Classical algebraic geometry dealt with complex varieties defined by a system of polynomial equations in several variables. Several approaches to nonarchimedean geometry brian conrad 9 introduction 9 1. The purpose of this book is to introduce a new notion of analytic space over a nonarchimedean field. My phd advisor was brian conrad at stanford university. Developed beginning in the seventeenth century, it is also known as cartesian geometry or coordinate geometry. Georgia standards of excellence framework gse analytic geometry unit 4 mathematics gse analytic geometry unit 4. Analytic geometry based on the georgia standards of excellence and effective beginning with winter 2015 endofcourse administration. Then, there is a functor from the category of strictly kanalytic spaces berkovich. Seminar on rigid analytic space and berkovich space. What is the difference between total integral closure and integral closure geometrically, in the context of rigid analytic geometry. He is an expert in arithmetic geometry, a subfield of number theory which provides geometric formulations of the abc conjecture the viewpoint studied in mochizukis work. Following invaluable introductions by john tate and vladimir berkovich, two pioneers of nonarchimedean geometry, brian conrad s chapter introduces the general theory of tates rigid analytic spaces, raynauds view of them as the generic fibers of formal schemes, and berkovich spaces. Derived hom spaces in rigid analytic geometry 5 347070, and from the ky fan and yufen fan membership fund and the s.
Rigid geometry of curves and their jacobians werner. Analytic geometry for colleges, universities, and technical. We define a new category of nonarchimedean analytic spaces over a complete discretely valued field, which we call uniformly rigid. Introduction complex analytic geometry is a powerful tool in the study of algebraic geometry. This is a classical result due to artin 70, theorem 7. We give a rigid analytic version of hironakas embedded resolution of singularities over an algebraically closed field of characteristic zero, complete with respect to a nonarchimedean norm.
This makes it impossible for a student of analytic geometry to get a correct sense of what a proof is. The basic concepts of analytic geometry are the simplest geometric representationspoints, straight lines, planes, curves, and secondorder surfaces. The purpose of these notes is to introduce the basic notions of rigid analytic geometry, with the aim of discussing the nonarchimedean uniformizations of certain abelian varieties. We explain the basic ideas and facts in rigid geometry from todays point of view. Half a century of rigid analytic spaces siegfried bosch dedicated to john tate on the occasion of his eightieth birthday. In fact, if you are a math major in college, you probably know that there are a number of disciplines within geometry. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space. The group places a long piece of wood on the edge of the bookshelf and angles it down so that it touches kalebs head and the ground. A generalization of formal schemes and rigid analytic varieties 535 b f. Valuation seminorm algebraicnumbers analytic functions number. The use of a coordinate system to relate geometric points. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Berkovich, spectral theory and analytic geometry over nonarchimedean fields, mathematical surveys and monographs, vol. We prove that the category of coadmissible modules supported on a closed smooth subvariety of is naturally equivalent to the category of coadmissible modules and use this result to construct a large family of pairwise nonisomorphic simple coadmissible.
Rigid geometry is one of the modern branches of algebraic and arithmetic geometry. Classical rigid geometry may be viewed as a theory of analytic functions over local. I can be reached at warner at math dot columbia dot edu or in office 512 in the math building. In this first lecture, we discuss the commutative algebra that forms the foundation for the local theory of rigidanalytic spaces, much as. Starting out from tates harvard notes, the classical approach by the school of grauert and remmert is covered, as well as the. If it be said that analytic geometry is not concerned with proof, i. Several approaches to nonarchimedean geometry stanford. Review of analytic geometry the points in a plane can be identi. This site is to help you keep up with assignments and dates as well as your overal progress throughout the school year. I do number theory and algebraic geometry, especially nonarchimedean analytic geometry. Every affinoid space may be endowed with a canonical rigid structure, which is preserved under morphisms.
In the early 1990s, huber developed the theory of adic spaces, so far the most general and powerful version of rigid analytic geometry. Gse analytic geometry unit 1 mathematics 1gse analytic geometry unit. Analytic geometry is a branch of mathematics that uses algebraic equations to describe the size and position of geometric figures. An introduction to berkovich analytic spaces and non. The approach here is due to tate, who introduced the notion of rigid analytic spaces. Tates rigid analytic geometry, which aimed at developing an analytic geometry over nonarchimedean valued fields. Tate and vladimir berkovich, two pioneers of nonarchimedean geometry, brian conrads chapter introduces the general theory of tates rigid analytic spaces. The purpose of this work is to give a concise treatment on the existence of blowing up in rigid analytic geometry.
