Wei Zhang's Arithmetic Fundamental Lemma for artinian unitary group elements

Let S(E0)S(E_0) be the relevant symmetric space, let U(J1)U(J_1) be the unitary group attached to the odd hermitian form, and let Ω(γ)Ω(γ), Oγ(1S(OE0))\partial O_γ(1_{S(\mathcal{O}_{E_0})}), and Int(g)\mathrm{Int}(g) denote respectively the transfer factor, the derivative of the orbital integral of the characteristic function of S(OE0)S(\mathcal{O}_{E_0}), and the intersection number associated with an artinian element gg. For every element γS(E0)rsγ\in S(E_0)_{\mathrm{rs}} that matches an artinian element gU(J1)rsg\in U(J_1)_{\mathrm{rs}}, Arithmetic Fundamental Lemma. there is an equality

Ω(γ)Oγ(1S(OE0))=Int(g)log(q).Ω(γ)\partial O_γ(1_{S(\mathcal{O}_{E_0})})=-\mathrm{Int}(g)\log(q).

This is the arithmetic counterpart of the Jacquet–Rallis Fundamental Lemma, relating derivatives of orbital integrals to intersection numbers on relative unitary Rapoport–Zink spaces. The source does not provide a resolution status; the general statement is attributed to Wei Zhang.

Sources & referencesView supporting material

Primary source

Andreas Mihatsch, “Relative unitary RZ-spaces and the Arithmetic Fundamental Lemma”, arXiv:1611.06520 (2019).

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