Fontaine–Mazur conjecture

In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Template:Harvs about when p-adic representations of Galois groups of number fields can be constructed from representations on étale cohomology groups of varieties. Some cases of this conjecture in dimension 2 have been proved by Template:Harvtxt.

The first conjecture stated by Fontaine and Mazur assumes that $\rho :\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}|\mathbb{Q})\to \mathrm{G}\mathrm{L}({\bar{\mathbb{Q}}}_p)$ 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 $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}|\mathbb{Q})$. It claims that in this case, $\rho$ is associated to a cuspidal newform if and only if $\rho$ is potentially semi-stable at $p$.

References

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External links

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

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