LongMarchAcrossGaloisTheory 2009-03-09 17:58:47 2009-03-11 00:05:53 Topic "The Long March across Galois Theory" % this is the default PlanetPhysics preamble. as your % almost certainly you want these \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{A. Grothendieck's Long March across the Theory of (\'Evariste) Galois} ``La Longue Marche \'a travers la th\'eorie de Galois'' (``The Long March Through Galois Theory'') is an approximately 1600--page handwritten manuscript produced by Grothendieck during the years 1980--1981, containing many of the ideas leading to the \PMlinkexternal{``Esquisse d'un Programme''}{http://en.wikipedia.org/wiki/User_talk:Bci2#Esquisse_d.27_un_Programme}. ``Typed in Tex, it comes out to about 600 pages. It goes together with a further 1,000 pages or so of additional notes and sections which have not yet been read or typed. Many of the major themes were summarised in the 1983 manuscript {\em Esquisse d'un Programme.}'', and in particular studying the Teichm\"uller theory. The Table of Contents for this important work by Alexander Grothendieck was originally compiled in French by the author and is reproduced here after the English Translation of the major parts of the Long March. \subsection{Table of Contents for the Long March across Galois Theory} \begin{enumerate} \item Multi-Galois Toposes (topoi) \item Applications to topos coverings \item Pro-multi-Galois variants \item Complements \item Introducing the arithmetic context; an `anabelian' (non-Abelian) fundamental conjecture \item Local analysis of $(X, S)$ for $s \in S$ \item Reformulation of the conjecture (the necessary `purgatorium'...) \item A taxonomic reflexion \item Tangential structure at $s \in S$ (sections of second type extensions) \item Adjusting the hypotheses \item Conditions on the groupoid systems originating from geometric considerations (in the nonabelian case, the groupoid system can be expressed in terms of outer groups) \item Returning to the arithmetic case: the Galois--type formulation, p. 53 \item A cohomological digression, p.58 \item Returning to the topological case: critical orbits \item Returning to the concept of cyclic group \item Application to the finite subgroups of $Aut_{ext}$ (the discrete case, para.18) \item Tour of Teichm\"uller (spaces) \item Digression: the description of 2-isotopic categories of algebraic curves p.116 \item 21. Teichm\"uller spaces p.126 \item 23. Returning to the surfaces of (finite) groups of operators (`formulating the equations' of the problem) \item ``Special'' Teichm\"uller Groups \item The case of ``two groups of operators'' \item {\bf 26.} Profinite Teichm\"uller Groups, connection with the modular Teichm\"uller topos, conjecture \item 29. Critique of the previous approach \item 31. Digression: a finite group $G$ over a profinite cyclic group $\pi$ \item {\bf 32} Returning to the arithmetic aspects: a remarkable reconstruction of all of the \'etale topos of a complete algebraic curve starting from an open nonabelian space... \item 33. A topological digression: anti-involutions of compact, oriented surfaces \item 35. Injectivity of $\Gamma_Q \to Autext_{lac}(\mathcal{T}^+_{1,1}) = Autext_{lac} SL(2,\mathcal{Z}^)$ \item 36. The isomorphism $\Gamma_Q \cong \Gamma_{1,1}$ and the injectivity of $\Gamma_Q \to Autext_{lac}(\mathcal{T}^+_{1,1}) = Autext_{lac} SL(2,\mathcal{Z}^)$ \item 37. Modules of elliptic curves via Legendre functions, or $M_{1,1}[2]' \cong \mathcal{U}_{0,3} \cong M^!_{0,4} .