Timeline for Irreducible global Galois representation with weights 0, 1, 3?
Current License: CC BY-SA 4.0
11 events
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| Feb 3, 2020 at 22:25 | comment | added | SashaP | Yes, you're completely right, I erroneously assumed that the Weil restrictions of these pieces split over $\mathbb{Q}$. So are you saying that in general even if an imaginary field is needed to split off the relevant motive, for some reason the symmetry will continue to hold on each piece? | |
| Feb 3, 2020 at 22:13 | comment | added | David Loeffler | I don't think I agree with your analysis of this CM elliptic curve example. It splits into 2 motives with $K$-coeffs, each of which has weight 1 and Hodge types \{ (0, 1), (1, 0) \}$. Both summands split further when you restrict them to motives over $K$ with K-coefficients but that's a different thing. | |
| Feb 3, 2020 at 20:41 | comment | added | SashaP | Could you elaborate on your motivic argument? I don't quite see how to deduce the symmetry of weights because we might be given an object of the category of motives over $\mathbb{Q}$ with coefficients in a field $F$ that has no real embeddings, so we are not getting a real Hodge structure out of it. For instance, if $K$ is a quadratic imaginary field of class number $1$ and $E$ is an elliptic curve over $K$ with CM by $K$ then the motive $h^1(Res_{K/\mathbb{Q}}E)$ splits into 4 motives of weights $0,0,1,1$ in the category of motives with coefficients in $K$. | |
| Feb 3, 2020 at 16:29 | comment | added | user145520 | then do you know historically why is it attributed to these two guys? Did they state in a talk? | |
| Feb 2, 2020 at 10:35 | history | edited | David Loeffler | CC BY-SA 4.0 |
fixed formatting
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| Feb 2, 2020 at 10:29 | comment | added | David Loeffler | Fontaine and Mazur only stated the automorphic conjecture for $n = 2$. For the general version, see Conjecture 1.2.1 (4) of Patrikis' monograph "Variations on a theorem of Tate" (AMS Memoirs, 2019) -- you can find a pdf of it on Patrikis' website math.utah.edu/~patrikis/variationsrevision.pdf. | |
| Feb 2, 2020 at 1:30 | history | bounty ended | CommunityBot | ||
| Feb 2, 2020 at 1:30 | vote | accept | CommunityBot | ||
| Feb 2, 2020 at 1:29 | comment | added | user145520 | thank you for your incredible answer! would you happen to have a precise reference (book or article, page number) for the automorphic Fontaine-Mazur conjecture? I have trouble locating it in the literature. | |
| Feb 1, 2020 at 13:10 | history | edited | David Loeffler | CC BY-SA 4.0 |
added 13 characters in body
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| Feb 1, 2020 at 12:10 | history | answered | David Loeffler | CC BY-SA 4.0 |