Abstract class field theory

Skip to main content

\(\def\AA{\mathbb{A}} \def\CC{\mathbb{C}} \def\FF{\mathbb{F}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\kbar{\overline{k}} \def\gotha{\mathfrak{a}} \def\gothb{\mathfrak{b}} \def\gothm{\mathfrak{m}} \def\gotho{\mathfrak{o}} \def\gothp{\mathfrak{p}} \def\gothq{\mathfrak{q}} \def\gothr{\mathfrak{r}} \DeclareMathOperator{\ab}{ab} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\Cor}{Cor} \DeclareMathOperator{\cyc}{cyc} \DeclareMathOperator{\disc}{Disc} \DeclareMathOperator{\fin}{fin} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\Inf}{Inf} \DeclareMathOperator{\inv}{inv} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\Real}{Re} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\sep}{sep} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\smcy}{smcy} \DeclareMathOperator{\Trace}{Trace} \DeclareMathOperator{\unr}{unr} \DeclareMathOperator{\Ver}{Ver} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Chapter 5 Abstract class field theory

We now turn to an alternate method for deriving the results of local class field theory, particularly the local reciprocity law (Theorem 4.1.2). This method, based on a presentation of Artin and Tate (the method of “class formations” introduced in [1]), isolates the main cohomological inputs in the local case and gives an outline of how to proceed to global class field theory. We conclude with a preview of how the method will apply in the global case; see Section 5.4.

Caveat.

In the context of abstract class field theory, we will assign certain words (e.g., unramified) new meanings that will coincide with their existing definitions when the abstract setup is specialized to local class field theory. We will then transfer these meanings to the global application of abstract class field theory, where we will usually use scare quotes (i.e., “unramified”).