Lang's weak and strong conjectures for varieties of general type

Let $k$ be a number field and let $X$ be a smooth variety of general type over $k$, meaning that its Kodaira dimension equals its dimension.

Lang's conjecture. (a) Weak form: The set $X(k)$ of $k$-rational points is not Zariski dense in $X$. (b) Strong form: There exists a proper Zariski-closed subset $Z\subset X$ such that, for every finite extension $K$ of $k$, the set of $K$-rational points on $X\setminus Z$ is finite.

The weak form predicts nondensity of rational points, while the strong form gives a uniform exceptional subset and finiteness over every finite extension. The paper invokes these forms in establishing finiteness and uniformity results for rational points on the varieties arising from the curves under study.