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1. 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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2. Shimura varieties for unitary groups and the doubling method

   https://doi.org/10.1007/978-3-030-68506-5_6  

   Harris, Michael (January 2021, Relative Trace Formulas)

   Müller, Werner; Shin, Sug Woo; Templier, Nicolas (Ed.)

   ThetheoryofGaloisrepresentationsattachedtoautomorphicrepresenta- tions of GL(n) is largely based on the study of the cohomology of Shimura varieties of PEL type attached to unitary similitude groups. The need to keep track of the similitude factor complicates notation while making no difference to the final result. It is more natural to work with Shimura varieties attached to the unitary groups themselves, which do not introduce these unnecessary complications; however, these are of abelian type, not of PEL type, and the Galois representations on their cohomology differ slightly from those obtained from the more familiar Shimura varieties. Results on the critical values of the L-functions of these Galois representations have been established by studying the PEL type Shimura varieties. It is not immediately obvious that the automorphic periods for these varieties are the same as for those attached to unitary groups, which appear more naturally in applications of relative trace formulas, such as the refined Gan-Gross-Prasad conjecture (conjecture of Ichino-Ikeda and N. Harris). The present article reconsiders these critical values, using the Shimura varieties attached to unitary groups, and obtains results that can be used more simply in applications. 

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3. Arithmetic fundamental lemma for the spherical Hecke algebra

   https://doi.org/10.1007/s00229-024-01572-0  

   Li, Chao; Rapoport, Michael; Zhang, Wei (June 2024, manuscripta mathematica)

   Abstract We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case$$\textrm{U} (1)\times \textrm{U} (2)$$ $\text{U}\left(1\right)\times \text{U}\left(2\right)$. 

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4. From sum of two squares to arithmetic Siegel–Weil formulas

   https://doi.org/10.1090/bull/1786  

   Li, Chao (July 2023, Bulletin of the American Mathematical Society)

   The main goal of this expository article is to survey recent progress on the arithmetic Siegel–Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel–Weil formula. We then motivate the geometric and arithmetic Siegel–Weil formula using the classical example of the product of modular curves. After explaining the recent result on the arithmetic Siegel–Weil formula for Shimura varieties of arbitrary dimension, we discuss some aspects of the proof and its application to the arithmetic inner product formula and the Beilinson–Bloch conjecture. Rather than being intended as a complete survey of this vast field, this article focuses more on examples and background to provide easier access to several recent works by the author with W. Zhang and Y. Liu. 

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5. Hodge classes and the Jacquet–Langlands correspondence

   https://doi.org/10.1017/fmp.2023.20  

   Ichino, Atsushi; Prasanna, Kartik (September 2023, Forum of Mathematics, Pi)

   We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in ℓ-adic cohomology is Galois invariant for all ℓ. 

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