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Reciprocity for Gauss sums on finite Abelian groups. (English) Zbl 1048.11501
Summary: A version of the classical reciprocity formula (CRF) for Gauss sums due to Cauchy, Dirichlet, and Kronecker for multivariate Gauss sums was first obtained by A. Krazer in 1912. Krazer’s formula generalizes the case $$w=0$$ of CRF via replacing one of the numbers $$a, b$$ by an integer quadratic form of several variables.
Recently, F. Deloup [Trans. Am. Math. Soc. 351, 1895–1918 (1999; Zbl 0938.57012)] 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 CRF.
In this paper we establish a more general reciprocity for Gauss sums including the Krazer and Deloup formulas and CRF 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 CRF 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 CRF.

##### MSC:
 11T24 Other character sums and Gauss sums 11L05 Gauss and Kloosterman sums; generalizations 11E39 Bilinear and Hermitian forms
Zbl 0938.57012
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