p-adic Cohomology and Applications

p-进上同调及其应用

• 批准号: 0400727
• 负责人: Kiran Kedlaya
• 金额: $ 12.74万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400727&HistoricalAwards=false
• 关键词: adic; Cohomology; Applications

Abstract for the award of Kiran Kedlaya DMS-0400272p-adic analysis, initiated by Hensel at the turn of the last century,seeks to bring the "continuous" techniques of calculus to bear on"discrete" problems in number theory. We study applications of p-adicanalysis in arithmetic algebraic geometry; a typical problem is to countsolutions of systems of polynomial equations. We work on one hand onimproving our theoretical understanding of this problem, and on the otherhand on developing practical algorithms for treating important specialcases. Some of these cases occur in applications to cryptography, errorcorrecting codes, and other areas of computer science.More specifically, we are developing Berthelot's rigid cohomology in parallel with the older theory of etale cohomology, which is better understood theoretically but ill-equipped for explicit computations.One long-term goal is to extend Lafforgue's work on the function field Langlands correspondence to p-adic sheaves (crystals). In another direction, we are investigating algorithms for computing in p-adic cohomology, which may have mathematical applications on top of the practical ones mentioned above.

Hensel在上个世纪初发起的Kiran Kedlaya DMS-0400272P-ADIC分析奖励摘要，试图将微积分的“连续”技术带入数字理论中的“离散”问题。我们研究了p- adicanalysis在算术代数几何形状中的应用；一个典型的问题是多项式方程系统的计数。一方面，我们努力研究了我们对这个问题的理论理解，而另一方面，我们就开发了用于治疗重要特种商品的实用算法。其中一些情况发生在密码学，错误正确的代码以及计算机科学领域的应用中。更具体地说，我们正在与较旧的Etale同胞理论平行地开发Berthelot的刚性刚性共同体，这是理论上更好地理解的，但在理论上是不合适的，但要进行详细的计算。一个长期的目标是扩展了Lafforgue在lafforgue的工作范围，即在lafforgue的工作中扩展了该功能（PRAFFINGE）。在另一个方向上，我们正在研究用于计算P-ADIC共同体计算的算法，该算法可能在上面提到的实用方法上具有数学应用。