Joint IAS/PU Number Theory

A group is said to have bounded generation (BG) if it is a finite product of cyclic subgroups. We will survey the known examples of groups with (BG) and their properties. We will then report on a recent result (joint with P. Corvaja, J. Ren and U...

Up to a finite covering, a sequence of nested subvarieties of an affine algebraic variety just looks like a flag of vector spaces (Noether); understanding this « up to » is a primary motivation for a fine study of finite coverings.

The aim of...

Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for...

Sums of Dirichlet characters ?n?x?(n)?n?x?(n) (where ?? is a character modulo some prime rr, say) are one of the best studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya...

I will explain some recent work on special cases of the Bloch-Kato conjecture for the symmetric cube of certain modular Galois representations. Under certain standard conjectures, this work constructs nontrivial elements in the Selmer groups of...

A classical branching theorem of Weyl describes how an irreducible representation of compact U(n+1)U(n+1) decomposes when restricted to U(n)U(n). The local Gan-Gross-Prasad conjecture provides a conjectural extension to the setting of...

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...