A global proof of the arithmetic fundamental lemma for unitary groups over Q_p (odd p >= n) via an SL_2-equivariant relative trace formula and derived CM cycles, which also yields the Jacquet-Rallis fundamental lemma for residue field size q >= n.
Kudla--Rapoport cycles and derivatives of local densities
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abstract
We prove the local Kudla--Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport--Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla--Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia--Sankaran, we also prove cases of the arithmetic Siegel--Weil formula in any dimension.
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Weil representation and Arithmetic Fundamental Lemma
A global proof of the arithmetic fundamental lemma for unitary groups over Q_p (odd p >= n) via an SL_2-equivariant relative trace formula and derived CM cycles, which also yields the Jacquet-Rallis fundamental lemma for residue field size q >= n.