Long March across the Theory of Galois

A. Grothendieck's Long March across the Theory of (Évariste) Galois

“La Longue Marche á travers la théorie 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 “Esquisse d'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 “Esquisse d'un Programme”, and in particular studying the Teichmüller 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.

Table of Contents for the Long March across Galois Theory

  1. Multi-Galois Toposes (topoi)
  2. Applications to topos coverings
  3. Pro-multi-Galois variants
  4. Complements
  5. Introducing the arithmetic context; an `anabelian' (non-Abelian) fundamental conjecture
  6. Local analysis of $ (X, S)$ for $ s \in S$
  7. Reformulation of the conjecture (the necessary `purgatorium'...)
  8. A taxonomic reflexion
  9. Tangential structure at $ s \in S$ (sections of second type extensions)
  10. Adjusting the hypotheses
  11. Conditions on the groupoid systems originating from geometric considerations (in the nonabelian case, the groupoid system can be expressed in terms of outer groups)
  12. Returning to the arithmetic case: the Galois-type formulation, p. 53
  13. A cohomological digression, p.58
  14. Returning to the topological case: critical orbits
  15. Returning to the concept of cyclic group
  16. Application to the finite subgroups of $ Aut_{ext}$ (the discrete case, para.18)
  17. Tour of Teichmüller (spaces)
  18. Digression: the description of 2-isotopic categories of algebraic curves p.116
  19. 21. Teichmüller spaces p.126
  20. 23. Returning to the surfaces of (finite) groups of operators (`formulating the equations' of the problem)
  21. “Special” Teichmüller Groups
  22. The case of “two groups of operators”
  23. 26. Profinite Teichmüller Groups, connection with the modular Teichmüller topos, conjecture
  24. 29. Critique of the previous approach
  25. 31. Digression: a finite group $ G$ over a profinite cyclic group $ \pi$
  26. 32 Returning to the arithmetic aspects: a remarkable reconstruction of all of the étale topos of a complete algebraic curve starting from an open nonabelian space...
  27. 33. A topological digression: anti-involutions of compact, oriented surfaces
  28. 35. Injectivity of $ \Gamma_Q \to Autext_{lac}(\mathcal{T}^+_{1,1}) = Autext_{lac} SL(2,\mathcal{Z}^)$
  29. 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}^)$
  30. 37. Modules of elliptic curves via Legendre functions, or $ M_{1,1}[2]' \cong \mathcal{U}_{0,3} \cong M^!_{0,4} .$

Alexander Grothendieck's original document in French:

Bibliography

1
Allyn Jackson. March 1999. The IHÉS at Forty., 9 pp. (``The IHÉS was founded in 1958 by mathematician/ mathematical physicist Léon Motchane, and followed for many years Robert Oppenheimer council. Léon was born in St. Petersburg in 1900 to Swiss parents.'')

2
DAVID AUBIN, Un pacte singulier entre mathématiques et industrie, La Recherche, No. 313 (October 1998), 98-103.

3
PIERRE CARTIER, La folle journée, de Grothendieck á Connes et Kontsevich, Les Relations entre les Mathématiques et la Physique Théorique, Festschrift for the 40th anniversary of the IHÉS, Publications de l'IHÉS, October 1998.



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