Local models in the ramified case. I: The EL-case

Abstract

Local models are schemes defined in linear algebra terms that describe the étale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study modifications of the local models for G = \textrm R E / F \textrm G L ( n ) G=\text \textrm {R}_{E/F}\text \textrm {GL}(n) , with E / F E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We give applications to the structure of Schubert varieties in the affine Grassmannian and to the calculation of sheaves of nearby cycles and describe a relation with geometric convolution. In the general EL case, we replace the conjecture of Rapoport-Zink with a conjecture about the modified local models.