Brauer--Manin obstruction for Weil restrictions

Source abstract

For a quasi-projective variety $X$ defined over a number field $k$, let $X'$ be its Weil restriction with respect to a finite field extension $k/k'$. We prove that the Brauer--Manin set of $X$ agrees with the Brauer--Manin set of $X'$. We also prove the agreement for Brauer--Manin set of them given by $\mathrm{Br}_1$.