# Joint IAS/PU Number Theory

## The Lucky Logarithmic Derivative

## Irreducible components of affine Deligne-Lusztig varieties and orbital integrals

## Explicit formulae for Stark Units and Hilbert's 12th problem

## Towards a p-adic Deligne--Lusztig theory

The seminal work of Deligne and Lusztig on the representations of finite reductive groups has influenced an industry studying parallel constructions in the same theme. In this talk, we will discuss recent progress on studying analogues of Deligne--Lusztig varieties attached to p-adic groups.

## A converse to a theorem of Gross--Zagier, Kolyvagin and Rubin, II

Let $E$ be a CM elliptic curve over a totally real number field $F$ and $p$ an odd ordinary prime. If the ${p^{\infty}\mbox{-}\mathrm{Selmer}}$ group of $E$ over $F$ has ${\mathbb{Z}_{p}\mbox{-}\mathrm{corank}}$ one, we show that the analytic rank of $E$ over $F$ is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy.

## Algorithms for the topology of arithmetic groups and Hecke actions II

At the November workshop, I described a new algorithm to cover compact, congruence locally symmetric spaces by balls. I’ll discuss how to compute the nerve of such a covering and Hecke actions on its cohomology. Joint work with Aurel Page.At the November workshop, I described a new algorithm to cover compact, congruence locally symmetric spaces by balls. I’ll discuss how to compute the nerve of such a covering and Hecke actions on its cohomology.

Joint work with Aurel Page.

## Non-spherical Poincaré series, cusp forms and L-functions for GL(3)

## Summation formulae and speculations on period integrals attached to triples of automorphic representations

Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces $(V_i,Q_i)$ of even dimension by the equation

$Q_1(v_1)=Q_2(v_2)=Q_3(v_3)$.

## The Weyl law for algebraic tori

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.