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Supercuspidal representations in the cohomology of the Rapoport-Zink space for the unitary group in three variables (保型表現とその周辺 : RIMS研究集会報告集)
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ito, Tetsushi |
| Copyright Year | 2013 |
| Abstract | This is a summary of the author’s talk at the RIMS workshop “Automorphic Representations and Related Topics” on January 23, 2013. We report on a recent joint work with Yoichi Mieda on supercuspidal representations appearing in the -adic cohomology of the Rapoport-Zink space for the unramified unitary similitude group in three variables over $\mathbb{Q}_{p}$ for $p\neq 2$ . Details will appear elsewhere ([IM2]). Rapoport-Zink spaces are certain formal schemes $\mathscr{M}$ parameterizing quasi-isogenies of $p$-divisible groups with additional structures introduced by M. Rapoport and Th. Zink in the $1990$ ’s ([RZ], [Ra]). These spaces are generalizations of Lubin-Tate spaces and Drinfeld upper half spaces. They play an important role in the theory of $p$-adic uniformization of Shimura varieties, which has many striking applications to number theory and automorphic forms. It is widely believed that the -adic cohomology of the Rapoport-Zink spaces realize the local Langlands and Jacquet-Langlands correspondences in a rather mysterious way. Let us explain a rough outline of the story. For the background on Lubin-Tate spaces and Drinfeld upper half spaces, see Carayol’s paper [Ca]. (Note that the definition of general Rapoport-Zink spaces was not known at that time.) Let $M:=\mathscr{M}^{rig}$ be the rigid analytic space associated to the generic fiber of the formal scheme $\mathscr{M}$ . We have a tower of finite \’etale coverings $M_{r}arrow M$ defined by the level $p^{r}$-structures on the universal pdivisible group on M. The pro-object $M_{\infty}=\{M_{r}\}_{r}$ is sometimes called the Rapoport-Zink tower or the Rapoport-Zink space at infinite level. If the linear algebra datum (RapoportZink datum) defining the Rapoport-Zink space satisfies certain technical conditions, we have a $p$-adic reductive group $G$ , an inner form $J$ of $G$ , and a finite extension $E$ of $\mathbb{Q}_{p}$ (local reflex field). We have a natural action of the product of three groups $G(\mathbb{Q}_{p})\cross J(\mathbb{Q}_{p})xW_{E},$ where $W_{E}$ is the Weil group of $E$ , on the -adic cohomology with compact support |
| Starting Page | 105 |
| Ending Page | 116 |
| Page Count | 12 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1871-14.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
