Diophantine problems and p-adic period mappings

By Brian Lawrence, Akshay Venkatesh. Inventiones Mathematicae, 2018

69 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7ac80a549e5c9a2f8de4908e01cc3cca403767d9

Citations in the corpus

• On the distribution of the Hodge locus (2021, unknown): “As a simple geometric illustration of Theorem 1.5: Let us also mention that Theorem 1.5, combined with the main result of [39], provides a nice first result in the direction of the so called refined Bombieri-Lang conjecture .”