Applications and extensions of p-adic Hodge theory

数论是所有数学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多年前。虽然整数在某种意义上是离散的对象，但在现代，它们经常使用连续数学的技术来研究，例如微积分。PI的工作特别关注于p-adic数的使用，p-adic数是在世纪早期引入的，作为一种类似于真实的数的数字系统，但记录了整数整除性的信息。除了理论上的结果，p-adic数引起了计算机科学中的一些相关的计算技术，特别是在现代cryptography.Building最近的突破，在p-adic霍奇理论，我们在几个不同的方向扩展该领域。我们扩展了理论的p-adic表示，使系数在较大的环，着眼于应用程序的自守形式的p-adic插值。我们还允许替换的p-adic伽罗瓦群与etale基本群，这有望揭示新的光的p-adic朗兰兹对应。最后，我们考虑用整数的完整环替换p-adic数的方法;这涉及到在任意环上系统地使用Witt向量。

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.