Opened 12 years ago
Closed 12 years ago
#11163 closed defect (fixed)
documentation of p-adic L-function order_of_vanishing is very wrong
Reported by: | William Stein | Owned by: | William Stein |
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Priority: | minor | Milestone: | sage-4.7 |
Component: | elliptic curves | Keywords: | |
Cc: | Merged in: | sage-4.7.alpha5 | |
Authors: | William Stein | Reviewers: | David Loeffler |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description
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.
Attachments (1)
Change History (5)
comment:1 Changed 12 years ago by
Status: | new → needs_review |
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Changed 12 years ago by
Attachment: | trac_11163.patch added |
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comment:2 Changed 12 years ago by
Authors: | → William Stein |
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Component: | number theory → elliptic curves |
Reviewers: | → David Loeffler |
Status: | needs_review → positive_review |
Yes, this looks fine. (However: William, would you mind using the more specific trac ticket categories "elliptic curves", "modular forms", etc, rather than just bunging everything in "number theory"?)
comment:3 Changed 12 years ago by
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.
comment:4 Changed 12 years ago by
Merged in: | → sage-4.7.alpha5 |
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Resolution: | → fixed |
Status: | positive_review → closed |
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The attached patch is ready for review. It does *nothing* but changed the docstring for the order_of_vanishing function, by deleting the offending and incorrect paragraph, and a related incorrect doctest. I also added a rank 2 and rank 3 doctest to replace having deleted a rank 2 doctest.