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.

数论是数学所有分支中最古老的分支之一，其第一个重要成果可以追溯到两千年前。虽然整数在某种意义上是离散的对象，但在现代，人们经常使用连续数学技术（例如微积分）来研究它们。 PI 的工作特别关注 p 进数的使用，该数于 20 世纪初作为类似于实数的数字系统引入，但记录了有关整数整除性的信息。除了理论结果之外，p 进数还引发了与计算机科学，特别是现代密码学相关的计算技术。基于 p 进 Hodge 理论的最新突破，我们在几个不同的方向上扩展了该领域。我们扩展了 p 进表示理论以允许更大环中的系数，着眼于自守形式的 p 进插值的应用。我们还允许用 etale 基本群替换 p 进伽罗瓦群；这有望为 p-adic Langlands 通信提供新的线索。最后，我们考虑用整环整数代替 p 进数的方法；这涉及在任意环上系统地使用维特向量。