Title: Demushkin groups of uncountable rank.

Abstract:
Demushkin groups play an important rule in Number Theory, being the maximal pro-p Galois groups of local fields. In 1996 Labute presented a generalization of the theory for infinite countably rank pro-p groups, and proved that the p-Sylow subgroups of the absolute Galois groups of local fields are Demushkin groups of infinite countable rank. These results were extended by Minac&Ware, who gave necessary and sufficient conditions for demushkin groups of infinite countable rank to occur as absolute Galois groups.

In a joint work with Prof. Nikolay Nikolov, we extended this theory to Demushkin groups of uncountable rank. Since for uncountable cardinals   there are  pairwise nonisomorphic nondegenerate bilinear forms, the class of Demushkin groups of uncountable rank is much richer, and in particular, the groups are not determined completely by their invariants, in contrary to the countable case.

We present some results regarding the structure of Demushkin groups of uncountable rank, compute their invariants, and count the number of Demushkin groups of a given uncountable rank with specific combinations of invariants. In addition, we give some equivalent conditions for a pro- group to be a Demushkin group, and investigate the ability of a Demushkin group of uncountable rank to be realized as an absolute Galois group.

The talk can be watched live via Panopto, and will be available later using the same address: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=4d670252-c2b3-4269-af6c-b06d008d7e01