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For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G_K$ of $K$ that is almost everywhere unramified comes from algebraic geometry (i.e., is a subquotient of the etale cohomology of some variety over $K$, up to Tate twist). As far as I can see, the only cases where any progress has been made concerns the case that $K$ is totally real or CM.

This made me wonder: Is the Fontaine-Mazur conjecture known to be true for $1$-dimensional representations for any number field $K$? For CM fields, the theory of CM abelian varieties gives varieties whose cohomology realizes nontrivial characters (and I guess that easy variations should produce all characters). What are the geometric objects appearing for other fields?

[edit: The word 'geometric' is avoided now, see the comments.]

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I thought Fontaine and Mazur defined a $p$-adic representaion of $G_K$ to be geometric if it is almost everywhere unramified and potentially semistable (at every place $\mathfrak{p}|p$ of $K$), and conjectured that every geometric representation is ... –  Chandan Singh Dalawat Jul 25 '10 at 13:25
    
... for example modular if moreover it is odd and $K={\bf Q}$ –  Chandan Singh Dalawat Jul 25 '10 at 13:34
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The terminology used in Fontaine and Mazur's paper is that "geometric" means "almost everywhere unramified and potentially semistable at places |p" and an irreducible p-adic Galois representation is said to "come from algebraic geometry" if it is isomorphic (up to Tate twist) to a subquotient of the etale cohomology of an algebraic variety. Their conjecture is then that an irreducible p-adic rep comes from algebraic geometry if and only if it is geometric. –  jnewton Jul 25 '10 at 15:02
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Oh and I guess the proposition in section 6 of Fontaine-Mazur is relevant to the question? –  jnewton Jul 25 '10 at 15:50
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Congratulations!(about the Clay.) –  Unknown Feb 15 '11 at 0:08
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1 Answer

up vote 13 down vote accepted

Let $\chi$ be a one-dimensional geometric (in the sense of FM) $p$-adic Galois representation of $G_K$ and let $\psi$ be the Hecke character of $K$ associated to $\chi$ by class field theory. The fact that $\chi$ is de Rham (=pst) at all primes above $p$ imples that $\psi$ is an algebraic Hecke character. Generally, the only algebraic Hecke characters of $K$ are of the form $(\text{finite order})\cdot\mathcal{N}^n$ where $\mathcal{N}$ is the norm character. Under class field theory, $\mathcal{N}$ corresponds to the cyclotomic character, so it comes from geometry; additionally, any finite order character comes from geometry (it arises as the subquotient of the $H^0$ of a zero-dimensional variety). The only time there are more algebraic Hecke characters is when $K$ contains a CM field. Denoting $L$ the maximal CM field in $K$, every algebraic Hecke character of $K$ is of the form $(\text{finite order})\cdot(\psi_L\circ\mathcal{N}_{K/L})$ where $\psi_L$ is an algebraic Hecke character of $L$ and $\mathcal{N}_{K/L}$ is the norm from $K$ to $L$. Again, finite order characters come from geometry, so this case is reduced to the CM case. As you've mentioned the CM case has been dealt with, so Fontaine–Mazur is true for $\mathrm{GL}(1)$.

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Thanks for your answer! I wasn't aware that algebraic characters have this special form. –  Peter Scholze Jul 25 '10 at 20:21
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Rob, should mention refs, such as Ch. III (esp. App. A) in Serre's book on abelian $\ell$-adic repn's, for why pst (or HT) at $p$-adic places implies global algebraicity up to finite-order, with finite-order part coming from ramification away from $p$ (finite, by global CFT) and finiteness of Hilbert class fields. Main Thm of CM relates Tate mods of CM abelian var. to certain alg. Hecke characters factoring through reflex norm of a CM type, but to conversely relate all alg. Hecke characters to CM abelian varieties is totally not obvious (& there's no "modern" published ref.), right? –  BCnrd Jul 25 '10 at 20:26
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"but to conversely relate all alg. Hecke characters to CM abelian varieties is totally not obvious (& there's no "modern" published ref.), right?" $$ $$ Do you just mean Casselman's theorem as in Shimura's paper? That is, Theorem 6 of jstor.org/stable/1970768 $$ $$ Maybe section 4 of chapter 1 of Schappacher is also of interest, for it deals with motives. springer.com/mathematics/numbers/book/978-3-540-18915-2 –  Junkie Jul 26 '10 at 7:16
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Dear Rob H. & Junkie: It is kind of embarrassing for the field that with all of the recent activity around the F-M conjecture there isn't a place in the recent (= last 15 years) literature where the 1-dimensional case is specifically addressed and proved (even if the "proof" amounts to gathering appropriate earlier references and explaining why they are sufficient for the desired conclusion). Hmm, maybe I should put this in the Hawaii lecture notes (oy vey, just 2 more months till that's due to be done...). –  BCnrd Jul 27 '10 at 23:00
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Peter, see math.u-psud.fr/~fargues/Motifs_abeliens.ps section 8 where you can find the references for FM conjecture for GL_1 (in particular you have to use a transcendence result in general). –  Laurent F. Jul 29 '10 at 8:53
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