arXiv subjects
Publications and source records attributed to Xu Shen.
We construct some $n$-dimensional eigenvarieties for finite slope overconvergent eigenforms over some unitary Shimura varieties with signature $(1,n-1)\times(0,n)\times\cdots\times(0,n)$ by adapting Andreatta-Iovita-Pilloni's method. We also show that there are some Galois pseudo-characters over our eigenvarieties by studying analytic continuation of finite slope eigenforms over these Shimura varieties.
In this paper we study the $p$-adic analytic geometry of the basic unitary group Rapoport-Zink spaces $\M_K$ with signature $(1,n-1)$. Using the theory of Harder-Narasimhan filtration of finite flat groups developed by Fargues in \cite{F2},\cite{F3}, and the Bruhat-Tits stratification of the reduced special fiber $\M_{red}$ defined by Vollaard-Wedhorn in \cite{VW}, we find some relatively compact fundamental domain $\D_K$ in $\M_K$ for the action of $G(\Q_p)\times J_b(\Q_p)$, the product of the associated $p$-adic reductive groups, and prove that $\M_K$ admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda's main theorem in \cite{Mi2}.
We prove that, for a $p$-divisible group with additional structures over a complete valuation ring of rank one $O_K$ with mixed characteristic $(0,p)$, if the Newton polygon and the Hodge polygon of its special fiber possess a non trivial contact point, which is a break point for the Newton polygon, then it admits a "Hodge-Newton filtration" over $O_K$. The proof is based on the theories of Harder-Narasimhan filtration of finite flat group schemes and admissible filtered isocrystals. We then apply this result to the study of some larger class of Rapoport-Zink spaces and Shimura varieties than those studied previously by Mantovan, and confirm some new cases of Harris's conjecture.
We reprove the Lefschetz trace formula for Lubin-Tate spaces, based on the locally finite cell decompositions of these spaces obtained by Fargues, and Mieda's theorem of Lefschetz trace formula for certain open adic spaces (\cite{Mi1} theorem 3.13). This proof is rather different from those of Strauch in \cite{St} (theorem 3.3.1) and of Mieda in \cite{Mi1} (example 4.21), and is quite hopeful to generalized to some other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, we proved a Lefschetz trace formula for some unitary Rapoport-Zink spaces in \cite{Sh} by using similar ideas here.