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 建议在代数簇的不同上同调理论之间发展新的显式关系，概括 Betti 和 de Rham 上同调（霍奇理论）之间以及 etale 和 de Rham 上同调（p-adic Hodge 理论）之间的现有关系。一个特别的项目是对非阿基米德分析空间的基本基本群的描述；这将有助于更好地理解 p-adic Hodge 理论中的周期域和周期映射。其他潜在结果包括 de Rham 上同调和代数 K 理论之间的联系、适合机器计算的 etale 上同调的明确描述，以及通过与晶体上同调类比而产生 L 函数谱解释的 de Rham 上同调变体。，这个项目可能具有相当大的变革性，它将历史上并行进行但缺乏连贯综合的研究领域汇集在一起。从长远来看，这些方法可能会带来解决数论经典问题的新技术，例如素数的分布和聚合性质。正如 PI 之前的调查所证明的那样，它们还可能导致数论中计算和数值方法的改进，这可能会对数论在计算机科学中的应用领域（特别是信息安全）产生影响。