Timeline for The Sato-Tate conjecture for hypersurfaces?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2014 at 20:00 | vote | accept | Qiaochu Yuan | ||
| Dec 17, 2013 at 21:37 | comment | added | ACL | A good, recent, reference is Serre's new book $N_X(p)$, math.cts.nthu.edu.tw/Mathematics/… | |
| Dec 17, 2013 at 21:30 | answer | added | Will Sawin | timeline score: 6 | |
| Dec 17, 2013 at 11:27 | comment | added | naf | As I said before, if you know a reference for arbitrary abelian varieties it should not be difficult to work out the statement for arbitrary varieties (except that some things that are known for abelian varieties are still conjectural for general varieties, e.g., 13.4? in Serre's article.) | |
| Dec 17, 2013 at 11:16 | comment | added | naf | Sorry, I didn't have the article with me when I wrote the comment. The precise statement is 13.5?; however, you will probably have to read some of the earlier sections for this to mean anything. | |
| Dec 16, 2013 at 23:35 | comment | added | Qiaochu Yuan | @ulrich: unfortunately my mathematical French is quite poor. Can you at least indicate which statement in that section I should be looking at? | |
| Dec 16, 2013 at 4:26 | comment | added | naf | See, for example, the last section of the article by Serre in "Motives", PSMP 55, for a formulation for arbitrary motives. | |
| Dec 15, 2013 at 22:08 | comment | added | Qiaochu Yuan | @eric: yes, it's called the Sato-Tate group in the literature. But again I've only seen discussion of curves and abelian varieties. | |
| Dec 15, 2013 at 21:25 | comment | added | eric | As you probably know, even for elliptic curves the form of the conjecture depends on whether you have CM or not. I guess in general there will be some sort of Mumford-Tate group (that's where things like the cup product will come in) and the conjecture should say that Frobenii are equidistributed within the real points of that group. | |
| Dec 15, 2013 at 20:06 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Dec 15, 2013 at 20:04 | comment | added | Qiaochu Yuan | @ulrich: that's certainly plausible. Do you know a reference where a conjecture of this form is asserted? I was only able to find conjectures about abelian varieties and curves. What's expected for more general varieties? | |
| Dec 15, 2013 at 5:46 | comment | added | naf | There is no significant difference between higher dimensional abelian varieties and hypersurfaces: replace $H^1$ of the abelian variety with the middle dimensional cohomology of the hypersurface. | |
| Dec 15, 2013 at 0:02 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |