# Computing Invariants of the Weil representation

## TL;DR

This paper introduces an algorithm to compute invariants of Weil representations linked to finite quadratic modules, establishing their integral basis and stable dimensions over finite fields.

## Contribution

It provides a novel algorithm for computing invariant spaces of Weil representations and proves their integral structure and dimension stability over finite fields.

## Key findings

- Spaces of invariants are defined over integers.
- Dimensions of these spaces are stable over certain finite fields.
- The paper offers an effective computational method for these invariants.

## Abstract

We propose an algorithm for computing bases and dimensions of spaces of invariants of Weil representations of $\mathrm{SL}_2(\mathbb{Z})$ associated to finite quadratic modules. We prove that these spaces are defined over $\mathbb{Z}$, and that their dimension remains stable if we replace the base field by suitable finite prime fields.