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
20 events
| when toggle format | what | by | license | comment | |
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| S Jan 12, 2015 at 5:39 | history | bounty ended | joro | ||
| S Jan 12, 2015 at 5:39 | history | notice removed | joro | ||
| Jan 12, 2015 at 5:39 | vote | accept | joro | ||
| Jan 10, 2015 at 6:09 | answer | added | Noam D. Elkies | timeline score: 10 | |
| Jan 8, 2015 at 12:13 | answer | added | joro | timeline score: 3 | |
| S Jan 5, 2015 at 12:58 | history | bounty started | joro | ||
| S Jan 5, 2015 at 12:58 | history | notice added | joro | Authoritative reference needed | |
| Jan 5, 2015 at 12:57 | history | edited | joro | CC BY-SA 3.0 |
Partial answer, asked about arbitrary large powers
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| Jan 3, 2015 at 14:12 | history | edited | joro | CC BY-SA 3.0 |
added 79 characters in body
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| Jan 3, 2015 at 13:59 | history | edited | joro | CC BY-SA 3.0 |
tried to clarify
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| Jan 3, 2015 at 13:53 | comment | added | Todd Trimble | The problem statement should be clarified (polarized elliptic curves carry abelian group structures, and that's what I thought "squares" referred to at first before reading the comments). There are two votes to close as "unclear what you're asking". I assume the following is meant? 1. Does there exist an irreducible elliptic curve $f(x, y) = 0$ over the rationals with infinitely many rational points $(x, y)$ of the form $(u^2, v^2)$? 2. Does there exist such an $f$ such that for all $N$ there exist $k, m \geq N$ and a point $(x, y) = (u^k, v^m)$ on the curve? | |
| Jan 3, 2015 at 13:47 | comment | added | joro | @MichaelStoll $f(x^k,y^m)$ might be reducible, keeping the genus. | |
| Jan 3, 2015 at 13:30 | comment | added | joro | @MichaelStoll Thank you. I don't require Weierstrass model in the question. | |
| Jan 3, 2015 at 13:29 | comment | added | Michael Stoll | Usually the cover will be ramified, so there will be only finitely many solutions. For example, if you have a Weierstrass model $y^2 = x^3 + ax + b$, then $y^2 = x^6 + ax^2 + b$ defines a curve of genus 2 unless $b = 0$. | |
| Jan 3, 2015 at 13:20 | comment | added | joro | @FelipeVoloch If you say so. The problem is I don't know $f$ and there are many of them. | |
| Jan 3, 2015 at 13:19 | review | Close votes | |||
| Jan 5, 2015 at 12:58 | |||||
| Jan 3, 2015 at 13:15 | comment | added | Felipe Voloch | You are essentially looking for rational points on a curve $f(x^2,y)=0$ or $f(x^2,y^2)=0$ depending on your condition. So you have to work out if the curve has genus 1 or > 1, which depends on whether the obvious cover is unramified or not. This is not an MO question. | |
| Jan 3, 2015 at 13:06 | comment | added | joro | @abx I mean squares of rationals $x=u^2,y=v^2$. I don't ask about rational function at all. | |
| Jan 3, 2015 at 13:04 | comment | added | abx | Squares of what? $x$ is a rational function on the curve with divisors of zeroes (or poles) of degree 3, it cannot be a square of a rational function. | |
| Jan 3, 2015 at 12:56 | history | asked | joro | CC BY-SA 3.0 |