On the Parshin-Arakelov theorem and integral sections on elliptic surfaces

It is well-known that the Parshin-Arakelov theorem implies the Mordell conjecture over complex function fields by a covering construction of Parshin. Via a similar map in the context of integral points on elliptic curves over function fields, we explain how to obtain a short geometric proof of a uniform version of Siegel's theorem. Our technique also allows us to establish a uniform quantitative result on the set-theoretic intersection of curves with the singular divisor in the compact moduli space of stable curves.