The ramified even-dimensional arithmetic transfer conjecture

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Assume that F/F0F/F_0 is ramified and that n≥2n\geq2 is even. Let K0♭K_0^{{{{{{{{\flat}}}}}}}} be the stabilizer of an almost π\pi-modular lattice and let K0+K_0^+ and K0−K_0^- be the stabilizers of the two π\pi-modular lattices in the relevant hermitian space. Let Int⁡(g)\operatorname{Int}(g) be the associated intersection number. Even ramified arithmetic transfer conjecture. (a) For every f′∈Cc∞(Sn(F0))f'\in C_c^\infty(S_n(F_0)) transferring to (1K0♭K0++1K0♭K0−,0)(\mathbf{1}_{K_0^{{{{{{{{\flat}}}}}}}} K_0^+}+\mathbf{1}_{K_0^{{{{{{{{\flat}}}}}}}} K_0^-},0), there exists fcorr′f'_{\mathrm{corr}} such that

2ω(γ)∂Orb⁡(γ,f′)=−Int⁡(g)⋅log⁡q+ω(γ)Orb⁡(γ,fcorr′)2\omega(\gamma)\operatorname{\partial Orb}(\gamma,f')=-\operatorname{Int}(g)\cdot\log q+\omega(\gamma)\operatorname{Orb}(\gamma,f'_{\mathrm{corr}})

for every matching regular semisimple pair. (b) There exists such an f′f' for which

2ω(γ)∂Orb⁡(γ,f′)=−Int⁡(g)⋅log⁡q2\omega(\gamma)\operatorname{\partial Orb}(\gamma,f')=-\operatorname{Int}(g)\cdot\log q

for every matching pair. This is the even-dimensional ramified arithmetic transfer statement formulated in the source.

References

Primary source

Michael Rapoport, Brian Smithling and Wei Zhang, “Regular formal moduli spaces and arithmetic transfer conjectures”, arXiv:1604.02419 (2017).

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