Seminars Sorted by Series

The classical Linnik problems are concerned with the equidistribution of adelic torus orbits on the homogeneous spaces attached to inner forms of GL2, as the discriminant of the torus gets large. When specialized, these problems admit beautiful...

One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new...

Fekete (1923) discovered the notion of transfinite diameter while studying the possible configurations of Galois orbits of algebraic integers in the complex plane. Based purely on the fact that the discriminants of monic integer irreducible...

Let $A$ be an abelian variety over a number field $E\subset \mathbb{C}$ and let $v$ be a place of good reduction lying over a prime $p$. For a prime $\ell\neq p$, a result of Deligne implies that upon replacing $E$ by a finite extension, the Galois...

A subset $D$ of a finite cyclic group $Z/mZ$ is called a "perfect difference set" if every nonzero element of $Z/mZ$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that...

Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. I will explain some recently established properties of these correspondences...

In recent years, o-minimality has found some striking applications to diophantine geometry. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the...

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the...

Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and...

A classical result identifies holomorphic modular forms with highest weight vectors of certain representations of $SL_2(\mathbb{R})$. We study locally analytic vectors of the (p-adically) completed cohomology of modular curves and prove a p-adic...

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any $p$-adic group $G$ that splits over a tame extension, provided $p$ does not divide the order of the Weyl group. This construction matches...

Following Bourgain, Gamburd, and Sarnak, we say that the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$ satisfies strong approximation at a prime $p$ if its integral points surject onto its $F_p$ points. In 2016, Bourgain, Gamburd, and Sarnak were...

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...

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 $G$ be a reductive group over a number field $F$ and $H$ a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on $G$ over $H(F)\backslash H(A_F)$. They are often related to special values of certain L functions. One...

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

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...

How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in...

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

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...

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

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

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

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

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...

Sums of Dirichlet characters $\sum_{n \leq x} \chi(n)$ (where $\chi$ is a character modulo some prime $r$, 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...

For certain automorphic representations $\pi$ on unitary groups, we show that if $L(s, \pi)$ vanishes to order one at the center $s=1/2$, then the associated $\pi$-localized Chow group of a unitary Shimura variety is nontrivial. This proves part 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...

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 this...

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

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 $|\chi(g)| \le \chi(1)^\alpha$, where $\chi$ is irreducible and $...

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...

I will discuss some recent progress in analytic number theory for polynomials over finite fields, giving strong new estimates for the number of primes in arithmetic progressions, as well as for sums of some arithmetic functions in arithmetic...

We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group. We present a proof of a subconvex bound in the t-aspect. More generally, we address the spectral aspect in the case of uniform parameter...

We study CM cycles on Kuga-Sato varieties over $X(N)$ via theta lifting and relative trace formula. Our first result is the modularity of CM cycles, in the sense that the Hecke modules they generate are semisimple whose irreducible components are...

For a polynomial $f\in \mathbb Q[x]$, Hilbert's irreducibility theorem asserts that the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a "thin" set of exceptions $R_f$. The problem of describing $R_f$ is...

In this talk, we prove an upper bound on the average number of 2-torsion elements in the class group of monogenised fields of any degree $n \geq 3$ and, conditional on a widely expected tail estimate, compute this average exactly. As an application...

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of $SL_2(\mathbb Z)$ whose Fourier coefficients have bounded denominators must be a modular form for some...

In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number...

Katz and Oort raised the following question: Given an algebraically closed field $k$, and a positive integer $g > 3$, does there exist an abelian variety over k not isogenous to a Jacobian over $k$? There has been much progress on this question...

We introduce a new orbit parametrization for square roots of the class of the inverse different in rings cut out by binary forms. This parametrization has many applications of interest in arithmetic statistics; for example, we use it to prove that...