#11163: documentation of p-adic L-function order_of_vanishing is very wrong
-----------------------------+----------------------------------------------
Reporter: was | Owner: was
Type: defect | Status: new
Priority: minor | Milestone: sage-4.7
Component: number theory | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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{{{
To: Chris Wuthrich
From: William Stein
We have a function order_of_vanishing on p-adic L-functions with
docstring:
-------------
Definition: L.order_of_vanishing(self)
Source:
def order_of_vanishing(self):
r"""
Return the order of vanishing of this `p`-adic L-series.
The output of this function is provably correct, due to a
theorem of Kato [Ka]. This function will terminate if and only if
the Mazur-Tate-Teitelbaum analogue [MTT] of the BSD conjecture
about
the rank of the curve is true and the subgroup of elements of
`p`-power order in the Tate-Shafarevich group of this curve is
finite. I.e. if this function terminates (with no errors!),
then you may conclude that the `p`-adic BSD rank conjecture is
true and that the `p`-part of Sha is finite.
-------------
The actual code doesn't call p-adic regulator anywhere. However, in
our paper we claim that not only does
one have to verify that the order of vanishing of the analytic p-adic
L-function equals the rank of the curve,
but one also needs that the p-adic regulator is nonzero, in order to
get the claimed conclusion that Sha is finite.
That makes sense to me, since I don't think Schneider gets anywhere
without knowing the height pairing
is nondegenerate.
So... is this a bug... I guess only in the documentation. We could
simply add to the documentation that if
you call this function, get a number, and then *also* call
padic_regulator and get a nonzero number, then
you can conclude that Sha(p) is finite.
William
----
From: Chris Wuthrich
To: William Stein
I agree that the documentation is wrong. I would simply delete to
mention Sha at all. This is probably better explained in p_primary
bound. So a reference to this function could be included.
Chris.
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11163>
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