Generalized Hasse-Witt Invariants for Some Shimura Varieties

The classical Hasse invariant is defined by using the determinant of
the Hasse-Witt matrix. It allows cutting out the ordinary locus within the
special fiber of a modular curve: this is the locus where the Hasse invariant is
invertible. For more general Shimura varieties, the ordinary locus may be empty,
and the Hasse invariant thus conveys no information. This defect is already
visible in dimension one for Shimura curves at primes dividing the discriminant.
Also, except in rare circumstances, there do not exist generalized Hasse
invariants that cut out the strata of a given stratification. We shall explain
how, starting instead from the Hasse-Witt matrix, we can define generalized
Hasse-Witt invariants for Shimura varieties of PEL type.