On the classification of quadratic forms over an integral domain of a global function field.

Let C be a smooth projective curve defined over the finite field F_q (q is odd) and let K=F_q(C) be its function field. Any (non-empty) finite set S of closed points of C gives rise to an integral domain O_S := F_q[C-S] in K. We show that given an O_S-regular quadratic space (V,q) of rank n ?oo 3, the set of genera in the proper classification of quadratic O_S-spaces isomorphic to (V,q) in the flat or étale topology, is in 1:1 correspondence with 2.Br(O_S), thus there are 2|S|-1 genera. Furthermore, if (V,q) is isotropic, then the abelian group Pic(O_S)/2 classifies the forms in the genus Cl_S(O_q) of (V,q). For n ?oo 5, this is true for all genera, hence the full classification is via the abelian group H2_ét(O_S,?_2). If time permits, we shall see when V is split by a hyperbolic plane H(L_0), an explicit isomorphism Pic(O_S)/2 –> Cl_S(O_q), and in case C is an elliptic curve and S={?} where ? is F_q-rational, an algorithm producing representatives of classes in Cl_S(O_q).