Timeline for Concrete Examples of Shimura Surfaces
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 27, 2014 at 20:06 | vote | accept | Lennart Meier | ||
| Dec 4, 2014 at 7:53 | comment | added | eric | I've just noticed that you're demanding that the deformation theory is controlled by a 1-dimensional p-divisible group in your question -- so you can rule out e.g. Siegel modular varieties (in general) -- you want to look at the unitary Shimura varieties used to great effect by people like Harris and Taylor in their orange book. These are higher-dimensional but their deformation theory is still controlled by a 1-dimensional p-divisible group (and sometimes they're even compact, so life is wonderful). | |
| Dec 4, 2014 at 7:50 | comment | added | eric | In my mind, one key difference in the arithmetic when you leave the world of Shimura curves defined as above, is that when considering higher-dimensional Shimura varieties parametrising abelian varieties plus (blah), if the associated formal groups in characteristic p aren't one-dimensional then it's much harder to give a well-behaved notion of a level structure at p -- Drinfelds trick as explained in Katz-Mazur in the elliptic curve case does not work. A key difference in the geometry is compactification -- the story is far more complicated when one leaves the world of curves. | |
| Dec 4, 2014 at 0:45 | answer | added | Tyler Lawson | timeline score: 20 | |
| Dec 3, 2014 at 21:45 | comment | added | anon | Most of what you ask for (except the equations) holds for all Shimura varieties of PEL-type, of which there are many examples. See the comprehensive book Arithmetic Compactifications of PEL-Type Shimura Varieties Kai-Wen Lan. You haven't needed adeles because you have only looked connected Shimura varieties. | |
| Dec 3, 2014 at 21:14 | comment | added | Dylan Wilson | en.wikipedia.org/wiki/Picard_modular_surface , en.wikipedia.org/wiki/Hilbert_modular_surface | |
| Dec 3, 2014 at 20:49 | history | asked | Lennart Meier | CC BY-SA 3.0 |