Abstract
The aim of Bogomolov’s programme is to prove birational anabelian conjectures for function fields K|k of varieties of dimension ≥ 2 over algebraically closed fields. The present article is concerned with the 1-dimensional case. While it is impossible to recover K|k from its absolute Galois group alone, we prove that it can be recovered from the pair (Aut(\(\bar K\)|k), Aut(\(\bar K\)|K)), consisting of the absolute Galois group of K and the larger group of field automorphisms fixing only the base field.
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Lüdtke, M. A birational anabelian reconstruction theorem for curves over algebraically closed fields in Arbitrary Characteristic. Isr. J. Math. 227, 987–1011 (2018). https://doi.org/10.1007/s11856-018-1757-2
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DOI: https://doi.org/10.1007/s11856-018-1757-2