Between ordinary and p-adic Hodge theory

普通 Hodge 理论与 p-adic Hodge 理论之间

• 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

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建议在代数簇上的不同上同调理论之间建立新的显式关系，推广Betti和de Rham上同调（Hodge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particira项目是一个描述的etale基本群nonarchimedean解析空间，这将导致更好地理解周期域和周期映射的p-adic霍奇理论。其他潜在的结果包括de Rham上同调和代数K理论之间的联系，适用于机器计算的etale上同调的明确描述，以及de Rham上同调的变体，通过与晶体上同调的类比来产生L函数的谱解释。在技术方面，该项目可能具有相当大的变革性，将历史上平行进行但缺乏连贯综合的研究领域结合在一起。从长远来看，这些方法可能会导致新的技术来攻击数论的经典问题，如素数的分布和聚合性质。正如PI先前的调查所证明的那样，它们也可能导致数论中计算和数值方法的改进，这可能会对数论应用于计算机科学（特别是信息安全）的领域产生影响。