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

יום ד', 16/11/2016 - 10:30
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Abstract:

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).