Theta Functions and Reciprocity Laws

TL;DR

This paper explores the properties of generalized quadratic Gauss sums as finite analogues of theta functions, establishing their transformation laws, functional equations, and a general reciprocity law for exponential sums of quadratic forms.

Contribution

It introduces a new framework linking Gauss sums with theta functions and proves a broad reciprocity law for quadratic exponential sums in multiple variables.

Findings

Gauss sums exhibit a functional equation and Euler product similar to zeta functions.

Finite Dirichlet series attached to Gauss sums have roots on the critical line.

A general reciprocity law for sums of quadratic exponential functions is established.

Abstract

In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using a M\"obius transform, and show they have a functional equation, Euler product factorization, and roots only on the critical line. In the second part, we prove a general reciprocity law for sums of exponentials of rational quadratic forms in any number of variables, using the transformation formula of the Riemann theta function on the Siegel upper half-space.