Reciprocity for Gauss sums on finite abelian groups

Abstract

A classical reciprocity formula for Gauss sums due to Cauchy, Dirichlet, and Kronecker states thatformula herewhere a, b are nonzero integers and w∈ℚ such that ab+2aw∈2ℤ. For a detailed proof and historical background, see [4, chapter IX]. Various versions of formula (1) have been extensively studied in connection with transformation properties of theta functions. A version of (1) for multivariate Gauss sums was first obtained by A. Krazer [10] in 1912, see also [2, 5, 11, 14]. Krazer's formula generalizes the case w=0 of (1) via replacing one of the numbers a, b by an integer quadratic form of several variables.Recently, Florian Deloup [6] found a new and most beautiful reciprocity formula for multivariate Gauss sums. His formula is a far reaching generalization of Krazer's result. Roughly speaking, Deloup replaces both numbers a, b with quadratic forms. Deloup involves Wu classes of quadratic forms which allows him to remove the evenness condition appearing in Krazer's formulation. However, Deloup's formula covers only the cases w=0 and w=b/2 of (1).In this paper we establish a more general reciprocity for Gauss sums including the Krazer and Deloup formulas and formula (1) in its full generality. Our reciprocity law involves two quadratic forms and a so-called rational Wu class, as defined below.The original proofs of Cauchy, Dirichlet, Kronecker and Krazer are analytical and involve a study of a limit of a transformation formula for theta-functions. Deloup's proof goes by a reduction to the case w=b/2 of (1) based on a careful study of Witt groups of quadratic forms. Our proof is more direct and uses only (a generalization of) the van der Blij computation of Gauss sums via signatures of integer quadratic forms. In particular, our argument provides a new proof of (1).