Principal polarizations of supersingular abelian surfaces
Let \(A\) be a (fixed) supersingular abelian surface over a field \(k\) of characteristic \(p>0\), which is not superspecial, and let \(\overline{k}\) be an algebraic closure of \(k.\) The main problem that was dealt with in the paper under review is on the number of isomorphism classes of principal polarizations of any fixed such an abelian surface \(A\). (Recall that it was known in [the author et al., Compos. Math. 57, 127--152 (1986; Zbl 0589.14028)] that the number of principal polarizations of \(E^n\) (\(n \geq 2\)) for a supersingular elliptic curve \(E\) is equal to the class number of quaternion hermitian lattices of rank \(n\) that belong to the principal genus, and explicit such values were obtained for \(n=2\) and for \(n=3\) in other papers.) In this regard, the first main result of this paper is an explicit formula for the numbers of principal polarizations of \(A\) up to \(\overline{k}\)-isomorphism. More precisely, the number of principal polarizations of \(A\) up to \(\Aut(A)\) was obtained depending on two factors: i) the condition whether the tangent of the embedding \(\iota : \alpha_p \rightarrow \alpha_p^2\), which gives an isomorphism \(A \cong E^2 / \iota (\alpha_p )\) lies outside \(\mathbb{F}_{p^4}\) (called Case I) or is an element of \(\mathbb{F}_{p^4} \setminus \mathbb{F}_{p^2}\) (called Case II), and ii) of course, the value of the characteristic \(p>0\). (For a concrete description, see Theorem 1.1.) Another interesting result that the author gave was on the description of the automorphism group \(\Aut(A,\lambda)\) of a principally polarized abelian surface \((A, \lambda)\). In Case I, the cardinality of the group \(\Aut(A, \lambda)\) (with the unique principal polarization \(\lambda\) on \(A\)) is equal to \(32\) if \(p=2,\) and equals \(2\) (that is, \(\Aut(A, \lambda) = \{ \pm 1 \})\) for any principal polarization \(\lambda\) on \(A\) if \(p \geq 3\). Similarly, in Case II, the cardinality of the group \(\Aut(A, \lambda)\) (with the unique principal polarization \(\lambda\) on \(A\)) is equal to \(160\) if \(p=2\), and equals roughly either \(2\) or \(10\) depending on the congruence of \(p\) modulo \(5\). (For a concrete description, see Theorem 1.2.) One important feature of this theorem is that it provides a case where a conjecture of Oort, saying that the automorphism group \(\Aut(A, \lambda) = \{\pm 1 \}\) for any generic principally polarized supersingular abelian variety \((A,\lambda)\) is false (especially when \(p=2\) and \(\dim(A)=2\)), while the conjecture is true when \(p\) is any odd prime and \(\dim(A)=2.\) The proofs of these theorems are based on the description of endomorphisms of \(A\), mass formulas of principal polarizations of \(A\), and one of the known results on the automorphism groups of the pullbacks \((E^2 , \lambda_0)\) of \((A, \lambda)\), where \(\lambda_0 = \psi^* (\lambda)\) with \(\psi : E^2 \rightarrow A \cong E^2 / \iota (\alpha_p )\) being the natural map.
scientific article; zbMATH DE number 7290145
scientific article