In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.
Contents
Definition
Let L/k be a finite extension of fields, and X a variety defined over L. The functor
(In particular, the k-rational points of
From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L
Properties
For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension.
Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism
Examples and applications
1) Let L be a finite extension of k of degree s. Then
2) If X is an affine L-variety, defined by
we can write
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
5) The Weil restriction
6) Restriction of scalars on abelian varieties (e.g. elliptic curves) yields abelian varieties, if L is separable over k. James 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.
7) In elliptic curve cryptography, the Weil descent attack uses the Weil restriction to transform a discrete logarithm problem on an elliptic curve over a finite extension field L/K, into a discrete log problem on the Jacobian variety of a hyperelliptic curve over the base field K, that is potentially easier to solve because of K's smaller size.
Weil restrictions vs. Greenberg transforms
Restriction of scalars is similar to the Greenberg transform, but does not generalize it, since the ring of Witt vectors on a commutative algebra A is not in general an A-algebra.