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a. Kedlaya, fall 2004 more corrections on berkovich spaces ill get this right eventually, i promise. In this chapter the author discusses the construction of the triangulated category of rigid analytic motives, and an extension to rigid analytic geometry of a large number of voevodskys fundamental results such as his theory of homotopy invariants presheaves with transfers. Moishezon spaces in rigid geometry 3 berkovich spaces we allow for the possibility that the absolute value on kis trivial.
Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete nonarchimedean valuation base fields. Brian conrad, several approaches to nonarchimedean geometry, lectures at arizona winter school 2007, pdf. Number theory and representation theory seminar analytic number theory, algebraic number theory, arithmetic geometry, automorphic forms, and even some things not beginning with the letter a. The aim of this paper is to develop a rigid analytic theory of relative ampleness for line bundles, and to record some applications to rigid analytic faithfully flat descent for morphisms and for proper geometric objects equipped with a relatively ample line bundle. The purpose of my lectures at the conference was to introduce the newcomer to the field of rigid analytic geometry. Tate and others developed the theory of rigid analytic geometry in order to at least make coherent sheaf theory including. Choose from used and new textbooks or get instant access with etextbooks and digital materials. I am a ritt assistant professor at columbia university. One might, for example, study the surface in threedimensional space defined by. We use the word quadrant for each of the four regions into which the plane is divided by the axes. We start by drawing two perpendicular coordinate lines that intersect at the origin on each line. There is a classical theory of k analytic manifolds often used in the theory of algebraic groups with ka local. These notes1 are intended to be a short course in rigid analytic geometry, without, however, providing always proofs. Moreover, under suitable conditions analytification is a fully faithful functor.
Spectral theory and analytic geometry over nonarchimedean. Recommend this journal email your librarian or administrator to recommend adding this journal to your organisations collection. Symplectic structures in rigid geometry mathoverflow. It extends the category of rigid spaces, and it can be described in terms of bounded functions on.
It encodes the hidden deformationtheoretic informations of the underlying classical moduli space. Compatibility of formal completion and rigid analytic. Mathematica provides an attractive environment for studying analytic geometry. Jan 23, 2018 we construct a derived enhancement of hom spaces between rigid analytic spaces.
The overflow blog socializing with coworkers while social distancing. Rigid analytic geometry and abelian varieties mihran papikian abstract. A textbook account of the proof is in neeman 07, section 10. This theory is a padic analogue of complex analytic geometry. 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. An introduction to berkovich analytic spaces and nonarchimedean potential theory on curves matthew baker1 introduction and notation this is an expository set of lecture notes meant to accompany the authors lectures at the 2007 arizona winter school on padic geometry. The basic rigid analytic object is the n dimensional unit polydisc, whose ring of functions is the tate algebra, made of power series in n variables whose coefficients approach zero in some complete nonarchimedean field k.
Second, the basic algebras of rigid geometry are the tate algebras kt1. Abstract brian conrad modular curves and rigid analytic. Citeseerx relative ampleness in rigidanalytic geometry. However, the examples will be oriented toward applications and so will take some thought. In addition to defining analytic functions over nonarchimedean fields, berkovich spaces also have a nice underlying topological space. Following invaluable introductions by john tate and vladimir berkovich, two pioneers of nonarchimedean geometry, brian conrads chapter introduces the general theory of tates rigid analytic spaces, raynauds view of them as the generic fibers of formal schemes, and berkovich spaces. Boschs booklectures on formal and rigid geometry berkovichs note on berkovich space. Analytic geometry and calculus i exam 1 practice problems. Extending the number system july 2019 page 4 of 41 use properties of rational and irrational numbers. 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. Nicases note on the introduction on formal and rigid geometry. In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. The text is written for the oneterm analytic geometry course, often taught in sequence with college algebra, and is designed for students with a reasonably sound background in algebra, geometry, and trigonometry.
Schedule and notes for the 201718 seminaire godement. Conrad, moishezon spaces in rigid geometry, in preparation 2009. The representability theorem provides us sufficient and necessary. Conrad s note on minimal models of elliptic curves. In this course, we will study the basics of hubers theory and the connection with the other variants of rigid analytic geometry mentioned above. Half a century of rigid analytic spaces international press. Kaleb stands parallel to the bookshelf at a distance of 2. The tate algebra is the completion of the polynomial ring in n variables under the gauss norm.
