Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of. Spiess, class formations and higherdimensional local class. For higher dimensional fields we study the case of higher local fields of positive characteristic. The book grew as an extended version of talks given at the conference in m.
By studying norm groups of cohomological objects and using cohomological approach to higher local class field theory the existence theorem is proved. Department of mathematics university of nottingham nottingham ng7 2rd england email. Local class field theory pan yan summer 2015 these are notes for a reading course with d. This is a concise survey of links between galois module theory and class field theory cft. Class field theory abstract these notes are based on a course in class eld theory given by freydoon shahidi at purdue university in the fall of 2014.
A brief summary of the main statements of class field theory. A refinement of the local class field theory of serre and. Suppose that c x is the wiesend id ele class group of x. Introduction class eld theory is the description of abelian extensions of global elds and local elds. For a polynomial fx 2zx, kronecker considered the number n. Class field theory describes the abelian extensions of a local or global. This form is called the categorical geometric langlands conjecture since dmodules on moduli stacks of gprincipal bundles play a central role in gauge quantum field theory in particular as hitchin connections on bundles of conformal blocks of g gchernsimons theory holographically dual to the wzw model 2d conformal field theory and since the.
Along the way we prove new results on space filling curves on arithmetic schemes and on the class field theory of local rings. The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i. The first axiom in class formation is again hilbert 90. For a local field with characteristic 0, a large number of studies have been made even for higher dimensional open varieties over the local field e. From theory to practice by george gras should have many examples. Class formations and higher dimensional local class field. In this lecture we give a brief overview of local class field theory, following.
The following survey of connesmarcolli work has a quick introduction to algebraic number theory including basic notions of cft. Class formations and higher dimensional local class field theory. The theory is referred to as higher local class field theory and higher global class field theory. Higher local reciprocity homomorphism from this quotient of the nth milnor kgroup to the galois group of the maximal abelian extension of the higher local field has many features similar to those of the onedimensional local class field theory. The label \class eld refers to a eld extension satisfying a technical property that is historically related to ideal class groups, and one of the main theorems is that class elds. This thesis investigates class field theory for one dimensional fields and higher dimensional fields. We show how one can deduce the more classical version of higher global class field theory due to kato and saito from wiesends version. Fesenkolocal fields, local class field theory, higher local class field theory via algebraick. Introduction to local class field theory shuji saito university of tokyo local class eld theory is a fundamental pillar of the number theory. Roquette received december 9, 1995 the reciprocity law of higher dimensional local class field theory is proved with the help of class. Invitation to higher local fields school of mathematical sciences. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. The aim of this book is to provide an introduction to higher local.
Viewing higher local fields as ring objects in the category of iterated proindobjects, a definition of open subgroups in milnor kgroups of the fields is given. Im wondering what are peoples opinions of the book above, can you give me some suggestions on learning class field theory, and could you recommend some good books on class field theory. Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. Seminar on higher dimensional class field theory fall 2014 wednesdays 67.
The main content of the thesis is divided into two parts. Roughly speaking, its main theorem identi es the galois group of a. The main theorem of local class field theory consists of the isomorphism theorem and existence. This was first done by artin and tate using the theory of group cohomology. When general theory proves the existence of some construction, then doing it explicitly is a useful exercise that helps one to keep a grip. Normal subgroups in the cremona group cantat, serge, lamy, stephane, and cornulier, yves, acta mathematica, 20. In this post we discuss local class field theory specifically looking at adic fields with a focus on the broader picture, and the multiple approaches goal of post.
Class formations ans higher dimensional local class field theory. This theory is about extensionsprimarily abelianof local i. Chow group of 0cycles with modulus and higherdimensional class field theory kerz, moritz and saito, shuji, duke mathematical journal, 2016. A classic reference that rewards the effort you put into it. Pdf invitation to higher local fields, part i, section 5. Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. What does tate mean when he wrote higher dimensional class. Im not sure which tate is actually referring to, since i dont have a copy of the new preface.
If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your. There are undoubtably mistakes in these notes, and they are the authors alone. Galois cohomology and brauer group of local fields, trudy mat. It explores various uses of cft in galois module theory, it comments on the absence of cft in contexts where it might be expected to play a role and. A nonarchimedean local eld of characteristic p0 is isomorphic to f qt for some power qof p.
Bell mathematics department, the cooper union school o enuineerinii, 41 cooper square, new york. Therelationbetween the galois cohomology group and the brauer group is well known, so we consider only the ppart. What does tate mean when he wrote higher dimensional. The goal of local class field theory is to classify all finite abelian. Kurihara, is a p resentation of main ingr edients of kato s higher local class. The selfduality of the additive group of a higher local field is proved. It begins with the structure theory of local fields, develops group cohomology from scratch, and then proves the main theorem of local class field theory.
