#7096: bug in dual isogeny computation
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Reporter: cremona | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-4.2
Component: elliptic curves | Keywords: elliptic curve isogeny
Work_issues: | Author:
Reviewer: | Merged:
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Comment(by wuthrich):
I started implementing some more related to this ticket. Especially the
{{{formal()}}} for isogeny. Then there are two ideas how to compute the
dual.
* I can take the implementation as it is now. This yields '''a''' isogeny
of the correct degree
in the opposite direction. Then I can compute the leading term of the
composition in the
formal expansion (or simply check to what multiple the differential is
pulled-back to). This
gives me the {{{WeierstrassIsomorphism}}} to use. I am not 100 % sure
if this will work in
all cases. Say the elliptic curve is defined over a finite field and
has a cyclic isogeny of
degree ''n^2^'' to itself. It is certain that our current
implementation gives back a cyclic
isogeny and not just ''[n]''. I fear one could find counterexamples...
I have to do some
testings.
* Otherwise, I will try to implement the full computation of the dual via
the formal group. I
believe that there is an algorithm with running time ''O(n)'' for an
isogeny of prime degree
''n''. Though I have not checked this in details. It would only involve
to compute the first
''2n'' coefficients in ''[n]'' and the {{{division_polynomial}}} in the
formal expansion. The
one example I have computed so far by hand was a failure :(. One
obstacle here will be the
fact that {{{.reversion()}}} is only defined for power-series with
coefficients in '''Q'''.
à suivre.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7096#comment:6>
Sage <http://www.sagemath.org>
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