Timeline for Curves which are not covers of P^1 with four branch points

4 events

when toggle format	what		by	license	comment
Aug 3, 2011 at 2:51	comment	added	Chandan Singh Dalawat		Çiperiani, Mirela and Wiles, Andrew, Solvable points on genus one curves. Duke Math. J. 142 (2008), no. 3, 381–464. $$ $$ This important article shows that any genus one curve $C$ over ${\bf Q}$ with a rational point over every $p$-adic completion of ${\bf Q}$ and semistable Jacobian has a point defined over a solvable extension of ${\bf Q}$. $$ $$ Reviewed by Henri Darmon
Aug 2, 2011 at 19:05	comment	added	Jason Starr		Yes, Pal works over $\mathbb{Q}(t)$, not over $\overline{\mathbb{Q}}(t)$. There are also examples over $\overline{\mathbb{Q}}(s,t)$. I think I can also see how to produce curves over $\overline{\mathbb{Q}}(s,t)$ which admit no morphism to $\mathbb{P}^1$ branched over four (rational) points. But of course that is also not the question you asked!
Aug 2, 2011 at 17:34	comment	added	JSE		But by the way, I like the question about solvable points, too, and I hasten to point out that over a global field (Q or F_p(t)) we have NO IDEA whether there are genus-g curves without solvable points. Even the genus 1 case is very hard! (It turns out there are always solvable points in that case -- Mirela Ciperiani proved this in her Ph.D. thesis.) Pal, if I remember right, always works over a "big" base field like Q(T), which gives you much more Galois-theoretic wiggle room.
Aug 2, 2011 at 17:27	history	answered	Jason Starr	CC BY-SA 3.0