Timeline for Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 12, 2015 at 15:23 | comment | added | joro | Just to clarify about the genus 1 factor: by "sure ..." you mean very likely all factors of q(x^k,y^m) for all $k,m$ will be of genus at most 1? | |
| Jan 12, 2015 at 9:02 | comment | added | joro | OK, I see. So far couldn't find higher genus in the factorization. (For me in sage computing the genus is slower than the factorization). | |
| Jan 12, 2015 at 5:39 | history | bounty ended | joro | ||
| Jan 12, 2015 at 5:39 | vote | accept | joro | ||
| Jan 11, 2015 at 15:33 | comment | added | Noam D. Elkies | (About the older comment: sure, but at some point it becomes infeasible to factor bivariate polynomials whose degree grows with $km$ while it's still possible to compute resultants of degrees $k$ and $m$.) | |
| Jan 11, 2015 at 15:30 | comment | added | Noam D. Elkies | That's covered by the $(u',v')$ test I gave. When $m$ is even you can't take $u,v$ to be the $X,Y$ of the classical Weierstrass equation because then the map $(u,v) \mapsto (u^k,v^m)$ is not generically $1:1$ (try $(u',v') = (u,-v)$). But $(X,Y+1)$ works. (Also for $Y^2=X^3+2$ if $3 \mid k$ you must tweak $u$ to avoid triplication on the $X$ side.) | |
| Jan 11, 2015 at 8:49 | comment | added | joro | By the way, for $k=m=2$ the resultant is square and is genus 0. | |
| Jan 11, 2015 at 7:08 | comment | added | joro | This indeed works. Better proof than checking is to factor q(x^11,y^7) and see it is divisible by the curve. Is it coincidence that in the factorization the degree 18 factor is genus 1 too? | |
| Jan 10, 2015 at 16:53 | comment | added | Noam D. Elkies | You're welcome. I think the resultant is right. I just edited to add the gp code. I also checked that a few multiples of $(-1,1)$ yield rational multiples whose coordinates are the expected $11$th and $7$th powers. | |
| Jan 10, 2015 at 16:52 | history | edited | Noam D. Elkies | CC BY-SA 3.0 |
add requirement that u,v have no zero or pole at the test point, and a line of gp code; fix typos;
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| Jan 10, 2015 at 7:06 | comment | added | joro | Is typo in the resultants possible? You don't appear to use $F$, might be wrong. | |
| Jan 10, 2015 at 6:09 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |