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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?

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This always true if you 'define' your parameter $t$ by $dt = dx/y$. In each case, this defines $t$ as a multivalued function on the Riemann surface (= algebraic curve) (minus the poles of the differential $dx/y$, of course), or, if you prefer, it defines $t$ on a cover of the Riemann surface. You might ask when this $t$ has some special properties, such as, for example, that $dx/y$ be of the second kind, or have no poles, or some such condition, which might be of interest. – Robert Bryant Aug 4 '12 at 14:48

I would say it's a consequence of the Riemann-Roch theorem. Let's take a look at what happens in the case of an elliptic curve. The $\wp$ function has a double pole at a prescribed point $\mathcal{O}$. The function $\wp'$ thus has a triple pole there. By the Riemann-Roch theorem, the vector space $H^0(6\mathcal{O})$ has dimension $6+1-g=6$. Thus, the functions ${1, \wp, \wp^2, \wp^3, \wp\wp', \wp', \wp'^2}$ must be linearly dependent. A nontrivial relation of linear dependence for this set translates into a parametrization of the curve using $\wp$ and $\wp'$ - this is pretty much a tautology.

Note that the integers $n$ and $n+1$ are always relatively prime. This implies that every sufficiently large integer can be written as a linear combination of $n$ and $n+1$ with non-negative integer coefficients (this is the so-called postage stamp theorem). Take any meromorphic differential $\omega$ on a curve $X$, with a pole at $\mathcal{O}$. The derivative $d\omega$ has a pole of order $n+1$ at $\mathcal{O}$. The exact same construction can be carried out!

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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.

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In the sense of having a common form of differential dx/y, which when integrated gives the inverse function of the trigonometric/elliptic function, this is a classical analogy. As far as I know it is reasonable to extend this approach to hyperelliptic curves, but not general curves of higher genus. And even there you have to look more closely at some theory. For example in genus 2 the theory of the Jacobian relates to integrating both dx/y and xdx/y, where y is a square root of a sextic.

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Both are Calogero-Moser systems of algebraic groups. See From solitons to many–body sytems by David Ben-Zvi and Tom Nevins section 2

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