Chow’s Theorem for Semi-abelian Varieties and Bounds for Splitting Fields of Algebraic Tori

Abstract

A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.

On Tamagawa numbers of CM tori

“…It is shown in [12,Proposition 1.2.1] (also see [23] for other proofs) that every algebraic k-torus splits over a finite separable field extension K/k. The action of Γ k on X(T ) gives a continuous representation (2.1)…”