Abstract
Katsura and Oort obtained an explicit description of the supersingular locus S3,1 of the Siegel modular variety of degree
3 in terms of class numbers. In this paper we study an alternative stratification of S3,1, the so-called mass stratification.
We show that when p = 2, there are eleven strata (one of
a-number 3, two of a-number 2 and eight of a-number 1). We
give an explicit mass formula for each stratum and classify
possible automorphism groups on each stratum of a-number
one. On the largest open stratum we show that every automorphism group is {±1} if and only if p = 2; that is, we prove
that Oort’s conjecture on the automorphism groups of generic
supersingular abelian threefolds holds precisely when p > 2.
3 in terms of class numbers. In this paper we study an alternative stratification of S3,1, the so-called mass stratification.
We show that when p = 2, there are eleven strata (one of
a-number 3, two of a-number 2 and eight of a-number 1). We
give an explicit mass formula for each stratum and classify
possible automorphism groups on each stratum of a-number
one. On the largest open stratum we show that every automorphism group is {±1} if and only if p = 2; that is, we prove
that Oort’s conjecture on the automorphism groups of generic
supersingular abelian threefolds holds precisely when p > 2.
| Original language | English |
|---|---|
| Article number | 107812 |
| Number of pages | 52 |
| Journal | Advances in Mathematics |
| Volume | 386 |
| DOIs | |
| Publication status | Published - 6 Aug 2021 |
Bibliographical note
Funding Information:Parts of this work were carried out when the first author visited the Academia Sinica, and when the first and third authors visited RIMS and Kyoto University. They would like to thank these institutes for their hospitality and excellent working conditions. A part of this paper is contained in the second author's master's thesis written at Tohoku University; he thanks his advisor Nobuo Tsuzuki for enlightening comments, advice and encouragement. The authors are grateful to Ming-Lun Hsieh and Akio Tamagawa for useful discussions, and for proving Propositions A.2 and A.3, respectively. They would like to thank Tomoyoshi Ibukiyama and Jiangwei Xue for useful discussions and helpful comments on an earlier manuscript, and the anonymous referee for their comments which improved the exposition. The second author is supported by JSPS grants 15J05073 and 19K14501. The third author is partially supported by MoST grants 107-2115-M-001-001-MY2 and 109-2115-M-001-002-MY3.
Funding Information:
Parts of this work were carried out when the first author visited the Academia Sinica, and when the first and third authors visited RIMS and Kyoto University. They would like to thank these institutes for their hospitality and excellent working conditions. A part of this paper is contained in the second author's master's thesis written at Tohoku University; he thanks his advisor Nobuo Tsuzuki for enlightening comments, advice and encouragement. The authors are grateful to Ming-Lun Hsieh and Akio Tamagawa for useful discussions, and for proving Propositions A.2 and A.3 , respectively. They would like to thank Tomoyoshi Ibukiyama and Jiangwei Xue for useful discussions and helpful comments on an earlier manuscript, and the anonymous referee for their comments which improved the exposition. The second author is supported by JSPS grants 15J05073 and 19K14501 . The third author is partially supported by MoST grants 107-2115-M-001-001-MY2 and 109-2115-M-001-002-MY3 .
Publisher Copyright:
© 2021 Elsevier Inc.
Keywords
- Automorphism groups
- Mass formulae
- Supersingular abelian varieties