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

Let C be a smooth projective curve defined over a finite field F_q (q is odd), and let K=F_q(C) be its (global) function field.  This field is considered as the geometric analogue of a number field.

 

Removing one closed point from C results in an affine curve C^af.  The ring of regular functions over C^af is an integral domain, over which we consider a non-degenerate bilinear and symmetric form f of any rank n. 

 

We express the number c(f) of isomorphism classes in the genus of f in cohomological terms and use it to present a sufficient and necessary condition depending only on C^af, under which f admits the Hasse local-global principle.  We say that f admits the Hasse local-global principle if c(f)=1, namely, |Pic(C^\af)| is odd for any n other than 2 and equal to 1 for n=2.  

 

This result emphasizes the difference between Galois cohomology and etale cohomology. Examples are provided.