Stratifications of Hilbert modular varieties. (English) Zbl 0973.14010

In this paper the authors study a stratification of the moduli space of \(g\)-dimensional abelian varieties in characteristic \(p\) with a given action of \({\mathfrak O}_L\), the ring of integers of a totally real field \(L\) of degree \(g\) in which \(p\) is inert. The stratification is in terms of “types”, i.e., the set of characters \(\tau\) of \({\mathfrak O}_L/(p)\) which correspond to field embeddings, that intervene in the action of \({\mathfrak O}_L\) on the “local-local part” of the \(p\)-torsion of the abelian varieties that are classified. The corresponding stratum which consists of points such that the type of the corresponding abelian variety contains \(\tau\) is denoted by \(W_{\tau}\), and a close study is made of these strata. For instance it is proved that the \(W_{\tau}\)’s are locally irreducible and locally linear, the generic point of every component of \(W_{\tau}\) has type \(\tau\), the dimension of \(W_{\tau}\) is \(g-|\tau|\): further their incidence properties are determined.
The method used is the theory of displays of formal \(p\)-divisible groups as in the work of Norman-Oort and T. Zink. The results of this paper play a crucial role in the first author’s study of mod \(p\) and \(p\)-adic Hilbert modular forms from a geometric viewpoint, and in particular the construction of a plethora of Hasse invariants whose divisors are related to the \(W_{\tau}\)’s.


14G35 Modular and Shimura varieties
14K15 Arithmetic ground fields for abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14G20 Local ground fields in algebraic geometry