Twisted arithmetic transfer conjecture for regular semisimple intersections

Let L0=ZpnL_0=\mathbb Z_p^n and L0=p1ZptZpntL_0^\vee=p^{-1}\mathbb Z_p^t\oplus\mathbb Z_p^{n-t}, and let GL(L0,L0)\mathrm{GL}(L_0,L_0^\vee) be the corresponding parahoric subgroup. For a regular semisimple pair (g,u)(GL(V)/U(V)×V)(Qp)rs(g,u)\in(\mathrm{GL}(\mathbb V)/\mathrm U(\mathbb V)\times\mathbb V)(\mathbb Q_p)_{\mathrm{rs}}, let IntHerm,V(g,u)\mathrm {Int}^{\mathrm {Herm},\mathbb V}(g,u) be the derived intersection number defined in the supplied context. For a matching regular semisimple triple (γ,u1,u2)(GLn(Qp)×Qpn×(Qpn))rs(\gamma,u_1,u_2)\in(\mathrm {GL}_n(\mathbb Q_p)\times\mathbb Q_p^n\times(\mathbb Q_p^n)^*)_{\mathrm{rs}}, let Orb\mathrm{Orb} be the twisted orbital integral defined there. Twisted arithmetic transfer conjecture. For every such matching pair and triple,

IntHerm,V(g,u)logp=ddss=0Orb((γ,u1,u2),1GL(L0,L0)×1L0×1(L0),s)Qlogp.\mathrm {Int}^{\mathrm {Herm},\mathbb V}(g,u)\log p=-\left.\frac{d}{ds}\right|_{s=0}\mathrm {Orb}\left((\gamma,u_1,u_2),1_{\mathrm {GL}(L_0,L_0^\vee)}\times1_{L_0}\times1_{(L_0^\vee)^*},s\right)\in\mathbb Q\log p.

This is an arithmetic transfer identity comparing derived intersection numbers on a unitary Rapoport--Zink space with derivatives of twisted orbital integrals; the supplied text gives no resolution of the conjecture.

Sources & referencesView supporting material

Primary source

Zhiyu Zhang, “Non-reductive cycles and twisted arithmetic transfers for Shimura curves”, arXiv:2502.16754 (2025).

Additional references

2 papers in this index state this conjecture (2024–2025). The statement above is taken from the most recent of them; the others are arXiv:2406.00986.

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