# Thursday 20 Jul 2017: Ramification, Galois scaffolds and Hopf orders associated with the Heisenberg group modulo $p$.

### Griff Elder - University of Nebraska at Omaha

H101 14:30-15:30

Let H denote de Heisenberg group modulo p, namely the nonabelian group of order $p^3$. Let K be a local field o characteristic p with finite residue field. I will discuss the embedding of a totally ramified $C_p\times C_p$-extension into H-extensions and what this means for the conductor of the H-extension. Then I will exhibit a family of totally ramified H-extensions with a Galois scaffold and explain the strategy I use when searching for such extensions. For these extensions, Galois module structure is known, and this, based upon a wild guess, suggests a form that the Hopf orders in KH may take. I will then explain how I verified that this is indeed a family of Hopf orders in KH, and discuss their duals.