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Moduli of supersingular abelian varieties. (English) Zbl 0920.14021
Lecture Notes in Mathematics. 1680. Berlin: Springer. 116 p. DM 37.00; öS 271.00; sFr 34.50; £14.00; $ 27.00 (1998).
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 K. Hashimoto and 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 \(S_g\).

MSC:
14K10 Algebraic moduli of abelian varieties, classification
14G15 Finite ground fields in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
11G10 Abelian varieties of dimension \(> 1\)
11R29 Class numbers, class groups, discriminants
14M17 Homogeneous spaces and generalizations
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14B05 Singularities in algebraic geometry
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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