When is the Fourier transform of an elementary function elementary?

TL;DR

This paper investigates when the Fourier transform of elementary functions on vector spaces over local fields remains elementary, providing classifications and examples related to quadratic, cubic forms, and their connections to algebraic varieties and covering systems.

Contribution

It characterizes elementary functions whose Fourier transforms are elementary, especially for quadratic and cubic forms, and links these classifications to algebraic geometry and covering systems.

Findings

Classified elementary functions with elementary Fourier transforms for quadratic and cubic forms.

Connected the classification to Zak's Severi varieties.

Provided explicit examples over real, complex, and p-adic fields.

Abstract

Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method.