Orbital exponential sums for prehomogeneous vector spaces

TL;DR

This paper introduces a new combinatorial and linear algebra-based method to explicitly evaluate Fourier transforms of functions on prehomogeneous vector spaces, revealing structural properties and potential for improved number-theoretic applications.

Contribution

It develops an efficient computational approach for Fourier transforms in prehomogeneous vector spaces and applies it to five cases, uncovering structural features and cancellation phenomena.

Findings

Fourier transforms exhibit significant structure

Some transforms show more than square root cancellation

Method applied successfully to five prehomogeneous vector spaces

Abstract

Let (G, V) be a prehomogeneous vector space, let O be any G(F_q)-invariant subset of V(F_q), and let f be the characteristic function of O. In this paper we develop a method for explicitly and efficiently evaluating the Fourier transform of f, based on combinatorics and linear algebra. We then carry out these computations in full for each of five prehomogeneous vector spaces, including the 12-dimensional space of pairs of ternary quadratic forms. Our computations reveal that these Fourier transforms enjoy a great deal of structure, and sometimes exhibit more than square root cancellation on average. These Fourier transforms naturally arise in analytic number theory, where explicit formulas (or upper bounds) lead to sieve level of distribution results for related arithmetic sequences. We describe some examples, and in a companion paper we develop a new method to do so, designed to exploit the particular structure of these Fourier transforms.