Seminars Sorted by Series

In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity had only been proved for degrees d=1,2,3. We prove the hyperbolicity of all (but possibly...

The singularities in the reduction modulo $p$ of the modular curve $Y_0(p)$ are visualized by the famous picture of two curves meeting transversally at the supersingular points. It is a fundamental question to understand the singularities which...

The classical Hasse-Herbrand function is an important object in ramification theory, related to higher ramification groups. In this talk, I will discuss generalizations of the Hasse-Herbrand function and go over some of their properties. These...

We parametrise elements in the full Hecke algebra in a way such that the parametrisation represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities...

Bhargava showed that 100% of quintic fields have non-solvable Galois closure. For this reason, the arithmetic of elliptic curves over such fields is beyond the reach of methods based on Heegner points. In this talk we will report on a joint work in...

In their seminal work from the 80’s, Lubotzky, Phillips and Sarnak gave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS...

The stacky approach was originated by Bhatt and Lurie. (But the possible mistakes in my talk are mine.)

Let X be a scheme over F_p. Many years ago Grothendieck and Berthelot defined the notion of crystal on X; moreover, they defined the notion of...

This talk is about qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the...

Let G be a reductive group. Following Gross, and generalizing Serre's classical notion in the two-dimensional case, I will define what it means for a G-valued representation of the Galois group of a (totallyreal) number field to be odd. This notion...

Some 25 years ago, as announced at a previous version of this seminar, Vaughan and the speaker obtained asymptotic upper and lower bounds for the number of non-trivial integral points on the Segre cubic x31+...+x36=x1+...+x6=0x_1^3+...+x_6^3=x_1+...

We will discuss the problem of constructing and characterizing uniquely, integral models of Shimura varieties over some primes where non-smooth reduction is expected.

We discuss some recent results on singularity properties of word maps and Lie algebra word maps on semisimple algebraic groups and semisimple Lie algebras, generalizing a work of Aizenbud-Avni done in the case of the commutator map. These...

The Farey subdivision of the real line describes the action of PSL(2,Z) and gives a continued fraction algorithm approximating real numbers by rational numbers. Asmus Schmidt defined an analogue for the complex plane, depending on a choice of...

In a recent work, Matomäki, Radziwill and Tao showed that the Möbius function is discorrelated with linear exponential phases on almost all intervals of length $X^{\varepsilon}$. I will discuss joint work where we generalize this result to...

This is an update on the talk I gave about 10 years ago on this seminar. The arithmetic Gan-Gross-Prasad (AGGP) conjecture, a high dimensional generalization of the Gross–Zagier theorem, relates the height pairing of arithmetic diagonal cycles on...

The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties...

We will discuss how to study the cubic moment of any family of automorphic L-functions on PGL(2) using regularized diagonal periods of Eisenstein series, following a strategy suggested by Michel--Venkatesh. Applications include generalizations to...

In this talk, I formulate and prove a new Rubin-type Iwasawa main conjecture for imaginary quadratic fields in which p is inert or ramified, as well as a Perrin-Riou type Heegner point main conjecture for certain supersingular CM elliptic curves...

We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some...

In this talk, we will introduce a new technique of isolating the cuspidal part, or more general cuspidal components, from the $L^2$ spectrum on the spectral side of the trace formula. As an application, we prove the global Gan-Gross-Prasad and...

Every smooth proper algebraic variety over a p-adic field is expected to have semistable model after passing to a finite extension. This conjecture is open in general, but its analogue for Galois representations, the p-adic monodromy theorem, is...

I will discuss how the use of the p-adic Langlands correspondence for GL_2(Q_p) allows to study Eisenstein congruences of various weight, level and slopes in order to construct Euler systems. I will discuss the GL(2)-case with some details and give...

**Please note: All in-person IAS seminars have been cancelled. Seminar will be livestreamed. Please follow link: https://youtu.be/vZz1SFWHiog

Recently, Cai, Friedberg, Ginzburg and Kaplan generalized the doubling method of Piatetski-Shapiro and...

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->T to be an isomorphism of complete intersections. In addition to proving modularity theorems, this numerical criterion also implies a connection between the...

Let G be a real semi-simple Lie group with an irreducible unitary representation \pi. The non-temperedness of \pi is measured by the parameter p(\pi) which is defined as the infimum of p\geq 2 such that \pi has matrix coefficients in L^p(G). Sarnak...

The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. It is a key local...

Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with p-torsion coefficients. These Galois representations are expected to satisfy local-global...

Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2), thought of as the L-group...

The factorization of Fourier coefficients of automorphic forms plays an important role in a wide range of topics, from the study of L-functions to the interpretation of scattering amplitudes in string theory. In this talk I will present a transfer...

Establishing the conjectured analytic properties of triple product L-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on GL_3; in...

We will investigate the geometry of the p-adic eigencurve at classical points where the Galois representation is locally trivial at p, and will give applications to Iwasawa and Hida theories.

We describe a new method to study Eisenstein family and Iwasawa theory on unitary groups over totally real fields of general signatures. As a consequence we prove that if the central L-value of a cuspidal eigenform on the unitary group twisted by a...

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...