Let S be a split unitary Shimura variety of signature (g,g) at a split prime p, and let S_basic denote its basic locus. Outside the degenerate cases, there should exist a dense open subset U⊆S_basic such that, for every geometric point x∈ U, the automorphism group of the associated abelian variety with its endomorphism structure and polarization is operatornameAut(A_x,iota_x,lambda_x)=pm1.

Oort’s conjecture for split unitary Shimura varieties

Oort posed the underlying conjecture in 2001. The split-unitary analogue predicts the expected generic automorphism group in the basic locus; the new preprint claims this generically, apart from stated degenerate cases.

Known results

Genus $2$ , $p>2$ : proved independently by Ibukiyama and by Karemaker–Pries; failure at $(g,p)=(2,2)$ .
Genus $3$ , $p>2$ : generic group $\{\pm1\}$ , by Karemaker, Yobuko, and Yu; for $p=2$ , it has $8$ elements.
Genus $4$ : proved for all $p$ by Karemaker and Yu, with an independent proof for $p>2$ by Dragutinović.

September 2026 generic split-unitary claim

The preprint’s Theorem 5.1 claims that, on a dense open subset of the basic locus for split-unitary signature $(g,g)$ at a split prime $p$ , every finite-order automorphism of the associated $p$ -divisible group is a unit-root scalar in $\mathbb{Z}_p^\times$ . This is presented as the expected generic analogue of Oort’s conjecture, but remains an unrefereed and unverified claim.

Current status (as of September 2026): The generic split-unitary case is claimed solved by an unrefereed preprint, while independent verification and treatment of excluded degenerate cases remain outstanding.

Oort’s conjecture for split unitary Shimura varieties

Primary source

Additional references

Progress summary

Solutions 0