For a Shimura datum of the relevant parahoric type, with maximal parahoric level at an odd prime p, let M^loc be the corresponding canonical local model over the ring of integers O_E of the reflex field. The conjecture asks whether there exists a proper birational equivariant modification widetildeM→ M^loc that is an isomorphism on the generic fiber, such that widetildeM is regular and its special fiber is a normal-crossings divisor. In the corresponding global formulation, the modification should produce a regular integral model of the Shimura variety with normal-crossings special fiber.

Pappas’s equivariant modification conjecture

For a Shimura datum of the relevant parahoric type, with maximal parahoric level at an odd prime $p$ , let $M^{\mathrm{loc}}$ be the corresponding canonical local model over the ring of integers $\mathcal{O}_E$ of the reflex field. The conjecture asks whether there exists a proper birational equivariant modification $\widetilde{M}\to M^{\mathrm{loc}}$ that is an isomorphism on the generic fiber, such that $\widetilde{M}$ is regular and its special fiber is a normal-crossings divisor. In the corresponding global formulation, the modification should produce a regular integral model of the Shimura variety with normal-crossings special fiber.

References

Primary source

arXiv

Additional references

Regular integral models of Shimura varieties and a conjecture of Pappas — arXiv — Jie Yang, Ioannis Zachos, Zhihao Zhao

Posted 9 hours ago · edited 9 hours ago

Progress summary

Refreshed 9 hours ago

Claimed solved

A new preprint claims the conjecture is proved for several Shimura-variety groups and levels, but the claim has not been independently checked.

Pappas’s conjecture asks for equivariant modifications producing regular integral models with normal-crossings special fibers at specified maximal parahoric levels. The newly claimed result covers selected unitary, symplectic, and even orthogonal similitude groups at odd primes.

Known results

Pappas and Zachos, 2022: regular models for orthogonal and spin Shimura varieties at specified parahoric levels, with normal-crossings special fiber.
Kisin and Pappas, 2018: integral models with parahoric level for abelian-type Shimura data under tameness assumptions, plus related nearby-cycle results.

Claimed proof in On semi-stable integral models for Shimura varieties

The preprint claims proper birational equivariant modifications, unchanged on generic fibers, yielding regular models with normal-crossings special fibers for Weil restrictions of unramified unitary similitude groups, symplectic similitude groups, and even orthogonal similitude groups. It further claims potentially semistable models and consequences for inertia and nearby cycles. No independent verification, correction, or referee assessment is reported.

Current status (as of October 2026): A preprint claims the conjecture in the stated cases, while the claim remains unverified and broader cases remain open.

Sources

No solutions have been posted yet.