We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $mathbb{G}_m$ and abelian varieties.
In this paper we study tensor products of affine abelian group schemes over a perfect field $k.$ We first prove that the tensor product $G_1 otimes G_2$ of two affine abelian group schemes $G_1,G_2$ over a perfect field $k$ exists. We then describe t
he multiplicative and unipotent part of the group scheme $G_1 otimes G_2$. The multiplicative part is described in terms of Galois modules over the absolute Galois group of $k.$ We describe the unipotent part of $G_1 otimes G_2$ explicitly, using Dieudonne theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.
We compute the etale cohomology ring $H^*(text{Spec } mathcal{O}_K,mathbb{Z}/nmathbb{Z})$ where $mathcal{O}_K$ is the ring of integers of a number field $K.$ As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.
We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For t
he proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse
Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $text{Aut}(text{PSL}(2,q^2))$ for $q$ an odd prime power, cannot be realized as an unramified Galois group over $K,$ but its maximal solvable quotient can. To prove this result, we determine the ring structure of the etale cohomology ring $H^*(text{Spec }mathcal{O}_K;mathbb{Z}/ 2mathbb{Z})$ where $mathcal{O}_K$ is the ring of integers of an arbitrary totally imaginary number field $K.$
