The Witt Ring of a Smooth Projective Curve over a Finite Field

In this paper we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field of characteristic different from 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. We show that the triviality of the Clifford algebra of a bilinear space over C gives the main relation. The calculation is then completed using classical results for bilinear spaces over fields.