Maximum generalized Hasse-Witt invariants and their applications to anabelian geometry
Maximum generalized Hasse-Witt invariants and their applications to anabelian geometry
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最大广义哈斯维特不变量及其在阿贝尔几何中的应用
DOI：
10.1007/s00029-021-00720-8
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Yu Yang
中科院分区：
文献类型：
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作者：
Ito Kazuhiro;Ito Tetsushi;Koshikawa Teruhisa;中島規博;Yota Shamoto;竹島康博;Yu Yang
设为代数闭特征域上拓扑型的一条点稳定曲线。在一定的假设下，我们证明了，如果分支是一般的，则每个素数到周期的允许覆盖的第一个广义Hasse-Witt不变量达到最大值。这一结果将S.Nakajima关于光滑射影一般曲线的素数到周期覆盖的一个结果推广到(可能是分支的)指向稳定曲线的(可能是奇异的)容许覆盖的情况。此外，我们还证明了，如果是一条任意点的稳定曲线，则存在一个素数到周期的允许覆盖，它的第一个广义Hasse-Witt不变量达到最大值。这一结果推广了M.Raynaud关于光滑射影曲线的素数到周期覆盖的新常规性的结果，推广到(可能是分支的)(可能是奇异的)点稳定曲线的容许覆盖的情形。作为应用，我们得到了一个关于标记点的惯性子群的场结构的Anabelian公式，并证明了从理论上讲，与标记点的惯性子群相关的场结构可以由允许的基本群的开连续同态重构。这些结果将A.Tamagawa关于光滑点稳定曲线标记点的惯性子群的拓扑类型和场结构重构的Anabelian公式推广到任意点稳定曲线的情况。
Letbe a pointed stable curve of topological typeover an algebraically closed field of characteristic. Under certain assumptions, we prove that, ifiscomponent-generic, then the first generalized Hasse–Witt invariant ofeveryprime-to-pcyclic admissible covering ofattains maximum. This result generalizes a result of S. Nakajima concerning the ordinariness of prime-to-pcyclic étale coverings of smooth projective generic curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. Moreover, we prove that, ifis anarbitrarypointed stable curve, then thereexistsa prime-to-pcyclic admissible covering ofwhose first generalized Hasse–Witt invariant attains maximum. This result generalizes a result of M. Raynaud concerning the new-ordinariness of prime-to-pcyclic étale coverings of smooth projective curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. As applications, we obtainan anabelian formula for, and prove that the field structures associated to inertia subgroups of marked points can be reconstructed group-theoretically from open continuous homomorphisms of admissible fundamental groups. Those results generalize A. Tamagawa’s results concerning an anabelian formula for topological types and reconstructions of field structures associated to inertia subgroups of marked points of smooth pointed stable curves to the case ofarbitrarypointed stable curves.