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For a $p$-adic local field K, the Fontaine-Mazur conjecture characterises the $p$-adic representations of the Galois group of $K$ which arise from geometry. Is there a conjectural generalization to higher local fields? or to the $p$-adic representations of the etale fundamental group of a variety defined over $K$?


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The Fontaine-Mazur conjecture is a global statement---well, I always thought it was. Can you be more precise about what you mean by the conjecture? – Kevin Buzzard Apr 8 '11 at 19:54
Not three weeks ago, I heard Fontaine said that he didn't know how to formulate the local Fontaine-Mazur conjecture (that is , he didn't know how to characterize local representations coming from varieties over local fields). – Olivier Apr 9 '11 at 5:53
@Olivier: right! That sounds like a really hard question. The problem is that a local representation coming from geometry has to satisfy all sorts of "arithmeticity" statements---for example if you have a variety with good reduction then the eigenvalues of Frobenius on the cohomology will all be algebraic numbers whose norms under embeddings into the complexes all have to satisfy various size constraints because of the Weil conjectures. In the bad reduction case there is still a similar, but more complicated, story. The miracle about F-M is that in the global setting... – Kevin Buzzard Apr 9 '11 at 6:30
...these constraints are conjecturally forced upon you by a natural-looking local assumption ("de Rham") plus a natural global assumption ("unramified outside a finite set of primes"). [The reason these conjectures don't appear earlier in the folklore is that it took until the 80s for complicated notions like de Rham to be formulated] – Kevin Buzzard Apr 9 '11 at 6:34
Many thanks for all the very informative comments. @Buzzard: You are right that the Fontaine-Mazur conjecture is global So the question as stated does not make any sense. @Olivier: thanks for generously answering the question that should have been asked! Thanks! – SGP Apr 9 '11 at 11:02

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