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The supersingular loci and mass formulas on Siegel modular varieties. (English) Zbl 1118.11031
Let \({\mathcal A}_{g,1,N}\) be the moduli space over \(\mathbb Z_{(p)}[J_N]\) of \(g\)-dimensional principally polarized Abelian varieties \((A, \lambda, \eta)\) which symplectic level-\(N\) structure (\(p\) is a prime number and \(N\geq 3\) prime to \(p\)). \({\mathcal A}_{2,1,N,(p)}\) should be the cover of \({\mathcal A}_{2,1,N}\) which parametrizes isomorphism classes of objects \((A,\lambda,\eta,H)\), where \((A,\lambda,\eta)\) is an object in \({\mathcal A}_{2,1,N}\) and \(H\subset{\mathcal A}[p]\) a finite flat subgroup scheme of rank \(p\). If \(S_{2,1,N,(p)}\) denotes the supersingular locus of the moduli space \({\mathcal A}_{2,1,N,(p)}\otimes \overline{\mathbb F}_p\) then this scheme is equi-dimensional and each irreducible component is isomorphic to \(\mathbb P^1\), has \[ |S_{p_4}(\mathbb Z/N\mathbb Z)|\cdot {(-1)\zeta(-1)\zeta(-3)\over 4}\cdot [(p^2- 1)+ (p-1)(p^2+1)] \] irreducible components, has only double singular point and there are \[ |S_{p_4}(\mathbb Z/N\mathbb Z)|\cdot {(-1)\zeta(-1)\zeta(-3)\over 4} (p-1)(p^2+ 1)(p+ 1) \] of them and the natural morphism \(S_{2,1,N,(p)}\to S_{2,1,N}\) contracts \[ |S_{p_4}(\mathbb Z/N\mathbb Z)| \cdot{(-1)\zeta(-1)\zeta(-3)\over 4} (p-1)(p^2\neq 1) \] projective lines onto the superspecial points of \(S_{2,1,N}\). The result is an extension of previous work of T. Katsura and F. Oort [Compos. Math. 62, 107–167 (1987; Zbl 0636.14017)] concerning \(S_{2,1,N}\).

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14K12 Subvarieties of abelian varieties
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