#16456: Bug in descend_to method for elliptic curves
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Reporter: cremona | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.3
Component: elliptic | Keywords: elliptic curve base change
curves | Authors:
Merged in: | Report Upstream: N/A
Reviewers: | Branch:
Work issues: | Dependencies:
Commit: |
Stopgaps: |
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The function descend_to which was implemented in #9384 is incorrect. The
twisting parameter d in L* is only defined modulo (L*)^2 and one has to
determine whether there is a representative which lies in K*, which the
implementation does not do, so some "positive" results are missed. I will
post an example of this. The heart of the problem being solved here is to
determine whether (given that j(E) is in K of course) the twisting
parameter in L*/(L*)^2 is in the image of K*/(K*)^2 or not, and the
implementation ignores this. (For j=0 or 1728 the principle is the same
with squares replced by 6th or 4th powers respectively.)
I can see how to fix this for number fields (one can restrict from
K*/(K*)^2 to a finite K(S,2) for an easily determined set S of primes) or
for finite fields, but it may not be possible to have this implemented for
arbitrary fields, which will cause a problem with the patching.
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Ticket URL: <http://trac.sagemath.org/ticket/16456>
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