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  • Online publication date: January 2012

Lectures on anabelian phenomena in geometry and arithmetic

[An] André, Y., On a geometric description of Gal(Qp|ℚ) and a p-adic avatar of, Duke Math. J. 119 (2003), 1–39.
[Ar] Artin, E., Geometric Algebra, Interscience Publishers, New York 1957.
[Be] Belyi, G. V., On Galois extensions of a maximal cyclotomic field, Mathematics USSR Izvestija, Vol. 14 (1980), no. 2, 247–256. (Original in Russian: Izvestiya Akademii Nauk SSSR, vol. 14 (1979), no. 2, 269–276.)
[Bo] Bogomolov, F. A., On two conjectures in birational algebraic geometry, in Algebraic geometry and analytic geometry, ICM-90 Satellite Conference Proceedings, ed. A., Fujiki et al., Springer Verlag, Tokyo 1991.
[B–T1] Bogomolov, F. A., and Tschinkel, Y.Commuting elements in Galois groups of function fields, in: Motives, Polylogs and Hodge Theory, International Press 2002, 75–120.
[B–T2] Bogomolov, F. A., and Tschinkel, Y., Reconstruction of function fields, Geometric and Functional Analysis 18 (2008), 400–462.
[BOU] Bourbaki, , Algèbre commutative, Hermann, Paris 1964.
[C–P] Corry, S. and Pop, F., Pro-p hom-form of the birational anabelian conjecture over sub-p-adic fields, Journal reine angew. Math. 628 (2009), 121–128.
[De] Deligne, P., Le groupe fondamental de la droite projective moins trois points, in: Galois groups over Q, Math. Sci. Res. Inst. Publ. 16, 79–297, Springer 1989.
[Dr] Drinfeld, V. G., On quasi-triangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/ℚ) (Russian), Algebra i Analiz 2, no. 4 (1990), 149–181; translation in: Leningrad Math. J. 2, no. 4 (1991), 829–860.
[Ef1] Efrat, I., Construction of valuations from K-theory, Mathematical Research Letters 6 (1999), 335–344.
[Ef2] Efrat, I., Recovering higher global and local fields from Galois groups–an algebraic approach, Invitation to higher local fields (Münster, 1999), 273–279 (electronic), Geom., Topol. Monogr. 3, Geom. Topol. Publ., Coventry, 2000
[E–K] Engler, A. J. and Koenigsmann, J., Abelian subgroups of pro-p Galois groups, Trans. AMS 350 (1998), no. 6, 2473–2485.
[E–W] Esnault, H. and Wittenberg, O., On abelian birational sections, J. AMS, 23 (2010), 713–724.
[F1] Faltings, G., p-adic Hodge theory, Journal AMS 1 (1988), 255–299.
[F2] Faltings, G., Hodge–Tate structures and modular forms, Math. Annalen 278 (1987), 133–149.
[F3] Faltings, G., Curves and their fundamental groups [following Grothendieck, Tamagawa, Mochizuki], Séminaire Bourbaki, Vol 1997-98, Exposé 840, Mars 1998.
[F–J] Fried, M. D. and Jarden, M., Field arithmetic, in: Ergebnisse der Mathematik und ihre Grenzgebiete, 3. Folge, Vol 11, Springer Verlag 2004.
[G1] Grothendieck, A., Letter to Faltings, June 1983. See [GGA].
[G2] Grothendieck, A., Esquisse d'un programme, 1984. See [GGA].
[GGA] ,Geometric Galois Actions I, LMS LNS Vol. 242, eds. L. Schneps and P. Lochak, Cambridge University Press 1998.
[Hn] Hain, R., Rational points of universal curves, preprint, January 2010; see arXiv:mathNT/1001.5008v1.
[H–M] Hain, R. and Matsumoto, M., Tannakian fundamental groups associated to Galois groups, in: Galois Groups and Fundamental Groups, ed. L., Schneps, MSRI Pub. Series 41, 2003, pp.183–216.
[H–Sz] Harari, D. and Szamuely, T., Galois sections for abelianized fundamental groups, and Appendix by E. V., Flynn, Math. Annalen 344 (2009), 779–800.
[Ha1] Harbater, D., Abhyankar's conjecture on Galois groups over curves, Invent. Math. 117 (1994), 1–25.
[Ha2] Harbater, D., Fundamental groups of curves in characteristic p, in: Proceedings of the ICM (Zürich, 1994), Birkhauser, Basel, 1995, 656–666.
