If you know about the padic numbers qp for primes p if you dont know about them. Ive also bene ted a lot from bhatts lecture notes for a class on perfectoid spaces. In the context of inner product spaces of ini nite dimension, there is a di erence between a vector space basis, the hamel basis of v, and an orthonormal basis for v, the hilbert basis for v, because though the two always exist, they are not always equal unless dimv adic group representations over a p adic vector space. Nt 18 jan 2019 cohomology of p adic stein spaces pierre colmez, gabriel dospinescu, and wiesl awa niziol abstract. Scholzes perfectoid spaces and etale cohomology of diamonds, together with a tiny amount of bhattm.
Lecture 1 is an introduction, explaining the motivation coming from the langlands correspondence and moduli spaces of shtukas. Generalized semiinvariant distributions on p adic spaces 1729 we suggest in this paper that, to describe all semiinvariant distributions on the spaces. Our main theorem is a product formula for these jhomomorphisms, saying what happens when you start with a rational vector. Chapter 8 vector spaces in quantum mechanics we have seen in the previous chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states. Alternatively, one sometimes defines vector space as a twosorted notion. Peter scholzes lectures on p adic geometry, fall 2014 last updated on 29 jan. A locally convex kvector space is called a kbanach space if its topology. One is the existence of absolute moments of operatorsemistable probability measures and another is an answer to the question whether one can get semistability of a probability measure from that of all its projections. Peter scholzes lectures on padic geometry, fall 2014. The answer is that there is a solution if and only if b is a linear combination of the columns column vectors of a.
We introduce the notion of a formal k vector space. In homotopy theory, this transformation gets encoded in a map called the real jhomomorphism. Moreover, we assume that p 1 and p 2 are dual in the sense of abelian varieties. In characteristic p, this statement is true in greater generality. Gwgiven by wextends to a wvalued rigid analytic function on g. Abstract vector spaces, linear transformations, and their coordinate representations contents 1 vector spaces 1.
The category of such objects is one possible padic. Finitedimensional topological vector spaces keith conrad uconn. However, the difference between two points can be regarded as a vector, namely the motion also called displacement or translation. Of course, there is no continuous action of galq on yc. Here we define and investigate p adic jhomomorphisms, which sort of turn p adic vector spaces into spheres. Generalized semiinvariant distributions on padic spaces. Note that the values of the padic value are bounded by 1 on z. Let v be a finitedimensional normed vector space over k. In this case a continuous representation may have in. Mirror symmetry for moduli spaces of higgs bundles via p adic integration 3 there exist families of abelian varieties p 1.
Quotient spaces oklahoma state universitystillwater. A vector space v is a collection of objects with a vector. A p 2, which act faithfully and transitively on the bres of m i. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. In the papers 46 the author used such measures to represent the dual spaces of certain spaces of continuous scalarvalued or vector valued functions. Special thanks to brian conrad for his thorough comments on every lecture. Let k be a complete ultrametric field, for instance k p, let e be a kvector space and u. The hodgetate decomposition via perfectoid spaces arizona. The study of norms on padic vector spaces it not new. Abstract vector spaces, linear transformations, and their. In many ways padic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of padic numbers is much simpler. Observables are linear operators, in fact, hermitian operators acting on this complex vector space. In this latter situation, the topology of dp is mostly incompatible with that of an.
Key examples are 1 the orthogonal group sov where v is a. Quotient spaces in all the development above we have created examples of vector spaces primarily as subspaces of other vector spaces. These authors were concerned with the metric topology. Completely continuous endomorphisms of adic banach spaces.
You have met or you will meet the concept of a normed vector space both in algebra and analysis courses. Inverse limits in the category of adic spaces 15 3. The hodgetate linear algebra datum is convenient, since it is only a graded vector. One key observation is that the local structures of adic spaces in the pro. Throughout this course k is a finite extension of the field qp of padic numbers. Formal vector spaces over a field of positive characteristic, preprint. In the proetale case, the main input is a comparison theorem for padic stein spaces. Moments and projections of semistable probability measures on. One of the goals of this course is to understand the proof of the following theorem, the hodgetate decomposition. We demonstrate that latticebased methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. Show that w is a subspace of the vector space v of all 3. Subspaces a subspace of a vector space v is a subset h of v that has three properties. There are further technical conditions that are omitted for the sake of brevity. In many ways p adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p adic numbers is much simpler.
Recently i have found a need to learn more about p adic group representations over a p adic vector space. Several other authors studied p adic measures which are either scalar valued or have values in the dual of a nonarchimedean locally convex space. V is a finite dimensional vector space over a ground field k of characteristic 2. Any vector in the three dimensional space can be written as a linear combi. Oct 17, 2006 in this paper, two topics on semistable probability measures on p adic vector spaces are studied. Locally analytic vectors in representations of padic groups 3 representation of g, then we say that this representation is ganalytic if for each w.
