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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).

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  • $\begingroup$ It would be best if you could define what you consider trivial, to make this a precise question. $\endgroup$ Nov 15, 2014 at 1:44
  • $\begingroup$ I would like to know the faith of classical diophantine equations over the join of all cyclotomic fields. For example, Fermat equations like $x^4 +y^4 = 1$ - will it have infinitely many solutions? $\endgroup$
    – Pablo
    Nov 15, 2014 at 8:01
  • $\begingroup$ @BjornPoonen : There are equations known to have finitely many solutions either by Faltings' theorem or by some other very intricate argument (this may be considered nontrivial). I would like to see equations with an argument of this type (say, a uniform bound by Faltings' thorem on all cyclotomic fields at once) showing that the number of solutions is finite over $\mathbb{Q}^{\text{ab}}$. $\endgroup$
    – Pablo
    Nov 15, 2014 at 8:07
  • $\begingroup$ The quartic Fermat equation $x^{4} + y^{4} = 1$ will have infinitely many points on it over $\mathbb{Q}^{{\rm ab}}$, simply because $x^{2} + y^{2} = 1$ has infinitely many points on it over $\mathbb{Q}$. $\endgroup$ Nov 15, 2014 at 18:10
  • $\begingroup$ Oh thats right. What about 5 or 6 instead of 4? $\endgroup$
    – Pablo
    Nov 15, 2014 at 20:45

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