In the recent breakthrough on the uniform Mordell-Lang problem by Dimitrov-Gao-Habegger and Kuhne, their key result is a uniform Bogomolov type of theorem for curves over number fields. In this talk, we introduce a refinement and generalization of...

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Joint IAS/PU Number Theory

This is intended to complement the recent talk of Pham Huu Tiep in this seminar but will not assume familiarity with that talk. The estimates in the title are upper bounds of the form |χ(g)|≤χ(1)α, where χ is irreducible and α depends on the size of...

I will discuss recent work with Harald Helfgott in which we establish roughly speaking that the graph connecting nn to n±pn±p with pp a prime dividing nn is almost "locally Ramanujan". As a result we obtain improvements of results of Tao and Tao...

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...

### On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)

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≤xχ(n)∑n≤xχ(n) (where χχ 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...