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 varieties. arXiv preprint Some cases of this conjecture in dimension 2 have been proved by .
The first conjecture stated by Fontaine and Mazur assumes that 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 . It claims that in this case, is associated to a cuspidal newform if and only if is potentially semi-stable at .
References
External links Robert Coleman's lectures on the Fontaine–Mazur conjecture