Moduli of supersingular abelian varieties
This book contains a systematic study of the moduli spaces of supersingular abelian varieties. An abelian variety \(X\) of dimension \(g\) defined over a field \(K\) of characteristic \(p\) is said to be supersingular if and only if \(X\otimes\overline K\sim E^g\) where \(E\) is a supersingular elliptic curve over \(\overline K\), \(\overline K\) is an algebraically closed field containing \(K\) and \(\sim\) means isogeny equivalence. Let \({\mathcal A}_{g,d} \otimes \mathbb{F}_p\) be the moduli space of abelian varieties of dimension \(g\) with a polarization of degree \(d^2\) over a field of characteristic \(p\). The subset, \(S_{g,d}\), of \({\mathcal A}_{g,d} \otimes\mathbb{F}_p\) representing supersingular polarized abelian varieties is a closed algebraic subset. The main results proved in the book are: (a) The dimension of \(S_{g,1}\) is \([g^2/4]\) (b) The number of irreducible components of \(S_{g,1}\) is: \[ H_g(p,1)\text{ if }g \text{ is odd};\quad H_q(1,p) \text{ if }g\text{ is even}, \] \(H_g(p,1)\) (resp., \(H_g(1,p))\) being the class number of the principal genus (resp. the non-principal genus) defined by \textit{K. Hashimoto} and \textit{T. Ibukiyama} [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 549-601 (1980; Zbl 0452.10029) and 28, 695-699 (1981; Zbl 0493.10030)]. The main ingredient in the description of the moduli spaces \(s_g\) of supersingular abelian varieties is the study of polarized flat type quotients: A flat type quotient is a sequence of isogenies of abelian varieties: \[ E^g=X_{g-1} @>\rho_{g-1}>> \cdots @>\rho_i>> X_i/ \alpha_p^i\simeq X_{i-1} @>\rho_{i-1}>> \cdots @>\rho_1>> X_0 \] where \(E\) is a supersingular elliptic curve and \(\alpha_p\) the simple finite group scheme. A polarized flag type quotient with respect to a polarization \(\eta\) of degree \(d^2p^{g(g-1)}\) is a family of polarized abelian varieties \(\{(X_i\eta_i)\), \(0\leq i\leq g-1\}\) and isogenies \(\rho_i:X_i\to X_{i-1}\) compatible with the polarizations such that \(\{X_i, \rho_i\}\) is a flag type quotient and: (a) \(\eta_{g-1}=\eta\) (b) \(\text{Ker} \eta_i \subset\text{Ker}(F^{i-j} \circ V^j)\), where \(F\) and \(V\) are the Frobenius and the Verschiebung. It is proved that there exists a moduli space \(P_{g,\eta}\) parametrizing polarized flag type quotients of type \(\eta\). A flag type quotient \(\{X_i,\rho_i\}\) is called rigid if it verifies the condition: \[ \text{Ker}(X_{g-1} \to X_i)=\text{Ker} (X_{g-1}\to X_0) \cap X_{g-1}[F^{g-1-i}] (0<i<g). \] The rigid polarized flag type quotients define an open subscheme \(P_{g,\eta}' \subset P_{g, \eta}\). The authors prove that there exists a quasi-finite and surjective morphism \(\Psi:P'= \coprod_\eta P_{g, \eta}'\to S_g\). The properties of \(P'\) and \(\Psi\) determine in some sense the structure of the moduli spaces \(S_g\).
moduli spaces of supersingular abelian varieties
scientific article; zbMATH DE number 1116599