Curves with many torsion points
(Jit here.) I am going to write about some recent ideas in diophantine geometry. Consider a curve $C$ inside $\mathbb{C}^2$ given by some polynomial $P(x,y) = 0$. Can there be infinitely many points $(x,y)$ lying on our curve $C$ such that both $x$ and $y$ are roots of unity? <--!more--> If we just want one of the coordinates to be a root of unity, it is clear that there are infinitely such points, although they should be "sparse". The idea is then that if we impose this condition on both coordinates, there should somehow only be finitely many such points, unless $C$ is a special curve. For example, if $C$ was of the form $\{\omega \} \times \mathbb{C}$ where $\omega$ is a root of unity, then clearly it has infinitely many such points on it. In general, if our curve $C$ were given by $x^i y^j = \omega$, where $\omega$ is a root of unity, then $y$ will be a root of unity if $x$ is and so we get infinitely many points with both coordinates roots of unity. We can unders
