Lecture 3: Reduction Theory Ii Lecture by Brian Conrad Stanford Number Theory Learning Seminar

Example 3. For k′/k a finite separable extension and H any linear algebraic k-group, we have H ↪−→ Rk′/k(Hk′) =: H ′, where Rk′/k(Hk′) is the Weil restriction functor (which is representable by a linear algebraic k-group). This functor is defined by Rk′/k(Hk′)(A) = Hk′(A⊗k k′), so the injection H ↪−→ Rk′/k(Hk′) is given by sending a k-morphism SpecA→ H to its base change by k′. Then we have [H ′] = Hk′(k)\Hk′(Ak′), since Rk′/k(X )(Ak) = X (Ak′ = k′ ⊗k Ak) as topological spaces: see [2, Example 2.4]. The fact that this identification sends Rk′/k(X )(Ak) to X (Ak′) 1 comes from the following proposition: