Timeline for Does the Mordell conjecture imply the Shafarevich conjecture
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 15, 2012 at 21:18 | comment | added | Felipe Voloch | @Harized. Embed the curve in its Jacobian and get covers using multiplication by $n$ on the Jacobian. But there's a lot more. The computation of the fundamental group of a (topological) surface of genus g is a standard exercise in topology. | |
| Mar 15, 2012 at 20:48 | comment | added | Harized | Is $\mathbf{P}^1$ the only simply connected curve? I know elliptic curves have nontrivial etale covers (multiplication by $n$), but why is the etale fundamental group of a hyperbolic curve nontrivial? | |
| Mar 15, 2012 at 18:44 | comment | added | M P | The moduli space of curves of genus $g$ is not a Weil restriction of a curve, at least when it is non-rational (e.g. for $g \geq 24$). Indeed, if it were, then it would be the Weil restriction of a curve of genus zero, since it is simply connected: geometrically, a Weil restriction is a product. But then it would be rational. | |
| Mar 15, 2012 at 16:30 | comment | added | Felipe Voloch | I very much doubt that the moduli space is the Weil restriction of a curve, but I don't see immediately how to prove that. | |
| Mar 15, 2012 at 16:18 | comment | added | Bobby | Although it's not difficult to see, I have to admit that I didn't know of this interpretation. As a naive idea, couldn't one try to use that the moduli space is the Weil restriction of some curve over a suitable number field to obtain the finiteness of the set of S-integral points of the moduli space? | |
| Mar 15, 2012 at 15:56 | history | answered | Felipe Voloch | CC BY-SA 3.0 |