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.

中文翻译：

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Diophantine 问题和 p-adic 周期映射

我们给出了法尔廷斯定理（莫德尔猜想）的另一种证明：在数域上至少有两个属的曲线具有有限多个有理点。我们的论证利用了 Faltings 原始证明的设置，但在精神上更接近 Chabauty 和 Kim 的方法：我们通过更详细地分析 $p$-adic Galois 表示在一个代数簇族。该分析的关键输入是 $p$-adic Hodge 理论的比较定理和单向性的显式拓扑计算。通过相同的方法，我们表明，在足够大的维数和度数上，射影空间中的超曲面集与固定的素数集有很好的缩减，包含在所有超曲面的模空间的适当 Zariski 闭子集中.