On emertons padic banach spaces oxford academic journals. Completely continuous endomorphisms of padic banach spaces. Completely continuous endomorphisms of p adic banach spaces by jeanpierre serre publ. Analytic padic banach spaces and the fundamental lemma of. Note that there is no natural involution in the padic case. Let v be a banach space, and let u v be a closed vector subspace. A perfect duality between padic banach spaces and compactoids by w. Thanks to arthur ogus and jay pottharst for helpful comments. Existence of invariant norms in padic representations of. The complementation property of linfinity in padic banach spaces. In lectures 1215, we study shtukas with one leg, and their connection to pdivisible groups and padic hodge theory. To this end, we introduce the category of spectral banach spaces, which are padic banach spaces with a c palgebra of analytic functions, and the subcategory of e.
Introduction in the construction of eigenvarieties, the space of overconvergent modular forms arises as an in nitedimensional p adic banach space. The equalization of padic banach spaces and compactoids. This example corresponds to a point in the world of perfectoid kspaces. However, we will not need in nite dimensional spaces here. Schikhof mathematical institute, university ofnijmegen, toernooiveld, 6525 ed nijmegen, the netherlands communicated by prof. Springer at the meeting of november 29, 2004 abstract. Given a linear algebraic group g defined over a number field, emerton 10 has defined a sequence of padic banach spaces. The closely related adeles and ideles were introduced in the 1930s by claude chevalley. Funtional analysis lecture notes for 18 mit mathematics. Isometries in banach spaces download ebook pdf, epub. Throughout, f will denote either the real line r or the complex plane c.
The number p is a real number with 1 p adic banach spaces and families of modular forms robert f. Abstractly, banach spaces are less convenient than hilbert spaces, but still su ciently simple so many important properties hold. The usual meaning taken for p adic analysis is the theory of p adic valued functions on spaces of interest. Banachhecke algebras and adic galois representations. Completely continuous endomorphisms of adic banach spaces. It is the application of padic analysis to quantum mechanics. A padic jacquetlanglands correspondence abstract in this paper, we construct a candidate padic jacquetlanglands correspondence. Let v be a finitedimensional normed vector space over k.
Throughout this course k is a finite extension of the field qp of padic numbers, denotes the absolute. The u p operator acts on this space as a compact operator, and cuts out a nitedimensional locus on which it acts with a bounded slope. On irreducible representations of compact padic analytic. A p adic shimura isomorphism and p adic periods of modular forms. Springer at the meeting of september 26,1994 abstract. Introduction let pbe a prime, cp the completion of an algebraic closure of the padicnumbers qp and ka.
Padic quantum mechanics is a relatively recent approach to understanding the nature of fundamental physics. Such a correspondence has been shown for g gl2 when the. Nonarchimedean banach spaces over trivially valued fields. Padic banach spaces and families of modular forms robert f. The fundamental lemma states the surjectivity of certain analytic maps and describes their kernel.
In mathematics, p adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p adic numbers the theory of complexvalued numerical functions on the p adic numbers is part of the theory of locally compact groups. Piotrowo 5, 618 poznah, poland communicated by prof. Part i background from p adic functional analysis throughout this course kis a nite extension of the eld q p of p adic numbers, jjdenotes. All vector spaces are assumed to be over the eld f. Proceedings of the international conference held in trento, italy, may 29june 2, 1989.
But as there are no hilbert spaces we have to work in the category of banach spaces over the. Schikhof mathematical institute, university of nijmegen, toernooiveld, 6525 ed nijmegen. Coleman this paper is dedicated to bernard dwork who has been a friend and an inspiration for many years. Introduction in the construction of eigenvarieties, the space of overconvergent modular forms arises as an in nitedimensional padic banach space. A presentation of results in padic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, padic harmonic analysis, and related areas. The purpose of the current paper is to introduce some new methods for studying the spaces. Springer at the meeting of september 26,1994 abstract in the set up of nonarchimedean functional analysis, banach spaces as well as. A little padic functional analysis part 1 of 2 im going to start with a little bit of terminology and notation about nonarchimedean banach spaces trusting that you can. Introduction let pbe a prime, cp the completion of an algebraic closure of the p adicnumbers qp and ka. Riesz lemma is an elementary result often an adequate substitute in banach spaces for the lack of sharper.
In part a, which we entitle families of banach spaces, we show how serres p adicbanachfredholmriesz theory s works in a family, i. On orthocomplemented subspaces in padic banach spaces by albert kubzdela institute of civil engineering, poznatl university of technology, ui. Weak and strong ccompactness in non archimedean banach spaces. Given a linear algebraic group g defined over a number field, emerton has defined a sequence of padic banach spaces these spaces are representations of the group of adele points of g, and may be regarded as spaces of padic automorphic forms. The author begins with a discussion of weak topologies, weak compactness and isomorphisms of banach spaces before proceeding to the more detailed study of particular spaces. To this end, we introduce the category of spectral banach spaces, which are p adic banach spaces with a. A very basic fact about banach spaces is the following so called open mapping theorem. Exercise 1 sequence spaces p are banach spaces given a sequence x 1. Theres a lot more where this came from, but we wont need the rest of it just yet. Several standard results true in greater generality have simpler proofs for banach spaces.
A padic shimura isomorphism and padic periods of modular forms, contemporary mathematics 165. Complemented subspaces of padic second dual banach spaces article pdf available in international journal of mathematics and mathematical sciences 183 january 1995 with 26 reads. Completely continuous endomorphisms of padic banach spaces by jeanpierre serre. Berkovich, spectral theory and analytic geometry over nonarchimedean fields, ams, 1990 the above conditions are in corollary 9. The padic numbers are a counterintuitive arithmetic system that was discovered by the german mathematician kurt hensel in about 1899. Thanks to abhinav kumar for providing the corrections incorporated into this version. This is a correspondence between unitary continuous admissible representaitons of gl2q pvaluedinpadic banach spaces, and unitary continuous representations of d.
P adic banach spaces and families of modular forms. It showcases research results in functional analysis. Complemented subspaces of padic second dual banach spaces. The book is intended to be used with graduate courses in banach space theory, so the prerequisites are a background in functional, complex and real analysis. It is online only to give the audience of our lecture easy access to it. On orthocomplemented subspaces in padic banach spaces. Bhargav bhatt perfectoid spaces are a class of algebrogeometric objects living in the realm of p adic geometry that were introduced by peter scholze sch12 in his.
Escassut, ultrametric banach algebras, world scientific, 2003. Request pdf on orthocomplemented subspaces in padic banach spaces for linear subspaces of finitedimensional normed spaces over k, where k is a nonarchimedean complete valued field which is. In this case, the representations of g which took part in the correspondence were banach spaces with a ginvariant norm. There is a beautiful theory of representations over such and so there is no serious problem. Mathematically it is natural to work over the padic. Peter scholzes lectures on padic geometry, fall 2014 last updated on 29 jan. The complementation property of linfinity in p adic banach spaces.
It showcases research results in functional analysis over nonarchimedean valued complete fields. Peter scholzes lectures on padic geometry, fall 2014. We recall basic notions and conventions in banach spaces and banach. Padic banach spaces and families of modular forms robert.
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