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

• Galois theory
• Number theory
• Conjectures
• Abstract algebra stubs

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