Anabelian Geometry and Field Arithmetic II

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This research project concerns the study of anabelian phenomena in arithmetic and algebraic geometry as well as questions in field arithmetic. The PI plans to continue his work on an anabelian program initiated by Bogomolov, which aims at recovering function fields of transcendence degree at least two from their pro-l abelian-by-central Galois theory in a functorial way. The PI completed that program for function fields over algebraic closures of finite fields, and he plans to complete that program for function fields over algebraic closures of global fields and more general algebraically closed base fields. The PI plans to exploit the relation of the anabelian program under discussion with the Ihara/Oda-Matsumoto conjecture, and to use these methods to give generalizations in several directions of the Ihara/Oda-Matsumoto conjecture; in particular, to prove this conjecture for arbitrary base fields. This would have a major impact on understanding the Galois structure of the field of rational numbers in particular, and of arbitrary fields in general. The PI (jointly with collaborators) expects as well to make progress on Grothendieck's (p-adic) section conjecture and its relation to (an effective) Mordell conjecture --Faltings' Theorem. Finally, the PI expects to make progress on better understanding how localization processes --in particular, which such processes-- lead to large fields. In particular, to gain a better understanding of how localization processes relate via local-global principles to large fields and the Freeness Conjecture. The PI plans to simplify and prove stronger results concerning the solvability of non-trivial split embedding problems over large fields in classical Galois, as well as differential Galois, theoretical context, both by developing new tools and by using results of general type proved previously and used successfully in other context.Positive answers to the questions mentioned above would have a significant impact on the progress of modern Galois theory and on some of the very fundamental questions in arithmetic geometry and algebraic geometry. The results will be widely disseminated to the mathematical community via talks and publications in scientific journals. The PI is co-organizer of, and senior invited researcher at, activities which aim to do both: first, to create a broad basis for international cooperation, training, and scientific exchange at all levels; and second, to have special activities for graduate students and young researchers, thus enhancing teaching and technological understanding.