Timeline for Order of torsion group
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 1, 2015 at 4:46 | comment | added | Vesselin Dimitrov | @BobbyGrizzard: What you say about $\mathbb{G}_m$ remains true for the Neron-Tate height on an elliptic curve. This in particular implies the finiteness statement here. But what I'd first written here (using heights) was something simpler and much cruder. You can look at my initial answer by clicking on the edits; I believe this is also what René had in mind. | |
| May 1, 2015 at 4:29 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
added 52 characters in body
|
| May 1, 2015 at 4:27 | comment | added | Vesselin Dimitrov | @BobbyGrizzard: Indeed, it has to be the chapter on the formal group. Will edit to correct. | |
| May 1, 2015 at 4:27 | comment | added | Bobby Grizzard | @René when you say "the argument with heights," I think of the argument going back to Bombieri & Zannier that, in a Galois extension of $\mathbb{Q}$ which sits in a finite extension of some $\mathbb{Q}_p$, the lim inf of the logarithmic height values (on $\mathbb{G_m}$) is positive. Is this what you mean? | |
| May 1, 2015 at 4:21 | comment | added | Bobby Grizzard | @vesselinDimitrov by the way, do you possibly mean a different Chapter (IV?) of Silverman? | |
| May 1, 2015 at 4:07 | comment | added | Vesselin Dimitrov | @BobbyGrizzard: Yes, the torsion group is finite for every elliptic curve (or abelian variety) over $\mathbb{Q}^{(d)}$. However, its size is not uniformly bounded unless the elliptic curve is defined over $\mathbb{Q}$ (or over a number field of a bounded global degree), in which case, as you note, Mazur's theorem applies to give a uniform bound. | |
| May 1, 2015 at 4:02 | comment | added | Bobby Grizzard | Nice! I did state in my "answer" that I assumed the curve was defined over $\mathbb{Q}$, and in fact as you'll see from my comment on the accepted answer, I assumed (based on the fact that that answer was accepted) that OP had intended that. If OP wanted to know what torsion could be defined over $\mathbb{Q}^{(2)}$ for an elliptic curve which is just know to be defined over that field, I don't think we know the answer, although your argument apparently shows the group is still finite in this case. | |
| Apr 30, 2015 at 22:46 | comment | added | R.P. | Yes, I also knew the argument with heights, but I had never seen this one before. Very nice! | |
| Apr 30, 2015 at 22:43 | comment | added | Vesselin Dimitrov | @René: Thank you, indeed, I negleced to give this reference in addition to Theorem II 6.4. Another proof is also possible using height functions, and the same observation that $K^{(2)}$ has bounded $p$-adic degrees. I outlined it in my initial answer to this question, but I then replaced it by this simpler local argument. | |
| Apr 30, 2015 at 22:24 | history | edited | R.P. | CC BY-SA 3.0 |
fixed reference to Silverman
|
| Apr 30, 2015 at 22:22 | comment | added | R.P. | One also needs the fact that the kernel of reduction $A_1(F)$ is of finite index in $A(F)$. (Additionally, for small $p$, one may need to consider kernels of reduction modulo higher powers of $p$ in order to be in a position to deploy the theory of formal groups, but these are readily seen to be of finite index in $A_1(F)$.) This follows essentially from local compactness of $F$; this is the approach suggested in Chapter VII, Exercise 7.6 in Silverman's book. | |
| Apr 30, 2015 at 22:15 | history | edited | R.P. | CC BY-SA 3.0 |
added 12 characters in body
|
| Apr 30, 2015 at 22:12 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
deleted 1526 characters in body
|
| Apr 30, 2015 at 21:57 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
added 280 characters in body
|
| Apr 30, 2015 at 21:42 | history | answered | Vesselin Dimitrov | CC BY-SA 3.0 |