Nilpotent Sections

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 These quotients have been studied in the realm of the section conjecture by Ellenberg around 2000, unpublished, and later by Wickelgren in her thesis (Wickelgren, Lower central series obstructions to homotopy sections of curves over number fields, Thesis, Stanford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.