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1. THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

   https://doi.org/10.1017/S1474748020000535  

   Meli, Alexander Bertoloni (January 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract We study the l -adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $$\mathrm {GL_n}$$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $$\mathrm {Mant}_{b, \mu }$$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $$\mathrm {Mant}_{b, \mu }(\rho )$$ for $$\rho $$ a supercuspidal representation. In this paper, we give a conjectural formula for $$\mathrm {Mant}_{b, \mu }(\rho )$$ for $$\rho $$ an admissible representation and prove it when $$\rho $$ is essentially square-integrable. Our proof works for general $$\rho $$ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup. 

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2. 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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3. 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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4. 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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5. Linear invariance of intersections on unitary Rapoport–Zink spaces

   https://doi.org/10.1515/forum-2019-0023  

   Howard, Benjamin (September 2019, Forum Mathematicum)

   Abstract We prove an invariance property of intersections of Kudla–Rapoport divisors on a unitary Rapoport–Zink space. 

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