Weierstrass elliptic functions and ordinary differential equations

I think this simply means the following: let $P(X,Y)\in\mathbb C[X,Y]$. This defines an algebraic curve $P(X,Y) = 0$, and it also defines a differential equation $P(f,f') = 0$. A solution to the differential equation automatically defines a (local) parameterization of the algebraic curve by $t\mapsto (f(t), f'(t))$.

The Weierstrass $\wp$-function for a given lattice $\Lambda$ satisfies such a differential equation:

$$\wp'(z)^2 = 4\wp(z)^3 - g_2\wp(z) - g_3$$

for some $g_2, g_3\in\mathbb C$, where $Y^2 = 4X^3 - g_2X - g_3$ defines a non-singular curve of genus 1, i.e. an elliptic curve.

The statement you are asking about concerns the converse: whenever $P$ defines an elliptic curve, the differential equation $P(f,f') = 0$ is solved by a Weierstrass $\wp$-function for some lattice. This is Abel's inversion problem from 1827.

Much of this material can be found in chapter 6 of Silverman's beautiful The Arithmetic of Elliptic Curves, but it states this specific theorem without a proof. It does go into detail about another ingredient, how to transform any plane equation defining an elliptic curve into Weierstrass form: $Y^2 = X^3 + aX + b$, and more generally how to obtain this form for any elliptic curve, which is the form of the differential equation satisfied by Weierstrass $\wp$-functions.

Some places where the inversion theorem is proved:

• In chapter 6 of Knapp's Elliptic Curves this is proved explicitly through elliptic integrals.

• In 1.4 of A First Course in Modular Forms by Diamond and Shurman (and in the other references below) this is proved, how else, using modular forms.

• In the classics on modular forms by Shimura, in 4.2, and by Serre, chapter VII.

• Finally, proofs can be found in the excellent and free, ever evolving lecture notes by Milne, on Modular Forms and in his very affordable book on Elliptic Curves as theorem 3.10.