# Joint IAS/PU Number Theory

## Eisenstein series and the cubic moment for PGL(2)

## Canonical integral models of Shimura varieties

We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.

## A slice or two of a diagonal cubic: arithmetic stratification via the circle method

## Extremal cases of Rapoport-Zink spaces

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.

## A stacky approach to crystalline (and prismatic) cohomology.

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.

## Golden Gates in PU(n) and the Density Hypothesis.

## Taking the Hecke algebra to its limits

## Singularities in reductions of Shimura varieties

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