#15299: Incorrect results for analytic Sha due to low precision
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Reporter: jdemeyer | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-5.13
Component: elliptic curves | Resolution:
Keywords: | Merged in:
Authors: Jeroen Demeyer | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by pbruin):
Replying to [comment:19 jdemeyer]:
> Do you propose that this change should be made, or is it just an
observation? Given that the function `deriv_at1()` is in practice only
called when we know that `L(E,1) = 0`, I personally think the warning
suffices.
I agree, it was more an observation that we could in principle use
`L1_vanishes()` here than a proposal to actually do it.
There is a formula for the ''r''-th derivative which is valid when all
lower derivatives vanish. As far as I know, only for the 0-th derivative
is there a known way to prove that it vanishes by a numerical computation.
For parity reasons (the root number), this means that if the order of
vanishing is 0, 1 or 2, then we can prove this. If the order of vanishing
is 3, then in general we don't know how to prove that it is not 1.
This means that if and when the formula mentioned above is implemented, we
won't be able to verify the condition "all lower derivatives are 0" when
''r'' is at least 3. Hence we should probably not insist on verifying it
when ''r'' = 1.
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Ticket URL: <http://trac.sagemath.org/ticket/15299#comment:20>
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