Timeline for Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Sep 22, 2010 at 14:43 | comment | added | stankewicz | Here are several more examples: warwick.ac.uk/staff/J.E.Cremona/ecegr/ecegrqf.html | |
| Sep 22, 2010 at 9:15 | comment | added | Kevin Buzzard | As for the Shimura variety you mention, from what you write I think you are talking about the Hilbert modular surface attached to the data: these are built over the complexes precisely by acting on two copies of the upper half plane rather than one and have canonical models over $\mathbf{Q}$. I believe that one can check that the cohomology of $E$ does not show up in the cohomology of these surfaces. The interesting cohomology of the surface will be in the middle dimension, and the weights in the middle dimension exclude the possibility that the curve can show up there. | |
| Sep 22, 2010 at 9:13 | comment | added | Kevin Buzzard | Hey William. In fact there are lots of conductor 1 examples in the literature. The earliest one I know about is $y^2+xy+e^2y=x^3$ over $\mathbf{Q}(\sqrt{29})$ discovered by Tate, with $e=(5+\sqrt{29})/2$; this is mentioned in Serre's 1972 Inventiones article on the image of Galois in the Tate module. Richard Pinch's thesis has a bunch in, IIRC, and one of these was proved to be modular by an explicit computation, by Socrates and Whitehouse, in 2004. | |
| Sep 22, 2010 at 0:00 | comment | added | James Weigandt | P.S.: I would upvote William, but he's currently at 389 reputation, and I have the feeling he wants to stay there. | |
| Sep 21, 2010 at 23:59 | comment | added | James Weigandt | @James Borger: A Weierstrass model $y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ for an elliptic curve is written $[a_1, a_2, a_3, a_4, a_6]$. | |
| Sep 21, 2010 at 23:23 | comment | added | JBorger | Hi William. This is a nice example to have here, but unfortunately I and probably many people don't know what the bracket notation means. I guess they're the coefficients of a Weierstrass equation, but in which order? | |
| Sep 21, 2010 at 19:18 | history | answered | William Stein | CC BY-SA 2.5 |