Identities on quadratic Gauss sums

Gérardin, Paul; Li, Wen-Ch’ing Winnie

doi:10.1090/S0002-9947-1990-0974516-1

Identities on quadratic Gauss sums
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Trans. Amer. Math. Soc. 321 (1990), 159-182 Request permission

Abstract:

Given a local field $F$, each multiplicative character $\theta$ of the split algebra $F \times F$ or of a separable quadratic extension of $F$ has an associated generalized Gauss sum $\gamma _\theta ^F$. It is a complex valued function on the character group of ${F^ \times } \times F$, meromorphic in the first variable. We define a pairing between such Gauss sums and study its properties when $F$ is a nonarchimedean local field. This has important applications to the representation theory of $GL(2,F)$ and correspondences $[{\text {GL}}3]$.

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Fonctions zeta et distributions