On the rationality of the Weil Representation and the local theta correspondence

We prove that the Weil representation over a non-archimedean local field can be realised with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\mathrm{\ell }$-modular coefficient fields, as well as coefficient rings such as rings of integers i.e. in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational.