On the Quadratic Structure of Torsors over Affine Group Schemes

Abstract

Let $\mathcal{G}=\mathrm{Spec}(A)$ be a finite and flat group scheme over the ring of algebraic integers $R$ of a number field $K$ and suppose that the generic fiber of $\mathcal{G}$ is the constant group scheme over $K$ for a finite group $G$. Then the $R$-dual $A^D$of $A$ identifies as a Hopf $R$-order in the group algebra $K[G]$. If $B$ is a principal homogeneous space for $A$, then it is known that $B$ is a locally free $A^D$-module. By multiplying the trace form of $B_K/K$ by a certain scalar we obtain a $G$-invariant form $T{r}_B^{\prime }$ which provides a non-degenerate $R$-form on $B$. If $G$ has odd order, we show that the $G$-forms $(B,T{r}_B^{\prime })$ and $(A,T{r}_A^{\prime })$ are locally isomorphic and we study the question of when they are globally isomorphic. Suppose now that $K$ is a finite extension of ${\mathbb{Q}}_p$ with valuation ring $R$. In the course of our study we are led to consider the extension of scalars map ${\phi }_K:G_0(A^D)\to G_0(A_K^D)=G_0(K[G])$. When $A^D$ is the group ring $R[G]$, Swan showed that ${\phi }_K$ is an isomorphism. Jensen and Larson proved that ${\phi }_K$ is also an isomorphism for any Hopf $R$-order $A^D$ of $K[G]$ when $G$ is abelian and $K$ is large enough. Here we prove that $\ker {\phi }_K$ is at most a finite abelian $p$-group. However, numerous examples lead us to conjecture that Swan's result extends to all Hopf $R$-orders in $K[G]$, i.e. $\ker {\phi }_K$ is always trivial.