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 pointof 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 groupsis an invariant of C af . We then deduce that any such f of rank n > 2 admits the local-global Hasse principal if and only if |Pic (C af )| 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.
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. Any finite set S of closed points of C gives rise to an integral domain OS := Fq[C − S] in K. We show that given an OS-regular quadratic space (V, q) of rank n ≥ 3, the set of genera in the proper classification of quadratic OS-spaces isomorphic to (V, q) in the flat orétale topology, is in 1 : 1 correspondence with 2Br(OS ), thus there are 2 |S|−1 such. If (V, q) is isotropic, then Pic (OS)/2 classifies the forms in the genus of (V, q). For n ≥ 5 this is true for all genera, hence the full classification is via the abelian group H 2 et (OS, µ 2 ).
Let G be a semisimple group defined over a global function field K of a rational curve, not anistropic of type An. We express the (relative) Tamagawa number of G in terms of local data including the number t∞(G) of types in one orbit of a special vertex in the Bruhat-Tits building of G∞(K∞) for some place ∞ and the class number h∞(G) of G at ∞. denote suitable Cartan subgroups of G sc and G respectively; cf. Proposition 7.8. These concrete computations allow us to also provide a wealth of examples in Section 6 for which we compute the relative Tamagawa numbers. We also demonstrate the result in a case of a split group defined over the function field of an elliptic curve (Remark 7.11).This article is organized as follows: In the preliminary Section 2 we fix the relevant notions from Bruhat-Tits theory. In Section 3 we compute volumes of parahoric subgroups over local fields, their maximal unramified extensions, and their valuation rings. In Section 4 we revise the definition of the Tamagawa number of semisimple K-groups and establish a decomposition of G(A)/G(K) enabling us to express τ (G) in terms of a global invariant and a local one. In Section 5 we compute cohomology groups over rings of S-integers with |S| = 1, use Bruhat-Tits theory and Serre's formula ([Ser1, p. 84], [BL, Corollary 1.6]) in order to derive the above-mentioned formula (13) for computing the Tamagawa number. In Section 6 we express the number t ∞ (G) of types in the orbit of a special point in terms of F ∞ , accomplishing the proof of our Main Theorem. The final Section 7 addresses the above-mentioned application and examples.
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