Fontaine–Mazur conjecture

In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur (1995) about when p-adic representations of Galois groups of number fields can be constructed from representations on étale cohomology groups of varieties.^[1][2] Some cases of this conjecture in dimension 2 have been proved by Dieulefait (2004).

The first conjecture stated by Fontaine and Mazur assumes that ${\displaystyle \rho \colon \mathrm {Gal} ({\overline {\mathbb {Q} }}|\mathbb {Q} )\to \mathrm {GL} ({\overline {\mathbb {Q} }}_{p})}$ 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 ${\displaystyle \mathrm {Gal} ({\overline {\mathbb {Q} }}|\mathbb {Q} )}$. It claims that in this case, ${\displaystyle \rho }$ is associated to a cuspidal newform if and only if ${\displaystyle \rho }$ is potentially semi-stable at ${\displaystyle p}$.