Abstract
Galois groups of infinite p-extensions of number fields unramified at p are a complete mystery. We find by computer a family of pro-p groups that satisfy everything that such a Galois group must, and give evidence for the conjecture that these are the only such groups. This suggests that these mysterious Galois groups indeed have a specific form of presentation. There are surprising connections with knot theory and quantum field theory. Finally, the Fontaine-Mazur conjecture reduces to a purely group-theoretic conjecture, and evidence for this conjecture and an extension of it is given.
To John Thompson with thanks on the occasion of his 70th birthday
The author thanks Y. Barnea, L. Bartholdi, M. Bush, R. Grigorchuk, F. Hajir, J. Klüners, T. Kuhnt, and B. Mazur for useful discussions. He was supported by NSF DMS 99-70184
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Boston, N. (2005). Reducing the Fontaine-Mazur Conjecture to Group Theory. In: Voelklein, H., Shaska, T. (eds) Progress in Galois Theory. Developments in Mathematics, vol 12. Springer, Boston, MA. https://doi.org/10.1007/0-387-23534-5_3
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DOI: https://doi.org/10.1007/0-387-23534-5_3
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