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Hasse invariants for Hilbert modular varieties. (English) Zbl 1066.11018
Summary: Given a totally real field $$L$$ of degree $$g$$, we construct $$g$$ Hasse invariants on Hilbert modular varieties in characteristic $$p$$ and characterize their divisors. We show that these divisors give the type stratification defined by the action of $${\mathcal 0}_L$$ on the $$\alpha_p$$-elementary subgroup. Under certain conditions, involving special values of zeta functions, the product of these Hasse invariants is the reduction of an Eisenstein series of weight $$p-1$$.

##### MSC:
 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties
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