Weil restriction

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 Res"L"/"k""X", defined over "k". It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.
Definition.
Let "L/k" be a finite extension of fields, and "X" a variety defined over "L". The functor formula_1 from "k"-schemesop to sets is defined by
(In particular, the "k"-rational points of formula_3 are the "L"-rational points of "X".) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists.
From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism formula_4 and is right adjoint to fiber product of schemes, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on "X" can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars.
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 formula_5 of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability.
Examples and applications.
1) Let "L" be a finite extension of "k" of degree s. Then formula_6 and formula_7 is an "s"-dimensional affine space formula_8 over Spec "k".
2) If "X" is an affine "L"-variety, defined by
we can write formula_3 as Spec formula_11, where "yi,j" 
(formula_12) are new variables, 
and "gl,r" (formula_13) are polynomials in formula_14 given by taking a "k"-basis formula_15 of "L" and setting formula_16 and formula_17.
3) Restriction of scalars over a finite extension of fields takes group schemes to group schemes.
In particular:
4) The torus
where formula_19 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 formula_20 The real points have a Lie group structure isomorphic to formula_21. See Mumford–Tate group.
5) The Weil restriction formula_22 of a (commutative) group variety formula_23 is again a (commutative) group variety of dimension formula_24 if "L" is separable over "k". Aleksander Momot applied Weil restrictions of commutative group varieties with formula_25 and formula_26 in order to derive new results in transcendence theory which were based on the increase in algebraic dimension.
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.
References.
The original reference is Section 1.3 of Weil's 1959-1960 Lectures, published as:
Other references: