Abstract
The next difficult characteristic quotient of a profinite group beyond the maximal abelian quotient might be the maximal pro-nilpotent quotient or its truncated versions of bounded nilpotency. These quotients have been studied in the realm of the section conjecture by Ellenberg around 2000, unpublished, and later by Wickelgren in her thesis (Wickelgren, Lower central series obstructions to homotopy sections of curves over number fields, Thesis, Stanford University, 2009), and in Wickelgren (2-nilpotent real section conjecture, http://arxiv.org/abs/1006.0265v1 arXiv: 1006.0265v1 [math.AG], 2010; Stix, J. (ed.), Contributions in Mathematical and Computational Science, vol. 2, 2012; Nakamura, H., Pop, F., Schneps, L., Tamagawa, A. (eds.) Proceedings for Conferences in Kyoto “Galois-Teichmueller theory and Arithmetic Geometry”, 2010) with special emphasis on the interesting case \({\mathbb{P}}^{1} -\{ 0,1,\infty \}\).The (relative) pro-algebraic version has played an important role in at least two strands of mathematics: (1) on the Hodge theoretic side in the study conducted by Hain of the Teichmüller group and the section conjecture for the generic curve (Hain, J. Am. Math. Soc. 24:709–769, 2011), and (2) on the arithmetic side in the non-abelian Chabauty method of Kim ( Invent. Math. 161(3):629–656, 2005) for Diophantine finiteness problems.We will examine in detail the Lie algebra associated to the maximal pro-\(\mathcal{l}\) quotient of the geometric fundamental group, see Sect. 14.3, and in particular prove Proposition 207 about the sub Lie algebra of invariants under a finite abelian group action. This will be crucial for counting pro-\(\mathcal{l}\) sections over a finite field in Sect. 15.3.The nilpotent section conjecture is known to fail by work of Hoshi ( Publ. RIMS Kyoto Univ. 46:829–848, 2010). We try to explain that examples for this failure should be seen as accidents due to an accidental coincidence of very special properties. In Sect. 14.7, we extend the range of examples, show that in most of these examples the spaces of pro-p sections are in fact uncountable, and suggest a way of reviving the pro-p version of the section conjecture by asking a virtually pro-p section conjecture.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
I thank Yuichiro Hoshi for bringing Tamagawa’s observation to my attention.
References
Asada, M., Nakamura, H.: On graded quotient modules of mapping class groups of surfaces. Israel J. Math. 90, 93–113 (1995)
Benson, D.J.: Lambda and psi operations on Green rings. J. Algebra 87, 360–367 (1984)
Brandt, A.J.: The free Lie ring and Lie representations of the full linear group. Trans. Am. Math. Soc. 56, 528–536 (1944)
Bryant, R.M.: Free Lie algebras and Adams operations. J. Lond. Math. Soc. (2) 68, 355–370 (2003)
Gross, B.H., Rohrlich, D.E.: Some results on the Mordell–Weil group of the Jacobian of the Fermat curve. Invent. Math. 44, 201–224 (1978)
Hain, R.: Remarks on Non-Abelian Cohomology of Proalgebraic Groups. J. Algebraic Geom. http://arxiv.org/abs/1009.3662v2arXiv: 1009.3662v2 [math.AG] (2011)
Hain, R.: Rational points of universal curves. J. Am. Math. Soc. 2, 709–769 (2011)
Hall, M. Jr.: The Theory of Groups, xiii + 434 pp. Macmillan, New York (1959)
Hoshi, Y.: Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic curves over finite fields. Publ. Res. Inst. Math. Sci. 45, 661–744 (2009)
Hoshi, Y.: Existence of nongeometric pro-p Galois sections of hyperbolic curves. Publ. RIMS Kyoto Univ. 46, 829–848 (2010)
Kim, M.: The motivic fundamental group of \({\mathbb{P}}^{1} \setminus \{ 0, 1,\infty \}\) and the theorem of Siegel. Invent. Math. 161, 629–656 (2005)
Kim, M., Tamagawa, A.: The \(\mathcal{l}\)-component of the unipotent Albanese map. Math. Ann. 340, 223–235 (2008)
Labute, J.: Demuškin groups of rank \({\aleph }_{0}\). Bull. Soc. Math. France 94, 211–244 (1966)
Labute, J.: Algèbres de Lie et pro-p-groupes définis par une seule relation. Invent. Math. 4, 142–158 (1967)
Labute, J.: On the descending central series of groups with a single defining relation. J. Algebra 14, 16–23 (1970)
Labute, J.: Groups and lie algebras: The Magnus theory. In: The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions. Contemporary Mathematics, vol. 169, pp. 397–406. American Mathematical Society, Providence (1994)
Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of number fields, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 323, xvi + 825 pp. Springer, Berlin (2008)
Serre, J.-P.: Linear representations of finite groups. Graduate Text in Mathematics, vol. 42, x + 188 pp. Springer, Berlin (1977)
Grothendieck, A.: Séminaire de Géométrie Algébrique du Bois Marie (SGA 1) 1960–1961: Revêtements étales et groupe fondamental. Documents Mathématiques vol. 3, xviii + 327 pp. Société Mathématique de France (2003)
Stix, J.: Projective anabelian curves in positive characteristic and descent theory for log étale covers, Thesis, Bonner Mathematische Schriften 354, xviii+118 (2002)
Stix, J.: A monodromy criterion for extending curves. Intern. Math. Res. Notices 29, 1787–1802 (2005)
Wickelgren, K.: Lower central series obstructions to homotopy sections of curves over number fields. Thesis, 97 pp. Stanford University (2009)
Wickelgren, K.: 2-nilpotent real section conjecture. http://arxiv.org/abs/1006.0265v1arXiv: 1006.0265v1 [math.AG] (June 2010)
Wickelgren, K.: On 3-nilpotent obstructions to π1 sections for \({\mathbb{P}}_{\mathbb{Q}}^{1} -\{ 0, 1,\infty \}\). In: Stix, J. (ed.) The Arithmetic of Fundamental Groups. PIA 2010. Contributions in Mathematical and Computational Science, vol. 2, pp. 281–328. Springer, Berlin (2012)
Wickelgren, K.: n-Nilpotent obstructions to π1 sections of \({\mathbb{P}}^{1} -\{ 0, 1,\infty \}\) and Massey products. In: Nakamura, H., Pop, F., Schneps, L., Tamagawa, A. (eds.) Galois-Teichmller Theory and Arithmetic Geometry. Proceedings for a Conferences in Kyoto (October 2010). Advanced Studies in Pure Mathematics, vol. 63, pp. 579–600 (2012)
Wolfrath, S.: Die scheingeometrische étale Fundamentalgruppe. Thesis, 62 pp., Universität Regensburg, preprint no. 03/2009
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Stix, J. (2013). Nilpotent Sections. In: Rational Points and Arithmetic of Fundamental Groups. Lecture Notes in Mathematics, vol 2054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30674-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-30674-7_14
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30673-0
Online ISBN: 978-3-642-30674-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)