Seminar on Arithmetic Algebraic Geometry: A Hasse invariant for the \muordinary locus of Shimura varieties
 Date

2012727 14:00 
2012727 16:00
 Place
 Faculty of Science Building #3 Room 210

Speaker/Organizer
 MarcHubert Nicole (AixMarseille II (Luminy))

 Abstract:
The classical Hasse invariant is defined by using the determinant of the HasseWitt 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 PELtype socalled generalized HasseWitt invariants which are vectorvalued, 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 PELtype Shimura variety is empty, Wedhorn has shown the existence of
an open, dense stratum called the \muordinary locus. If the ordinary locus is nonempty, 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 \muordinary locus, alongside new numbertheoretical applications.