#15299: Incorrect results for analytic Sha due to low precision
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Reporter: jdemeyer | Owner:
Type: defect | Status: new
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 cremona):
Replying to [comment:3 jdemeyer]:
> So this is plain wrong then (I don't know enough mathematics to judge
this):
> {{{
> if self.__E.root_number() == 1:
> return 0
> }}}
The root number is the sign of the functional equation so is +1 for even
analytic rank and -1 for odd. This function computes the first
derivative. *In practice* this is something one would only want to do if
the 0'th derivative was already known to be 0, in which case the code you
quote would be OK since if the value is 0 and the order is even then the
order is at least 2 so the first derivative is exactly 0. But of course
this function then lies in wait for the user who decides they want the
first derivative's value even when the value is nonzero (as for 11a1).
The trouble is that (1) Formulas for the r'th derivative which are
implemented are *only* valid under the assumption that all previous
derivatives are 0; and of course (2) proving the earlier derivatives are
exactly 0 is usually impossible with current theory.
Where does that leave this deriv_at1 function? At the very least it
should come with a huge warning about all this. And it really should
return 0 when the root number is +1 unless the user has made an explicit
assumption (assume_order_of_vanishing_is_positive=True, say) and otherwise
raise a NotImplemented error (or attempt to prove that L(1)=0).
--
Ticket URL: <http://trac.sagemath.org/ticket/15299#comment:5>
Sage <http://www.sagemath.org>
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