On fields of definition of components of the Siegel supersingular locus

Author:
Chia-Fu Yu

Journal:
Proc. Amer. Math. Soc. **145** (2017), 5053-5058

MSC (2010):
Primary 11G15, 11G10

DOI:
https://doi.org/10.1090/proc/13741

Published electronically:
August 30, 2017

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Abstract | References | Similar Articles | Additional Information

Abstract: Recently Ibukiyama proved an explicit formula for the number of certain non-principal polarizations on a superspecial abelian surface, extending his earlier work with Katsura for principal polarizations [Compos. Math. **91** (1994), 37-46]. As a consequence of Ibukiyama's formula, there exists a geometrically irreducible component of the Siegel supersingular locus which is defined over the prime finite field. In this note we give a direct proof of this result.

- [1]
T. Ibukiyama,
*Quinary lattices and binary quaternion hermitian lattices*, preprint, 2016. To appear in Tohoku Math. J. - [2]
T. Ibukiyama,
*Type numbers of quaternion hermitian forms and supersingular abelian varieties*, preprint, 2016. To appear in Osaka J. Math. **[3]**Tomoyoshi Ibukiyama and Toshiyuki Katsura,*On the field of definition of superspecial polarized abelian varieties and type numbers*, Compositio Math.**91**(1994), no. 1, 37–46. MR**1273924****[4]**Toshiyuki Katsura and Frans Oort,*Families of supersingular abelian surfaces*, Compositio Math.**62**(1987), no. 2, 107–167. MR**898731****[5]**Ke-Zheng Li and Frans Oort,*Moduli of supersingular abelian varieties*, Lecture Notes in Mathematics, vol. 1680, Springer-Verlag, Berlin, 1998. MR**1611305****[6]**Chia-Fu Yu,*The supersingular loci and mass formulas on Siegel modular varieties*, Doc. Math.**11**(2006), 449–468. MR**2288077**- [7]
C.-F. Yu,
*On arithmetic of the superspecial locus*, arXiv:1210.1120v2. To appear in Indiana Univ. Math. J.

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Additional Information

**Chia-Fu Yu**

Affiliation:
Institute of Mathematics, Academia Sinica, 6th Floor, Astronomy Mathematics Building, No. 1, Roosevelt Road Section 4, Taipei, Taiwan, 10617 – and – National Center for Theoretical Sciences, No. 1 Roosevelt Road Section 4, National Taiwan University, Taipei, Taiwan, 10617

Email:
chiafu@math.sinica.edu.tw

DOI:
https://doi.org/10.1090/proc/13741

Keywords:
Superspecial abelian surfaces,
supersingular locus,
field of definition

Received by editor(s):
December 5, 2016

Published electronically:
August 30, 2017

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2017
American Mathematical Society