Arithmetic Fundamental Lemma for regular semi-simple self-adjoint artinian pairs

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Let (x,j)(x,j) be a pair in the odd hermitian setting. It is regular semi-simple and self-adjoint as defined in the source, and it is artinian when the schematic intersection Z(x)∩Z(j)\mathcal{Z}(x)\cap\mathcal{Z}(j) is artinian; write Int(x,j)\mathrm{Int}(x,j) for its length. Let ∂O(x,j)\partial O(x,j) be the derivative of the signed lattice-counting quantity. Arithmetic Fundamental Lemma. For every regular semi-simple, self-adjoint and artinian pair (x,j)(x,j),

∂O(x,j)=−Int(x,j).\partial O(x,j)=-\mathrm{Int}(x,j).

This is the lattice formulation of the Arithmetic Fundamental Lemma for the odd hermitian form. The source states it as a conjectural identity and does not give a resolution status.

References

Primary source

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

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