[H–Sch] Harbater, D. and Schneps, L., Fundamental groups of moduli and the Grothendieck–Teichmüller group, Trans. AMS, Vol. 352 (2000), 3117–3148.
[Ho] Hoshi, Y., Existence of non-geometric pro-p Galois sections of hyperbolic curves, preprint, January 2010, RIMS preprint ∑ 1689.
[Hr] Hrushovski, E., The Mordell–Lang conjecture for function fields, Journal AMS 9 (1996), 667–690.
[I1] Ihara, Y., On Galois represent. arising from towers of covers of P1\{0, 1, 8}, Invent. Math. 86 (1986), 427–459.
[I2] Ihara, Y., Braids, Galois groups, and some arithmetic functions, Proceedings of the ICM'90, Vol. I, II, Math. Soc. Japan, Tokyo, 1991, pp.99–120.
[I3] Ihara, Y., On beta and gamma functions associated with the Grothendieck–Teichmüller group II, J. reine angew. Math. 527 (2000), 1–11.
[I–M] Ihara, Y. and Matsumoto, M., On Galois actions on profinite completions of braid groups, in: Recent developments in the inverse Galois problem (Seattle, WA, 1993), 173–200, Contemp. Math. 186 AMS Providence, RI, 1995.
[I–N] Ihara, Y., and Nakamura, H., Some illustrative examples for anabelian geometry in high dimensions, in: Geometric Galois Actions I, 127–138, LMS 242, Cambridge University Press, Cambridge, 1997.
[Ik] Ikeda, M., Completeness of the absolute Galois group of the rational number field, J. reine angew. Math. 291 (1977), 1–22.
[J–W] Jannsen, U., and Wingberg, K., Die Struktur der absoluten Galoisgruppe p-adischer Zahlkörper, Invent. Math. 70 (1982/83), no. 1, 71–98.
[J] de Jong, A. J., Families of curves and alterations, Annales de l'institute Fourier, 47 (1997), pp. 599–621.
[Ka] Kato, K., Logarithmic structures of Fontaine–Illusie, Proceedings of the First JAMI Conference, Johns Hopkins Univ. Press (1990), 191–224.
[K–L] Katz, N., and Lang, S.Finiteness theorems in geometric class field theory, with an Appendix by K. Ribet, Enseign. Math. 27 (1981), 285–319.
[K–P–R] Kuhlmann, F.-V., Pank, M., and Roquette, P., Immediate and purely wild extensions of valued fields, Manuscripta Math. 55 (1986), 39–67.
[K1] Kim, M., The motivic fundamental group of P\{0, 1, ∞} and the theorem of Siegel, Inventiones Math. 161 (2005), 629–656.
[K2] Kim, M., The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), 89–133.
[K3] Kim, M., Massey products for elliptic curves of rank one, J. AMS 23 (2010), 725–747.
[K4] Kim, M., A remark on fundamental groups and effective Diophantine methods for hyperbolic curves, in: Number theory, analysis and geometry – In memory of Serge Lang, eds: Goldfeld, Jorgenson, Jones, Ramakrishnan, Ribet, Tate; Springer-Verlag 2010.
[Ko1] Koenigsmann, J., From p-rigid elements to valuations (with a Galoischaracterization of p-adic fields). With an appendix by Florian Pop, J. Reine Angew. Math. 465 (1995), 165–182.
[Ko2] Koenigsmann, J., Solvable absolute Galois groups are metabelian, Invent. Math. 144 (2001), no. 1, 1–22.
[Ko3] Koenigsmann, J., On the ‘section conjecture’ in anabelian geometry, J. reine angew. Math. 588 (2005), 221–235.
[Ko] Koch, H., Die Galoissche Theorie der p-Erweiterungen, Math. Monographien 10, Berlin 1970.
[La] Lang, S., Algebra, Springer-Verlag 2001.
[L–T] Lang, S. and Tate, J., Principal homogeneous spaces over Abelian varieties, Am. J. Math. 80 (1958), 659–684.
[LS1] The Grothendieck Theory of Dessins d'Enfants, ed. Leila, Schneps, LMS LNS 200, Cambridge Univ Press, 1994.
[LS2] Around Grothendieck's Esquisse d'un Programme, eds. Schneps, & Lochak, , LMS LNS 242, Cambridge University Press, 1997.
[L–Sch] Lochak, P. and Schneps, L., A cohomological interpretation of the Grothendieck-Teichmüller group. Appendix by C. Scheiderer, Invent. Math. 127 (1997), 571–600.