Brian conrad several approaches to nonarchimedean geometry lecture. Question 2 eliminate the parameter t to nd a cartesian equation of the curve given parametrically by the relations. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. What are some good analytic geometry books for beginners. Rigid analytic geometry often just rigid geometry for short is a form of analytic geometry over a nonarchimedean field k k which considers spaces glued from polydiscs, hence from maximal spectra of tate algebras quotients of a k kalgebra of converging power series.
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, thereby formalizing the similarity ideas of same. The basic means of analysis in analytic geometry are the method of coordinates and the methods of elementary algebra. Constancy of generalized hodgetate weights of a local. In general, an analytic extension eld kkis one that is complete with respect to a xed absolute value extending the one on k. Algebraic spaces are locally of nite type over kand maps between them are kmaps unless otherwise. Siegfried bosch lectures on formal and rigid geometry. Notes on the oxford iut workshop by brian conrad mathbabe. The basic definition is fibral, but pointwise arguments from the algebraic and complex analytic cases do not apply, so we use cohomological. Several approaches to nonarchimedean geometry the basic properties of t n are summarized in the following result that is analogous to properties of polynomial rings over a. Morphisms of spaces with a rigid structure, and also the concept of the rigid structure induced on a subspace, are defined by analogy with these concepts for ringed spaces. The second chapter deals with voevodskys approach based on transfers. In the x,y coordinate system we normally write the xaxis horizontally, with positive numbers to the right of the origin, and the yaxis vertically, with positive numbers above. Georgia standards of excellence curriculum frameworks mathematics. Brian conrad gave another proof as an application of his results of relative ampleness in the rigid analytic setting see relative ampleness in rigid geometry, ann.
Several approaches to nonarchimedean geometry brian conrad1 introduction let kbe a nonarchimedean eld. Several approaches to nonarchimedean geometry brian conrad1 introduction let k be a nonarchimedean. Please visit regularly for updates, and for easy access, please use the qr code you were provided. 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. The proofs of these properties are inspired by the local study of complexanalytic spaces via weierstrass preparation techniques to carry out induction on n. We give a rigid analytic version of hironakas embedded resolution of singularities over an algebraically closed. Such spaces were introduced by john tate in 1962, as an outgrowth of his work on uniformizing padic elliptic curves with bad reduction using the multiplicative group.
This respected text makes extensive use of applications and features items such as historical vignettes to make the material useful and interesting. Rigid analytic geometry and its applications jean fresnel springer. Discussion in more general analytic geometry is in conradtemkin 09, section 2. Students, engineers and mathematicians alike who are interested in analytic geometry can use this book and software for the study, research or just plain enjoyment of analytic geometry. In the fall we will discuss adic spaces and their relation with rigidanalytic geometry, in the winter we will discuss the basics of perfectoid spaces especially the tilting functor and the almost purity theorem, and in the spring we will discuss some applications to padic hodge theory. Nonarchimedean analytification of algebraic spaces. This form of mathematics refers to the geometry of analytic variables, also known as analytic functions. Berkovich gave a definition which gives nice analytic spaces over such, and also gives back the usual definition over. For an introduction into rigid analytic geometry, we refer to the book bgr. Brian conrad, several approaches to nonarchimedean geometry.
Jan 08, 2017 if you pull up your textbook list and discover that one or more of these are on it, you can breathe a sigh of relief and know that they will provide you with a comprehensive overview of the material so that you can succeed in your classes. Analytic geometry article about analytic geometry by the. Analytic geometry study guide 8 a group of students are indirectly calculating the height of a bookshelf. In recent decades, padic geometry and padic cohomology theories have become indispensable. Analytic geometry matematik bolumu, mimar sinan guzel. This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their jacobians, and of abelian varieties all of them defined over a complete nonarchimedean valued field. Brian conrad is a math professor at stanford and was one of the participants at the oxford workshop on mochizukis work on the abc conjecture.
Rigid analytic geometry often just rigid geometry for short is a form of analytic geometry over a nonarchimedean field k which considers. Pdf geometriccinematic synthesis of planetary mechanisms. Rigid analytic spaces vs berkovich spaces vs formal schemes. Textbook writers of recent times do not make it clear which way they are going. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Rigid patching this section, like section 3, discusses an approach to carrying over the ideas of section 2 from complex curves to more general curves. We develop a rigid analytic theory of relative ampleness for line bundles and record some applications to faithfully flat descent for.
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