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Families of supersingular abelian surfaces. (English) Zbl 0636.14017
The purpose of the present paper is to study the set V of principally polarized supersingular abelian surfaces over an algebraically closed field k of characteristic \(p>0,\) in the way of the method of Oort and of Morret - Bailly.
The authors show that any component of V is the image of a family of supersingular abelian surfaces over the projective line \({\mathbb{P}}^ 1\) constructed in the papers by the aforementioned authors.
For any irreducible component W of V, from the above construction, the authors determine a group G, in \(Aut({\mathbb{P}}^ 1)\), which is a subgroup of the symmetric group of degree 6 for \(p\geq 3\). Then they show that the number of irreducible components of V is equal to the class number of the non-principal genus in \(B^ 2\), where B is a definite quaternion algebra over the field of rationals with discriminant p (theorem 5.7) and that V is irreducible if and only if \(p\leq 11\), under the computation by Hashimoto and Ibukiyama.
In section 6, the authors compute the number of automorphisms of abelian surfaces under a certain condition on polarization and further they determine all ramification groups appearing in the morphisms \({\mathbb{P}}^ 1\to {\mathbb{P}}^ 1/G\cong\) the normalization of W.
Reviewer: K.Katayama

MSC:
14K10 Algebraic moduli of abelian varieties, classification
14L30 Group actions on varieties or schemes (quotients)
14B05 Singularities in algebraic geometry
14H20 Singularities of curves, local rings
14J50 Automorphisms of surfaces and higher-dimensional varieties
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