The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field

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• 作者：Rony A. Bitan ^{rony.bitan@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Global function field ; Bilinear forms ; É ; tale cohomology ; Hasse principle
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：168
• 期：Complete
• 页码：346-359
• 全文大小：384 K

文摘

Let C   be a smooth projective curve defined over the finite field F_q${\mathbb{F}}_q$ (q   is odd) and let K=F_q(C)$K={\mathbb{F}}_q(C)$ be its function field. Removing one closed point C^af=C−{∞}$C^{\text{af}}=C-\{\infty \}$ results in an integral domain O_{∞}=F_q[C^af]${\mathcal{O}}_{\{\infty \}}={\mathbb{F}}_q[C^{\text{af}}]$ of K, over which we consider a non-degenerate bilinear and symmetric form f   with orthogonal group ${\underset{\_}{\mathbf{O}}}_V$. We show that the set ${\text{Cl}}_{\infty }({\underset{\_}{\mathbf{O}}}_V)$ of O_{∞}${\mathcal{O}}_{\{\infty \}}$-isomorphism classes in the genus of f   of rank n>2$n>2$ is bijective as a pointed set to the abelian groups $H_{\text{ét}}^2({\mathcal{O}}_{\{\infty \}},{\underset{\_}{\mu }}_2)\cong \text{Pic}(C^{\text{af}})/2$, i.e. it is an invariant of C^af$C^{\text{af}}$. We then deduce that any such f   of rank n>2$n>2$ admits the local-global Hasse principal if and only if |Pic (C^af)|$|\text{Pic}(C^{\text{af}})|$ is odd. For rank 2 this principle holds if the integral closure of O_{∞}${\mathcal{O}}_{\{\infty \}}$ in the splitting field of ${\underset{\_}{\mathbf{O}}}_V{\otimes }_{{\mathcal{O}}_{\{\infty \}}}K$ is a UFD.