[Ma] Matsumoto, M., Galois representations on profinite braid groups on curves, J. reine angew. Math. 474 (1996), 169–219.
[Mzk1] Mochizuki, Sh., The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Univ Tokyo 3 (1966), 571–627.
[Mzk2] Mochizuki, Sh., A version of the Grothendieck conjecture for p-adic local fields, Internat. J. Math. no. 4, 8 (1997), 499–506.
[Mzk3] Mochizuki, Sh., The local pro-p Grothendieck conjecture for hyperbolic curves, Invent. Math. 138 (1999), 319–423.
[Mzk4] Mochizuki, Sh., The absolute anabelian geometry of hyperbolic curves, Galois theory and modular forms, 77–122, Dev. Math. 11, Kluwer Acad. Publ., Boston, MA, 2004.
[Mzk5] Mochizuki, Sh., Absolute anabelian cuspidalizations of proper hyperbolic curves., J. Math. Kyoto Univ. 47 (2007), 451–539.
[Mzk6] Mochizuki, Sh., Topics in absolute anabelian geometry II: Decomposition groups and endomorphisms, RIMS Preprint 1625, March 2008.
[N] Nagata, M., A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85–91.
[Na1] Nakamura, H., Galois rigidity of the étale fundamental groups of punctured projective lines, J. reine angew. Math. 411 (1990) 205–216.
[Na2] Nakamura, H., Galois rigidity of algebraic mappings into some hyperbolic varieties, Int. J. Math. 4 (1993), 421–438.
[N–Sch] Nakamura, H. and Schneps, L., On a subgroup of the Grothendieck–Teichmüller group acting on the profinite Teichmüller modular group, Invent. Math. 141 (2000), 503–560.
[N1] Neukirch, J., Über eine algebraische Kennzeichnung der Henselkörper, J. reine angew. Math. 231 (1968), 75–81.
[N2] Neukirch, J., Kennzeichnung der p-adischen und endlichen algebraischen Zahlkörper, Inventiones math. 6 (1969), 269–314.
[N3] Neukirch, J., Kennzeichnung der endlich-algebraischen Zahlkörper durch die Galoisgruppe der maximal auflösbaren, Erweiterungen, J. für Math. 238 (1969), 135–147.
[O] Oda, T., A note on ramification of the Galois representation of the fundamental group of an algebraic curve I, J. Number Theory (1990) 225–228.
[Pa] Parshin, A. N., Finiteness Theorems and Hyperbolic Manifolds, in: The Grothendieck Festschrift III, ed P., Cartier et al., PM Series vol. 88, Birkhäuser, Boston Basel Berlin 1990.
[Po] Pop, F., Étale Galois covers of affine, smooth curves, Invent. Math. 120 (1995), 555–578.
[P1] Pop, F., On Grothendieck's conjecture of birational anabelian geometry, Ann. of Math. 139 (1994), 145–182.
[P2] Pop, F., On Grothendieck's conjecture of birational anabelian geometry II, preprint, 1995
[P3] Pop, F., MSRI talk notes, fall 1999. See,
[P4] Pop, F., Pro-ℓ birational anabelian geometry over alg. closed fields I, manuscript, Bonn 2003. See∅3∅7/∅3∅7∅76
[P5] Pop, F., Recovering fields from their decomposition graphs, in: Number theory, analysis and geometry – In memory of Serge Lang, Springer special volume 2010; eds: Goldfeld, Jorgenson, Jones, Ramakrishnan, Ribet, Tate.
[P6] Pop, F., On the birational anabelian program initiated by Bogomolov I, (to appear).
[P7] Pop, F., On I/OM, Manuscript 2010.
[P8] Pop, F., Galoissche Kennzeichnung p-adisch abgeschlossener Körper, dissertation, Heidelberg 1986.
[P9] Pop, F., On the birational p-adic section conjecture, Compositio Math. 146 (2010), 621–637.
[P10] Pop, F., Inertia elements versus Frobenius elements, Math. Annalen 348 (2010), 1005–1017.
[P–S] Pop, F., and Saidi, M.On the specialization homomorphism of fundamental groups of curves in positive characteristic, in: Galois groups and fundamental groups (ed. L., Schneps), MSRI Pub. Series 41, 2003, pp.107–118.
[P–St] Pop, F., and Stix, J., Arithmetic in the fundamental group of a p-adic curve, manuscript, Cambridge–Heidelberg 2009.
