Closed Intertwining Operators on locally algebraic principal series

数论研究有理数域Q上的多项式的解；p-进数论相反地研究Q上的多项式的解：除了通常的绝对值外，对于每个素数p，在Q上都有p-进绝对值，它衡量p除以一个整数的频率。用这个绝对值来补q，我们就得到了p进数域q_p。对于q上的有理多项式，解析论元在q之外给出了q_p的根；补全在算术上简化了q_p。一旦在每个Q_p上解决了可解性问题，那么它在许多方面都在Q(Hasse原理)上。伽罗瓦理论通过研究Q_p的域扩张的自同构来研究这个问题。绝对Galois群由Q_p的代数闭包的所有Q_p-代数自同构组成。它编码了Q_p上的所有算术信息，我们通过观察它在有限维向量空间上的作用来研究它。在这里，p-进的朗兰兹程序设想了神秘的算术Galois群和熟悉的线性群G=GL_n(Q_P)之间的桥梁，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。B上有一致连续的G-作用当且仅当B上有G-不变范数||。这在很大程度上是一个公开问题。Colmez等人利用单位球上的R次可微(或C^R-)函数刻画了实数R&gt；=0的情形GL_2(Q_P)，从而解决了GL_2(Q_P)的情形。对于更一般的情形，本文引入了p-进流形上的C^R-理论。最新的预印本计算了单位球上C^R-函数在Q_p的有限扩张F中的傅立叶系数。这有助于将Colmez的证明策略推广到GL_2(F)。如果G-作用是由有理函数局部给出的，那么C^R-理论允许我们将||看作是G上多元C^R-函数的先验半范。这一结果在我在S自同构的讲稿中概述。首先，我用泰勒多项式对C^R-理论进行了修正，并将它与最近的p-进位演算的其他方法进行了比较对于正规范数，C^R-函数空间上的特殊算子，虽然肯定不是连续的，但一定是闭的。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-进泛函分析中却是闻所未闻的。这个问题我将在Pierre Colmez和Ariane Mezard的协助下在特殊情况下进行调查。

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.