Some generalizations of Oort's conjecture

Abstract

For a prime p≥ 5, let 𝒮_g be the moduli space over 𝔽̅_p of g-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group {±1}: (i) certain supersingular Ekedahl--Oort strata when g is even, (ii) the loci in 𝒮_g with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when g≥ 3, and (iii) the locus in 𝒮_g with a-number at least $2$ when g≥ 4. Consequently, for g≥ 4, the complement in 𝒮_g of the open locus where the automorphism group is {±1} has codimension at least $2$. These results confirm Oort's conjecture for p≥ 5. We reduce them to statements about affine Deligne--Lusztig varieties for GSp_2g and prove analogues of (ii) for GL_2g and GSO_4m.

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BibTeX RIS

BibTeX

@misc{shimada2026some,
  title = {{Some generalizations of Oort's conjecture}},
  author = {Ryosuke Shimada and Teppei Takamatsu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03188},
  eprint = {2609.03188},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03188},
}

RIS

TY  - UNPB
TI  - Some generalizations of Oort's conjecture
AU  - Ryosuke Shimada
AU  - Teppei Takamatsu
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.03188
UR  - https://arxiv.org/abs/2609.03188
AB  - For a prime p≥ 5, let 𝒮_g be the moduli space over 𝔽̅_p of g-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group {±1}: (i) certain supersingular Ekedahl--Oort strata when g is even, (ii) the loci in 𝒮_g with non-supersingular Ekedahl--Oort invariants of positive Coxeter type when g≥ 3, and (iii) the locus in 𝒮_g with a-number at least $2$ when g≥ 4. Consequently, for g≥ 4, the complement in 𝒮_g of the open locus where the automorphism group is {±1} has codimension at least $2$. These results confirm Oort's conjecture for p≥ 5. We reduce them to statements about affine Deligne--Lusztig varieties for GSp_2g and prove analogues of (ii) for GL_2g and GSO_4m.
ER  -

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