On the Quadratic Structure of Torsors over Affine Group Schemes

Let G=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 G is the constant group scheme over K for a finite group G. Then the R-dual ADof 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 AD-module. By multiplying the trace form of BK/K by a certain scalar we obtain a G-invariant form Tr'B which provides a non-degenerate R-form on B. If G has odd order, we show that the G-forms (B, Tr'B) and (A, Tr'A) are locally isomorphic and we study the question of when they are globally isomorphic. Suppose now that K is a finite extension of Qₚ with valuation ring R. In the course of our study we are led to consider the extension of scalars map φK: G₀(AD)→ G₀(ADK)=G₀(K[G]). When AD is the group ring $R[G]$, Swan showed that φK is an isomorphism. Jensen and Larson proved that φK is also an isomorphism for any Hopf R-order AD of $K[G]$ when G is abelian and K is large enough. Here we prove that φK 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. φK is always trivial.