经典超曲面的扭曲形式
Twisted forms of classical hypersurfaces
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中文总结 AI 辅助
该研究在实数域和有限域上计数费马、德尔萨特、克莱因超曲面的扭曲形式，推广微分方法到正特征域，计算其正特征自同构群，重现了费马超曲面实形式的相关定理。
中文摘要 AI 辅助
我们在实数域和有限域上计数三类具有大自同构群的光滑超曲面的扭曲形式：费马（Fermat）超曲面、德尔萨特（Delsarte）超曲面和克莱因（Klein）超曲面。我们的主要工具是一个计数公式，适用于自同构群为对角阿贝尔群与其定义方程单项式置换群的半直积的光滑超曲面的伽罗瓦上同调集。为在有限域上应用该公式，我们将Oguiso和Yu的微分方法推广到正特征域，并在素数p的显式算术条件下，计算费马、德尔萨特和克莱因超曲面在特征为正的代数闭域上的自同构群。在实数域上，我们的计数结果重现了Sasaki关于费马超曲面实形式的最新定理。
英文摘要
We count the twisted forms, over the field of real numbers and over finite fields, of the three classical families of smooth hypersurfaces with large automorphism group: the Fermat, Delsarte and Klein hypersurfaces. Our main tool is a counting formula for the Galois cohomology set of a smooth hypersurface whose automorphism group is the semidirect product of a diagonal abelian group and a group of permutations of the monomials of its defining equation. In order to apply this formula over finite fields, we extend the differential method of Oguiso and Yu to positive characteristic, and we compute the automorphism groups of the Fermat, Delsarte and Klein hypersurfaces over algebraically closed fields of positive characteristic under explicit arithmetic conditions on p. Over the reals, our count recovers a recent theorem of Sasaki on the real forms of Fermat hypersurfaces.
