Twisting commutative algebraic groups

Details Twisting commutative algebraic groups Twisting commutative algebraic groups

2006-09-02

arxiv.org/abs/math/0609066v2

Mathematics

Algebraic Geometry

To appear in Journal of Algebra. Minor changes from original version

Scientific paper

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 $I \otimes_O V$ 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.

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