Integral Bases and Invariant Vectors for Weil Representations

TL;DR

This paper demonstrates that Weil representations linked to discriminant forms have bases with algebraic integer entries and provides explicit formulas for their action, aiding in invariant space dimension calculations.

Contribution

It introduces explicit bases for Weil representations with algebraic integer entries and simplifies the action of SL(2,Z) elements for invariant space analysis.

Findings

Existence of bases with algebraic integer entries for Weil representations

Explicit formulas for the action of SL(2,Z) elements on these bases

Determination of invariant space dimensions for certain discriminant forms

Abstract

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