# zbMATH — the first resource for mathematics

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
Full Text: