# Equivariant Gauss sum of finite quadratic forms

## TL;DR

This paper introduces an equivariant Gauss sum for finite quadratic forms, extending classical sums by incorporating group actions, with applications in modular form dimension formulas.

## Contribution

It defines a new equivariant Gauss sum for finite quadratic forms and establishes arithmetic formulas for basic classes, linking to modular form theory.

## Key findings

- Arithmetic formulas for basic quadratic forms
- Invariant's role in vector-valued modular forms
- Extension of classical Gauss sums to equivariant setting

## Abstract

The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulae hold for some basic classes of quadratic forms. In application, such invariant appears in the dimension formula for certain vector-valued modular forms.