The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field

Let C   be a smooth projective curve defined over the finite field FqFq (q   is odd) and let K=Fq(C)K=Fq(C) be its function field. Removing one closed point Caf=C−{∞}Caf=C−{∞} results in an integral domain O{∞}=Fq[Caf]O{∞}=Fq[Caf] of K, over which we consider a non-degenerate bilinear and symmetric form f   with orthogonal group O_V. We show that the set Cl∞(O_V) of O{∞}O{∞}-isomorphism classes in the genus of f   of rank n>2n>2 is bijective as a pointed set to the abelian groups Hét2(O{∞},μ_2)≅Pic (Caf)/2, i.e. it is an invariant of CafCaf. We then deduce that any such f   of rank n>2n>2 admits the local-global Hasse principal if and only if |Pic (Caf)||Pic (Caf)| is odd. For rank 2 this principle holds if the integral closure of O{∞}O{∞} in the splitting field of O_V⊗O{∞}K is a UFD.