#11163: documentation of p-adic L-function order_of_vanishing is very wrong
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Reporter: was | Owner: was
Type: defect | Status: positive_review
Priority: minor | Milestone: sage-4.7
Component: elliptic curves | Keywords:
Author: William Stein | Upstream: N/A
Reviewer: David Loeffler | Merged:
Work_issues: |
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Comment(by was):
It turns out that we were confused. However, I think the ticket should
still go in as is, since it's well... confusing have theorems without
references stated left and right in docstrings.
{{{
> Now I'm confused again. Theorem 6.1 in our paper says:
>
> "The order of vanishing of f_E(T) at T = 0 is at least equal to the
rank r. It
> is equal to r if and only if the p-adic height pairing is nondegenerate
(Conjec-
> ture 4.1) and the p-primary part of the Tate-Shafarevich group X(E=Q)(p)
is
> fi nite (Conjecture 1.2)."
>
> I'm assuming here that p>=5 is good ordinary.
>
> By Kato we know that f_e(T) divides L_p(E,T). For example, if E(Q)
> has rank r, and we compute and find that L_p(E,T) vanishes to order
> <=r, that proves that f_E(T) vanishes to order <= r, hence f_E(T)
> vanishes to order exactly r at T=0. This implies that the p-adic
> height pairing is nondegenerate and Sha(E/Q)(p) is finite.
>
> That said, I think the trac ticket is fine as is, since we probably
> shouldn't have docstrings with theorems like that in them anyways.
> (The new version is simple and clear.)
>
> I guess this means that if my project was only showing that
> Sha(E/Q)(p) is *finite* for many curves, the only calculation I have
> to do is of L_p(E,T) to enough precision to determine a good enough
> upper bound on the order of vanishing. Computing the p-adic
> regulator isn't necessary. That said, I did them all already, and
> having them will provide an excellent *double check* on the results,
> via p-adic BSD, and is also needed to show that Sha(E/Q)(p) = 0, and
> not just that Sha(E/Q)(p) is finite.
>
> If you agree with the above, I should post something to the trac
> ticket for the record. (I won't make another patch.)
Ooops, yes everything you say is correct. And I agree with your
conclusion. The confusion is all mine, sorry. It is a long time I have
not thought about shark.
The finiteness of Sha comes indeed very easily and without having to
compute p-adic regulators. But I still think that the possibility of
computing the order is the great thing. We can prove Hasse principles
for curves, without having to compute any Galois cohomology groups.
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11163#comment:3>
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