Diophantine Problems and p-adic Period mappings ( after Lawrence and Venkatesh )
Diophantine Problems and p-adic Period mappings ( after Lawrence and Venkatesh )
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丢番图问题和 p 进周期映射（劳伦斯和文卡特什之后）
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Milan Malčić
中科院分区：
文献类型：
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作者：
Milan Malčić
新的证明利用了Faltings原始证明的结构，但没有使用阿贝尔变体。相反，Lawrence和Venkatesh使用p进霍奇理论构造了一个p进周期映射，该映射将p进伽罗瓦表示的变化编码为代数变体族。他们论证的一个重要方面是p进周期映射和复周期映射之间的相互作用：关于p进周期映射的一个重要陈述可以在复周期映射上得到验证，这是作者通过在相关黎曼曲面上的显式单态拓扑计算得到的。在[LV18，§1.5]中，作者将他们的证明与Faltings的证明进行了比较，并评估其并不比后者更简单或更困难。该方法的真正优点在于它适用于高维变量。这就引出了他们论文的第二个主要结论。Lawrence和Venkatesh在高维情况下应用了相同的方法，证明了投影空间中的超曲面集包含在所有超曲面的模空间的适当的zariski闭子集中，并且与固定的素数集有很好的分离。更准确地说，他们得到：
The new proof utilizes the set-up of Faltings’ original proof, but makes no usage of abelian varieties. Instead, Lawrence and Venkatesh use p-adic Hodge theory to construct a p-adic period map, which encodes the variation of p-adic Galois representations in a family of algebraic varieties. One important aspect of their argument then is the interplay between the p-adic period map and the complex period map: a crucial statement about the former can be verified on the latter, which the authors do by explicit topological computations of monodromy on the relevant Riemann surfaces. In [LV18, §1.5], the authors compare their proof to Faltings’ and assess that it is not simpler or less difficult than the latter. The real gain of their method is its applicability to higher-dimensional varieties. This brings us to the second main result of their paper. Lawrence and Venkatesh apply the same methods in a higher-dimensional situation to show that the set of hypersurfaces in a 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. More precisely, they obtain:
丢番图问题和p进周期映射
影响因子：
3.1
作者：
Lawrence, Brian;Venkatesh, Akshay
通讯作者：
Venkatesh, Akshay