Higher level quadratically twisted Gauss sums and totally isotropic subspaces

TL;DR

This paper evaluates a generalized Gauss sum over matrices in a number field and links its value to the enumeration of totally isotropic subspaces in quadratic spaces, extending previous results.

Contribution

It introduces a new class of quadratically twisted Gauss sums and relates their values to counts of totally isotropic subspaces, generalizing existing theories.

Findings

Explicit evaluation of the generalized Gauss sum.

Connection established between Gauss sums and isotropic subspace counts.

Extension of classical results to more general quadratic spaces.

Abstract

We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of such a Gauss sum, realizing its value in terms of numbers of totally isotropic subspaces of related quadratic spaces.