Applications and extensions of p-adic Hodge theory

p进Hodge理论的应用和扩展

• 批准号: 1501214
• 负责人: Kiran Kedlaya
• 金额: $ 17万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501214&HistoricalAwards=false
• 关键词: Applications; extensions; adic; Hodge; theory

Number theory is one of the oldest of all branches of mathematics, with its first important results dating back more than two millennia. While whole numbers are in a sense discrete objects, in modern times they are often studied using techniques of continuous mathematics, such as calculus. The PI's work focuses in particular on the use of p-adic numbers, which were introduced early in the 20th century as a number system analogous to the real numbers, but recording information about divisibility of integers. In addition to theoretical results, the p-adic numbers give rise to computational techniques with some relevance in computer science, especially in modern cryptography.Building on recent breakthroughs in p-adic Hodge theory, we extend the field in several different directions. We extend the theory of p-adic representations to allow coefficients in larger rings, with an eye towards applications to the p-adic interpolation of automorphic forms. We also allow the replacement of p-adic Galois groups with etale fundamental groups; this is expected to shed new light on the p-adic Langlands correspondence. Finally, we consider ways to replace the p-adic numbers with the full ring of integers; this involves systematic use of Witt vectors over arbitrary rings.

数字理论是数学所有分支中最古老的理论之一，其首先重要的结果可以追溯到两千年以上。尽管整个数字在某种意义上是离散的对象，但在现代中，它们经常使用连续数学的技术（例如微积分）进行研究。 PI的工作尤其着重于使用P-Adic数字，这些数字在20世纪初引入了类似于实数的许多系统，但记录了有关整数的可分裂性的信息。除了理论结果外，P-ADIC数量还引起了计算机科学中有一定相关性的计算技术，尤其是在现代密码学中。建立P-Adic Hodge理论的最新突破，我们以几个不同的方向扩展了该领域。我们扩展了P-ADIC表示的理论，以允许在较大环中的系数，并着眼于应用自动形式的P-Adic插值。我们还允许用依托基本组替换P-Adic Galois组；预计这将为P-ADIC Langlands对应提供新的灯光。最后，我们考虑用整数环代替P-ADIC数字的方法。这涉及系统地使用Witt向量在任意环上。