If V is a commutative algebraic group over a field k, O is a
commutative ring that acts on V, and I is a finitely generated free
O-module with a right action of the absolute Galois group of k, then there
is a commutative algebraic group I⊗OV over k, which is a twist of
a power of V. These group varieties have applications to cryptography (in the
cases of abelian varieties and algebraic tori over finite fields) and to the
arithmetic of abelian varieties over number fields. For purposes of such
applications we devote this article to making explicit this tensor product
construction and its basic properties.Comment: To appear in Journal of Algebra. Minor changes from original versio