#9320: Implement root numbers for elliptic curves over number fields
-------------------------------------+-------------------------------------
Reporter: arminstraub | Owner: cremona
Type: enhancement | Status: needs_work
Priority: minor | Milestone:
Component: elliptic curves | Resolution:
Keywords: root number | Merged in:
Authors: Tim Dokchitser | Reviewers:
and group (Sage Days 22) | Work issues: fix ReST formatting,
Report Upstream: N/A | coverage
Branch: u/chapoton/9320 | Commit:
Dependencies: | d40bf637722cd2cb68b1ce0310290bbed2f7a1f5
| Stopgaps:
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Comment (by wuthrich):
I had a quick look at this and there are indeed major issues with the
local root number at places above 2 when the reduction is additive,
potentially good. That is in the function {{{_root_number_local_2}}} in
{{{ell_number_field.py}}}.
With the current commit above we get an error for
{{{
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([1+i, 1+i, 0, i, 0])
sage: P2 = K.ideal(1+i)
sage: E.root_number(P2)
}}}
because line 1966 tries to create and extension with a reducible
polynomial {{{K2 = K.extension(t**2+E.a2()*t+E.a4(), 'a2')}}}. I guess -
but it would take me some time to read the sources - that K2 should just
be K in this case.
However there are other issues. For instance
{{{
sage: E = EllipticCurve([1+i,0,0,0,i])
sage: E._root_number_local_2(P2)
}}}
goes "bang".
I would think the code without the very bad case at places above 2 is ok
and that could already go in. So I see two possible futures for this
ticket:
1. Someone fixes these issues above 2 or
2. We raise an {{{NotImplementedError}}} if the reduction is additive and
potentially good at a place above 2 and open a new ticket for fixing the
problem.
I can help with 2), but I won't have time to do 1). But maybe there is
someone out there who would.
--
Ticket URL: <http://trac.sagemath.org/ticket/9320#comment:20>
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