Timeline for What's known about complete split primes in Q(E[p])?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Nov 11, 2011 at 20:23 | answer | added | user12750 | timeline score: 3 | |
| Oct 21, 2011 at 3:26 | comment | added | Chandan Singh Dalawat | @Nicolas B, you are absolutely right. I was relying on my faltering memory; the equation should have been $$y^2+y=x^3-x^2,$$ which indeed has conductor $11$. Whoever sees this comment, please "upvote" it so that it is always visible. | |
| Oct 18, 2011 at 20:50 | comment | added | Nicolas B. | @Chandan Singh Dalawat: I think that there is a mistake in the equation of the elliptic curve in your first comment. The one displayed is the curve labeled 37a1 in Cremona's table, but if I compute the $a_{\ell}$'s of $E$ for $\ell=4831, 22051,\ldots$, I don't find $2\pmod{7}$. Am I mistaken? | |
| Oct 13, 2011 at 10:19 | comment | added | Tommaso Centeleghe | I meant to say $1/(p^2-1)(p^2-p)$ as density in my last comment... | |
| Oct 12, 2011 at 3:00 | comment | added | Chandan Singh Dalawat | For $E:y^2+y=x^3-x$, the representation on $E[p]$ is known to be surjective for $p\neq5$. | |
| Oct 12, 2011 at 0:08 | comment | added | Tommaso Centeleghe | Interesting. I suppose we know the density of this set, which should be something like $(p-1)/(p^2-1)(p^2-p)$, where $p=7$, if the representation on E[7] is surjective. | |
| Oct 11, 2011 at 3:16 | comment | added | Chandan Singh Dalawat | By the way, Shimura did not give any examples of primes which split in say ${\bf Q}(E[7])$. The first such examples $$ 4831, 22051, 78583, 125441, 129641, 147617, 153287, 173573, 195581, ... $$ were computed by Tim Dokchitser. | |
| Oct 10, 2011 at 7:55 | comment | added | Tommaso Centeleghe | Nicolas! I haven't. Yet. But will. soon! Thanks | |
| Oct 10, 2011 at 7:37 | comment | added | Nicolas B. | Have you looked at Adelmann's LNM (vol. 1761) "The Decomposition of Primes in Torsion Point Fields"? | |
| Oct 10, 2011 at 7:15 | vote | accept | Tommaso Centeleghe | ||
| Oct 12, 2011 at 0:11 | |||||
| Oct 10, 2011 at 5:22 | answer | added | Denis Chaperon de Lauzières | timeline score: 3 | |
| Oct 10, 2011 at 5:03 | comment | added | Chandan Singh Dalawat | See also mathoverflow.net/questions/11747/… | |
| Oct 10, 2011 at 3:35 | comment | added | Chandan Singh Dalawat | Shimura looked at $E:y^2+y=x^3-x$ and showed that a prime $l\neq11,p$ splits in ${\bf Q}(E[p])$ if and only if $$\rho(Frob_l)=\pmatrix{1&0\cr0&1\cr}$$ where $\rho$ is the mod-$p$ representation associated with $$ q\prod_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2. $$ See my arxiv.org/abs/1007.4426 for an elementary introduction. | |
| Oct 10, 2011 at 2:20 | history | asked | Tommaso Centeleghe | CC BY-SA 3.0 |