Number Theory Seminar
Almost 60 years ago Golod and Shafarevich gave examples of number fields with infinite Hilbert class field towers. These fields generally had quite large root discriminants. Forty and twenty years ago respectively, Martinet and Hajir-Maire gave examples of tamely ramified p-towers of number fields with relatively small root discriminant, though these are still far away from the GRH lower bounds. I will explain new recent records and work on better understanding tame Shafarevich groups, kernels of H^2 localization maps. This is joint with Hajir and Maire.