Let k be a complete ultrametric field, for instance k p, let e be a kvector space and u. 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. The answer is that there is a solution if and only if b is a linear combination of the columns column vectors of a. Throughout this course k is a finite extension of the field qp of padic numbers. We also analyze determinants, characteristic polynomials and lu factorization. This was originally proved by tate for abelian varieties with good reduction and more generally a result for pdivisible groups, and has since been. Several other authors studied padic measures which are either scalar valued or have values in the dual of a nonarchimedean locally convex space. Our main theorem is a product formula for these jhomomorphisms, saying what happens when you start with a rational vector. Such vectors belong to the foundation vector space rn of all vector spaces. This is an analogy to riesz theorem on the selfduality for hilbert spaces. We now construct the analogous sequence at the level of analytic spaces. Recently i have found a need to learn more about padic group representations over a padic vector space. A locally convex kvector space is called a kbanach space if its topology.
In the papers 46 the author used such measures to represent the dual spaces of certain spaces of continuous scalarvalued or vectorvalued functions. Moreover, very interesting examples of padic galois representations arise from geometry. We will give an explicit way to compute orthogonal bases in padic vector space and will define an ultrametric on the padic projective space. In spring 2020, i am teaching both sections of math 3527 number theory 1 and i am also teaching math 4571 advanced linear algebra in summerii 2020, i will be teaching both sections of math 3081 probability and statistics. There is a classical relation between the padic absolute value of the eigenvalues of frobenius on crystalline cohomology and hodge numbers, for a variety in charac. We observed last session that matrices can be added and multiplied by scalars in the same way as vectors.
One of the goals of this course is to understand the proof of the following theorem, the hodgetate decomposition. On certain extensions of vector bundles in padic geometry serin hong abstract. Let g be a connected reductive group over a padic local field f. Generalized semiinvariant distributions on padic spaces. Matrices also have a multiplicative structure, which is not commutative. A vector space with more than one element is said to be nontrivial. Peter scholzes lectures on padic geometry, fall 2014 last updated on 29 jan. The study of norms on padic vector spaces it not new. When 6 p, these actions can be understood2 in terms of algebraic geometry over the.
Onepoint compactification turns real vector spaces into spheres. Nt 18 jan 2019 cohomology of padic stein spaces pierre colmez, gabriel dospinescu, and wiesl awa niziol abstract. A lattice is a discrete subgroup of a euclidean vector space, and geometry of. In reality, linear algebra is the study of vector spaces and the functions of vector spaces linear transformations. Locally analytic vectors in representations of locally padic analytic. What about the other padic galois representations, not coming from algebraic geometry. In this case a continuous representation may have in. The aim of this chapter is to set up the theory of padic integration to an extent which is su. Generalized semiinvariant distributions on p adic spaces 1731. It is natural to hope that after tensoring with the padic version of c generally written c p they two vector spaces would naturally be isomorphic. These authors were concerned with the metric topology. We demonstrate that latticebased methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. The set of all vectors in 3dimensional euclidean space is.
Observables are linear operators, in fact, hermitian operators acting on this complex vector space. 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. The bernstein center of a padic unipotent group justin campbell september 4, 2019 abstract fran. In homotopy theory, this transformation gets encoded in a map called the real jhomomorphism. Examples of such representations are provided by finite. Generalized semiinvariant distributions on p adic spaces. Describing these representations proved to be crucial in the p. In general, p adic uniformization replaces complex symmetric spaces xwith p adic analytic spaces xone in each dimension. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every. Given three arbitrary vector bundles on the farguesfontaine curve where one of them is assumed to be semistable, we give an explicit and complete criterion in terms of hardernarasimha polygons on whether there exists a short exact sequence among them. Topological padic vector spaces and index theory numdam.
Gklocally convex topological vector spaces over qp. V is a finite dimensional vector space over a ground field k of characteristic 2. Topological vector spaces over padic fields show distinctive features. Peter scholzes lectures on padic geometry, fall 2014. Note the basics of padic analysis, padic mathematical physics are explained in 5,29. A eld is roughly a set of numbers with addition and multiplication operations that obey all the rules of arithmetic. I am an assistant teaching professor of mathematics at northeastern university. Since vector spaces are always given in terms of a scalar eld, we need to start with them. The padic general linear group acts on x, and, as with complex symmetric spaces, the quotients of x by discrete groups sometimes turn out to be algebraic varieties defined over number fields. The additive identity in this case is the zero polynomial, for which all coe.
The hodgetate decomposition via perfectoid spaces arizona. The p adic general linear group acts on x, and, as with complex symmetric spaces, the quotients of x by discrete groups sometimes turn out to be algebraic varieties defined over number fields. The theory has its beginnings in jeanpierre serre and john tates study of tate modules of abelian varieties and the notion of hodgetate representation. They form the fundamental objects which we will be studying throughout the remaining course. Strictly speaking, maybe we should underline it now, g a is what you expect in. The norm of a nonzero vector is always a positive real number, even for vector spaces whose scalars arent real numbers. In this course you will be expected to learn several things about vector spaces of course. Theory and practice observation answers the question given a matrix a, for what righthand side vector, b, does ax b have a solution. Aguayo, vector measures and integral operators, in ultrametric functional analysis, cont. We learned that some subsets of a vector space could generate the entire vector space.
Generalized semiinvariant distributions on padic spaces 1731. Some examples of valuations are the absolute value of real numbers, the modulus of complex numbers and the padic metric of padic numbers. Introduction to vector spaces this section introduces the concept of vector space. In quantum mechanics the state of a physical system is a vector in a complex vector space. In mathematics, padic hodge theory is a theory that provides a way to classify and study padic galois representations of characteristic 0 local fields with residual characteristic p such as q p. A vector space v is a collection of objects with a vector. 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. In general, padic uniformization replaces complex symmetric spaces xwith padic analytic spaces xone in each dimension. Thanks to arthur ogus and jay pottharst for helpful comments. But for this to be useful, should be as simple as possible, and. The vector spaces seem to have been first introduced in.
The vector space v in particular is an omodule so that we can speak about osubmodules of v. Special thanks to brian conrad for his thorough comments on every lecture. On padic vector measure spaces connecting repositories. The idea is to nd a big extension k 1k, which should be an in nite and deeply rami ed galois extension, and we write for its galois group.