$ \end{enumerate} \subsubsection{Alexander Grothendieck's original document in French:} \begin{itemize} \item 1. Topos multigaloisiens \item 2. Application aux $rev\widehat{e}tements$ des topos \item 3. Variantes pro-multigaloisiennes \item 4. Compl\'ements, remords \item 5. Introduction du contexte arithm\'etique; conjecture anab\'elienne fondamentale \item 6. Analyse locale de $(X,S)$ en un $s \in S$ \item 7. Reformulation `bord\'elique' de la conjecture (le purgatoire n\'ecessaire...) \item 8. R\'eflexion taxonomique (distinction des cas o\'u le purgatoire s\'am\'enage un peu...) \item 9. Structure tangentielle en les $s \in S$ (sections d'extensions ``de deuxi\'eme type'') \item 10. Ajustement des hypoth\'eses (remords) \item 11. Conditions sur les syst\'emes de groupo\"ides obtenus \'a partir de situations g\'eom\'etriques (o\'u on se convaincu aussi que le bordel groupo\"idal peut s\'exprimer compl\'etement, dans les cas anab\'eliens, par les groupes ext\'erieurs \'a lacets) \item 13. Retour au cas arithm\'etique; formulation `galoisienne'' . 53 \item 13 bis. Retour sur la notion de groupe 'a lacets . . 56 \item 14. Digression cohomologique (sur le ```bouchage de rous'') .. 58 \item 15. Retour sur le cas topologique: orbites critiques des scindages d'extensions; application aux sous--groupes finis de $Autext_{lac}$ (cas discret; cf. aussi para.18) \item 16. Bouchage et forage de trous: pr\'eliminaires topologiques g\'en\'eraux . . . . 79 \item 17. Compl\'ement au para. 15; sous--groupes de groupes \'a lacets . . . . . . . . 89 \item 18. Forage de trous; applications aux sous--groupes finis de Autextlac() . . . 91 (cas discret) \item 19. Tour de Teichm\"uller . . . . . . . . . . . . . . . . . . 102 \item 20. Digression: description 2--isotopique de la cat\'egorie des isomorphismes topologiques \item 21. Les espaces de Teichm\"uller . . . . . . . . . . . . . . . . 126 Manque le para 22. \item 22 [notation en marge] \item 23. Retour sur les surfaces \'a groupes (finis)d’op\'erateurs (``mise en \'equations'' du probl\'eme) \item 24. Essai de d\'etermination de $A^{0 \Gamma}$; lien avec les relations $\pi_{(g,\nu,\nu + n -1)}^{\Gamma} = {1}$ programme de travail \item 25. Groupes de Teichm\"uller ``sp\'eciaux'' . . . . . . . . . . . . . 148 \item 25bis. ``Cas des deux groupes'' d'op\'erateurs; retour sur les notations . . . . 155 \item 26. Groupes de Teichm\"uller profinis, discr\'etifications et pr\'ediscr\'etifications; lien avec le topos modulaire de Teichm\"uller, conjecture $h\widehat{a}tive$ \item 27. Changement de type $(g,\nu): a)$ bouchage de trous . 172 \item 28. Changement de type $(g,\nu): b)$ passage \'a un $rev\widehat{e}tements$ fini (la conjecture $h\widehat{a}tive$ grince...) \item 29. Critique de l'approche pr\'ec\'edente . . . . 186 (on rajuste les notions et les conjectures) \item 30. Propri\'et\'es des $\mathcal{N}_g, \Gamma_{g,\nu}$ : . . . . . . . 192 a) Propri\'et'es li\'ees aux sous-groupes finis de Teichm\"uller \item 31. Digression sur les rel\'evements d'une action ext\'erieure . . . . . . . . 198 d'un groupe fini $G$ sur un groupe profini \'a lacets \item 32. Retour sur les aspects arithm\'etiques du bouchage de trous: relations entre $\Gamma_{g,\nu}$ et $\Gamma_{g,\nu-1}$ (o\'u on reconstitue, sans le dire, tout le topos \'etale d'une courbe alg\'ebrique compl\'ete, \'a partir du $\pi_1$ d'un ouvert anab\'elien...) \item 33. Digression topologique: anti--involutions des surfaces orient\'ees compactes . 221 (en laissant de $c\widehat{o}t$\'e d'abord le cas ``a trous'') \item 33 bis. Relation entre les $\mathcal{N}_{g,\nu} , \Gamma_{g,\nu}$, pour g variable .239 (\'etude des $rev\widehat{e}tements$ finis) \item 34. Description heuristique profinie de la cat\'egorie des courbes alg\'ebriques d\'efinies sur des sous--extensions finies $K$ de $Q_0/Q$ \item 35. L'injectivit\'e de $\Gamma_Q \to Autext_{lac}(\widehat{\pi}(0,3)$. 249 \item 36. L'isomorphisme $\Gamma_Q \cong \Gamma_{1,1}$. et l'injectivit\'e de $\Gamma_Q \to Autext_{lac}(\mathcal{T}^+_{1,1}) = Autext_{lac} SL(2,\mathcal{Z}^)$ \item 37. Modules des courbes elliptiques via Legendre, ou $M_{1,1}[2]' \cong \mathcal{U}_{0,3} \cong M^!_{0,4} .$ \end{itemize}