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 a 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 ${\displaystyle p}$.

• Fontaine, Jean-Marc; Mazur, Barry (1995), "Geometric Galois representations", in Coates, John; Yau., S.-T. (eds.), Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Series in Number Theory, vol. 1, Int. Press, Cambridge, MA, pp. 41–78, ISBN  978-1-57146-026-4, MR  1363495
• Dieulefait, Luis V. (2004). "Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2004 (577). arXiv: math/0304433. Bibcode: 2003math......4433D. doi: 10.1515/crll.2004.2004.577.147. S2CID  16949796.

• Robert Coleman's lectures on the Fontaine–Mazur conjecture

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