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I apologize for the title. I myself conined it...

I am referring to Conjecture 1.2 (page 7) of Richard Taylor's paper Galois representations (Annales de la Faculte des Sciences de Toulouse 13 (2004) ) The pdf of the paper is here:

http://www.math.ias.edu/~rtaylor/longicm02.pdf

My question is:

What is the reference nowadays for this conjecture? What has been proved so far? I am particularly interested about part 3, i.e., the existence of a "common" set of Hodge-Tate weights.

And of course, I really would like to know the correct name of this conjecture...

Thanks!

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Dear rntb, The $H^1$ of any smooth projective variety coincides (up to twist) with the $\ell$-adic Tate module of its Albanese, and so the conjecture for $H^1$ reduces to the case of abelian varieties, and amounts to showing that the Tate module of a simple abelian variety is an irreducible Galois rep'n, which follows from Faltings's proof of the "Tate conjecture" (that endomorphisms of ab. varieties can be detected on the level of Tate modules) in this case. Regards, –  Emerton Dec 21 '12 at 4:25
    
Dear Prof. Emerton, Thank you! So, the "common" HT weights in the H^1 case are just 0 and 1, both repeated g times? –  rntb Dec 21 '12 at 4:31
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Dear rntb, Regarding Illusie's paper, you seem to misunderstand the meaning of statement 3 for the whole of $H^i$. What the Illusie reference will show is that (by the $\ell$-adic comparison theorem) the HT weights of the $\ell$-adic comparison at any prime of $\overline{\mathbb Q}_{\ell}$ over $\ell$ are given by the jumps in the Hodge filtration of $H^i_{dR}(X_{/\overline{\mathbb Q}}_{\ell})$; but these are the same as the jumps in the Hodge filtration for $H^i_{dR}(X)$ istelf, and so are independent of the choice of embedding of $\overline{\mathbb Q}$ into $\overline{\mathbb Q}_{\ell}$. –  Emerton Dec 21 '12 at 4:34
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P.S. For an example of coming to grips with the relationship between HT wts. at $\ell$ and HT wts. at infinity in a context which is not an $H^i$, you could look at Calegari's recent papers on even two-dimensional Galois reps. (The context is not quite that of Tate's conjecture, but rather the Fontaine--Mazur conjecture, but there is a fairly tight relationship.) Also, before considering the general conjecture, one could ask whether e.g. the cohomology of Shimura varieties gives semisimple Galois reps. This is already a tricky question, on which people have worked and are working. –  Emerton Dec 21 '12 at 4:38
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Dear rntb, As I wrote, the HT wts. for $H^i$ are intrinsically determined by the Hodge filtratino on $H^i_{dR}(X)$, and so are independent of $\ell$ and $\mathbb C$. Regards, –  Emerton Dec 21 '12 at 4:39
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1 Answer

Dear mtb, I don't think that anything (at least that anything major) has been proved about this conjecture since Taylor wrote this paper about ten years ago. So what is known is essentially what Taylor says, that is the case of the whole $H^i$, and the components of $H^0$ and $H^1$.

Note also that one cannot separates the part 3 of the conjecture from the rest, especially from part 1.

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Dear Joël, thanks for the answer. But I am still confused about the references. In Taylor's paper, he mentioned a paper by Illusie to treat the whole H^i. And a paper of Faltings to treat H^1. Do you know any other reference in English or French telling the story in Faltings' paper? Thank you again! –  rntb Dec 21 '12 at 2:46
    
Also, it seems to me that the (review) paper of Illusie only considers schemes over local fields, but the statement of the Conjecture in Taylor's paper is about schemes over Q... –  rntb Dec 21 '12 at 2:54
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The Faltings' paper was translated into English, see "Arithmetic Geometry" by Cornell and Silverman, Springer-Verlag, 1984. See also "Rational Points" by Faltings, Wuestholz et al. (Vieweg, several rditions), Proceedings of Szpiro S\'eminaire (Ast\'erisque, 127 (1985) and my survey paper with Parshin, arXiv:0912.4325. –  Yuri Zarhin Dec 21 '12 at 3:26
    
@Prof. Zarhin, Thank you for pointing out the references. But do you know how the results of Faltings imply the Conjecture in Taylor's paper? Thanks again! –  rntb Dec 21 '12 at 3:45
    
@mtb, You are welcome. So, we deal with $H^1$ and replacing $X$ by its Albanese variety, assume that $X$ is an abelian variety of positive dimension over the field $Q$ of rational numbers. Let me restrict myself to Part 1 of Conjecture 1.2. If $End_{Q}X$ is the ring of integers $Z$ then by Faltings' theorem $H^1(X(C),Q_{\ell}$ is an absolutely irreducible representation of the absolute Galois group $G_Q$ of $Q$ and we put $M_1=H^1(X(C),\bar{Q})$. Clearly, tensoring $M_1$ by $\bar{Q}_{\ell}}$, we obtain an irreducible representation $H^1(X(C),\bar{Q}_{\ell}$ of $G_Q$. –  Yuri Zarhin Dec 22 '12 at 0:56
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