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.

Some 25 years ago, as announced at a previous version of this seminar, Vaughan and the speaker obtained asymptotic upper and lower bounds for the number of non-trivial integral points on the Segre cubic $$x_1^3+...+x_6^3=x_1+...+x_6=0$$ with naive...

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...

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...

In their seminal work from the 80’s, Lubotzky, Phillips and Sarnak gave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS...

We parametrise elements in the full Hecke algebra in a way such that the parametrisation represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities...

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...

On an elliptic curve $y^2=x^3+ax+b$, the points with coordinates $(x,y)$ in a given number field form a finitely generated abelian group. One natural question is how the rank of this group changes when changing the number field. For the simplest...

The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the...

The thesis of Akshay Venkatesh obtains a ``Beyond Endoscopy'' proof of stable functorial transfer from tori to ${\rm SL}(2)$, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work...