Joint IAS/PU Number Theory

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

Samit Dasgupta
Duke University
October 11, 2018
Hilbert’s 12th problem is to provide explicit analytic formulae for elements generating the maximal abelian extension of a given number field. In this talk I will describe an approach to Hilbert’s 12th that involves proving exact p-adic formulae for Gross-Stark units. This builds on prior joint work with Kakde and Ventullo in which we proved Gross’s conjectural leading term formula for Deligne-Ribet p-adic L-functions at s=0. This is joint work with Mahesh Kakde.

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

Ashay Burungale
Universite Paris 13; Member, School of Mathematics
May 1, 2018

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

Michael Lipnowski
Member, School of Mathematics
April 24, 2018

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.

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

Jayce Getz
Duke University; Member, School of Mathematics
March 27, 2018

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

Ian Petrow
ETH Zurich
March 13, 2018

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.

Abstract homomorphisms of algebraic groups and applications

Igor Rapinchuk
Michigan State University
February 13, 2018

I will discuss several results on abstract homomorphisms between the groups of rational points of algebraic groups. The main focus will be on a conjecture of Borel and Tits formulated in their landmark 1973 paper.
 
Our results settle this conjecture in several cases; the proofs make use of the notion of an algebraic ring. I will mention several applications to character varieties of finitely generated groups and representations of some non-arithmetic groups.

Automorphy for coherent cohomology of Shimura varieties

Jun Su
Princeton University
December 5, 2017
We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings’ B-G-G spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.