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Presentations of Galois groups of maximal extensions with restricted ramifications

Algebra and Number Theory Seminar

Presentations of Galois groups of maximal extensions with restricted ramifications
Series: Algebra and Number Theory Seminar
Location: Online
Presenter: Yuan Liu, University of Michigan

In this talk, we are going to discuss how to use Galois cohomology to study the presentation of Galois groups of maximal extensions with restricted ramifications. In previous work with Melanie Matchett Wood and David Zureick-Brown, we conjecture that an explicitly-defined random profinite group model can predict the distribution of the Galois groups of maximal unramified extension of global fields that range over $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, our conjecture is supported by the moment computation, but very little is known in the number field case. Interestingly, our conjecture suggests that the random group should simulate the maximal unramified Galois groups, and hence suggests some particular requirements on the structure of these Galois groups. In this talk, we will prove that the maximal unramified Galois groups are always achievable by our random group model, which verifies those interesting requirements.