If a function is analytic and single valued within a given region, we call it \regular. Rigid analytic modular symbols glenn stevens april 21, 1994 0. In general simple functions of z itself, not jzj, have regions where they are analytic. 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. Given a prime number p and the galois orbit ox of an element x of. Rigid analytic spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. 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. University of groningen essential singularities of rigid. While this is not something we will spend a lot of time on, it becomes much more important in some other classes, in particular complex analysis. This example explains the analytic function on the complex plane. 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.
We give an overview of the theory of dumodules on rigid analytic spaces and its applications to admissible locally analytic representations of padic lie groups. Cn i vanishing loci of set i complex analytic spaces. The editors of inventiones mathematicae believe that it is in the general interests of the mathematical community to make these notes available to everyone editorial board. Also known as holomorphic function explanation of rigidity theorem for analytic functions. Kedlaya and ruochuan liu november 21, 2016 abstract we prove that the cohomology groups of an. This work, a revised and greatly expanded new english edition of an earlier french text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, points of rigid spaces. Find out information about rigidity theorem for analytic functions. Locally analytic vectors in representations of padic groups 3 representation of g, then we say that this representation is ganalytic if for each w. To add items to a personal list choose the desired list from the selection box or create a new list. To close, click the close button or press the esc key.
Classical rigid geometry may be viewed as a theory of analytic functions over local. Till now, these notes were only distributed privately, via ihes. Analytical methods for dynamic simulation of nonpenetrating. 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. Rigidity theorem for analytic functions definition of. Rigid analytic geometry and its applications jean fresnel. Lecture notes on classical mechanics a work in progress. Rigidity theorem for analytic functions article about.
In mathematics, an analytic function is a function that is locally given by a convergent power series. An overview when they were distributed, tates fundamental notes on rigid analytic spaces t immediately received strong attention for several reasons. Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. However, to even define a structure sheaf or, equivalently, local analytic functions on them, we have to bring in the notion of grothendieck topology see.
The proofs of these properties are inspired by the local study of complex analytic spaces via weierstrass preparation techniques to carry out induction on n. Springer at the meeting of october 25, 1980 introduction the picard theorem for a complex analytic function can be formulated as follows. Apr 16, 2018 let k be a field of characteristic zero complete with respect to a non. The main lack of the latter approach is connected with the fact that the usual local definition of an analytic function as a powerseries expansion in a neighbourhood. Analytical methods for dynamic simulation of nonpenetrating rigid bodies david baraff program of computer graphics cornell university ithaca, ny 14853 abstract a method for analytically calculating the forces between systems of rigid bodies in resting noncolliding contact is presented. A locally ringed topological space which is locally isomorphic to zn p, sheaf of locally analytic function is called a q p analytic manifold, and it is not a rigid analytic space in the sense of tate. A function which can be represented by a convergent taylor series. In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Request pdf representation results for equivariant rigid analytic functions given a prime number p and the galois orbit ox of an element x of. Aug 04, 2004 for the big picard type theorem, we prove that if y is a nonsingular rigid analytic space, e is an effective simple normal crossings divisor on y, and if x is a smooth projective variety with a simple normal crossings divisor d such that x, d is of loggeneral type, then any dominant rigid analytic map f. What is the difference between analytical rigid and discrete rigid in abaqus software. The naive answer would be to follow real or complex analysis and to call a function analytic if it has, locally. Then you can look at points of rigid analytic spaces over extensions of the base field, in a much more agebrogeometric flavour. Exact analytic solution for the rotation of a rigid body.
Rigid analytic flatificators 3 to a maximal ideal mbin b, then a bis. The proofs of these properties are inspired by the local study of complexanalytic spaces via weierstrass preparation techniques to carry out induction on n. It is for instance always totally disconnected as a topological space, which already prevents from. In this spirit we will often use measuretheoretic conventions and write. These functions arise by piecing together elements of affinoid kalgebras. Heres our basic algebraic object, the tate algebra. We build upon faltings theory of almost etale extensions, ampli.
An introduction to rigid analytic geometry websupport1. Rigidity theorem for analytic functions synonyms, rigidity theorem for analytic functions pronunciation, rigidity theorem for analytic functions translation, english dictionary definition of rigidity theorem for analytic functions. Equivariant dmodules on rigid analytic spaces 3 abelian category c x of coadmissible domodules on x and showed that it behaves in several ways analogously to the category of coherent dmodules on complex smooth. The derivative of the exponential function is the function itself.
Minimal models of rigid analytic surfaces contents 1. Kalgebra a acts as a function algebra on its maximal spectrum and the resulting space x is called an a. The book visual complex analysis gives a good explanation. First we recall the following facts, which are useful in their own right. Finiteness of cohomology of local systems on rigid analytic. A smooth function of two real variables may map disks to ellipses. Why do functions in complex analysis behave so well. Several approaches to nonarchimedean geometry stanford. Gwgiven by wextends to a wvalued rigid analytic function on g. Locally analytic vectors in representations of padic groups 3 representation of g, then we say that this representation is g analytic if for each w.
The vector spaces v v and w w may be generalised to analytic manifolds and more generally analytic spaces. Looking for rigidity theorem for analytic functions. Siegfried bosch lectures on formal and rigid geometry. The function is analytic throughout a region in the complex plane if f. Another new ingredient is the pro etale site, which makes all constructions completely functorial. Let p be a prime 3 and consider the tate algebra a. What is the difference between analytical rigid and. Analytic banach space valued functions let b be a banach space and d be an open subset of c. Notes on analytic functions northwestern university. Finiteness of cohomology of local systems on rigid analytic spaces kiran s.
There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. That is, a real valued function can distort disks in a way that analytic functions cannot. The space x carries a canonical topology induced by the topology of k. Oct 11, 2018 applications include the construction of \etale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic. Analytic and entire functions mercedes lueck march 4, 2004 contents 1 what is a complex function. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists. However, the main application of this theorem will appear in a future paper 7 with gardener on quanti. A function is said to be analytic in the region t of complex plane x if, fx has derivative at each and every point of x and fx has unique values that are it follows one to one function. Analytic function real and complex function properties. An analytical rigid does not need to be meshed, and instead can be described by an analytical function. First, there was the fascinating design of a new analytic theory which, for the. However, these are manifolds and varieties modelled on vector spaces using analytic transition functions, so the notion of analytic function between vector spaces is most fundamental. Bounded linear endomorphisms of rigid analytic functions. Representation results for equivariant rigid analytic.
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