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，其最早的重要结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，人们经常使用连续数学的技术来研究它们，比如微积分。PI的工作重点尤其是p-进制数的使用，它在20世纪初被引入，作为一种类似于实数的数字系统，但记录了关于整数的整除的信息。除了理论结果之外，p-ady数还产生了与计算机科学，特别是在现代密码学中有一定相关性的计算技术。基于p-adadyHodge理论最近的突破，我们从几个不同的方向扩展了这个领域。我们将p-进表示的理论推广到更大环中的系数，并着眼于应用于自同构形式的p-进插值。我们还允许用其他基本群替换p-进Galois群；这有望对p-进的朗兰兹对应有新的启示。最后，我们考虑用完整的整数环来替换p-进制数的方法；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.