From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Elliptic y^2=x^3+x
Date: 19 Jan 1997 08:47:36 GMT
In article <5ataeb$jb@rzsun02.rrz.uni-hamburg.de>,
Hauke Reddmann wrote:
>Can I parametrize this equation?
>Be aware that I don't work in Q but
>in Q+i+sqrt(whatevermaycomeup), so
>a parametrization which contains
>additional sqrt signs is OK and
>EC theory doesn't apply directly.
Sure it does. "EC theory" is pretty vast (as opposed to some of us
who use it, who are only half-vast).
First of all, note that _every_ elliptic curve can be parameterized:
that's the point of the Weierstrass Pe-functions; these establish a
one-to-one correspondence between the cosets C/L of a lattice in the
complex plane on the one hand, and the points of an elliptic curve, on
the other. Both domain and range are complex manifolds; the parameterization
is analytic. Even better, this map is an isomorphism of groups. It's
even fairly easy to perform explicit computations with them; for
example, the PARI / GP package includes function calls for this
parameterization (and its inverse).
I'm guessing however that what you wanted to know is whether there is
a _rational_ parameterization (x(t), y(t)) of this curve (evidently
you are willing to allow the functions to have coefficients in some
algebraic number field). This you cannot do. If you had such a
function, it would inter alia be a meromorphic function C -> E, hence
(if not constant) necessarily surjective, thus a covering of a torus
by a sphere, which is impossible. (Indeed, this argument precludes
the parameterization of _any_ [nonsingular] elliptic curve.)
Perhaps you were thinking of situations in which an argument concludes,
"...this curve has no rational points, but it has some over the field ...".
The nearest analogy I can suggest might be for elliptic curves
over function fields, that is, curves given as
y^2 = x^3 + A x + B
where A and B are _rational functions_ (of T, say). If you consider
all the combinations of (x, y, T), you see that this equation defines
a surface (i.e., a 2-dimensional manifold, if nonsingular). Now it
is more challenging to ask if there are rational functions from C
to the surface, since these need not be onto. In fact, given such a
surface, one can actually ask several related questions:
1) Is there a rational curve on the surface? (i.e., can you
find rational functions (x(t), y(t), T(t)) whose image
is contained in the surface?)
2) Is there such a curve for which T(t) is invertible? (This
requires finding a curve taking each T value once and
only once.)
3) Viewing the surface as an elliptic curve over the field C(T)
of rational functions, does this curve have a rational
point? (This is actually equivalent to question 2)
4) Is it true that for (almost) every T0 there is a point
(x,y,T) on the surface with T=T0 ?
In every case, one can ask the same questions with the complex field C
replaced by a number field K; a "yes" for K means a "yes" for C, too.
The questions are often quite hard, and the relationships among them are
not easy to clarify. Surely (1)=>(2)<=>(3)=>(4), but the two end
implications are not reversible. If (3) holds over C, it holds over a
number field, but it's not immediately clear how to find that field.
In fact, computationally, these questions are rather a mess. (For example,
the (weak) Mordell-Weil theorem guaranteeing that the group of points on the
elliptic curve is finitely generated is true over function fields such
as C(T), just as it is over number fields; but the proof is harder
because there are more issues involving units and factorization.
Consequently, it's harder to convert the theory into algorithms.)
Question (1), for example, is expected to have an affirmative answer if
question (4) does, in general, but that's not proven and I certainly
have seen no algorithm for finding an embedded rational curve in an
elliptic surface, even if it's known to have many points (and not be split).
By the way, you could try to "cheat" and view your original question in
this context (simply declare the coefficients of your curve to be
constant rational functions, rather than numbers). The resulting
surface would then be simply the product E x C. Such split elliptic
surfaces tend to behave differently from the general ones and give rise to
asterisks in the discussion of questions (1)-(4). So you get nowhere
by cheating.
Silverman's second book ("Advanced topics...") has a chapter on
elliptic surfaces which covers much of this.
dave
(I assume you know the curve has no _rational_ points apart from the
torsion subgroup { [0,0], O }. )
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.num-analysis,sci.math.symbolic
Subject: Re: Parametrization of a plane curve
Date: 5 Dec 1997 07:13:53 GMT
In article <3487639B.79F7@cs.purdue.edu>,
Cassiano Durand wrote:
>Anybody knows how to parametrize the curve
>
>x^4 + a*x^2 + b*y^2 + c = 0, where a,b,c are nonzero real numbers
>
>without using square roots? Maybe using only sin/cos?
>^^^^^^^^^^^^^^^^^^^^^^^^^^^
No can do. That's a curve of genus 1 (unless you're very lucky in your
choice of a,b,c). You could use the Weierstrass pe-function, I suppose
(this is the real locus of a Riemann surface). But from a numerical
analysis perspective, you'd be much better off with square roots!
dave