Integral bases and invariant vectors for Weil representations

Abstract

We construct, for the Weil representation associated with any discriminant form, an explicit basis in which the action of the representation involves algebraic integers over its field of definition. The action of a general element of $${\text {SL}}_{2}(\mathbb {Z})$$ on many parts of these bases is simple and explicit, a fact that we use for determining the dimension of the space of invariants for some families of discriminant forms.