June 2020
My sincerest sympathies to all those suffering loss and hardship due to the dreadful COVID-19 pandemic.Welcome to my math web site.
This series of web pages are based on notes I made about 30 years ago when studying the theory of elliptic functions. In them we compute the $g_2$, $g_3$ invariants for several genus 1 curves of low degree. The $g_2$ and $g_3$ invariants are not quite intrinsic to the curve itself, but are uniquely defined once a specific holomorphic differential on the curve is chosen.
One of the original motivations for carrying out these computations was to answer what seemed like a simple question.
What is the differential equation for the general elliptic function of order 3 and how do you calculate the $g_2$ and $g_3$ invariants of the corresponding genus 1 curve?
The differential equation for elliptic functions $f$ of order 2 is \begin{equation*} f'^2 = af^4 + bf^3 + cf^2 + df + e \end{equation*} and it is, more or less, the starting point of the theory of elliptic integrals. The associated curve of genus 1 is \begin{equation*} y^2 = ax^4 + bx^3 + cx^2 + dx + e \end{equation*} The $g_2$ and $g_3$ invariants are \begin{equation*} g_2 = 60 \sum {\frac 1 {\omega^4}} = \tfrac {1} {12} (12ae - 3bd + c^2) \end{equation*} \begin{equation*} g_3 = 140 \sum {\frac 1 {\omega^6}} = \tfrac {1} {432} (72ace - 27ad^2 - 27b^2e + 9bcd - 2c^3) \end{equation*} where the $\omega$ are the non-trivial periods of the integral of the holomorphic differential $\displaystyle \frac {dx} {y}$. \begin{equation*} \omega = \bigint_C {1 \over \sqrt{ax^4 + bx^3 + cx^2 + dx + e}} dx \end{equation*}
We want to carry out a similar computation for elliptic functions $f$ of order 3.
There is a special case answer to the above question:
The elliptic functions $f$ of order 3 with 3-fold rotational symmetry have differential equation \begin{equation*} f'^3 = (af^3 + bf^2 + cf + d)^2 \end{equation*} The associated curve of genus 1 is \begin{equation*} y^3 = (ax^3 + bx^2 + cx + d)^2 \end{equation*} The $g_2$ and $g_3$ invariants are \begin{equation*} g_2 = 60 \sum {\frac 1 {\omega^4}} = 0 \end{equation*} \begin{equation*} g_3 = 140 \sum {\frac 1 {\omega^6}} = \tfrac {1} {729} (27a^2d^2 - 18abcd + 4ac^3 + 4b^3d - b^2c^2) \end{equation*} where, as above, the $\omega$ are the non-trivial periods of the integral of the differential $\displaystyle \frac {dx} y$. \begin{equation*} \omega = \bigint_C {1 \over \sqrt[3]{(ax^3 + bx^2 + cx + d)^2}} dx \end{equation*} In both the above cases the $g_2$ and $g_3$ invariants are expressed in terms of the invariants of the curves coefficient polynomials. We seek something similar for the general order 3 elliptic function.
Editor's Note:
You can see the MathJax logo in the top corner of this page. Thanks to this excellent tool I have been able to write up these web pages in a simple text editor in plain HTML and $\TeX$.
I have written these notes in an expository style. You will find details missing, a few over-simplifications, and many unsubstantiated assertions. But hopefully you will also find a coherent thread of reasoning from start to finish.
It has been a pleasure writing up these notes. Please enjoy!
Gregg Kelly