AI 中文总结
推广井草局部p进单值化定理，研究\(\mathbb{F}_p\)上光滑拟射影簇X的超收敛F - 等晶体单位根子对象局部单值化，推广到志村簇，特殊情况也有成果，还证明了阿贝尔簇赫克轨道约化的有限性。
AI 中文摘要
我们推广了井草的一个局部p进单值化定理，该定理研究与模曲线超奇异点周围的泛椭圆曲线相关的单值化表示。设X是\(\mathbb{F}_p\)上的光滑拟射影簇。我们建立并证明了X上超收敛F - 等晶体的井草定理的类似物，使得弗罗贝尼乌斯作用在X的闭点处是代数的、p平凡的且半单的。我们研究了它们的单位根子对象在X中具有等斜牛顿斜率的点周围的局部单值化。特别地，我们的结果将井草定理推广到了来自志村簇的超收敛F - 等晶体，对于阿贝尔型志村簇是无条件的，对于特殊志村簇在弗罗贝尼乌斯半单性条件下（该性质未知）。在西格尔志村簇的特殊情况下，我们还证明了其紧化边界上一点的局部单值化的类似结果。作为我们结果的一个推论，我们证明了等特征局部域上阿贝尔簇的赫克轨道约化的有限性结果。
英文摘要
We generalise a local $p$-adic monodromy theorem of Igusa that studies the monodromy representation associated to the universal elliptic curve around a supersingular point of the modular curve. Let $X$ be a smooth quasi-projective variety over $\mathbb{F}_p$. We set up and prove the analog of Igusa's theorem for overconvergent F-isocrystals on $X$ such that the action of the Frobenius is algebraic, $p$-plain, and semisimple at the closed points of $X$. We study the local monodromy of their unit root sub-objects around a point in $X$ with isoclinic Newton slopes. In particular, our result generalises Igusa's Theorem to overconvergent F-isocrystals arising from Shimura varieties, unconditionally for Shimura varieties of abelian type and conditional on Frobenius semisimplicity for exceptional Shimura varieties where this property is not known yet. In the particular case of Siegel Shimura varieties, we also prove an analogous result for the local monodromy of a point in the boundary of its compactification. As a consequence of our results, we prove a finiteness result for the reduction of the Hecke orbit of abelian varieties over a local field of equicharacteristic.
