Integral models of Shimura varieties beyond parahoric level

The modular curve with $\Gamma_0(p)$-level structure has a
beautiful integral model over $\BZ_p$, and generalizations of this model
to any Shimura variety with $p$-parahoric level have been constructed in
the last 30 years. When passing from $\Gamma_0(p)$-level to
$\Gamma_1(p)$-level, the situation changes drastically. In the talk I
will explain a general method that potentially allows to deal with such
level structures. The method is based on the construction of torus
embeddings of the maximal torus of a reductive group scheme. Joint work
with G. Pappas.