The aim is to provide an introduction to higher local fields more generally complete discrete valuation fields with arbitrary residue field and render the main ideas of this theory part i, as. Kato introduced certain complexes cr,sx of gerstenblochogus type whose. On the properties of the vostokov and parshin pairing in. It brings together the quadratic and higher reciprocity laws of gauss, legendre, and others, and vastly generalizes them. Historically, local class field theory branched off from global, or classical. Finally, the key to the proof of theorem 3 is the reciprocity law for 2dimensional complete local. So, before i begin actually writing about local class field theory, id like to make clear what my goals for this post are. The notes were typed by graduate students daniel shankman and dongming she. This is a graduated student seminar on higher dimensional class field theory held in harvard. Local number fields, brauer groups, galois cohomology. The reciprocity law of higher dimensional local class field theory is proved with the help of class formations. Here is a more precise outline of the contents of these four parts. It uses algebraic k theory and appropriate milnor kgroups replace which is in use in onedimensional class field theory. These are notes for a course in local class field theory taught at caltech winter term of 2008.
Local class field theory university of british columbia. In this note we present a new selfcontained approach to the class field theory of arithmetic schemes in the sense of wiesend. But also its worth keeping in mind the following comment of miles reid from his undergraduate algebraic geometry p. Kato homology of arithmetic schemes and higher class field.
The remaining sections are of a global or semiglobal nature and do not rely on sections. Higher reciprocity laws 200 thepowerresiduesymbol,201. In global class field theory, the idele class group plays the role of in local class field theory. For local class field theory, there is local fields and their extensions by i. Localglobal principles for 1motives harari, david and szamuely, tamas, duke mathematical journal, 2008. The idele group, profinite groups, infinite galois theory pdf 25. Pdf invitation to higher local fields, part ii, section. Is there an english reference for the analytic construction of the hilbert class field of an imaginary quadratic field without using class field theory. Class formations ans higher dimensional local class field. In part i we learn the new approach to higher dimensional cft based on the ideas of wiesend.
Part of the applied mathematics commons, cosmology, relativity, and gravity commons, elementary. Pdf invitation to higher local fields, part ii, section 10. We say k is a local field if it is a topological field whose. Among the few books on class field theory i tried to read, weils basic number theory is the one i found most accessible. From a different perspective, it describes the local components of the global artin map. On subgroups of the additive group in differentially closed fields suer, sonat, journal of symbolic logic, 2012. A generalization of local class field theory by using. Journal of pure and applied algebra elsevier journal of pure and applied algebra 108 1996 716 on the properties of the vostokov and parshin pairing in higher local class field theory robert a. The basic one dimensional class field theory stems from the ideas of kronecker and weber, and results of hilbert soon after. Explicit local class field theory harvard math harvard university. Florian pop, advisor let kbe a nite eld, and suppose that the arithmetical variety x. Although i have learned cohomology of groups, i find that those theorems in the book are complicated and can not form a system. On properties of the ndimensional norm residue symbol in.
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. A generalization of local class field theory by using kgroups, iii. A standard method for developing global class field theory since the 1930s is to develop local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory. These notes are based on a course in class field theory given by. Nagoya mathematical journal latest issue cambridge core. Higher class field theory and the connected component. On a duality theorem of abelian varieties over higher. This question is really about fields in the sense of commutative division rings. By studying norm groups of cohomological objects and using cohomological approach to higher local class field theory the existence theorem is. In this section we consider the isomorphism theorem. On the mean number of 2 torsion elements in the class groups, narrow class groups, and ideal. The label \ class eld refers to a eld extension satisfying a technical property that is historically related to ideal class groups, and one of the main theorems is that class elds.
The main reference will be casselsfrolich and artintate. To send this article to your account, please select one or more formats and confirm that you agree to abide by our usage policies. By studying norm groups of cohomological objects and using cohomological approach to higher local class field theory the. I dont know if local class field theory is different from a local field theorybecause im not familiar with the former wording. The main theorems of global class field theory pdf 23.
A reference in chapter nto an assertion in chapter mdoes not state the number m explicitly if and only if m n. Local class eld theory says that there is a homomorphism. Kato homology of arithmetic schemes and higher class field theory over local fields dedicated to kazuya kato on the occasion of his 50th birthday uwe jannsen and shuji saito received. For the second axiom, we will prove is the the product of local artin maps and it kills all the global. Local class field theory, strictly speaking, does not appear until the fourth part. Class field theory for curves over local fields core.
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