Timeline for Mordell-Weil rank of an elliptic curve over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$?
Current License: CC BY-SA 3.0
5 events
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| Jul 13, 2016 at 15:18 | comment | added | Wojowu | @TimoKeller If the set of points generated by these points of infinite order (each one in $\Bbb Q(\sqrt{d})\setminus\Bbb Q$ for distinct primes (say) $d$) were finitely generated, then all the generators would be elements of a single finite extension of $\Bbb Q$, which would imply that a finite extension contains infinitely many square roots of primes, which isn't the case. | |
| Jul 13, 2016 at 11:41 | comment | added | Chris Wuthrich | Actually, I would think that this is proven. Anayltic methods should give a large density of negative $d$ for which the twist vanishes to order $1$. Then Heegner point constructions should yield a new point of infinite order. | |
| Jul 13, 2016 at 10:50 | vote | accept | The Thin Whistler | ||
| Jul 13, 2016 at 10:50 | |||||
| Jul 13, 2016 at 10:41 | comment | added | user19475 | "and, by doing that for infinitely many $d$, you get infinite rank over your field." Can you elaborate a bit on this, please? | |
| Jul 13, 2016 at 9:15 | history | answered | Felipe Voloch | CC BY-SA 3.0 |