Joint IAS/Princeton University Number Theory Seminar

I will report on the construction of p-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing ramification (including deep levels) at p, with good behaviors over the loci where certain (multiplicative) ordinary level structures are defined. (We know almost nothing about the non-ordinary loci when p is ramified, but this theory is still sufficient for at least one important application.) This is a somewhat notorious subject, but I will try to begin with motivations and qualitative descriptions that might (it is hoped) make things easier.