#5976: [with patch; needs work] Add an Elliptic Curve Isogeny object
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Reporter: shumow | Owner: shumow
Type: enhancement | Status: assigned
Priority: major | Milestone: sage-4.0
Component: number theory | Keywords: Elliptic Curves
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Comment(by cremona):
I realized this morning that I had been saying confusing things about this
"normalization" question. This is really superseded by David's
suggestions, but anyway: while it is true that E and the kernel G
uniquely determine the isogenous curve E' = E/G up to isomorphism, if we
replace E' by an isomorphic curve E", isomorphic to E' via [u,r,s,t], then
the scaling constant gets multiplied (possibly divided) by u.
This means that if we do not care in advance about the model we use for E'
we can always replace E' by a suitable E" such that the scaling constant
is 1; then it seems to me that E" and the isogeny are uniquely determined
up to translations (as David says), i.e. [u,r,s,t] with u=1. (And _not_
up to general automorphisms of E", since these may multiply the scaling
constant by a root of unity (or order at most 6 of course).
In the context of finite fields, there is no reason to choose any one
scaling of a Weierstrass model over another, so it is harmless to insist
that isogenies are normalized (or at least to reduce to that case).
Indeed, the only place I have seen the adjective "normalized" applied to
isogenies was precisely in this context (e.g. papers of Morain).
The situation is different over number fields: when the class number is 1
and global minimal models exist we may well want to insist that E' is a
minimal model, which fixes the scaling constant up to a unit in the field
only. We cannot therefore normalize the isogeny without sacrificing
integrality or minimality of the model. For example, given an isogeny of
prime degree ell, and its dual, the product of the two scaling constants
is ell. Over Q with minimal (or at least ell-minimal) models, one of them
has constant ell and the other has constant 1. (Or they could be -ell and
-1, but we could then compose with [u,r,s,t]=[-1,0,0,0], the negation
map.) In this context I think one says that the isogeny with constant 1
is etale, rather than normalized.
I think David's proposed framework allows for all these possibilities.
Over finite fields, given E and G the default would be to return a
normalised isogeny. Over Q the default would be (I think?) to return an
isogeny whose codomain is a minimal model. I'm not sure about other
number fields. And teh user could override this if they provide their own
codomain.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5976#comment:16>
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