zbMATH — the first resource for mathematics

The moduli spaces of polarized abelian varieties. (English) Zbl 0839.14036
In this article we study the moduli spaces \({\mathcal A}_{g,d}\) over \(\text{Spec} (\mathbb{Z})\) parametrizing abelian varieties of dimension \(g\) with a polarization of degree \(d^2\). We use the language of algebraic stacks. – Although these moduli spaces are in a sense the most basic moduli spaces of abelian varieties not much seems to be known about their local and global structure. D. Mumford, in Actes Congr. internat. Math. 1970, Vol. 1, 457-465 (1971; Zbl 0222.14023), determined the components of \({\mathcal A}_{g,d}\) using the spaces \({\mathcal A}_{g, \delta}\) parametrizing the polarized abelian varieties with a fixed sequence of “elementary divisors” \(\delta=(\delta_1, \dots, \delta_g)\) of positive integers \(\delta_1 |\delta_2 \dots |\delta_g\) such that \(\prod^g_{i=1} \delta_i=d\) associated to the polarization. S. E. Crick jun. [Am. J. Math. 97, 851-861 (1975; Zbl 0321.14020)] determined the local structure of \({\mathcal A}_{g,d}\) at a point of \({\mathcal A}_{g, \delta}\). Finally, P. Norman and F. Oort [Ann. Math., II. Ser. 112, 413-439 (1980; Zbl 0483.14010)] studied the stratification by \(p\)-rank \({\mathcal V}_0 \subset {\mathcal V}_1 \subset \cdots \subset {\mathcal V}_{g-1} \subset {\mathcal V}_g={\mathcal A}_{g,d} \otimes \mathbb{F}_p\); it is shown there that the dimension of \({\mathcal V}_{g-i}\) is equal to \({1 \over 2} g(g+1)-i\).
In section 1 we (re)define the locally closed substacks \({\mathcal A}_{g,\delta} \subset {\mathcal A}_{g,d}\) whose closures are the components of \({\mathcal A}_{g,d}\). We prove that all geometric fibres of the morphisms \({\mathcal A}_{g, \delta} \to \text{Spec} (\mathbb{Z})\) are irreducible; this answers a question of Mumford (loc. cit.). It follows that the closures of \({\mathcal A}_{g, \delta} \otimes \overline \mathbb{F}_p \) are the irreducible components of \({\mathcal A}_{g,d} \otimes \overline \mathbb{F}_p\). – In section 2 we determine how the fibres of the components of \({\mathcal A}_{g,d}\) over the finite primes split up into irreducible components. – In section 3 we continue the study of the stratification by \(p\)-rank of \({\mathcal A}_{g,d} \otimes \overline \mathbb{F}_p\) which was started by P. Norman and F. Oort (loc. cit.).

14K10 Algebraic moduli of abelian varieties, classification
14G15 Finite ground fields in algebraic geometry
Full Text: DOI EuDML
[1] EGA Grothendieck, A., Dieudonn?, J.: El?ments de g?om?trie alg?briques. Berlin Heidelbeg New York: Springer 1971; Publ. Math. Inst. Hautes ?tud. Sci.4, 8, 11, 17, 20, 24, 28, 32 (1961-1967)
[2] SGA Grothendieck, A., et al.: S?minaire de g?om?trie alg?brique. Springer Lecture Notes and Masson-North Holland. Especially: SGA 3: Demazure, M., Grothendieck, A.: Sch?mas en groupes I, II, III. (Lect. Notes Math., vols. 151, 152, 153) Berlin Heidelberg New York: Springer 1971
[3] [An] Anantharaman, S.: Sch?mas en groupes, espaces homog?nes et espaces alg?briques. Bull. Soc. Math. Fr.33, 5-79 (1973) · Zbl 0286.14001
[4] [AV] Mumford, D.: Abelian varieties. (Stud. Math., Tata Inst. Fundam. Research, vol. 5) Oxford: Oxford University Press 1970 · Zbl 0223.14022
[5] [Cr] Crick, S.E.: Local moduli of abelian varieties. Am. J. Math97, 851-861 (1975) · Zbl 0321.14020
[6] [DM] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. Inst. Hautes ?tud. Sci.36, 75-109 (1969) · Zbl 0181.48803
[7] [Ek] Ekedahl, T.: The action of monodromy on torsion points of Jacobians. In: van der Geer, G., Oort, F., Steenbrink, J.H.M. (eds.) Arithmetic algebraic geometry. (Prog. Math., vol. 89, pp. 41-49) Boston Basel Stuttgart: Birkh?user 1991
[8] [EO] Ekedahl, T., Oort, F.: Connected subsets of a moduli space of abelian varieties (to appear)
[9] [FC] Faltings, G., Chai, C.-L.: Degeneration of abelian varieties. (Ergeb. Math. Grenzgeb., 3. Folge, Bd. 22) Berlin Heidelberg New York: Springer 1990 · Zbl 0744.14031
[10] [GIT] Mumford, D.: Geometric invariant theory, second enlarged edition. (Ergeb. Math. Grenzgeb., 3. Folge, Bd. 34) Berlin Heidelberg New York: Springer 1982
[11] [Ka] Katz, N.: Serre-Tate local moduli. Surfaces alg?briques (?dit? par J. Giraud, L. Illusie et M. Raynaud), expos? 5bis, pp. 138-202. (Lect. Notes Math., vol. 868) Berlin Heidelberg New York: Springer 1981
[12] [KO] Katsura, T., Oort, F.: Families of supersingular abelian surfaces. Compos. Math.62, 107-167 (1987) · Zbl 0636.14017
[13] [MB1] Moret-Bailly, L.: Pinceaux de vari?t?s ab?liennes. (Ast?risque, vol. 129) Paris: Soc. Math. Fr. 1985
[14] [MB2] Moret-Bailly, L.: Familles de courbes et de vari?t?s ab?liennes sur ?1 I, II. In: Szpiro, L. (ed.) S?minaire sur les pinceaux de courbes de genre au moins deux. (Ast?risque, vol. 86) Paris: Soc. Math. Fr. 1981
[15] [Me] Messing, W.: The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. (Lect. Notes Math., vol. 264) Berlin Heidelberg New York: Springer 1972 · Zbl 0243.14013
[16] [Mu] Mumford, D.: The structure of the moduli spaces of curves and abelian varieties. In: Actes, Congr?s international math. 1970, tome 1, pp. 457-465. Paris: Gauthier-Villars 1971
[17] [No1] Norman, P.: Lifting abelian varieties. Invent. Math.64, 431-443 (1981) · Zbl 0472.14021
[18] [No2] Norman, P.: Intersections of the components of the moduli space of abelian varieties. J. Pure Appl. Algebra13, 105-107 (1978) · Zbl 0399.14028
[19] [No3] Norman, P.: An algorithm for computing local moduli of abelian varieties. Ann. Math.101, 499-509 (1975) · Zbl 0309.14031
[20] [NO] Norman, P., Oort, F.: Moduli of abelian varieties. Ann. Math.112, 413-439 (1980) · Zbl 0483.14010
[21] [Od] Oda, T.: The first de Rham cohomology groups and Dieudonn? modules. Ann. Sci. ?c. Norm. Sup?r.2, 63-135 (1959)
[22] [Oo] Oort, F.: Finite group schemes, local moduli for abelian varieties and lifting problems. In: Algebraic geometry. Oslo, 1970, pp. 223-254, also in: Compos. Math.23, 265-296 (1972)
[23] [Ra] Raynaud, M.: Schemas en groupes de type (p, ...,p). Bull. Soc. Math. Fr.102, 241-280 (1974) · Zbl 0325.14020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.