Xuhua He (University of Maryland)
Good and semi-stable reduction of Shimura varieties

Abstract: The problem of the reduction modulo p of a Shimura variety has a long and complicated history, perhaps beginning with Kronecker. 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. Are there other level structures such that the reduction modulo $p$ is good, resp. is semi-stable? A more precise interpretation of the question is to ask for good, resp. semi-stable, reduction of a specific class of $p$-integral models of Shimura varieties. It is known for a long time that the Drinfeld case has the semi-stable reduction. 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 the good and semi-stable reductions. It is based on a joint work with G. Pappas and M. Rapoport.