Examples and Applications
1) Let L be a finite extension of k of degree s. Then (Spec L) = Spec(k) and is an s-dimensional affine space over Spec k.
2) If X is an affine L-variety, defined by
we can write as Spec, where yi,j are new variables, and gl,r are polynomials in given by taking a k-basis of L and setting and .
3) Restriction of scalars over a finite extension of fields takes group schemes to group schemes.
In particular:
4) The torus
where Gm denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of S. The real points have a Lie group structure isomorphic to . See Mumford–Tate group.
5) The Weil restriction of a (commutative) group variety is again a (commutative) group variety, if L is separable over k. Aleksander Momot applied restriction of scalars on group varieties and obtained numerous generalizations of classical results from transcendence theory.
6) Restriction of scalars on abelian varieties (e.g. elliptic curves) yields abelian varieties, if L is separable over k, and Milne used this to reduce the Birch and Swinnerton-Dyer conjecture for abelian varieties over all number fields to the same conjecture over the rationals.
Read more about this topic: Weil Restriction
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