Timeline for Tate Conjecture on decomposition of motives(?)
Current License: CC BY-SA 3.0
10 events
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Dec 21, 2012 at 4:39 | comment | added | Emerton | 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, | |
Dec 21, 2012 at 4:39 | comment | added | rntb | What I am saying is that Illusie's paper deals with a fixed prime. But here in Taylor's statement, there is the problem of independence of primes, right? | |
Dec 21, 2012 at 4:38 | comment | added | Emerton | 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. | |
Dec 21, 2012 at 4:37 | comment | added | rntb | @Prof. Emerton. But are the HT weights independent of ℓ? | |
Dec 21, 2012 at 4:35 | comment | added | Emerton | This gives (a) of part 3. And then part (b) just follows from the fact that these jumps are the same when we basechange from $\overline{\mathbb Q}$ to $\mathbb C$ (for any embedding). And yes, in the $i = 1$ case, you get $g$ zeroes and $g$ ones. Regards, | |
Dec 21, 2012 at 4:34 | comment | added | Emerton | 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 $\overline{\mathbb Q}_{\ell}$. | |
Dec 21, 2012 at 4:31 | comment | added | rntb | Dear Prof. Emerton, Thank you! So, the "common" HT weights in the H^1 case are just 0 and 1, both repeated g times? | |
Dec 21, 2012 at 4:25 | comment | added | Emerton | 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, | |
Dec 21, 2012 at 2:04 | answer | added | Joël | timeline score: 2 | |
Dec 21, 2012 at 1:37 | history | asked | rntb | CC BY-SA 3.0 |