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Operations on Point Sets






To each elliptic curve E is associated a series of point sets of
E indexed by coefficient rings.  These point sets, not E, are the
objects that points lie in.
If K is the base ring of E and L is some
extension of K, then the point set E(L) has as elements all
points lying on E whose coordinates are in L.


There is a special point set E(K) of E which is called the base
point set of E.  Many intrinsics
(such as #, or TorsionSubgroup)
strictly speaking only make sense when applied to point sets;
as a convenience, when E is passed to these functions the behaviour
is the same as if the base point set E(K) were passed instead.  It
is important to remember, however, that E and E(K) are different
objects and will not always behave the same.


The above statements are equally valid if the elliptic curve E is
replaced by some subgroup scheme G.  Moreover, the types of the
point sets of G and E are the same, and similarly for points.
(They may be distinguished by checking the type of the scheme that they
are point sets of.)




Subsections



Creation of Point Sets





E(L) : CrvEll, Rng -> SetPtEll


PointSet(E, L) : CrvEll, Rng -> SetPtEll



Given an elliptic curve E (or a subgroup scheme thereof)
and an extension L of its base ring,
returns a point set whose elements are points on E with
coefficients in L.



E(m) : CrvEll, Map -> SetPtEll


PointSet(E, m) : CrvEll, Map -> SetPtEll



Given an elliptic curve E (or a subgroup scheme thereof)
and a map m from the base ring of E
to a field L, returns a point set whose elements are points on E
with coefficients in L.  The map is retained to manage coercions
between point sets.



Associated Structures





Category(H) : SetPtEll -> Cat


Type(H) : SetPtEll -> Cat



Returns the category of point sets of elliptic curves, SetPtEll.



Scheme(H) : SetPtEll -> CrvEll



Returns the associated scheme (either an elliptic curve or a subgroup
scheme) that H is a point set of.



Curve(H) : SetPtEll -> CrvEll



Returns the associated elliptic curve that contains Scheme(H).



Ring(H) : SetPtEll -> Rng



The ring that contains the coordinates of points in H.



Predicates on Point Sets





H1 eq H2 : SetPtEll, SetPtEll -> BoolElt



Returns whether the two point sets are equal.  That is, whether the
point sets have equal coefficient rings and elliptic curves (subgroup
schemes).



H1 ne H2 : SetPtEll, SetPtEll -> BoolElt



The logical negation of eq.





Example CrvEll_PointSets (H91E13)

We create an elliptic curve over GF(5) and two point sets of it:





> K := GF(5);
> E := EllipticCurve([K | 1, 0]);
> H := E(K);
> H;
Set of points of E with coordinates in GF(5)
> H2 := E(GF(5,2));
> H2;
Set of points of E with coordinates in GF(5^2)


We see that although these are point sets of the same curve, they are
not equal because the rings are not equal.





> Scheme(H) eq Scheme(H2);
true
> Ring(H) eq Ring(H2);
false
> H eq H2;
false


Similarly, we see that a point set of a subgroup scheme is not the same
object as the point set of the curve because the schemes are different.





> P<t> := PolynomialRing(K);
> G := SubgroupScheme(E, t - 2);
> HG := G(K);
> Scheme(HG) eq Scheme(H);
false
> Ring(HG) eq Ring(H);
true
> HG eq H;
false


Also note that the scheme and the curve of point sets of G are different:





> Scheme(HG);
Subgroup scheme of E defined by x + 3
> Curve(HG);
Elliptic Curve defined by y^2 = x^3 + x over GF(5)






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