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

TL;DR

This paper computes the Witt ring of a smooth projective curve over a finite field, revealing its structure through Clifford algebra relations and classical bilinear space results.

Contribution

It provides the first explicit calculation of the Witt ring for such curves, linking geometric properties to algebraic invariants.

Findings

W(C) is a subring of W(k(C))

Clifford algebra triviality determines main relations

Classical bilinear space results complete the calculation

Abstract

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.