  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.
