完美oid志村簇与Calegari-Emerton猜想

经由Sen理论的完美oid性及其在Shimura簇中的应用

Perfectoidness via Sen Theory and Applications to Shimura Varieties

本文通过发展一般赋值环上的p进Hodge理论，建立Sen判据的几何类比，证明一般Shimura簇在p处无穷层的完备化茎具有完美oid性，并由此验证Calegari–Emerton关于整值完备化上同调高次消失的猜想。

局部解析完成上同调

Locally analytic completed cohomology

研究通过旗形簇的等变向量丛和Hodge-Tate周期映射计算Shimura簇的几何Sen算子，并验证Calegari-Emerton猜想中的有理消失性。

幺幂水平下志村簇的消没定理

Vanishing theorems for Shimura varieties at unipotent level

本文研究Hodge型无限$Γ_1(p^∞)$水平志村簇的上同调消没问题，在群于p处分裂的假设下证明中间次数以上紧支上同调消没，推广了已有结果，并应用于常义完成同调余维数的相关结论。

Class groups of Kummer extensions via cup products in Galois cohomology

We use Galois cohomology to study the $p$-rank of the class group of $\mathbf{Q}(N^{1/p})$, where $N \equiv 1 \bmod{p}$ is prime. We prove a partial converse to a theorem of Calegari--Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case $p = 5$, we prove a complete characterization of the $5$-rank of the class group of $\mathbf{Q}(N^{1/5})$ in terms of whether or not $\prod_{k=1}^{(N-1)/2} k^{k}$ and $\frac{\sqrt{5} - 1}{2}$ are $5$th powers mod $N$.