#16456: Bug in descend_to method for elliptic curves
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Reporter: cremona | Owner: cremona
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.3
Component: elliptic curves | Resolution:
Keywords: elliptic curve | Merged in:
base change | Reviewers: Peter Bruin
Authors: John Cremona | Work issues:
Report Upstream: N/A | Commit:
Branch: u/cremona/16456 | 3db6c28c3ffb0c9d29eb5fd1e1fe5cd1acbd907f
Dependencies: | Stopgaps:
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Changes (by pbruin):
* reviewer: => Peter Bruin
Comment:
I remember looking at the Selmer group code some time ago (and again just
now) and getting the impression that the group ''K''(''S'', ''m'')
returned by `selmer_group(S, m)` is the one fitting in a natural exact
sequence (warning: unicode experiment ahead)
1 ⟶ O,,''K'',''S'',,^×^/O,,''K'',''S'',,^×''m''^ ⟶ ''K''(''S'', ''m'') ⟶
Cl,,''K,S'',,[''m''] ⟶ 0.
(I.e., I think ''K''(''S'', ''m'') should be canonically isomorphic to the
flat cohomology group H^1^(O,,''K'',,, '''''μ''',,m,,''). This can be
replaced by étale cohomology if ''S'' contains all places dividing ''m'',
and in that case I also think that the above H^1^, and ''K''(''S'',
''m''), should be isomorphic to H^1^(Gal(''K^S^''/''K''),
'''''μ''',,m,,'') with ''K^S^'' the maximal extension of ''K'' that is
unramified outside ''S''.)
Is the above exact sequence correct, and should it perhaps be mentioned in
the documentation? And is there a comparable exact sequence for the
"true" Selmer group consisting of the elements giving unramified
extensions of ''K''?
--
Ticket URL: <http://trac.sagemath.org/ticket/16456#comment:8>
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