Timeline for Extra automorphisms of curves and definability over \bar Q
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Apr 29, 2010 at 4:54 | comment | added | Tom Church | For those not familiar with Hurwitz's theorem, it says that the 84(g-1) here is the maximal number of automorphisms of a curve of genus g. This bound can be pushed down to 12(g-1) using the fact that the (2,2,2,3) orbifold, with area pi/3, has the smallest area of any non-triangular hyperbolic orbifold. Thus any curve with more than 15% of the possible number of automorphisms is defined over a number field. | |
| Apr 29, 2010 at 4:52 | comment | added | naf | The point of Belyi's theorem is that a curve defined over a number field is a cover of P^1 ramified over only three points. The converse is essentially trivial (assuming elementary algebraic geometry). | |
| Apr 29, 2010 at 4:45 | comment | added | Jonah Sinick | Thanks! I'm still curious about other questions connected with my original one. (For example, does the presence of any extra automorphisms whatsoever imply that the curve is defined over a number field?) | |
| Apr 29, 2010 at 4:43 | vote | accept | Jonah Sinick | ||
| Apr 29, 2010 at 3:52 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |