On Grothendieck's Conjecture of Birational Anabelian Geometry

Reconstruction of Function Fields

“…In this paper we implement the program outlined in [1] and [2] describing the correspondence between higher-dimensional function fields and their abelianized Galois groups. For results concerning the reconstruction of function fields from their (full) Galois groups (the birational Grothendieck program) we refer to the works of Pop, Mochizuki and Efrat (see [8], [7] and [5]). second author was employed by the Clay Mathematics Institute.…”

On the Hom-form of Grothendieck’s birational anabelian conjecture in positive characteristic

“…Later, Pop generalized their results to the case of finitely generated fields of higher transcendence degree ( [Pop 1994;; see also [Szamuely 2004] for a survey on Pop's results).…”

“…Remark 1.6. For other characterizations of decomposition groups -applicable to much more general situations -see, for example, Theorem 1.16 of [Pop 1994], Theorem 2 of [Koenigsmann 2003], or the results in [Engler and Koenigsmann 1998;Engler and Nogueira 1994].…”

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“…Later, Pop generalized their results to the case of finitely generated fields of higher transcendence degree ( [Pop 1994;2002]; see also [Szamuely 2004] for a survey on Pop's results).…”

“…Remark 1.6. For other characterizations of decomposition groups -applicable to much more general situations -see, for example, Theorem 1.16 of [Pop 1994], Theorem 2 of [Koenigsmann 2003], or the results in [Engler and Koenigsmann 1998;Engler and Nogueira 1994].…”