Tohoku Mathematical Journal

Quinary lattices and binary quaternion hermitian lattices

Tomoyoshi Ibukiyama

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In our previous papers, we defined the $G$-type number of any genera of quaternion hermitian lattices as a generalization of the type number of a quaternion algebra. Now we prove in this paper that the $G$-type number of any genus of positive definite binary quaternion hermitian maximal lattices in $B^2$ for a definite quaternion algebra $B$ over $\mathbb Q$ is equal to the class number of some explicitly defined genus of positive definite quinary quadratic lattices. This is a generalization of a part of the results in 1982, where only the principal genus was treated. Explicit formulas for this type number can be obtained by using Asai's class number formula. In particular, in case when the discriminant of $B$ is a prime, we will write down an explicit formula for $T$, $H$ and $2T-H$ for the non-principal genus, where $T$ and $H$ are the type number and the class number. This number was known for the principal genus before. In another paper, our new result is applied to polarized superspecial varieties and irreducible components of supersingular locus in the moduli of principally polarized abelian varieties having a model over a finite prime field, where $2T-H$ plays an important role.

Article information

Source
Tohoku Math. J. (2), Volume 71, Number 2 (2019), 207-220.

Dates
First available in Project Euclid: 21 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1561082596

Digital Object Identifier
doi:10.2748/tmj/1561082596

Mathematical Reviews number (MathSciNet)
MR3973249

Zentralblatt MATH identifier
07108037

Subjects
Primary: 11E41: Class numbers of quadratic and Hermitian forms
Secondary: 14K10: Algebraic moduli, classification [See also 11G15] 11E12: Quadratic forms over global rings and fields

Keywords
quinary lattice quaternion hermitian lattice supersingular abelian variety

Citation

Ibukiyama, Tomoyoshi. Quinary lattices and binary quaternion hermitian lattices. Tohoku Math. J. (2) 71 (2019), no. 2, 207--220. doi:10.2748/tmj/1561082596. https://projecteuclid.org/euclid.tmj/1561082596


Export citation

References

  • T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli numbers and zeta functions. With an appendix by Don Zagier. Springer Monographs in Mathematics. Springer, Tokyo, 2014. xii+274 pp.
  • T. Asai, The class number of positive definite quadratic forms, Japan. J. Math. 3 (1977), 239–296.
  • M. Eichler, Quadratische Formen und orthogonale Gruppen. Zweite Auflage. Die Grundlehren der mathematischen Wissenschaften, Band 63. Springer-Verlag, Berlin-New York, 1974. xii+222 pp.
  • K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms (I), J. Fac. Sci. Univ. Tokyo, Sec. IA 27 (1980), 549–601; (II) ibid.28 (1982), 695–699 (1982); (III) ibid.30 (1983), 393–401.
  • T. Ibukiyama, Type numbers of quatenion hermitian forms and supersingular abelian varieties, Osaka J. Math. 55 (2018), 369–382.
  • T. Ibukiyama and T. Katsura, On the field of definition of superspecial polarized abelian varieties and type numbers, Compositio Math. 91(1994), 37–46.
  • T. Katsura and F. Oort, Families of supersingular abelian surfaces, Compositio Math. 62(1987), 107–167.
  • T. Katsura and F. Oort, The class number of the principal genus of a positive definite quaternion hermitian space of dimension two and three, Algebraic geometry, Sendai, 1985, 1, 253–281, Adv. Stud. Pure Math. 10, North-Holland Publishing Co., Amsterdam; Kinokuniya Company Ltd., Tokyo, 1987.
  • Y. Kitaoka, Arithmetic of quadratic forms. Cambridge Tracts in Mathematics, 106. Cambridge University Press, Cambridge, 1993. x+268 pp.
  • M. Kneser, Quadratische Formen. Revised and edited in collaboration with Rudolf Scharlau. Springer-Verlag, Berlin, 2002. viii+164 pp.
  • K. Z. Li and F. Oort, Moduli of supersingular abelian varieties. Lecture Notes in Mathematics, 1680. Springer-Verlag, Berlin, 1998. iv+116 pp.
  • F. Oort, Newton polygon strata in the moduli space of abelian varieties. Moduli of abelian varieties (Texel Island, 1999), 417–440, Progr. Math., 195, Birkhäuser, Basel, 2001.
  • M. Peters, Ternäre quadratische Formen und Quaternionenalgebra, Acta Arith. 15, (1968/69), 329–365.
  • T. R. Shemanske, Ternary quadratic forms and quaternion algebras. J. Number Theory 23 (1986), 203–209.
  • G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms. J. Math. Soc. Japan 15 (1963), 33–65.
  • A. Weil, Basic Number Theory, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band 144, Springer-Verlag Berlin-Heidelberg 1967, xviii+294 pp.