Closed Intertwining Operators on locally algebraic principal series

数论研究有理数域Q上多项式的解; p-adic数论则研究Q的拓扑完备上多项式的解：除了通常的绝对值外，Q上每个素数p都有一个p-adic绝对值，它测量p整除整数的频率。对于Q上的有理多项式，解析式可以给出Q_p中Q_p以外的根，从而简化了Q_p的运算。一旦在每一个Q_p上的可解性问题得到解决，那么在Q上的可解性问题也就在许多方面得到解决（Hasse原理），伽罗瓦理论通过考察Q_p的域扩张的自同构来研究这个问题。绝对伽罗瓦群由Q_p的代数闭包的所有Q_p-代数自同构组成。它将所有算术信息编码在Q_p上，我们通过研究它在有限维向量空间上的作用来研究它。在这里，p-adic Langlands纲领设想了一个连接神秘的算术Galois群和熟悉的线性群G=GL_n（Q_p）的桥梁，它建立了n维向量空间上的连续Galois作用和Banach空间上的一致连续G-作用之间的对应关系。存在B上的一致连续G-作用当且仅当存在G-不变范数||在B。这在很大程度上是一个开放的问题。Colmez等人通过描述真实的数R>=0的B，解决了GL_2（Q_p）的情形。对于更一般的情形，本文引入了p-adic流形上的C^R-理论，并对其进行了更详细的研究。最新的预印本计算了单位球上的C^R-函数在Q_p的有限扩张F上的Fourier系数，这是Colmez的证明策略在GL_2（F）上的推广.如果G-作用量是由有理函数局部给定的，那么C^R-理论允许我们考虑||作为G上多元C^R-函数的先验证明。这个结果在我在Séminaire Automorphe的演讲笔记中概述。首先，我正在修改C^R理论，使其通过泰勒多项式进行方便的描述，并将其与最近的其他p-adic演算方法进行比较。更重要的是，对于||作为一个适当的规范，C^R函数空间上的特殊算子虽然肯定不是连续的，但必须是封闭的。闭算子的概念在经典泛函分析中是一个熟悉的概念，但在p-adic泛函分析中却闻所未闻。我将在皮埃尔·科尔梅兹和阿丽亚娜·梅扎德的协助下，在特殊情况下调查这个问题。

Number Theory studies the solutions of polynomials over the field of rational numbers Q; p-adic Number Theory instead studies the solutions of polynomials over Q's topological completions:Besides the usual absolute value, there is on Q for each prime number p the p-adic absolute value that measures how often p divides an integer. Completing Q by this absolute value, we obtain the field of p-adic numbers Q_p. For a rational polynomial over Q, analytical arguments provide roots in Q_p outside of Q; the completion simplified Q_p arithmetically. Once the question of solvability is settled over every Q_p, so it is in many ways over Q (Hasse principle).Galois Theory studies this question by looking at the automorphisms of Q_p's field extensions. The absolute Galois group consists of all Q_p-algebra automorphisms of Q_p's algebraic closure. It encodes all arithmetic information on Q_p.We study it by looking at its actions on finite dimensional vector spaces. Here the p-adic Langlands program envisions a bridge between the mysterious arithmetical Galois group and the familiar linear group G=GL_n(Q_p); it formulates a correspondence between continuous Galois actions on n-dimensional vector spaces and uniformly continuous G-actions on Banach spaces.My research revolves around the construction of such a Banach space. There is a uniformly continuous G-action on B if and only if there is a G-invariant norm || on B. This is largely an open problem. The case GL_2(Q_p) was solved by Colmez and others by describing B, for a real number R>=0, through R-times differentiable (or C^R-)functions on the unit ball of Q_p.For more general cases my thesis introduced a C^R-Theory over p-adic manifolds, which has been examined in more detail. The newest preprint computes the Fourier coefficients of the C^R-functions on the unit ball in a finite extension F of Q_p. This serves towards a generalization of Colmez`s proof strategy to GL_2(F).If the G-action is locally given by rational functions, then the C^R-theory allows us to regard || as, a priori, seminorm on C^R-functions in many variables on G. This result is outlined in the notes of my lecture at the Séminaire Automorphe.Firstly I am revising the C^R-theory towards a handy description via Taylor polynomials, and compare it with other recent approaches to p-adic calculus.What is more, for || being a proper norm, special operators on C^R-function spaces, though surely not continuous, nevertheless must be closed. The notion of a closed operator is a familiar notion in classical Functional Analysis, but yet unheard-of in p-adic Functional Analysis. This problem I will investigate in special cases under assistance by Pierre Colmez and Ariane Mezard.