Mass formula and Oort's conjecture for supersingular abelian threefolds (Q2039566)

The paper deals with mass stratification and conjecture by \textit{S. J. Edixhoven} et al. [Bull. Sci. Math. 125, No. 1, 1--22 (2001; Zbl 1009.11002)] on the automorphism groups of generic (supersingular) abelian threefolds for supersingular locus of the Siegel modular variety of degree 3. The authors of the paper under review give the number of strata and obtaine the explicit mass formula for each stratum. The classification of possible automorphism groups on each strata of \(\alpha\)-number one is also given. These give the Oort conjecture for investigated polarized abelian threefolds in the case \(p > 2.\) For supersingular abelian surfaces for any odd prime \(p\) the Oort conjecture is proved by \textit{T. Ibukiyama} [J. Math. Soc. Japan 72, No. 4, 1161--1180 (2020; Zbl 1471.14091)]. Let \((X, \lambda)\) be the \(p\)-power degree polarized abelian variety over an algebraically closed field \(k\) of characteristic \(p\) and \(x = (X_0, \lambda_0)\) be polarized supersingular abelian variety of \(p\)-power degree over \(k\). An abelian variety defined over \(k\) is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. The authors of the paper under review first define for any integer \(d \ge 1\) the (coarse) moduli space \({\mathcal A}_{g,d}\) over \(\overline{\mathbb F}_p\) of \(g\)-dimensional abelian varieties \((X, \lambda)\) with polarization degree \(\deg \lambda = d^2\) and for any \(m \ge 1\) the supersingular locus \({\mathcal G}_{g,p^m}\) of supersingular abelian varieties in \({\mathcal A}_{g,p^m}\). For abelian variety \((X,\lambda)\) let \(X^\bot\) be its dual and respectively let \(G\) and \(G^\bot\) their \(p\)-divicible groups. A polarization \(\lambda\) is an isogeny which is symmetric \((\lambda: X \to X^\bot)^\bot = \lambda\) with the identification \(X = X^{\bot \bot}\) by \textit{D. Mumford} [Abelian varieties. London: Oxford University Press (1970; Zbl 0223.14022)] and by \textit{T. Oda} and \textit{F. Oort} [in: Proc. int. Symp. on algebraic geometry, Kyoto. 595--621 (1977; Zbl 0402.14016)]. A quasi-polarization \(\lambda: G \to G^\bot\) of a \(p\)-divisible group \(G\) is a symmetric isogeny of \(p\)-divisible groups such that \((\lambda: G \to G^\bot)^\bot = \lambda\) by \textit{F. Oort} [Ann. Math. (2) 152, No. 1, 183--206 (2000; Zbl 0991.14016)]. Then the authors define the set \(\Lambda_x\) of isomorphim classes of \(p\)-power polarization degree polarized abelian varieties \((X, \lambda)\) over \(k\). The set \(\Lambda_x\) consists of those polarized abelian varieties whose assosiated quasi-polarized \(p\)-divisible groups satisfy \((X, \lambda)[p^\infty] = (X_0, \lambda_0)[p^\infty] .\) The set \(\Lambda_x\) is finite by \textit{C.-F. Yu} [J. Aust. Math. Soc. 78, No. 3, 373--392 (2005; Zbl 1137.11323)] The mass of \(\Lambda_x\) is defined by \(\mathrm{Mass}(\Lambda_x) = \sum_{(X, \lambda) \in \Lambda_x} \frac{1}{\#\mathrm{Aut}(X,\lambda)}.\) Sections 4 and 5 are concerned with computing the mass for principally polarized supersingular abelian threefolds respectivly for \(\alpha\)-number \(\ge 2\) (Theorem 4.3) and for \(\alpha\)-number 1 (Theorem 5.21). Section 6 deals with ``the automorphism groups of principally polarised abelian threefolds \((X, \lambda)\) over an algebraically closed field \(k \supseteq {\mathbb F}_p\) with \(\alpha(X) = 1\).'' The main result of the section is Theorem 6.4. The section includes also the discussion of arithmetic properties of definite quaternion algebras over rational numbers and the superspecial case. The section is ended with some open problems. The case of a set-theoretic intersection of the Fermat curve and a curve \(\Delta\) defined by authors in Section 5 is treated in the Appendix.