# Why does the parameterization (F:F':1) happen?

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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