#11905: Remove _splitting_field() from elliptic curves
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Reporter: jdemeyer | Owner: cremona
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.1
Component: elliptic curves | Resolution:
Keywords: | Merged in:
Authors: Jeroen Demeyer | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jdemeyer/ticket/11905 | 810ceecec7f6d507c5392fe2f2d00af18778f473
Dependencies: #2217, #15626, | Stopgaps:
#11271 |
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Comment (by cremona):
Replying to [comment:20 jdemeyer]:
> Replying to [comment:19 cremona]:
> > I know that this is not yet marked as ready for review, but I did look
at the code anyway. One minor remark: after adjoining all the
x-coordinates of the p-division points, *either* you have the full
division field already, i.e. you have all the y-coordinates, *or* you have
none, i.e. you have none of the y-coordinates. This could be used,
possibly, to simplify that last part of the division field code. But it
is not clear to me that there is a quicker way of determining that final
quadratic extension than picking any of the x-coordinates of a p-division
point and solving for y.
> Not that I don't believe you, but do you have a reference for this? At
least I could add it as a comment in the code.
Proof. For all P in E[p] and all Galois autos sigma (of K-bar over K) we
have sigma(P)=+/-P since x(sigma(P))=sigma(x(P))=x(P). Let P1, P2 be a
basis for E[p]. For each sigma we have sigma(P1)=+/-P1 and
sigma(P2)=+/-P2 and the signs are the *same* otherwise sigma(P1+P2) could
not equal either +(P1+P2) or -(P1+P2). So the image of sigma in the mod p
Galois representation is either the identity or minus the identity. In
the first case, sigma fixes all points in E[p], in the second case it
negates all of them. So if the Galois rep is not trivial, its image has
order 2 and its kernel cuts out a quadratic extension, say L/K. In the
latter case the non-trivial element sigma of Gal(L/K) maps to -1 which
means that sigma(P)=-P for *all* P in E[p].
I don't know a reference!
--
Ticket URL: <http://trac.sagemath.org/ticket/11905#comment:21>
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