Joint IAS/PU Number Theory

An important open problem is to classify rational points on all modular curves, or equivalently to classify the possible adelic Galois images of elliptic curves over Q, as envisioned in Mazur’s Program B. However, this problem is inherently infinite...

A G-function, in the sense of Siegel, is a power series with rational coefficients that on the one hand has nice arithmetic properties (the LCM of the denominators of the first \(n\) coefficients grows at most exponentially) and on the other hand...

The goal of this talk is to discuss a new local-global phenomenon in the Langlands program on automorphic forms and Galois representations. Spaces of modular forms are parametrized by weights, and those same weights parametrize local deformation...

The Schottky problem is a classical question asking for a characterization of Jacobians among abelian varieties.

In positive characteristic, one may asking whether there are values of discreet invariants known to occurs for abelian varieties which do...

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...