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SUPERSINGULAR LOCI OF LOW DIMENSIONS AND PARAHORIC SUBGROUPS

Ibukiyama, Tomoyoshi 大阪大学 DOI:10.18910/88493

2022.07

概要

The theory of polarized supersingular abelian varieties (A, λ) is often essentially related to the theory of quaternion hermitian lattices. In this paper, we add one more such relation by giving an adelic description of supersingular loci of low dimensions. Katsura, Li and Oort have shown that the supersingular locus in the moduli of principally polarized abelian varieties is not irreducible and that the number of its irreducible components is equal to the class number of certain maximal quaternion hermitian lattices. Now the locus has certain natural algebraic subsets consisting of A with fixed a-numbers that are defined as the dimensions of embeddings from α_p to A. For low dimensional cases when dim A ≤ 3, we describe configuration of these subsets in the locus by intersection properties of some adelic subgroups of quaternion hermitian groups corresponding to parahoric subgroups locally at characteristic p. In particular, we describe which components each superspecial point lies on. This is proved by using certain liftability property of isogenies of abelian varieties, where the isogenies are interpreted to cosets of GL_n and of parahoric subgroups of the quaternion hermitian groups acting on quaternion hermitian matrices.

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参考文献

[1] J. Faraut and A. Koranyi: Analysis on Symmetric Cones, Clarendon Press, Oxford University Press, New ´ York, 1994.

[2] K. Hashimoto and T. Ibukiyama: On class numbers of positive definite binary quaternion hermitian forms, J. Fac. Sci. Univ. Tokyo Sect IA. 27 (1980), 549–601; (II) ibid. 28 (1982), 695–699; (III) ibid. 30 (1983), 393–401.

[3] S. Harashita: The a-number stratification on the moduli space of supersingular abelian varieties, J. Pure Appl. Algebra 193 (2004), 163–191.

[4] T. Ibukiyama, T. Katsura and F. Oort: Supersingular curves of genus two and class numbers, Compositio Math. 57 (1986), 127–152.

[5] T. Ibukiyama: On automorphism groups of positive definite binary quaternion hermitian lattices and new mass formula; in Automorphic forms and geometry of arithmetic varieties, Advanced Studies in pure Math. 15, Kinokuniya, Tokyo, 1989, 301–349.

[6] T. Ibukiyama: On rational points of curves of genus 3 over finite fields, Tohoku Math. J. ˆ 45 (1993), 311– 329.

[7] T. Ibukiyama and T. Katsura: On the field of definition of superspecial polarized abelian varieties and type numbers, Compositio Math. 91 (1994), 37–46.

[8] T. Ibukiyama: Type numbers of quaternion hermitian forms and supersingular abelian varieties, Osaka J. Math. 55 (2018), 369–384.

[9] T. Ibukiyama: Supersingular abelian varieties and quatenion hermitian lattices, to appear in RIMS Koky ˆ uroku Bessatsu. ˆ

[10] V. Karemaker, F. Yobuko and C.-F. Yu: Mass formula and Oort’s conjecture for supersingular abelian threefolds, Adv. Math. 386 (2021), 107812, 52pp.

[11] T. Katsura and F. Oort: Supersingular abelian varieties of dimension two or three and class numbers; in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North Holland, Amsterdam, 1987, 253–281.

[12] T. Katsura and F. Oort: Families of supersingular abelian surfaces, Compositio Math. 62 (1987), 107–167.

[13] K. Li and F. Oort: Moduli of supersingular abelian varieties, Lecture Notes in Mathematics 1680, SpringerVerlag, Berlin, 1998.

[14] T. Oda and F. Oort: Supersingular abelian varieties; in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, 595–621.

[15] F. Oort: Which abelian surfaces are products of elliptic curves? Math. Ann. 214 (1975), 35–47.

[16] G. Shimura: Arithmetic of alternating forms and quaternion hermitian forms, J. Math. Soc. Japan 15 (1963), 33–65.

[17] G. Shimura: Introduction to the arithmetic theory of automorphic functions, Kano Memorial Lectures, ˆ No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J. 1971.

[18] T. Tamagawa: On the ζ-functions of a division algebra, Ann. Math, 77 (1963), 387–405.

[19] C.-F. Yu: The supersingular loci and mass formulas on Siegel modular varieties, Doc. Math. 11 (2006), 449–468.

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