Timeline for Torsion points of CM elliptic curves
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 7, 2011 at 15:01 | comment | added | monodromy | In fact, unless I am missing something, I believe the answer to the question is yes, although not every real model of $E$ as above (which exists by the stability assumption on $\mathfrak{f}$) will work. | |
| Dec 7, 2011 at 13:30 | comment | added | Keerthi Madapusi | There is the main Main theorem, which actually describes the action of the whole Galois group of $\mathbb{Q}$, though it seems a little excessive here. See Section 4 of jmilne.org/math/articles/2007c.pdf | |
| Dec 7, 2011 at 8:16 | comment | added | David Loeffler | @monodromy: the Main Theorem of CM is a statement about automorphisms of $\mathbb{C}$ which restrict to the identity on $K$, which complex conjugation does not. | |
| Dec 6, 2011 at 23:31 | comment | added | monodromy | @David: Doesn't the statement in the if clause of the last sentence in your question follow from the Main Theorem of Complex Multiplication? | |
| Dec 6, 2011 at 21:56 | comment | added | David Loeffler | @Dror: The j-invariant of any lattice invariant under complex conjugation is real, because the q-expansion coefficients of the j-invariant are in $\mathbb{R}$. One doesn't need the lattice to have CM for this to work. | |
| Dec 6, 2011 at 21:49 | comment | added | Dror Speiser | Small comment: since the ideal is stable under complex conjugation, it is of order two in the class group. Reducing it, we get a reduced ambiguous ideal, which under the assumptions isn't above $2$ (or $1$). So it is of the form $(m,\sqrt{\Delta})$, with $m|\Delta$. Normalizing, the lattice is homothetic to $(1,\sqrt{\Delta}/m)$. Hence, by V.2.1 in [AECII], the j-invariant is real. | |
| Dec 6, 2011 at 21:35 | history | edited | David Loeffler | CC BY-SA 3.0 |
trivial clarification
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| Dec 6, 2011 at 18:34 | history | asked | David Loeffler | CC BY-SA 3.0 |