Between ordinary and p-adic Hodge theory

普通 Hodge 理论与 p-adic Hodge 理论之间

基本信息

批准号：
1101343
负责人：
Kiran Kedlaya
金额：
$ 36万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2011
资助国家：
美国
起止时间：
2011-09-01 至 2015-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101343&HistoricalAwards=false
关键词：
Between ordinary adic Hodge theory

项目摘要

The PI proposes to develop new explicit relationships between different cohomology theories on algebraic varieties, generalize existing relationships between Betti and de Rham cohomology (Hodge theory), and between etale and de Rham cohomology (p-adic Hodge theory). One particulra project is a description of the etale fundamental groups of nonarchimedean analytic spaces; this will lead to better understanding of period domains and period mappings in p-adic Hodge theory. Other potential results include links between de Rham cohomology and algebraic K-theory, explicit descriptions of etale cohomology suitable for machine computations, and variants of de Rham cohomology giving rise to spectral interpretations of L-functions by analogy with crystalline cohomology.On the technical side, this project is potentially quite transformative, bringing together areas of research that have historically proceeded in parallel but lacking a coherent synthesis. In the long run, these methods may lead to new techniques for attacking classical question of number theory, such as the distribution and aggregate properties of prime numbers. As evidenced by the PI's prior investigations, they are also likely to lead to improvements in computational and numerical methods in number theory, which may have impacts in areas of application of number theory to computer science (notably information security).

PI提议在代数品种上建立不同的共同学理论之间建立新的显式关系，从而概括了Betti和De Rham共同体之间的现有关系（Hodge理论），以及Etale和De Rham的共同学（P-Adic Hodge理论）之间的现有关系。一个特殊的项目是对非架构分析空间的依托基本组的描述。这将使P-Adic Hodge理论中的时期域和周期映射更好地理解。其他潜在的结果包括DE RHAM共同体与代数K理论之间的联系，适合机器计算的依托共同学的明确描述以及De Rham同谋的变体通过与晶体的共同体相比，通过与晶体的共同体进行了相似的研究，这是一个潜在的研究，这是一定的，这是一定的研究，这是一项历史悠久的，这是一定的，这是一项历史悠久的研究，这是一项历史悠久的研究，这是一项历史悠久的研究，这是一项历史悠久的。合成。从长远来看，这些方法可能会导致用于攻击经典数字理论问题的新技术，例如素数的分布和汇总特性。正如PI先前的调查所证明的那样，它们也可能导致数字理论中计算和数值方法的改进，这可能会影响数字理论对计算机科学的应用领域（尤其是信息安全性）。