If we consider an elliptic curve projectively, it is a homogeneous form in $3$ variables say $x$, $y$ and $z$. How is this related to the Thue equations (homogeneous forms in $2$ variables)?
I'm looking for any information on how Thue's theorem can be generalized to equations of the form f(x,y,z) = A where f is a homogeneous form and futhermore if this may be related to Mordells' Theorem since an elliptic curve is such a homogeneous form in 3 variables (when considered in projective coordinates).
This generalisation of Thue's equation to ternary forms is well-known, see for example the article Ternary Form Equations by F. Beukers (2007). There in particular the case $T(x,y,z)=1$ in the integral unknowns $x,y,z$ is considered.