Joint IAS/PU Number Theory

Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension nn, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In...

Consider the function field F of a smooth curve over FqFq, with q>2q>2.

L-functions of automorphic representations of GL(2)GL(2) over FF are important objects for studying the arithmetic properties of the field FF. Unfortunately, they can be...

I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the...

Given a K3 surface XX over a number field KK, we prove that the set of primes of KK where the geometric Picard rank jumps is infinite, assuming that XX has everywhere potentially good reduction. This result is formulated in the general framework of...

Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points...

Let λλ be the Liouville function and P(x)P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n))λ(P(n)) changes sign infinitely often. We present a solution to this problem for...

Let GG be a reductive group over a number field FF and HH a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on GG over H(F)∖H(AF)H(F)∖H(AF). They are often related to special values of certain L functions. One of the...

The Generalized Ramanujan Conjecture (GRC) for GL(n) is a central open problem in modern number theory. Its resolution is known to yield several important applications. For instance, the Ramanujan-Petersson conjecture for GL(2), proven by Deligne...

Let \o be an order in a totally real field, say F. Let K be an odd-degree totally real field. Let S be a finite set of places of K. We study S-integral K-points on integral models H_\o of Hilbert modular varieties because not only do said varieties...