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Notes on class field theory
Kiran S. Kedlaya
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\(\def\AA{\mathbb{A}} \def\CC{\mathbb{C}} \def\FF{\mathbb{F}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\kbar{\overline{k}} \def\gotha{\mathfrak{a}} \def\gothb{\mathfrak{b}} \def\gothm{\mathfrak{m}} \def\gotho{\mathfrak{o}} \def\gothp{\mathfrak{p}} \def\gothq{\mathfrak{q}} \def\gothr{\mathfrak{r}} \DeclareMathOperator{\ab}{ab} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\Cor}{Cor} \DeclareMathOperator{\cyc}{cyc} \DeclareMathOperator{\disc}{Disc} \DeclareMathOperator{\fin}{fin} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\Inf}{Inf} \DeclareMathOperator{\inv}{inv} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\Real}{Re} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\sep}{sep} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\smcy}{smcy} \DeclareMathOperator{\Trace}{Trace} \DeclareMathOperator{\unr}{unr} \DeclareMathOperator{\Ver}{Ver} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Preface
1
Trailer: Abelian extensions of the rationals
1.1
The Kronecker-Weber theorem
Abelian extensions of
\(\QQ\)
A reciprocity law
Reduction to the local case
Exercises
1.2
Kummer theory
Theorem 90
Kummer extensions
The Kummer pairing
Cyclic extensions without roots of unity
Exercises
1.3
The local Kronecker-Weber theorem
Extensions of local fields
Proof of local Kronecker-Weber
Filling in the details
Exercises
2
The statements of class field theory
2.1
The Hilbert class field
An example of an unramified extension
Hilbert class fields
Exercises
2.2
Generalized ideal class groups and the Artin reciprocity law
An example (continued)
Generalized ideal class groups
The Artin reciprocity law
Exercises
2.3
The principal ideal theorem
Statement of the theorem
First steps of the proof
Translation into group theory
The final group-theoretic ingredient
Additional remarks
Exercises
2.4
Zeta functions and the Chebotaryov density theorem
The Dedekind zeta function of a number field
\(L\)
-functions of abelian characters
Nonvanishing of
\(L\)
-functions and consequences
The Chebotaryov density theorem
Exercises
3
Cohomology of groups
3.1
Cohomology of finite groups I: abstract nonsense
\(G\)
-modules and their invariants
Injective objects and resolutions
Right derived functors
Additional comments
Exercises
3.2
Cohomology of finite groups II: concrete nonsense
Induced
\(G\)
-modules
Group cohomology via homogeneous cochains
Fun with
\(H^1\)
Fun with
\(H^2\)
Extended functoriality
Exercises
3.3
Homology and Tate groups
Homology
The Tate groups
Extended functoriality revisited
Exercises
3.4
Cohomology of cyclic groups
The periodicity theorem
Herbrand quotients
Exercises
3.5
Profinite groups and infinite Galois theory
Profinite groups
Infinite Galois groups
Cohomology of profinite groups
Exercises
4
Local class field theory
4.1
Overview of local class field theory
The local reciprocity law
The local invariant map
Abstract class field theory
Exercises
4.2
Cohomology of local fields: some computations
Overview
The unramified case
The cyclic case
The general case
The local invariant map
Exercises
4.3
Local class field theory via Tate’s theorem
Tate’s theorem
Local reciprocity and norm limitation
The local existence theorem
Exercises
4.4
Ramification filtrations and local reciprocity
The lower numbering filtration
The Herbrand functions
The Hasse-Arf theorem
Exercises
4.5
Making the reciprocity map explicit
Initial setup
An explicit cocycle via periodicity
From a cocycle to reciprocity
5
Abstract class field theory
5.1
The setup of abstract class field theory
Abstract multiplicative groups and the class field axiom
Abstract ramification theory
Abstract valuation theory
Cohomology of units
Exercises
5.2
The abstract reciprocity map
Construction of the reciprocity map
Exercises
5.3
The theorems of abstract class field theory
Proof of the reciprocity law
5.4
A look ahead
Replacing the multiplicative group
Replacing the unramified extensions and the valuation
Further remarks
6
The adelic formulation
6.1
Adèles
Lattices of number fields
Profinite completions
The adèles (rational case)
The adèles (general case)
Adelic
\(S\)
-integers
The approximation theorem
Exercises
6.2
Idèles and class groups
Idèles
The idèle class group
Compactness and consequences
Aside: beyond class field theory
Exercises
6.3
Adèles and idèles in field extensions
Adèles in field extensions
Trace and norm
Idèle groups and class groups
Exercises
6.4
The adelic reciprocity law and Artin reciprocity
The adelic reciprocity law and existence theorem
More on the reciprocity map
6.5
Adelic reciprocity: what remains to be done
Abstract reciprocity
The existence theorem and local-global compatibility
Another approach via Brauer groups
6.6
Adelic Fourier analysis after Tate
Additive characters
Fourier inversion
The space of quasi-characters
Zeta functions and
\(L\)
-functions
7
The main results
7.1
Cohomology of the idèles I: the “First Inequality”
Some basic observations
Cohomology of the units: first steps
Cohomology of the units: a computation with
\(S\)
-units
Herbrand quotients of real lattices
Splitting of primes
Exercises
7.2
Cohomology of the idèles II: the “Second Inequality”
Back to ideals
A special case of Chebotaryov density
The Second Inequality
Aside: the Hasse norm theorem
The Albert-Brauer-Hasse-Noether theorem
Exercises
7.3
An “abstract” reciprocity map
Abstract unit groups and the class field axiom
Cyclotomic extensions and abstract ramification theory
An abstract henselian valuation
Consequences of abstract CFT
7.4
The existence theorem
A base case for the existence theorem
Proof of the existence theorem
An algebraic approach to the Second Inequality
Exercises
7.5
Local-global compatibility
Compatibility for cyclotomic extensions
Compatibility for general extensions
Globalization of local abelian extensions
Exercises
7.6
Brauer groups and the reciprocity map
The Brauer group of a field
The Brauer group of a number field
All Brauer classes are (cyclic) cyclotomic
Local-global compatibility via Brauer groups
Exercises
Back Matter
A
Parting thoughts
Bibliography
Colophon
Colophon
https://kskedlaya.org/cft
©2002–2021 Kiran S. Kedlaya
