Pop, Florian Pro-\(\ell\) abelian-by-central Galois theory of prime divisors. (English) Zbl 1309.12005 Isr. J. Math. 180, 43-68 (2010). From the text: In the present paper it is shown that one can recover much of the inertia structure of (quasi) divisors of a function field \(K| k\) over an algebraically closed base field \(k\) from the maximal pro-\(\ell\) abelian-by-central Galois theory of \(K\). The present paper is one of the major technical steps toward tackling a program initiated by F. A. Bogomolov [Algebraic geometry and analytic geometry, Proc. Conf., Tokyo/Jap. 1990, ICM-90 Satell. Conf. Proc., 26–52 (1991; Zbl 0789.14021)] at the beginning of the 1990s, whose final aim is to recover function fields from their pro-\(\ell\) abelian-by-central Galois theory. This program goes beyond A. Grothendieck’s birational anabelian Program as initiated in [Sketch of a programme. Lond. Math. Soc. Lect. Note Ser. 242, 5–48, English translation: 243–283 (1997; Zbl 0901.14001), Letter to G. Faltings, 49–58; English translation 285–293 (1997; Zbl 0901.14002)]. Cited in 2 ReviewsCited in 8 Documents MSC: 12F10 Separable extensions, Galois theory 14H05 Algebraic functions and function fields in algebraic geometry Keywords:function field; prime divisor; Galois theory Citations:Zbl 0789.14021; Zbl 0901.14001; Zbl 0901.14002 PDFBibTeX XMLCite \textit{F. Pop}, Isr. J. Math. 180, 43--68 (2010; Zbl 1309.12005) Full Text: DOI References: [1] E. Artin, Geometric Algebra, Interscience, New York, 1957. [2] F. A. Bogomolov, On two conjectures in birational algebraic geometry, in Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings (A. Fujiki et al., eds.), Springer-Verlag, Tokyo, 1991, pp. 26–52. · Zbl 0789.14021 [3] F. A. Bogomolov and Y. Tschinkel, Commuting elements in Galois groups of function fields, in Motives, Polylogarithms and Hodge Theory (F.A. Bogomolov and L. Katzarkov, eds.), International Press, Somerville, MA, 2002, pp. 75–120. · Zbl 1048.11090 [4] N. Bourbaki, Algèbre commutative, Hermann, Paris, 1964. · Zbl 0205.34302 [5] I. Efrat, Valuations, Orderings and Milnor K-Theory, AMS Mathematical Surveys and Monographs, Vol. 124, American Mathematical Society, Providence, RI, 2006. [6] O. Endler and A. J. Engler, Fields with Henselian valuation rings, Mathematische Zeitschrift 152 (1977), 191–193. · Zbl 0333.12104 [7] A. J. Engler and A. Prestel, Valued Fields, Springer Monographs in Mathematics Series, Springer-Verlag, Berlin, 2005. · Zbl 1128.12009 [8] L. Schneps and P. Lochak (eds.), Geometric Galois Actions I, London Mathematical Society Lecture Notes, Vol. 242, Cambridge University Press, Cambridge, 1998. [9] A. Grothendieck, Letter to Faltings, June 1983; see [GGA]. · Zbl 0901.14002 [10] A. Grothendieck, Esquisse d’un programme, 1984; see [GGA]. · Zbl 0901.14001 [11] H. Koch, Die Galoissche Theorie der p-Erweiterungen, Math. Monogr. 10, Berlin, 1970. · Zbl 0216.04704 [12] J. Koenigsmann, Solvable absolute Galois groups are metabelian, Inventiones Mathematicae 144 (2001), 1–22. · Zbl 1016.12005 [13] L. Mahé, J. Mináč and T. L. Smith, Additive structure of multiplicative subgroups of fields and Galois theory, Documenta Mathematica 9 (2004), 301–355. · Zbl 1073.11025 [14] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, Vol. 1358, 2nd edition, Springer-Verlag, Berlin, 1999. · Zbl 0945.14001 [15] J. Neukirch, Ü ber eine algebraische Kennzeichnung der Henselkörper, Journal für die Reine und Angewandte Mathematik 231 (1968), 75–81. · Zbl 0169.36502 [16] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd edition, Grundlehren der Mathematischen Wissenschaften, Vol. 323, Springer-Verlag, Berlin, 2008. · Zbl 1136.11001 [17] A. N. Parshin, Finiteness Theorems and Hyperbolic Manifolds, in The Grothendieck Festschrift III (P. Cartier et al., eds.), PM Series, Vol. 88, Birkhäuser, Boston, Basel, Berlin, 1990. · Zbl 0742.14018 [18] F. Pop, On Grothendieck’s conjecture of birational anabelian geometry, Annals of Mathematics 138 (1994), 145–182. · Zbl 0814.14027 [19] F. Pop, Glimpses of Grothendieck’s anabelian geometry, in Geometric Galois Actions I, London Mathematical Society Lecture Notes, Vol. 242, (L. Schneps and P. Lochak eds.), Cambridge University Press, 1998, Cambridge, pp 133–126. · Zbl 0917.14011 [20] F. Pop, Pro- birational anabelian geometry over algebraically closed fields I, Manuscript, Bonn, 2003; see: http://arxiv.org/pdf/math.AG/0307076 . [21] F. Pop, Pro- Galois theory of Zariski prime divisors, in Luminy Proceedings Conference, SMF No 13 (Débès et al., eds.), Hérmann, Paris, 2006. · Zbl 1177.12005 [22] F. Pop, Recovering fields from their decomposition graphs, Manuscript, 2007; see: http://www.math.upenn.edu/\(\sim\)pop/Research/Papers.html . [23] P. Roquette, Zur Theorie der Konstantenreduktion algebraischer Mannigfaltigkeiten, Journal für die Reine und Angewandte Mathematik 200 (1958), 1–44. · Zbl 0149.39202 [24] T. Szamuely, Groupes de Galois de corps de type fini (d’après Pop), Astérisque 294 (2004), 403–431. · Zbl 1148.12300 [25] J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin, 1965. [26] K. Uchida, Isomorphisms of Galois groups of solvably closed Galois extensions, The Tôhoku Mathematical Journal 31 (1979), 359–362. · Zbl 0422.12006 [27] R. Ware, Valuation Rings and rigid Elements in Fields, Canadian Journal of Mathematics 33 (1981), 1338–1355. · Zbl 0514.10015 [28] O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Springer-Verlag, New York, 1975. · Zbl 0313.13001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.