TITLE: Why does the parameterization (F:F':1) happen?
QUESTION [12 upvotes]: 1) To parameterize the conic $x^2+y^2=1$, we can use $(x,y)=(\sin t,\sin't)$ ($\sin'$ meaning the derivative of $\sin$, namely $\cos$).
2) To parameterize an elliptic curve $y^2=4x^3-g_2x-g_3$, we can use $(x,y)=(\wp(t),\wp'(t))$.
(I know how to prove (1) and (2), and that it is possible to parameterize curves of higher genus using the unit disc.)
Question: Is there a common explanation for why both parameterizations are of the form $(F(t),F'(t))$? (as opposed to $(F(t),G(t)$)  Does this phenomenon generalize to curves of higher genus?

REPLY [6 votes]: There is a very deep reason for this. The general principle goes back to Andre Bloch.
If you have a non-constant holomorphic map from the plane to a compact complex manifold,
then there is another such map satisfying a differential equation.
A nice popular explanation of this can be found in the survey of Demailly,
Variétés hyperboliques et équations différentielles algébriques. (French) [Hyperbolic manifolds and algebraic differential equations] Gaz. Math. No. 73 (1997), 3–23.