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1. A Tannakian Framework for G-displays and Rapoport–Zink Spaces

   https://doi.org/10.1093/imrn/rnz311  

   Daniels, Patrick (December 2019, International Mathematics Research Notices)

   Abstract We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport–Zink functors associated to triples $$(G,\{\mu \},[b])$$, where $$G$$ is a flat affine group scheme over $${\mathbb{Z}}_p$$ and $$\mu$$ is a cocharacter of $$G$$ defined over a finite unramified extension of $${\mathbb{Z}}_p$$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport–Zink spaces given by Bültel and Pappas. As an application, we prove a special case of a conjecture of Bültel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data. 

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2. The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

   https://doi.org/10.1515/crelle-2022-0077  

   Bertoloni Meli, Alexander; Nguyen, Kieu Hieu (January 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p {\operatorname{\mathbb{Q}}_{p}} in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider. 

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3. Chow groups and L -derivatives of automorphic motives for unitary groups, II.

   https://doi.org/10.1017/fmp.2022.2  

   Li, Chao; Liu, Yifeng (January 2022, Forum of Mathematics, Pi)

   Abstract In this article, we improve our main results from [LL21] in two directions: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla–Rapoport conjecture for exotic smooth Rapoport–Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomology of integral models of unitary Shimura varieties with Drinfeld level structures. 

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4. -DISPLAYS AND RAPOPORT–ZINK SPACES

   https://doi.org/10.1017/S1474748018000373  

   Bültel, O.; Pappas, G. (July 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$(G,\unicode[STIX]{x1D707})$$ be a pair of a reductive group $$G$$ over the $$p$$ -adic integers and a minuscule cocharacter $$\unicode[STIX]{x1D707}$$ of $$G$$ defined over an unramified extension. We introduce and study ‘ $$(G,\unicode[STIX]{x1D707})$$ -displays’ which generalize Zink’s Witt vector displays. We use these to define certain Rapoport–Zink formal schemes purely group theoretically, i.e. without $$p$$ -divisible groups. 

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5. FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

   https://doi.org/10.1017/fms.2019.45  

   HE, XUHUA; LI, CHAO; ZHU, YIHANG (January 2019, Forum of Mathematics, Sigma)

   We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic. 

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