Local Obstructions at a p-adic Place

Abstract

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

$${\mathcal{S}}_{{\pi }_{{}_{ 1}}(X/k)} \rightarrow {\prod \limits }_{v}{\mathcal{S}}_{{\pi }_{{}_{ 1}}(X{\times }_{k}{k}_{v}/{k}_{v})}$$

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 ${X}_{v} = X {\times }_{k}{k}_{v}$ does not admit a section of ${\pi }_{{}_{1}}({X}_{v}/{k}_{v})$, then we say that there is a local obstruction against sections for ${\pi }_{{}_{1}}(X/k)$ 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 Stix (Am. J. Math. 132(1):157–180, 2010) §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 ${\mathbb{Q}}_{p}$-linear form on $Lie ({{\mathrm{Pic}}}_{X}^{0})$ 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).