Joint IAS/PU Number Theory

The local intertwining relation (LIR) is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. It plays a central role in Arthur’s endoscopic classification for...

Jacobi's theta function Θ(q):=1+2q+2q4+2q9+…, and more generally the theta functions associated to positive-definite quadratic forms, have the property that they are modular forms of half-integral weight. The usual proof of this fact is completely...

Consider a collection of forms of odd degree with rational coefficients. Birch proved in 1957 that if the number of variables is sufficiently large, then the forms must have a nontrivial rational zero. The bounds resulting from Birch's proof...

Kudla and Millson proved in the 80's that the generating series of special cycles in orthogonal and unitary Shimura varieties are modular forms and a well-established conjecture of Kudla asks about extensions to toroidal compactifications. In this...

The past few years have seen rapid development in arithmetic statistics over function fields over finite fields, questions like:  what is the distribution of the ell-primary part of the class group of a random quadratic extension of F_q(t)?  (Cohen...

We construct the tame part of a split anticyclotomic Euler system in a setting where local multiplicity one does not hold. Instead of the traditional zeta integral approach, we prove the existence of the necessary test vectors using a new method...

In this talk, I will focus on simultaneous non-vanishing results for Dirichlet L-functions at the central point 1/2. Specifically, I will describe non-vanishing results (conditional on GRH) for two L-functions associated to Dirichlet characters in...

The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In...

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. In a series of papers, Venkatesh and his collaborators proposed an arithmetic reason for...

In characteristic zero, Castelnuovo proved that every unirational surface is rational. In positive characteristic, this fails dramatically: there exist many non-rational, often even general-type, surfaces that are nevertheless unirational. In 1977...