Timeline for On $a+b+c= abc = n$, elliptic curves, and solvable Galois groups
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Aug 12, 2015 at 14:49 | comment | added | Tito Piezas III | @NoamD.Elkies: Oh, I think I understand. I believe I just used $m=3$ in the group law in my post, but using $m=5$ will be a different matter. So I guess the answer to my second question is indeed "yes", as long as I stick to $m\leq4$. | |
| Aug 12, 2015 at 14:37 | comment | added | Noam D. Elkies | Sorry, I meant multiplication by 5 in the group law, as I see that Felipe Voloch already suggested. (And you must have meant g and h to have a non-solvable group.) | |
| Aug 12, 2015 at 13:20 | comment | added | Noam D. Elkies | Take a random elliptic curve and use multiplication by 5. | |
| Aug 12, 2015 at 13:02 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
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| Aug 12, 2015 at 12:58 | comment | added | Tito Piezas III | @FelipeVoloch: Thanks. Would you have at hand an example of an explicit elliptic curve and a transformation with $h(x)$ having a non-solvable group? (P.S. I've also added a question 2. Maybe the answer to the second is a "yes".) | |
| Aug 12, 2015 at 12:54 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Added question 2.
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| Aug 12, 2015 at 12:14 | comment | added | Felipe Voloch | Not in general. Your $f(x)$ is the $x$-coordinate of an endomorphism of the elliptic curve and $h$, for instance, has roots giving the $x$-coordinates of the kernel of the endomorphism and the Galois group is a subgroup of the automorphism group of said kernel. Sometimes this is solvable but if the endomorphism is multiplication by $m$ for some large $m$, it won't be. | |
| Aug 12, 2015 at 4:59 | history | asked | Tito Piezas III | CC BY-SA 3.0 |