Now we turn to cubic curves. The study of these was started by Isaac
Newton, but the subject didn't really flourish until somewhat later
after the advent of complex analysis. Typical cubics are referred to as
* elliptic curves*, which is bit confusing since they are not
actually ellipses. The reason for this name
comes from its connection to elliptic integrals
and functions, where an integral is called elliptic if the integrand
contains a square root of cubic polynomial. Below are a few references
in addition to the ones given earlier. A classical reference -- in spite of the name --
is Whittaker and Watson [3]. The first two books are more modern.

- Husemoeller, Elliptic curves,
- Silverman, The arithmetic of elliptic curves,
- Whittaker and Watson, A course in modern analysis.

After adding a point at infinity to the curve on the right, we get two circles topologically. Now let us treat the variables x and y are treated as complex. (In fact, historically signifcant progress in the study of elliptic integrals was made only after the introduction of complex analysis in the 19th century.) Now the above equation defines a complex elliptic curve which topologically is just a torus (after adding a point at infinity). The two real circles are marked in red here and below. We will also keep track of transevrse circle in yellow. It may be helpful to think of this as a "purely imaginary" curve, even though this isn't quite accurate.

The standard way to see this is by using elliptic
functions. These are doubly periodic functions with the mildest
possible (i.e. inessential) singularities. A function on the complex
plane is doubly periodic if its graph repeats itself in both the horizontal and vertical
directions. More precisely, we should be able to tile the plane into
equally sized parallelograms (called period
parallelograms), such that
the piece of the graph over each tile is the same.
The simplest nonzero example ( in spite of the complicated formula), with
period parallelogram having corners at
0, α_{1},α_{2} and α_{1} + α_{2},
is the Weierstrass ℘-function

At this point we can ask whether there exists a
rational parameterization of an elliptic curve. The answer is NO.
Suppose we could. Then it is possible to show that this extends to a
finite to one map of the Riemann sphere (C ∪∞) onto a torus. But
this is impossible. The justification of this last statement requires a bit of
*topology*. A version of the Jordan curve theorem says that
any closed curve on the sphere would seperate it into two parts (an
interior and exterior). If we could find a map as above, we
could conclude the Jordan curve holds for the torus: lift the curve up to
the sphere and then map the interior/exterior back down. However, it
is clear that this isn't true. Just look at the yellow or red curves above.

Even though, we now know the elliptic curve abstractly, we really want understand the
way it is embedded into C^{2}.
As above, we can get a
sense of it projecting to 3 real
dimensions.
The graph of

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