The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers
We resume the discussion of the space of sections from Chap. 4 under arithmetic assumptions. The space $${\mathcal{S}}_{{\pi }_{{}_{ 1}}(X/k)}$$ turns out to be a compact profinite space if X ∕ k is a smooth projective curve over an algebraic number field
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The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers
We resume the discussion of the space of sections from Chap. 4 under arithmetic assumptions. The space S1 .X=k/ turns out to be a compact profinite space if X=k is a smooth projective curve over an algebraic number field, see Proposition 97 or for any hyperbolic curve if k is a finite extension of Qp . We also recall the known relation between a weak form of the section conjecture, see Conjecture 100, and the genuine version, see Conjecture 2, and between the claim for affine versus proper curves. In the arithmetic situation, the centraliser of a section is always trivial, see Proposition 104. It follows that, just like rational points, sections obey Galois descent, see Corollary 107. Furthermore, we examine going up and going down for the section conjecture with respect to a finite e´ tale map. The discussion relies on Chap. 3.
9.1 Compactness of the Space of Sections in the Arithmetic Case We consider the space of sections in the case where k is either a number field, or a p-adic local field or R. Proposition 96. Let X be a geometrically connected variety over a p-adic or an archimedean local field k. Then the space of sections S1 .X=k/ is a profinite topological space and in particular compact. Proof. As Galk is topologically finitely generated by [Ja88] Theorem 5.1(c), see also [NSW08] Theorem 7.4.1, we can apply Corollary 45. u t
J. Stix, Rational Points and Arithmetic of Fundamental Groups, Lecture Notes in Mathematics 2054, DOI 10.1007/978-3-642-30674-7 9, © Springer-Verlag Berlin Heidelberg 2013
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9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers
Proposition 97. Let X be a proper, geometrically connected variety over an algebraic number field k. Then the space of sections S1 .X=k/ is a profinite topological space and in particular compact. Proof. By Lemma 44 we need to show the finiteness of the image of S1 .X=k/ ! SQn .1 .X=k// for each n. Let Bn Spec.ok Œ
1 / nŠ
be an open subset such that the extension Qn .1 .X=k// is induced from an extension of 1 .Bn / by Qn .1 .X //. Any section of Qn .1 .X=k// coming from a section of 1 .X=k/ will be unramified at places in Bn by Proposition 91, because the order of Qn .1 .X // is only divisible by primes n. Any two such sections thus differ by a non-abelian cohomology class in H1 Bn ; Qn .1 .X// ; which is a finite set by Hermite’s Theorem: the finiteness of the set of field extensions of an algebraic number field with bounded degree and places of ramification, see Lemma 146 below. t u Remark 98. (1) Unlike a countable profinite group, a countable profinite space need not be finite. In fact, the space M D f1=n I n 2 Ng [ f0g with the topology inherited as a subspace M R is a countable profinite set. Hence, contrary to a long term belief, the topological result of Proposition 97 does not reprove the Faltings–Mordell Theorem stating the finiteness of X.k/ for a smooth, projective curve X of genus at least 2 over a num
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