Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of unity (Kronecker-Weber).
The reasoning for showing that certain diophantine equations have finitely many solutions, sometimes consists of considering solutions in some cyclotomic fields, and deducing the finiteness of the set of rational solutions from this.
I would like to see as many examples as possible where a diophantine equation of the form $f(x,y) = 0$ (with $f \in \mathbb{Q}[x,y]$ a polynomial) can be shown to have a (nonempty) finite set of solutions by showing this for $\mathbb{Q}^{\text{ab}}$.
I expect the proof of the finiteness of the number of points to be somehow nontrivial, possibly involving some algebraic argument about ideals/primes/factorization in cyclotomic fields, or maybe some geometric idea.
To conclude, I am looking for affine plain curves $C$ defined over $\mathbb{Q}$ with a nonempty and finite set of $\mathbb{Q}^{\text{ab}}$-rational points such that the proof of this fact will not be too trivial.
One thing I consider trivial is: $f(x,y) = 2x^3 - y^3$ where the finiteness of solutions just follows from the fact that $t^3 - 2$ does not have solutions in $\mathbb{Q}^{\text{ab}}$ (and you don't need to go to $\mathbb{Q}^{\text{ab}}$ at all).