Seminar on Arithmetic Algebraic Geometry: A Hasse invariant for the \mu-ordinary locus of Shimura varieties

2012-7-27 14:00 - 2012-7-27 16:00
Faculty of Science Building #3 Room 210
Marc-Hubert Nicole (Aix-Marseille II (Luminy))
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 classical 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 every strata of a given stratification.
On the other hand, there exist for all Shimura varieties of PEL-type so-called generalized Hasse-Witt invariants which are vector-valued, but they are typically not robust enough to carry over the usual applications of the classical Hasse invariant.Even in cases where the ordinary locus of a PEL-type Shimura variety is empty, Wedhorn has shown the existence of
an open, dense stratum called the \mu-ordinary locus. If the ordinary locus is non-empty, the two loci coincide. We will report on work in progress joint with Wushi Goldring in which we construct a new Hasse invariant invertible over the \mu-ordinary locus, alongside new number-theoretical applications.