Abstract

Let F be a global field with char(F)≠2 and K an algebraic function field in one variable of genus zero over F. In this paper, we investigate two kinds of Hasse principles for Brauer classes on K. If Br(K) is the Brauer group of K and Br(K)′ is the subgroup of Br(K) whose elements have order relatively prime to char(F), then we precisely determine the kernels of the maps

where p runs over the prime spots of F and P runs over the places of K which are trivial over F, and F_p, K_P are the completions at p, P respectively. To facilitate the determination of these kernels, we compute the kernel of the map h : Br(K) → ∏ _P Br(KV _P) where V _P is the residue field with respect to P and show that the kernels of these three maps coincide. We then consider a more general version of the maps above by describing the 2-torsion subgroup of the kernel of h₁ when a finite number of prime spots in the product are omitted.

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