For every integer g≥ 2 and every prime p, let S_g denote the supersingular locus in the moduli space of principally polarized abelian varieties of dimension g over characteristic p. For every irreducible component C⊆S_g, the principally polarized abelian variety at the geometric generic point of C satisfies operatornameAut(A,lambda)=pm1.

Oort’s conjecture on supersingular abelian varieties

Oort’s conjecture asserts that a generic principally polarized supersingular abelian variety has automorphism group $\{\pm1\}$ . The full result is now claimed in a March 2026 preprint, while an independent August 2026 paper gives a complementary geometric proof for odd characteristic.

Known results

$g=2$ , $p>2$ : proved by Ibukiyama and independently by Karemaker–Pries; false for $(g,p)=(2,2)$ .
$g=3$ , $p>2$ : proved by Karemaker–Yobuko–Yu; for $p=2$ , the generic automorphism group has $8$ elements.
$g=4$ , all $p$ : proved by Karemaker–Yu, independently for $p>2$ by Dragutinović.
Every even $g$ and $p\ge5$ : proved by Karemaker–Yu.

2026 full and odd-characteristic proofs

Viehmann’s preprint proves the conjecture except for $(g,p)=(2,2)$ and $(3,2)$ . Karemaker and Yu independently prove all cases with $p>2$ , constructing a dimension- $g-1$ subvariety meeting every supersingular component densely in points with automorphism group $\{\pm1\}$ .

Current status (as of August 2026): The conjecture is resolved for every $g$ and $p$ except the known false cases $(g,p)=(2,2)$ and $(3,2)$ , with independent confirmation for all $p>2$ .

Oort’s conjecture on supersingular abelian varieties

Primary source

Additional references

Progress summary

Solutions 0