I am studying Elliptic functions for a University project with a particular focus on Weierstrass's theory. For the past few weeks I have been studying various basic properties of the $\wp$ function (the majority of the Elliptic functions section in Whittaker and Watson). Finally, I have come to the point were I want to choose a particular topic.
Ideally, I don't want my paper to become just a list of standard theorems that can already be found in the standard references so I have been trying to find something interesting and challenging. In searching for a project idea, I found that according to wikipedia:
"Genus one solutions of ordinary differential equations can be written in terms of Weierstrass's elliptic functions"
This sounds very attractive to me but I have been unable to find references to unpack exactly what is meant. Does a genus 1 ODE mean one whose associated curve is genus 1? If any one could explain the above statement and provide some references to further pursue this strain of knowledge I would be very grateful.
Thanks
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.