q}}_p)$$ is an irreducible representation that is unramified except at a finite number of primes and which is not the Tate twist of an even representation that factors through a finite quotient group of $$\mathrm{Gal}(\overline{\mathbb{Q}}|\mathbb{Q})$$. It claims that in this case, $$\rho$$ is associated to a cuspidal newform if and only if $$\rho$$ is potentially semi-stable at $$p$$.