Twisting Commutative Algebraic Groups

Abstract

If (V) is a commutative algebraic group over a field (k), [View the MathML] source is a commutative ring that acts on (V), and View the MathML source is a finitely generated free View the MathML source-module with a right action of the absolute Galois group of (k), then there is a commutative algebraic group [View the MathML source] 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.