Gauss sums for function fields

Abstract

In this paper, we study the behaviour of “Gauss sums” (introduced in [Th1, Th2]) for function fields of one variable over finite fields. We will show (Section 1) that their prime factorization is analogous to the classical case for the rational function field (at least when the infinite place chosen is of degree not more than two) and is quite interesting, though different for higher genus fields. Classically, the factorization of Gauss sums for composite modulus into those for its prime power factors, gives control on their absolute values and is quite important in “the local constants decomposition.” In our case, the situation is wildly different (Section 2). All these results should be seen in the light of various analogies established in [Th2].