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Fontaine–Mazur conjecture

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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 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

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  1. ^ Koch, Helmut (2013). "Fontaine-Mazur Conjecture". Galois theory of p-extensions. Springer Science & Business Media. p. 180. ISBN 9783662049679.
  2. ^ Calegari, Frank (2011). "Even Galois representations and the Fontaine–Mazur conjecture" (PDF). Inventiones Mathematicae. 185 (1): 1–16. arXiv:1012.4819. Bibcode:2011InMat.185....1C. doi:10.1007/s00222-010-0297-0. S2CID 8937648. arXiv preprint
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