Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. Threemanifolds class field theory homology of coverings for a nonvirtually haken manifold. Course notes and supplementary material pdf format. If you notice any mistakes or have any comments, please let me know. We have put artins and tates theory of class formations at the beginning. The rst half of this book deals with degree theory and the pointar ehopf theorem, the pontryagin construction, intersection theory, and lefschetz numbers. You should have a vague understanding of the use of complex multiplication to generate abelian extensions of imaginary quadratic fields first, in order. Suppose that c x is the wiesend id ele class group of x. Global class field theory note that when p is a prime ideal of o f and c j.
This is a digestible and excellent introduction to 3manifolds for the uninitiated. He laid the modern foundations of algebraic number theory by. One of the main questions to answer is to how many abelian extensions exists over a global or local field, and the numbertheoretic phenomena occurring in. Before discussing geometry, i will indicate some topological constructions yielding diverse threemanifolds, which appear to be very tangled. We introduce the notion of smooth cycle and then present some applications. The class of irreducible threedimensional manifolds differs from that of simple threedimensional manifolds by just three manifolds. Threemanifolds class field theory homology of coverings for a nonvirtually b 1positive manifold alexander reznikov 1, 2 selecta mathematica volume 3. This is the note for the class field theory seminar. For a certain set k of knots in a 3manifold m, we first present a local theory for each knot in k, which is analogous to local class field theory, and then, getting together over all knots in k, we give an analogue of idelic global class field theory for an integral homology sphere m. We will outline how class eld theory developed from these initial ideas through the work of kronecker, weber, hilbert, takagi. From a different perspective, it describes the local components of the global artin map. These results shed a surprising new light on conformal field theory in 1 f1 dimensions. Notes on class field theory notes from a onesemester course on class field theory uc berkeley, spring 2002.
In this talk, i want to rst prove some properties about the zeta functions and the l functions, and then use those properties to prove the universal norm inequality, and maybe the chebotarev density theorem. Computation of the norm residue symbol in certain local kummer fields 114 4. Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. Instead of the notion of a simple threedimensional manifold, it is often more useful to use the notion of an irreducible threedimensional manifold, that is, a manifold in which every sphere bounds a ball. This is a first in a series of papers, devoted to the relation betwwen threemanifolds and number fields. The theory had its origins in the proof of quadratic reciprocity by gauss at the end of the 18th century. A topological space x is a 3manifold if it is a secondcountable hausdorff space and if every point in x has a neighbourhood that is homeomorphic to euclidean 3space mathematical theory of 3manifolds. Unfortunately, it does not treat local class field theory.
Class field theory proofs and applications download. The present paper studies first homology of finite coverings of a threemanifold with primary interest in the. The topological, piecewiselinear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say. Local class field theory pan yan summer 2015 these are notes for a reading course with d. An original source for many of the ideas of global class field theory. In any case, the theory of geometry in threemanifolds promises to be very rich, bringing together many threads. The idele class group is a collection of multiplicative groups of local fields, giving a view of class field theory for global fields as a collection of class field theories for local fields. The chernsimons gauge theory on 3manifolds, its renormalization, geometric quantization, computation of partition functions by surgery, and relation with jones polynomials. Let each face be identi ed with its opposite face by a translation without twisting. In mathematics, class field theory is the branch of algebraic number theory concerned with the abelian extensions of number fields, global fields of positive characteristic, and local fields. Reznikov, threemanifolds class field theory homology of coverings.
You can imagine this as a direct extension from the 2torus we are comfortable with. In a lecture at the hermann weyl symposium last year 1, michael atiyah proposed two problems for quantum field theorists. For the additional material on intersection theory and lefschetz. A global function eld is a nite extension of f pt for some prime p, or equivalently is the function eld of a geometrically integral curve over a nite eld f q called the constant eld, where qis. Threedimensional manifold encyclopedia of mathematics. Class field theory local and global artin, emil, and john torrence tate. Find materials for this course in the pages linked along the left. The fourth section analyzes the role of symmetry in restricting the.
Su 3 chernimons field theory, threemanifolds 1991 msc. For the sake of formal simplicity we have used the notion of a. Class field theory in this chapter we will present unrami ed geometric abelian class eld theory which establishes a remarkable connection between the picard group and the abelianized etale fundamental group of a smooth projective curve over a nite eld. Threemanifolds class field theory homology of coverings for a nonvirtually haken manifold reznikov, alexander. Historically, local class field theory branched off from global, or classical, class field theory, which studies abelian extensions of global fieldsthat is.
We start with introducing the notion of a very admissible link k in m as an analogue of the set of primes of a number. The have been heavily revised and expanded from earlier versions. Geometric class field theory notes by tony feng for a talk by bhargav bhatt april 4, 2016 in the. Lectures on local fields pdf file 430k this is a very short introduction to local fields and local class field theory which uses an explicit description of the local reciprocity homomorphism and its inverse and does not use galois cohomology and the brauer group. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flatspace supermultiplet containing the rcurrent and the energymomentum tensor.
Class field theory studies finitedimensional abelian field extensions of number fields and of function fields, hence of global fields by relating them to the idele class group class field theory clarifies the origin of various reciprocity laws in number theory. Threemanifolds class field theory homology of coverings. These ideas were developed over the next century, giving rise to a set of conjectures by hilbert. Introduction in this article, we solve the nonabelian su 3 chernimons cs quantum field theory defined in a generic threemanifold a which is closed, connected and orientable. Notes on topological field theory xi yin harvard university introduction the notes give a survey of the basics of the following topological. In this part we follow closely the beautiful exposition of milnor in 14. Threemanifolds class field theory homology of coverings for a nonvirtually b 1positive manifold alexander reznikov 1,2. The present book is a mixture of an introductory text book on the geometrictopological theory of 3manifolds and a guide to some recent developments. Florian pop, advisor let kbe a nite eld, and suppose that the arithmetical variety x. Studied the riemann zeta function, and made the riemann hypothesis.
Introduction to 3manifolds arizona state university. The following theorem tells us exactly how the size of the ppart of the class group grows in a. We begin with stating the main theorem of the unrami ed theory in di erent. Citeseerx threemanifolds class field theory homology.