Irreducibility and \(p\)-adic monodromies on the Siegel moduli spaces
The author studies Siegel modular varieties in positive characteristic \(p\) with parahoric level structure at \(p\). These are moduli spaces of chains of isogenies of abelian varieties of fixed dimension \(g\). In a previous paper [\textit{C.-F. Yu}, Int. Math. Res. Not. 2004, No. 48, 2593--2597 (2004; Zbl 1065.14059)] the author has determined the number of irreducible components in the case of Iwahori level structure. In the paper at hand, this question is considered for the non-ordinary locus, and for smaller strata, in the more general case of arbitrary parahoric level structure. The author gives explicit formulas for the number of irreducible components of the almost ordinary locus, and also for the number of connected components of arbitrary \(p\)-rank strata in these spaces. As in loc.~cit., an important ingredient is the theorem by Faltings and Chai about the surjectivity of the \(p\)-adic monodromy. In the final sections, the author discusses the case \(g=2\) with Iwahori level structure in detail, and in particular gives a very explicit geometric description of the supersingular locus and of the Kottwitz-Rapoport stratification.
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