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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

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