Joint IAS/Princeton University Number Theory Seminar

It is known that the modular curve has good reduction at $p$ if the level structure is prime to $p$. If the level structure is of $\Gamma_0(p)$-type, then the modular curve has semi-stable reduction. For general Shimura varieties, one may ask for good, resp. semi-stable, reduction of the class of $p$-integral models of Shimura varieties defined recently by Kisin-Pappas. It was known for a long time that the Drinfeld case has semi-stable reduction, and a few more cases have been found since. Very recently, Faltings discovered a new case of semi-stable reduction. This triggered our interest in the classification of semi-stable local models. In this talk, I will discuss the classification of good and semi-stable reductions. Joint work with X. He and G. Pappas.