Twisting commutative algebraic groups (Q2382992)

The authors study in detail the properties of the following basic construction: Let \(k\) be a field, \(V/k\) a commutative algebraic group admitting the commutative ring \({\mathcal O}\) as a ring of endomorphisms and \({\mathcal I}\) a finite free \({\mathcal O}\)-module with a continuous action of the absolute Galois group \(G_k\) of \(k\). Then there is a commutative algebraic group \({\mathcal I}\otimes_{\mathcal O} V\) over \(k\), uniquely determined by the following property (Theorem 1.4): If \(k\subseteq L\) is a finite Galois-extension such that the \(G_k\)-action on \({\mathcal I}\) factors through Gal\((L/k)\) then there is an isomorphism of abelian groups, functorial in the \(k\)-algebra \(A\): \[ ({\mathcal I}\otimes_{\mathcal O} V)(A)\simeq ({\mathcal I}\otimes_{\mathcal O} V(A\otimes_k L))^{\text{ Gal}(L/k)}. \] Among the properties of the construction \[ (V,{\mathcal O},{\mathcal I})\mapsto {\mathcal I}\otimes_{\mathcal O} V \] discussed here one finds: Functoriality (Section 1), compatibility with torsion-points/Tate-modules (Theorem 2.2), decomposition up to isogeny (Section 2), relation with A. Weil's restriction of scalars (Sections 3 and 4), the special cases when the \(G_k\)-action factors through an abelian group (respectively a semi-direct product) (Section 5 resp. Section 6), actions of symmetric groups (Section 7) and numerous examples.