We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in $2^n$ variables over function fields of transcendence degree $n\ge 2$ over an algebraically closed field of characteristic $\ne 2$. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified cohomology of products of elliptic curves over discretely valued fields.

中文翻译：

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函数域上二次型各向同性的局部全局原理失效

我们证明了局部全局原理在离散估值方面的失败，对于二次形式的各向同性 $2^n$ 超越度函数域上的变量 $n\ge 2$ 在代数闭特征域上 $\ne 2$ 。我们的构造涉及 Borcea、Cynk 和 Hulek 考虑的广义 Kummer 簇，以及关于离散值域上椭圆曲线乘积的非分支上同调的非平凡性的新结果。