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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. Arithmetic diagonal cycles on unitary Shimura varieties

   https://doi.org/10.1112/S0010437X20007289  

   Rapoport, M.; Smithling, B.; Zhang, W. (September 2020, Compositio Mathematica)

   null (Ed.)

   We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension. 

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3. Intersection formulas on moduli spaces of unitary shtukas

   https://doi.org/10.1007/s40993-025-00638-6  

   Chen, Yongyi; Howard, Benjamin (June 2025, Research in Number Theory)

   Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a low-dimensional case, and deduce from it a Gross-Zagier style formula expressing intersection multiplicities of cycles in terms of higher derivatives of base-change L-functions. 

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4. Regular integral models for Shimura varieties of orthogonal type

   https://doi.org/10.1112/S0010437X22007370  

   Pappas, G.; Zachos, I. (April 2022, Compositio Mathematica)

   We consider Shimura varieties for orthogonal or spin groups acting on hermitian symmetric domains of type IV. We give regular $$p$$ -adic integral models for these varieties over odd primes $$p$$ at which the level subgroup is the connected stabilizer of a vertex lattice in the orthogonal space. Our construction is obtained by combining results of Kisin and the first author with an explicit presentation and resolution of a corresponding local model. 

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5. 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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