Timeline for minimal conductors among elliptic curves with a fixed CM type
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 15, 2014 at 14:57 | comment | added | Noam D. Elkies | A bit more precisely, an elliptic curve has CM not by $K$ but by some order in $K$, i.e. a finite-index subring of $O_K$. Usually that order must be $O_K$ itself, but there is one more choice for $d=-4$ and $d=-7$, and two more for $d=-3$. In all but one of these four extra cases, specifying the CM ring does not affect the answer, because there's always a minimal-conductor curve with endomorphisms by $O_K$ that's isogenous to $E$ (and isogenies preserve the conductor). But for $d=-3$ there's an index-2 subring ${\bf Z}[\sqrt{-3}]$ of $O_K$ whose minimal conductor is not $27$ but $36$. | |
| Jul 15, 2014 at 14:39 | history | edited | Jeremy Rouse |
edited tags
|
|
| Jul 15, 2014 at 14:27 | comment | added | NAME_IN_CAPS | Maybe yes and no for your purposes. I think they all differ by twisting by a class group character, which would preserve the conductor. Rohrlich's paper is projecteuclid.org/euclid.dmj/1077314180 I had the conditions wrong, either $d$ is odd, or $8|d$. In the first case there are $h$ choices, in the second case $2h$. He also takes $d<-4$. He says these canonical characters appear in the book of Gross. | |
| Jul 15, 2014 at 14:20 | comment | added | Hugo Chapdelaine | Thanks for the data. Yes my questions are closely related to asking for a "canonical Grossencharacter" of a quadratic CM field, but from what you say it seems that such an object does not really exist. | |
| Jul 15, 2014 at 14:08 | comment | added | NAME_IN_CAPS | See also the Lemma page 3 of link.springer.com/article/10.1007%2FBF01388657 | |
| Jul 15, 2014 at 14:02 | comment | added | NAME_IN_CAPS | There is a CM elliptic curve (over Q) of conductor $p^2$ for $p=7,11,19,43,67,163$, of conductor $256$ for $d=-8$, and $27$ and $32$ for $d=-3,-4$. I don't know if this helps you or not. I think Greenberg or Rohrlich has a "canonical" Grossencharacter (which is not canonical when $h\neq 1$, maybe need $8\not | d$ also) of infinity type (0,1) and the specified level, namely of norm $\frak p$ for $p=7..163$, and $\frak p^2$ for $d=-3$, and $\frak p^3$ for $d=-4$ and $\frak p^5$ for $d=-8$. | |
| Jul 15, 2014 at 13:28 | history | asked | Hugo Chapdelaine | CC BY-SA 3.0 |