Arithmetic in the fundamental group of a p-adic curve: On the p-adic   section conjecture for curves
  We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the function field of the curve. 
Florian Pop and Jakob Stix
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