Orbital exponential sums for prehomogeneous vector spaces

Let $(G,V)$ be a prehomogeneous vector space, let ${\cal O}$ be any $G({\Bbb F}_q)$-invariant subset of $V({\Bbb F}_q)$, and let $\Phi$ be the characteristic function of ${\cal O}$. In this paper we develop a method for explicitly and efficiently evaluating the Fourier transform $\widehat{\Phi}$, 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 {\it 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.