Let $\ldots \to X\_n \to X\_{n-1} \to \ldots \to X\_0$ be etale maps between smooth projective curves of genera $g(X\_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know that the sets $X\_n(K)$ are finite. 



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> Is the set $\bigcup\_{n=0}^\infty X\_n(K)$ always finite?
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Rational points on towers of curves
No. Start with a tower of Galois covers for which $X\_0(K) \ne \emptyset$. Then, inductively, replace each $X\_n$ by a twist such that the point from the previous layer lifts to a rational point on the twist. In this way, all $X\_n(K)$ are non-empty and their union is infinite.


