Klein forms and the generalized superelliptic equation

Let $F$ be a binary form over the integers and consider the exponential Diophantine equation $F(x,y)=z^n$ with $x$ and $y$ coprime. In general it seems very difficult to study this equation, but as we will explain in this talk, for so-called Klein forms $F$, the modular method can provide a good starting point. By combining this with a new method for solving infinite families of Thue equations, we can show in particular that there exist infinitely many (essentially different) cubic (Klein) forms $F$ for which the equation above has no solutions for large enough exponent $n$. This is joint work with Mike Bennett.