#18662: Implement (semi-)global minimal models for elliptic curves over number
fields with h>1
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Reporter: cremona | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.8
Component: elliptic curves | Keywords: Weierstrass models
Merged in: | Authors: John Cremona
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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Whe E is an elliptic curve defined over a number field K of class number
1, then it has a global minimal model, and we have a method to compute it,
namely E.global_minimal_model().
In the general case global minimal models may or may not exist. This
ticket will introduce a method E.has_minimal_model() which will determine
this, and to find it when it does -- one cannot proceed one prime at a
time as in the h=1 case. Moreover, when there is no global minimal model
the obstruction is that a certain ideal class is not principal, and we
will provide a function which returns that class. When the obstruction is
not trivial there exist models which are minimal at all primes except at
one prime in that class where the discriminant valuation is 12 more than
the minimal valuation, and we provide a method returning such a model.
The above functionality is implemented using work of Kraus which gives a
condition for when a pair of number field elements c4, c6 belong to an
integral Weierstrass model. This occurs if and only if it occurs locally
at each prime, and only primes dividing 2 or 3 are hard to deal with. In
order to compute the corresponding integral model one needs to combine
together the local transformations implicit in Kraus into a single global
one.
I have written code which does this and tested it on thousands of curves
defined over real quadratic fields of class numbers between 2 and 5, in
order to have nicer models for these curves in the LMFDB.
--
Ticket URL: <http://trac.sagemath.org/ticket/18662>
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