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Supersingular abelian varieties of dimension two or three and class numbers. (English) Zbl 0656.14025
Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 253-281 (1987).
[For the entire collection see Zbl 0628.00007.]
Let $$B$$ be a definite quaternion algebra over $${\mathbb{Q}}$$ admitting only one finite ramified place $$p$$, and, for each positive integer $$g$$, let $$H_ g$$ be the class number of the principal genus of the standard quaternionic space of dimension $$g$$ over $$B$$. In the first part of this paper, the authors compute, via the study of curves of genus 2, the number of principal polarizations on the square of a supersingular elliptic curve in characteristic $$p$$. This gives a new proof of an explicit formula for $$H_ 2$$, originally due to K. Hashimoto and T. Ibukiyama [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 549-601 (1980; Zbl 0452.10029) and 28, 695-699 (1982; Zbl 0493.10030)]. The process is reversed in the second part, where $$H_ 3$$ is interpreted as the number of connected components of the locus of supersingular abelian varieties of dimension 3 in the moduli space $${\mathcal A}_{3,1}$$; a known formula for $$H_ 3$$ then prevents this locus from being connected as soon as $$p$$ is odd.
Reviewer: D.Bertrand

##### MSC:
 14K05 Algebraic theory of abelian varieties 14K10 Algebraic moduli of abelian varieties, classification 15B33 Matrices over special rings (quaternions, finite fields, etc.) 16Kxx Division rings and semisimple Artin rings