Diophantine problems and $p$-adic period mappings

TL;DR

This paper offers an alternative proof of Faltings's theorem using $p$-adic Galois representations and extends the method to show that certain hypersurfaces form a proper subset in the moduli space, leveraging recent Ax--Schanuel results.

Contribution

It introduces a new proof of Faltings's theorem based on $p$-adic period mappings and applies similar techniques to moduli spaces of hypersurfaces, connecting Diophantine problems with $p$-adic Hodge theory.

Findings

Finiteness of rational points on high-genus curves over number fields.

Sufficiently large hypersurfaces with good reduction are contained in a proper Zariski-closed subset.

Application of Ax--Schanuel property to moduli space of hypersurfaces.

Abstract

We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of $p$-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of $p$-adic Hodge theory, and explicit topological computations of monodromy. By the same methods we show that, in sufficiently large dimension and degree, the set of hypersurfaces in projective space, with good reduction away from a fixed set of primes, is contained in a proper Zariski-closed subset of the moduli space of all hypersurfaces. This uses in an essential way the Ax--Schanuel property for period mappings, recently established by Bakker and Tsimerman.