The global arithmetic intersection formula at nontrivial level

Let f=pfpf=\otimes_p f_p be a completely decomposed pure tensor in the indicated partial Hecke algebra at nontrivial level, and let f=vfvf'=\otimes_v f'_v be a Gaussian test function on G=ResF/F0(GLn1×GLn)G'=\operatorname{Res}_{F/F_0}(\mathrm{GL}_{n-1}\times\mathrm{GL}_n) whose finite part is a smooth transfer of ff. Assume that ff has regular support at a place λ\lambda of F0F_0.

Global arithmetic intersection conjecture at nontrivial level. One has

Int(f)=J(f)J(fcorr),\operatorname{Int}(f)=-\partial J(f')-J(f'_{\mathrm{corr}}),

where fcorrCc(G(A))f'_{\mathrm{corr}}\in C_c^\infty(G'({\mathbb A})) is a correction function. Moreover, ff' may be chosen to have regular support at λ\lambda and to satisfy fcorr=0f'_{\mathrm{corr}}=0.

This extends the global arithmetic intersection formula to nontrivial level structure and asserts compatibility with correction terms. The source gives no evidence that it has been proved.

Sources & referencesView supporting material

Primary source

Michael Rapoport, Brian Smithling and Wei Zhang, “Arithmetic diagonal cycles on unitary Shimura varieties”, arXiv:1710.06962 (2020).

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