Title: Levels of distribution for sieve problems in prehomogeneous vector spaces,
Abstract: In a companion paper, we developed an efficient algebraic method for
computing the Fourier transforms of certain functions defined on prehomogeneous
vector spaces over finite fields, and we carried out these computations in a
variety of cases.
Here we develop a method, based on Fourier analysis and algebraic geometry,
which exploits these Fourier transform formulas to yield level of distribution
results, in the sense of analytic number theory. Such results are of the shape
typically required for a variety of sieve methods. As an example of such an
application we prove that there are $\gg$ X/log(X) quartic fields whose
discriminant is squarefree, bounded above by X, and has at most eight prime
factors.