[R1] Raynaud, M., Revêtements des courbes en caractéristique p > 0 et ordinarité, Compositio Math. 123 (2000), no. 1, 73–88.
[R2] Raynaud, M., Sur le groupe fondamental d'une courbe complète en caractéristique p > 0, in: Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), 335–351, Proc. Sympos. Pure Math., 70, AMS, Providence, RI, 2002.
[Ro] Roquette, P., Some tendencies in contemporary algebra, in: Perspectives in Math. Anniversary of Oberwolfach 1984, Basel 1984, 393–422.
[Sa1] Saidi, M., Revêtements modérés et groupe fondamental de graphes de groupes, Compositio Math. 107 (1997), 319–338.
[Sa2] Saidi, M., Good sections of arithmetic fundamental groups, manuscript 2009.
[Sa3] Saidi, M., Around the Grothendieck anabelian section conjecture. See arXiv:1∅1∅.1314v2[math.AG]
[S–T] Saidi, M. and Tamagawa, A, A prime-to-p version of the Grothendieck anabelian conjecture for hyperbolic curves over finite fields of characteristic p > 0, Publ. Res. Inst. Math. Sci. 45 (2009), 135–186.
[S1] Serre, J.-P., Cohomologie Galoisienne, 5th ed., LNM 5, Springer Verlag, Berlin, 1994
[S2] Serre, J.-P., Zeta and L functions, in: Arithmetical algebraic geometry (Proc. conf. Purdue Univ., 1963), pp.82–92; Harper & Row, New York 1965.
[S3] Serre, J.-P., Corps locaux, Hermann, Paris 1962.
[Sp] Spiess, M., An arithmetic proof of Pop's theorem concerning Galois groups of function fields over number fields, J. reine angew. Math. 478 (1996), 107–126.
[St1] Stix, J., Affine anabelian curves in positive characteristic, Compositio Math. 134 (2002), no. 1, 75–85
[St2] Stix, J., Projective anabelian curves in positive characteristic and descent theory for log-étale covers, dissertation, Bonner Mathematische Schriften 354, Universität Bonn, Math. Inst. Bonn, 2002; see\∼{}stix
[St3] Stix, J., On the period-index problem in light of the section conjecture, American J. Math. 132 (2010), 157–180.
[St4] Stix, J., The Brauer–Manin obstruction for sections of the fundamental group, preprint, Cambridge–Heidelberg, Oct. 2009; see arXiv:mathAG/∅91∅.5∅∅9v1.
[St5] Stix, J., A general Seifert–Van Kampen theorem for algebraic fundamental groups, Publications of RIMS 42 (2006), 763–786.
[Sz] Szamuely, T., Groupes de Galois de corpes de type finit [d'après Pop], Astérisque 294 (2004), 403–431.
[T1] Tamagawa, A., The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), 135–194.
[T2] Tamagawa, A., On the fundamental groups of curves over algebraically closed fields of characteristic > 0, Internat. Math. Res. Notices 1999, no. 16, 853–873.
[T3] Tamagawa, A., On the tame fundamental groups of curves over algebraically closed fields of characteristic > 0, in: Galois groups and fundamental groups, 47–105, MSRI Publ., 41, Cambridge Univ. Press, Cambridge, 2003.
[T4] Tamagawa, A., Fundamental groups and geometry of curves in positive characteristic, in: Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), 297–333, Proc. Sympos. Pure Math., 70, AMS, Providence, RI, 2002.
[T5] Tamagawa, A., Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups, J. Algebraic Geom. 13 (2004), no. 4, 675–724.
[T6] Tamagawa, A., Resolution of non-singularities of families of curves, Publ. Res. Inst. Math. Sci. 40 (2004), 1291–1336.
[U1] Uchida, K., Isomorphisms of Galois groups of algebraic function fields, Ann. of Math. 106 (1977), 589–598.
[U2] Tamagawa, A., Isomorphisms of Galois groups of solvably closed Galois extensions, Tôhoku Math. J. 31 (1979), 359–362.
[U3] Tamagawa, A., Homomorphisms of Galois groups of solvably closed Galois extensions, Journal Math. Soc. Japan 33, No.4, 1981.
[Ta1] Tate, J., Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.
[Ta2] Tate, J., p-divisible groups, Proceeding of a conference on local fields, Driebergen, Springer Verlag 1969, 158–183.
[W] Ware, R., Valuation rings and rigid elements in fields, Can. J. Math. 33 (1981), 1338–1355.