2025/09/05 by Philippe Cassou‐Noguès, M. J. Taylor, Cassou-Noguès, Philippe +1
Computer Science · Mathematics · #11E81 #16G30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT) #Polynomial and algebraic computation #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2509.04978
openalex publication_date 2025/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbb 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 ker 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. ker φK is always trivial.