#13266: Weil restriction for elliptic curves over number fields
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Reporter: Hester | Owner: cremona
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-5.3
Component: elliptic curves | Resolution:
Keywords: elliptic curve, weil restriction | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Hester Graves | Merged in:
Dependencies: | Stopgaps:
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Comment (by nbruin):
Another question: The current implementation makes
{{{
E.weilrestriction()
}}}
an alias for
{{{
E.defining_ideal().weilrestriction()
}}}
so it saves typing 17 characters, at the expense of implementing
"weilrestriction" for elliptic curves as a functor from the category of
elliptic curves over a number field K to the category of ideals over Q.
I'm afraid most people would expect the weil restriction of an elliptic
curve to be an abelian variety, not an ideal.
Plus, in what sense is the returned ideal describing the Weil restriction?
Since you're restricting a projective ideal, you have to be careful what
you divide out by. If you take Proj(..) of the ideal you return, you get a
semiabelian variety that is an extension of the abelian variety you're
interested in by a torus `Res_(K/Q) (G_m) / G_m`,
coming from the fact that projective equivalence for `(x:y:z) in P^2(K)`
is modulo `K^*` whereas for
`(x0:x1:y0:y1:z0:z1) in P^5(Q)` it's only modulo `Q^*`.
There's a reason why Weil restriction is described for affine varieties
and then extended via patches/sheaves!
I'm afraid that saving 17 keystrokes is not worth the misleading
terminology.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13266#comment:4>
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