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

יום ד', 16/11/2016 - 10:30

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 (global) function field. 

Any finite set S of closed points of C gives rise to a Dedekind domain O_S:=F_q[C-S] in K.  

We show that given an O_S-regular quadratic space (V,q) of rank n >= 3,  

the group Br(O_S)[2]  is bijective to the set of genera in the proper classification of quadratic O_S-spaces  

isomorphic to V,q for the \'etale topology, thus there are 2^{|S|-1} such.   

If (V,q) is isotropic, then Pic(O_S)/2 properly classifies the forms in the genus of (V,q). 

This is described concretely when V is split by an hyperbolic plane, 

including an explicit algorithm in case C is an elliptic curve.   

For n >= 5 this is true for all genera hence the full classification is via the abelian group H^2_et(O_S,\mu_2).