Recovering a local field from its Galois group

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• Marius Leonhardt (University of Cambridge)
• Tuesday 07 February 2017, 14:30-15:30
• MR13.

If you have a question about this talk, please contact G. Rosso.

What characteristics of a field can be deduced from its absolute Galois group? Does the Galois group uniquely determine the field? It turns out that the answer to this question depends on the “type” of field. For example, any two finite fields have isomorphic absolute Galois groups, whereas two number fields are isomorphic if and only if their Galois groups are. In the case of finite extensions of $\Q_p$, there are non-isomorphic fields with isomorphic Galois groups. However, if one requires the group isomorphism to respect the filtration given by the ramification subgroups, then S. Mochizuki has shown that one can fully reconstruct the field. In this talk I will give an overview of the methods involved in Mochizuki’s proof, focussing on the use of Hodge-Tate representations in the construction of an isomorphism between two given fields.

This talk is part of the Number Theory Seminar series.

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