#9320: Implement root numbers for elliptic curves over number fields
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Reporter: arminstraub | Owner: cremona
Type: enhancement | Status:
needs_review
Priority: minor | Milestone:
Component: elliptic curves | Keywords: root
number
Author: Tim Dokchitser and group (Sage Days 22) | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by newvalueoldvalue):
* cc: beankao (added)
* status: new => needs_review
* author: => Tim Dokchitser and group (Sage Days 22)
Old description:
> Root numbers for elliptic curves are currently only implemented via Pari
> (pari(E).ellrootno()) and only over the rational numbers.
>
> We (Tim Dokchitser's group at Sage Days 22) intend to add the possibility
> to compute local and global root numbers for elliptic curves over number
> fields. A first patch may not fully implement the case of additive
> reduction over primes dividing 2 or 3.
New description:
Root numbers for elliptic curves are currently only implemented via Pari
(pari(E).ellrootno()) and only over the rational numbers.
We (Tim Dokchitser's group at Sage Days 22) intend to add the possibility
to compute local and global root numbers for elliptic curves over number
fields. A first patch may not fully implement the case of additive
reduction over primes dividing 2 or 3.
Update: Attached is a first implementation which allows for instance:
{{{
sage: K.<a>=NumberField(x^4+2)
sage: E = EllipticCurve(K, [1, a, 0, 1+a, 0])
sage: E.root_number()
1
sage: E.root_number(K.ideal(a+1))
1
}}}
Note that the implementation needs the patches #9334 (Hilbert symbol) and
#9684 ("dirty model") to be applied.
To prevent incorrect results in some cases as well as crashes, the tickets
#9389 and #9417 need to be addressed.
--
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9320#comment:1>
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