Timeline for Variety acquiring rational point over any quadratic extension
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 7, 2015 at 4:16 | comment | added | Will Sawin | @MichaelStoll - Good point! So let me take $g \geq 3$. | |
| Mar 6, 2015 at 22:18 | comment | added | Michael Stoll | @JoeSilverman OK, I see. Thanks for the references! | |
| Mar 6, 2015 at 21:24 | comment | added | Joe Silverman | @MichaelStoll Yes. The proof (which isn't completely straightforward, but not hard) is in "Bielliptic curves and symmetric products." Proc. Amer. Math. Soc. 112 (1991). It was greatly generalized by Abramovich and Harris in "Abelian varieties and curves in $W_d(C)$." Compositio Math. 78 (1991). | |
| Mar 6, 2015 at 20:46 | comment | added | Michael Stoll | For curves of genus 3 or larger, you can consider the image of the symmetric square of the curve inside the Jacobian and apply Faltings' result to it. If the curve is not a doule cover of an elliptic curve of positive rank (this would give you a copy of the elliptic curve inside the symmetric square), then this tells you that there are only finitely many points on the symmetric square outside the "anti-diagonal", which means only finitely many quadratic points with irrational $x$-coordinate. | |
| Mar 6, 2015 at 20:42 | comment | added | Michael Stoll | Your first claim is false in general for curves of genus 2: if the Mordell-Weil group is infinite, then its elements will give you infinitely many quadratic points with irrational $x$-coordinate. (Let $P$ be a rational point on the Jacobian, then $P = [P_1 + P_2] - W$ where $P_1$ and $P_2$ are either two rational points or two conjugate quadratic points and $W$ is the canonical class. There are only finitely many rational points, so the second possibility occurs infinitely often, and for $P \neq 0$, $x(P_1) \notin \mathbb Q$.) | |
| Mar 6, 2015 at 20:28 | history | answered | Will Sawin | CC BY-SA 3.0 |