Prove that there exist nonconstant meromorphic functions \(f,g\) on \(\mathbb C\) satisfying \[ f(z)^3+g(z)^3=1. \]
Hint: show that the plane cubic \[ x^3+y^3=1 \] defines the same elliptic curve in \(\mathbb P^2\) as a Weierstrass cubic \[ y^2=4x^3-1. \]
Solution
We explicitly identify the Fermat cubic with the equianharmonic Weierstrass cubic and then uniformize the latter by a Weierstrass \(\wp\)-function.
1. On the affine Fermat cubic\[ x^3+y^3=1, \]set\[ U=\frac1{x+y}, \qquad V=\sqrt3\,\frac{x-y}{x+y}. \]Then\[ V^2=4U^3-1. \]
First, \(x+y\) never vanishes on the affine Fermat cubic, since \(x+y=0\) would imply \[ x^3+y^3=0. \] Thus \(U\) and \(V\) are well-defined there.
Put \[ s=x+y, \qquad d=x-y. \] Since \[ x^3+y^3=s^3-3xys=1, \] we have \[ xy=\frac{s^3-1}{3s}. \] Therefore \[ \begin{aligned} d^2 &=s^2-4xy\\ &=s^2-\frac{4(s^3-1)}{3s}\\ &=\frac{4-s^3}{3s}. \end{aligned} \] Hence \[ V^2 =3\frac{d^2}{s^2} =\frac{4-s^3}{s^3} =4U^3-1. \]
2. Conversely, from a point \((U,V)\) satisfying\[ V^2=4U^3-1 \]with \(U\ne0\), define\[ x=\frac{\sqrt3+V}{2\sqrt3\,U}, \qquad y=\frac{\sqrt3-V}{2\sqrt3\,U}. \]Then \(x^3+y^3=1\).
The formulas give \[ x+y=\frac1U, \qquad x-y=\frac{V}{\sqrt3\,U}. \] Thus \[ xy =\frac{(x+y)^2-(x-y)^2}{4} =\frac{3-V^2}{12U^2}. \] Using \(V^2=4U^3-1\), \[ \begin{aligned} x^3+y^3 &=(x+y)^3-3xy(x+y)\\ &=\frac1{U^3}-\frac{3-V^2}{4U^3}\\ &=\frac{1+V^2}{4U^3}\\ &=1. \end{aligned} \] Thus the two formulas are inverse birational transformations. Since the projective Fermat cubic and the projective Weierstrass cubic are smooth, this birational equivalence extends to an isomorphism of their smooth projective curves.
3. There is a lattice whose Weierstrass function satisfies\[ \wp'(z)^2=4\wp(z)^3-1. \]
For the hexagonal lattice, the rotation-symmetry computation in E-HM213A17-131 gives \[ G_2=\sum_{\lambda\ne0}\lambda^{-4}=0. \] Hence its Weierstrass invariant \[ g_2=60G_2 \] vanishes. Its cubic \[ y^2=4x^3-g_3 \] is smooth, so \(g_3\ne0\).
If the lattice is scaled by a nonzero complex number \(c\), then \[ g_2(c\Lambda)=c^{-4}g_2(\Lambda), \qquad g_3(c\Lambda)=c^{-6}g_3(\Lambda). \] Choose \(c\) with \[ c^6=g_3(\Lambda). \] For the scaled lattice we therefore have \[ g_2=0, \qquad g_3=1, \] and the Weierstrass differential equation \(\wp'^2=4\wp^3-g_2\wp-g_3\) becomes \[ \wp'^2=4\wp^3-1. \]
4. Define meromorphic functions on \(\mathbb C\) by\[ \boxed{ f(z)=\frac{\sqrt3+\wp'(z)}{2\sqrt3\,\wp(z)}, \qquad g(z)=\frac{\sqrt3-\wp'(z)}{2\sqrt3\,\wp(z)}. } \]Then\[ f^3+g^3=1. \]
The functions are meromorphic because they are rational expressions in the meromorphic functions \(\wp\) and \(\wp'\). Taking \[ U=\wp(z), \qquad V=\wp'(z) \] in 2 and using 3 gives exactly \[ f(z)^3+g(z)^3=1. \] The identity holds wherever the displayed rational expressions are finite, hence everywhere as an identity of meromorphic functions.
5. Both \(f\) and \(g\) are nonconstant.
If one of \(f,g\) were constant, the equation \[ f^3+g^3=1 \] would force the cube of the other to be constant, hence the other function itself to be constant. But then \[ f+g=\frac1\wp \] would be constant, contradicting the nonconstancy of \(\wp\). Therefore both \(f\) and \(g\) are nonconstant meromorphic functions.