Abstract
Let C be a smooth projective curve defined over the finite field Fq (q is odd) and let K=Fq(C) be its function field. Removing one closed point Caf=C−{∞} results in an integral domain 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{∞}-isomorphism classes in the genus of f of rank n>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 Caf. We then deduce that any such f of rank n>2 admits the local-global Hasse principal if and only if |Pic (Caf)| is odd. For rank 2 this principle holds if the integral closure of O{∞} in the splitting field of O_V⊗O{∞}K is a UFD.
Original language | English |
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Pages (from-to) | 346-359 |
Number of pages | 14 |
Journal | Journal of Number Theory |
Volume | 168 |
DOIs | |
State | Published - 1 Nov 2016 |
Bibliographical note
Publisher Copyright:© 2016 Elsevier Inc.
Keywords
- Bilinear forms
- Global function field
- Hasse principle
- Étale cohomology