Levels of distribution for sieve problems in prehomogeneous vector spaces

Abstract

In our companion paper [28], 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 ≫ X log X quartic fields whose discriminant is squarefree, bounded above by X, and has at most eight prime factors.

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

“…In §5, we first study the number of polynomials whose discriminants are almost prime. We prove an almost prime discriminant result for all n ě 3 that obtains discriminants with fewer prime factors than [TT20a] if n " 4.…”

“…discriminants with relatively few distinct prime factors. We draw inspiration from the following result of Taniguchi and Thorne [TT20a], who were in turn inspired by the folklore conjecture that there should be infinitely many fields of prime discriminant in every degree; this is known only for quadratic extensions, however.…”

Families of elliptic curves ordered by conductor

“…Indeed, the height of E ∈ E sf with C(E) = X can be as large as X 2 , in which case the Ekedahl sieve gives rise to an error term of O(X 4/3 ) which is much too large. Subsequent improvements to the Ekeshal sieve by Taniguchi-Thorne [30], in which the sieve is combined with equidistribution methods, are also insufficient for our purposes.…”

Galois groups of random integer polynomials and van der Waerden's Conjecture

“…A common ingredient in proving that sets are equidistributed modulo a prime p is Fourier analysis over F p (see, e.g., Taniguchi-Thorne [39] for a nice introduction to this topic where, in particular, the case of binary cubic forms is treated). Here, we apply Fourier analysis to show that integral binary forms (resp.…”