Diophantine problems and p-adic period mappings

• 成果类型: Article
• 署名作者: Lawrence, Brian; Venkatesh, Akshay
• 署名单位: University of Chicago; Institute for Advanced Study - USA
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00966-7
• 发表日期: 2020
• 页码: 893-999
• 关键词: abelian-varieties; lie-algebras; division; THEOREM

摘要

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.