Joint IAS/PU Number Theory
We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.
This talk is about qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.
The stacky approach was originated by Bhatt and Lurie. (But the possible mistakes in my talk are mine.)
Let X be a scheme over F_p. Many years ago Grothendieck and Berthelot defined the notion of crystal on X; moreover, they defined the notion of crystalline cohomology of a crystal.
The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a fundamental
question to understand the singularities which arise in the reductions
modulo $p$ of integral models of Shimura varieties. For PEL type
Shimura varieties with parahoric level structure at $p$, this question
has been studied since the 1990's. Due to the recent construction of