Timeline for Elliptic Curves over F_1?

5 events

when toggle format	what		by	license	comment
Dec 9, 2009 at 2:37	comment	added	JBorger		CM stands for "complex multiplication". It's a theorem that the endomorphism ring of an elliptic curve over a field of characteristic 0 is either Z or a rank 2 subring of an imaginary quadratic field. In the second case, one says it has complex multiplication. The point is that in the CM case, H^1, which is always 2-dimensional, now has an action of a 2-dimensional algebra, over which it is therefore 1-dimensional. So the "motives" of CM elliptic curves are essentially abelian. So CM elliptic curves are similar in some ways to G_m.
Dec 9, 2009 at 1:38	comment	added	Chris Schommer-Pries		Excuse my ignorance. What is a "CM elliptic curve"?
Dec 9, 2009 at 1:37	vote	accept	Chris Schommer-Pries
Dec 9, 2009 at 0:00	comment	added	JBorger		Yes, in my point of view there is "an F_1" for every Dedekind domain with finite residue fields. The one w.r.t Z gives the deepest one, and no elliptic curves are defined over it. But if R is the ring of integers in an imaginary quadratic of class number 1 and discriminant D, and E is an elliptic curve over R[1/D] with complex multiplication, then this scheme descends to the version of F_1 w.r.t R[1/D]. Without the class number restriction, it's probably still true, but you'd have to take E over the Hilbert class field and view it as a (non geom connected) scheme over R[1/D].
Dec 8, 2009 at 16:35	history	answered	Felipe Voloch	CC BY-SA 2.5