In this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to Bertin–Romagny, the Kodaira–Parshin families constructed by Lawrence–Venkatesh can themselves be assembled into a family. We then repeat Lawrence–Venkatesh's study of the (Formula presented.) -adic period map, together with the comparison of nearby fibres.
- Diophantine geometry
- p-adic geometry