Joint IAS/PU Number Theory

We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate...

Affine Deligne-Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport-Zink spaces; their irreducible components give rise to interesting algebraic cycles on the special fiber of Shimura varieties. We prove a...

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...

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-...

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...

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...

The analytic theory of Poincaré series and Maass cusp forms and their L-functions for $SL(3,Z)$ has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for $L^2(SL(3,Z)\PSL(3,R)/SO(3,R))$. I will describe the Maass...

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...

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...

Consider a pair $(X,\widehat{L})$ of a regular and projective algebraic variety over $\mathbb{Q}$ and a semipositive adelically metrized line bundle $\widehat{L}$ over $X$. Assume that all of Zhang's successive minima of the pair $(X,\widehat{L})$...