The set r of real numbers r is a vector space over r. The last bullet point is crucial and pretty amazing. In quantum mechanics the state of a physical system is a vector in a complex vector space. Dual spaces friday 3 november 2005 lectures for part a of oxford fhs in mathematics and joint schools linear functionals and the dual space dual bases annihilators an example the second dual. In this paper, two topics on semistable probability measures on p adic vector spaces are studied.
We compute padic etale and proetale cohomologies of drinfeld halfspaces. Background from padic functional analysis throughout this course kis a nite extension of the eld q p of padic numbers, jjdenotes the absolute value on k, o kthe ring of integers. Topological vector spaces over p adic fields show distinctive features. The theory has its beginnings in jeanpierre serre and john tates study of tate modules of abelian varieties and the notion of hodgetate representation. Theory and practice observation answers the question given a matrix a, for what righthand side vector, b, does ax b have a solution. The universal vector extension, and the universal cover 19 3. Vector spaces, bases, and dual spaces points, lines, planes and vectors. Pdf moments and projections of semistable probability. Moreover, very interesting examples of padic galois representations arise from geometry. The geometry of p adic symmetric spaces jeremy teitelbaum 1120 n otices of the ams v olume 42, number 10 m any of the geometric objects of interest to number theorists arise as. You will see many examples of vector spaces throughout your mathematical life. Thanks to arthur ogus and jay pottharst for helpful comments. A norm on a vector space which is defined over a field equipped with a discrete valuation a generalisation of p adic.
Notes by aleksander horawa these are notes from math 679 taught by serin hong in winter 2020, latexed by aleksander horawa who is the only person responsible for any mistakes that may be found in them. Moments and projections of semistable probability measures on p adic vector spaces article pdf available in journal of theoretical probability 192. Vector spaces and subspaces vector space v subspaces s of vector space v the subspace criterion subspaces are working sets the kernel theorem not a subspace theorem independence and dependence in abstract spaces independence test for two vectors v 1, v 2. The aim here is to use the example of the sterngerlach experiment to develop this idea further, and to show that the. Let v be a topological vector space over a nondiscrete locally compact field. The set r2 of all ordered pairs of real numers is a vector space over r. We prove that rapoportzink spaces at infinite level are perfectoid spaces, and give a description of these spaces purely in terms of p adic hodge theory. But it turns out that you already know lots of examples of vector spaces.
Another new ingredient is the pro etale site, which makes all constructions completely functorial. In fact when formulating the theta correspondence, one also needs a lemma amounting to the classical fact that two elements in the metaplectic group commute if and only if their images in the symplectic group commute. We also analyze determinants, characteristic polynomials and lu factorization. This category has vector spaces over k k as objects, and k klinear maps between these as morphisms multisorted notion. Onepoint compactification turns real vector spaces into spheres. When 6 p, these actions can be understood2 in terms of algebraic geometry over the. Such vectors belong to the foundation vector space rn of all vector spaces. The two key properties of vectors are that they can be added together and multiplied by scalars. This was originally proved by tate for abelian varieties with good reduction. Rudiments of this can be found in ols05, but, again, no. Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces.
For this purpose, ill denote vectors by arrows over a letter, and ill denote scalars by greek letters. Vector spaces and linear transformations beifang chen fall 2006 1 vector spaces a vector space is a nonempty set v, whose objects are called vectors, equipped with two operations, called addition and scalar multiplication. Topological padic vector spaces and index theory numdam. For topological spaces x and y, we denote by cx, y the kvector space of contin uous maps x y. Introduction to p adic hodge theory lectures by serin hong. Completely continuous endomorphisms of padic banach spaces by jeanpierre serre publ. Below well provide a construction which starts with a vector space v over a eld f and a subspace sof v, and which furnishes with an entirely new vector space. Consider the set fn of all ntuples with elements in f. Consider the variety xa1 0 over the eld c of complex numbers. In this course you will be expected to learn several things about vector spaces of course. Topological vector spaces over padic fields show distinctive features. It is online only to give the audience of our lecture easy access to it. Scholzes integral padic hodge theory in order to formulate some results in terms of integral perfectoid rings rather than perfectoid tate rings.
However, the difference between two points can be regarded as a vector, namely the. Moreover, very interesting examples of p adic galois representations arise from geometry. Completely continuous endomorphisms of p adic banach spaces by jeanpierre serre publ. We do not develop their theory in detail, and we leave the veri. A vector space is a set that is closed under addition and scalar multiplication. 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. In mathematics, p adic hodge theory is a theory that provides a way to classify and study p adic galois representations of characteristic 0 local fields with residual characteristic p such as q p. Metricandtopologicalspaces university of cambridge. Moments and projections of semistable probability measures. In brief, diamonds are to perfectoid spaces as algebraic spaces are to schemes.
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