On the Section Conjecture over Local Fields

Abstract

The analogue of the section conjecture can be explored over a local field k. In the archimedean case only $k = \mathbb{R}$ makes sense. In this case, the space of sections is in bijection with the set of connected components of real points, see Theorem 229. Several proofs of this fact are known and presented here.For a finite extension $k/{\mathbb{Q}}_{p}$ the results are less complete. A section in this case gives rise to a valuation of the function field that extends the p-adic valuation on k, such that the image of the section lies in the decomposition subgroup of the valuation, see Theorem 235. However, up to now we cannot exclude that this valuation might be an exotic valuation not corresponding to a rational point.