Anabelian Geometry and Field Arithmetic

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The Principal Investigator plans to continue his study of Anabelian geometry and of Field Arithmetic, as well as the interaction of these two subjects with other aspects of mathematics in general, and with arithmetic geometry and algebraic geometry in particular. This study will be guided in part by possible extensions of the known anabelian phenomena, and of the known facts from field arithmetic, to which the PI has made major contributions. The PI expects to prove the pro-l abelian-by-central birational anabelian Conjecture for function fields of tr.degree >1 over algebraic closures of finite fields, of global fields, and even more general algebraically closed fields. The PI expects to obtain the pro-l abelian-by-central form of the Ihara/Oda--Matsumoto conjecture, which would have a major impact on understanding the Galois structure of the field of rational numbers (and other fields). The PI expects as well to make progress on Grothendieck's (p-adic) Section Conjecture and its relation to (an effective) Mordell Conjecture --Faltings' Theorem. The PI expects to make progress on better understanding the class of large fields, and of the cohomology of fields as related to the Freeness Conjecture. Positive answers to the questions mentioned above would have a very 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 plan 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.