Joint PU/IAS Number Theory

In recent work with Hajir, Larsen and Maire, we have proved that a large class of finitely generated pro-p groups G can be realized as tamely ramified extensions of a number field K, though ramification at an infinite number of primes is required. We also find some nontrivial G that cannot be so realized.

In the second part of the talk, I will summarize duality results for tamely ramified pro-p extensions of the last 20 years and point to (hopeful) directions forward.