Nilpotent Sections
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
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Nilpotent Sections
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 These quotients have been studied in the realm of the section conjecture by Ellenberg around 2000, unpublished, and later by Wickelgren in her thesis [Wg09], and in [Wg10, Wg12a, Wg12b] with special emphasis on the interesting case P1 f0; 1; 1g. 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 Teichm¨uller group and the section conjecture for the generic curve [Ha11b], and (2) on the arithmetic side in the non-abelian Chabauty method of Kim [Ki05] for Diophantine finiteness problems. We will examine in detail the Lie algebra associated to the maximal pro-` 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-` sections over a finite field in Sect. 15.3. The nilpotent section conjecture is known to fail by work of Hoshi [Ho10]. 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.
14.1 Primary Decomposition Let X=k be a geometrically connected variety. We may push the extension 1 .X=k/ by the maximal pro-nilpotent quotient 1 .X / 1nilp .X / to obtain the maximal nilpotent extension 1nilp .X=k/. As any finite nilpotent group is canonically the J. Stix, Rational Points and Arithmetic of Fundamental Groups, Lecture Notes in Mathematics 2054, DOI 10.1007/978-3-642-30674-7 14, © Springer-Verlag Berlin Heidelberg 2013
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14 Nilpotent Sections
direct product of its unique p-Sylow groups we obtain in the limit a canonical isomorphism Y 1nilp .X / D 1pro-p .X / (14.1) p
and also a primary decomposition 1nilp .X=k/ D
Y
1pro-p .X=k/
(14.2)
p
that has to be read as a fibre product over Galk . For the corresponding section spaces and Kummer maps this leads to Y S pro-p .X=k/ : (14.3) nilp D .p /p W X.k/ ! S nilp .X=k/ D 1
1
p
If X=k moreover is abelian injective, then because ab .X=k/ is a quotient extension of 1nilp .X=k/ we have Y p .X.k//: X.k/ D nilp .X.k// p
Let for the moment k be an algebraic number field and X=k a smooth hyperbolic curve. Then, by Theorem 76, we have X.k/ D p .X.k//, and the section conjecture raises the question whether only diagonal tuples of pro-p sections lift to actual sections along S1 .X=k/ ! S nilp .X=k/ ; 1
or better: the section conjecture could be modified to ask for a definition of diagonal tuples as the image and
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