Local Obstructions at a p-adic Place
Let X be a geometrically connected variety over a number field k, and let k v denote the completion of k in a place v. Base change \(s\mapsto (s \otimes {k}_{v})\) as in Sect. 3.2 induces a localisation map $${\mathcal{S}}_{{\pi }_{{}_{ 1}}(X/k)} \rightar
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Local Obstructions at a p-adic Place
Let X be a geometrically connected variety over a number field k, and let kv denote the completion of k in a place v. Base change s 7! .s ˝ kv / as in Sect. 3.2 induces a localisation map Y S1 .X=k/ ! S1 .X k kv =kv / v
where v ranges over all places of k. We will address the section conjecture from this local to global point of view. More of the purely local problem will be addressed in Chap. 16, and the classical obstructions against the passage from local to global form the topic of Chap. 11. If the local curve Xv D X k kv does not admit a section of 1 .Xv =kv /, then we say that there is a local obstruction against sections for 1 .X=k/ at the place v. Any consequence of the existence of the section for a curve over a local field can be turned around to produce an example of a curve over a number field that satisfies the section conjecture: the example has neither a section nor a rational point, because sections are obstructed locally due to the arithmetic consequence not being true. Examples of this kind by considering period and index of p-adic curves with a section were first constructed in [Sx10b] 7 and are recalled below in Theorem 114. If a p-adic curve with a section fails to have index 1, then we can construct a Qp -linear form on Lie.Pic0X / for which no natural explanation exists and which we therefore consider bizarre (in the hope that this case does not exist in compliance with the local section conjecture).
10.1 Period and Index The following definitions work for arbitrary base fields K. Definition 113. Let X=K be a geometrically integral variety. J. Stix, Rational Points and Arithmetic of Fundamental Groups, Lecture Notes in Mathematics 2054, DOI 10.1007/978-3-642-30674-7 10, © Springer-Verlag Berlin Heidelberg 2013
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(1) The index of X is the positive number index.X / D gcdfdeg.a/ D Œ.a/ W K I a is a closed point of X g: (2) The period of X is the order period.X / of the universal Albanese torsor, see [Wi08], as a principal homogeneous space under the Albanese variety of X . (3) The relative Brauer group of X=K is the kernel Br.X=K/ of the pullback map Br.K/ ! Br.X /: The period–index result. We now put ourselves in a p-adic local framework and let K be a finite extension of Qp . ln the case of curves we can prove the following. Theorem 114. Let K be a finite extension of Qp and let X=K be a smooth, projective geometrically connected curve of genus > 0, such that 1 .X=K/ admits a section. (1) For p odd, we must have period.X / D index.X /; and both are powers of p. (2) For p D 2, both period.X / and index.X / are powers of 2. If we moreover assume that we have an even degree finite e´ tale cover X ! X0 , then also here period.X / D index.X /: Proof. This was proved in [Sx10b] Theorem 15C16. Another proof depending on the cycle class of the section was given later in [EsWi09] Corollary 3.6. t u It is the goal of the following Sect. 10.2 to connect Theorem 114 along the proof following [EsWi09] Corol
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