Nbasic notions of rigid analytic geometry bookshelf

At the outset the plane is just a set s where the elements p are called points. The notion of derived nonarchimedean analytic space is based on the. Schneider basic notions of rigid analytic geometry, appeared in. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. Then i will discuss floer cohomology of a number of singular torus fibrations, the nodal singularity in dimension two which is a self mirror, and, as time permits, the positive and negative singularities in dimension 3.

Tate and others developed the theory of rigid analytic geometry in order to at least make coherent sheaf theory including. Linear algebra and analytic geometry for physical sciences. Newest rigidanalyticgeometry questions mathoverflow. This book starts with a concise but rigorous overview of the basic notions of projective geometry, using straightforward and modern language. This makes it possible to apply the usual notions of homotopy and singular homology. Geometry and generality in freges philosophy of arithmetic jstor. Analytic geometryanalytic geometry basic conceptsbasic concepts 2. Citeseerx document details isaac councill, lee giles, pradeep teregowda. I will start by discussing floer cohomology on the cotangent bundle of the ntorus and its relation to rigid analytic geometry. The book will be useful to students taking a physics or engineer degree for a basic education as well as for students who wish to be competent in the subject and who may want to pursue a postgraduate qualification. Rigid analytic geometry often just rigid geometry for short is a form of analytic geometry over a nonarchimedean field k which considers spaces glued from polydiscs, hence from maximal spectra of tate algebras quotients of a k algebra of converging power series. We begin by agreeing that the basic ingredients in our study are points and lines in a plane.

Abstract brian conrad modular curves and rigid analytic spaces most geometers believe that one cannot really do geometry over a nonarchimedean complete eld, in contrast to the situation over the complex eld. Strong connections between tropical algebraic geometry and the theory of rigid analytic spaces allow one to prove theorems in one. The topology on affinoids is subtle, using notions of affinoid subdomains. Similarity, congruence, and proofs july 2019 page 4 of 202 similarity transformations rigid motions followed by dilations define similarity in the same way that rigid motions define congruence. Modeling, geometry, and statistics and probability. This is the original paper where the notion of rigid analytic spaces was introduced, and also the socalled \tates acyclicity theorem was rst proved.

The book which contains an extensive bibliography is a very valuable source for everyone wishing to learn about rigid geometry or its applications. We will ignore this technical, but absolutely crucial, point here, and assume that the notion of a perfectoid kspace, built by glueing together spectra of perfectoid k. Functions analytic in a neighborhood of a closed subset 44 chapter 3. Ek and an isomorphism of analytic spaces ean gan m. Overview of rigid analytic geometry this is very much a. Today rigid geometry is a discipline in its own right and has acquired vast and rich structures based on discoveries of its relationship with birational and formal geometries.

Rigid analytic geometry often just rigid geometry for short is a form of analytic geometry over a nonarchimedean field k which considers. Master mosig introduction to projective geometry chapter 1 introduction 1. Representability theorem in derived analytic geometry 5 the proof of theorem 1. This book requires a background at the level of basic graduate.

We develop a rigid analytic theory of relative ampleness for line bundles and record some applications to faithfully flat descent for morphisms and proper geometric objects. Riemann took as basic an antecedent of freges view that attributions of. Analytic geometryanalytic geometry a branch of mathematics which uses algebraic equations to describe the size and position of geometric figures on a coordinate system. Many of the subjects treated in this book are not easily available from the literature. Tates rigid analytic geometry, which aimed at developing an analytic geometry over nonarchimedean valued fields. Precise definitions of the key notions and precise statements of the basic facts were given. Rigid analytic geometry and its applications jean fresnel springer. Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and padic analytic geometry. The goal is not only to establish the notation and terminology used, but also to offer the reader a quick survey of the subject matter. Siegfried bosch lectures on formal and rigid geometry. Rigid analytic spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. Moduli spaces of pdivisible groups and period morphisms. The basic definition is fibral, but pointwise arguments from the algebraic and complex analytic cases do not apply, so we use cohomological. See below for a selection of the latest books from analytic geometry category.

The distance between two points in the plane is the length of the line segment joining the two points. This is in contrast to some modern approaches to nonarchimedean analytic geometry such as berkovich spaces which are glued from berkovichs analytic spectra and more recent hubers adic spaces. Rigid geometry analytic approach to arithmetic geometry. The basic definition is fibral, but pointwise arguments from the algebraic and complex analytic cases do not apply, so we use cohomological properties of. Georgia standards of excellence curriculum frameworks. Contents i rigid analytic geometry 3 1 introduction 3. The basic rigid analytic object is the ndimensional unit polydisc, whose ring of. These notes1 are intended to be a short course in rigid analytic geometry, without, however, providing always proofs. Dmodules on rigid analytic spaces i university of oxford. To do all of this we need rigid analytic spaces the subject of the next section. Galois representations in arithmetic algebraic geometry durham, 1996, 369378, london math. Tropical analytic geometry, newton polygons, and tropical. I will assume familiarity with the language of algebraic geometry math 631, math 632. Accelerated gse coordinate algebra analytic geometry a unit 5 mathematics accelerated gse coordinate algebra analytic geometry a unit 5.

The purp ose of m y lectures at the conference w as to in tro duce the new comer to the eld of rigid analytic geometry precise denitions of the k ey notions and precise statemen ts of the basic facts w ere giv en but of course the limited. Let us give a quick outline for the convenienceofthereader. Thefunctoraboveisinducedbytheunderlyingalgebrafunctorforgettingtheanalytic structure. The purpose of these notes is to introduce the basic notions of rigid analytic geometry, with the aim of discussing the nonarchimedean uni. Transformations in the coordinate plane july 2019 page 3 of 87 overview in this unit students will. Classical rigid geometry may be viewed as a theory of analytic functions over local. Hilbert also treats geometry in 3space, but we will only consider the 2dimensional case.

Rigid geometry is a new framework of geometry, introduced by j. Basic notions of rigid analytic geometry p eter sc hneider. Abstract brian conrad modular curves and rigid analytic. Looking for basic example in the theory of adic spaces. For a brief account see schneiders survey sch, for more detailed accounts see the lecture notes bo by. Although this framework has initially sprouted from the attempts to copy complex analytic geometry to the non. The author also studies the connection with the earlier notion of a rigid analytic space. Spectral theory and analytic geometry over nonarchimedean. Tate around 1960 first as an analytic geometry over nonarchimedean valued fields, such as padic fields. The interest of projective geometry arises in several visual computing domains, in particular computer vision modelling and computer graphics. The aim of global analytic geometry is to construct a category of spaces which contains these three geometries. The exposition is self contained, the authors only assume some familarity with basic algebraic geometry.

An appendix collects basic algebraic notions like group, ring and field. With expert reading recommendations made by people with a passion for books and some unique features lovereading will help you find great analytic geometry books and those from many. Gse analytic geometry unit 1 mathematics 1gse analytic geometry unit. The formulation and the proof of these criteria involve some basic notions in arakelov geometry, combined with complex and rigid analytic geometry notably, potential theory over complex and p. The standards in the traditional geometry course come fro the following conceptual categories. The purpose of this book is to introduce a new notion of analytic space over a nonarchimedean field.

In mathematics, a rigid analytic space is an analogue of a complex analytic space over a. Prior experience with some form of rigid analytic geometry say familiarity with the results discussed in bo or sch, or participation in mattias jonssons class on berkovich spaces last fall is not necessary, but will be very. Analytic geometry basic concepts linkedin slideshare. An affinoid is then a subset of the unit polydisc on which the elements of this ideal vanish, i. Spectral theory and analytic geometry over nonarchimedean fields.