Joint IAS/PU Number Theory

An asymptotic version of the prime power conjecture for perfect difference sets

Sarah Peluse
Institute for Advanced Study and Princeton University; Veblen Research Instructor, School of Mathematics
September 10, 2020
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 m=n2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in Z/(n2+n+1)Z whenever n is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists.

Independence of ℓ for Frobenius conjugacy classes attached to abelian varieties

Rong Zhou
Imperial College London
June 18, 2020
Let A be an abelian variety over a number field E⊂ℂ and let v be a place of good reduction lying over a prime p. For a prime ℓ≠p, a result of Deligne implies that upon replacing E by a finite extension, the Galois representation on the ℓ-adic Tate module of A factors as ρℓ:Gal(E⎯⎯⎯⎯/E)→GA, where GA is the Mumford--Tate group of Aℂ. For p>2, we prove that the conjugacy class of ρℓ(Frobv) is defined over ℚ and independent of ℓ. This is joint work with Mark Kisin.

New constraints on the Galois configurations of algebraic integers in the complex plane

Vesselin Dimitrov
University of Toronto
June 11, 2020
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 polynomials P(X)∈ℤ[X] are at least 1 in magnitude (since they are non-zero integers), he found that the incidences (,P) between these polynomials P(X) and compacts ⊂ℂ of transfinite diameter d()

Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions

Florian Richter
Northwestern University
June 4, 2020
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 dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases.

Joint equidistribution of adelic torus orbits and families of twisted L-functions

Farrell Brumley
Université Sorbonne Paris Nord
May 28, 2020
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 classical interpretations, such as the equidistribution of integer points on spheres, of Heegner points or packets of closed geodesics on the modular surface, or of supersingular reductions of CM elliptic curves.

Iwasawa theory and Bloch-Kato conjecture for unitary groups

Xin Wan
Morningside Center of Mathematics, Chinese Academy of Sciences
May 21, 2020
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 CM character is 0, then the corresponding Selmer group has positive rank. The method also has a byproduct the p-adic functional equations for p-adic L-functions and p-adic families of Eisenstein series on unitary groups.

On triple product L functions

Jayce Robert Getz
Duke University
May 7, 2020
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 some sense this is the smallest case that appears out of reach via standard techniques. The approach is based on the beautiful fibration method of Braverman and Kazhdan for constructing Schwartz spaces and proving analogues of the Poisson summation formula.

Eulerianity of Fourier coefficients of automorphic forms

Henrik Gustafsson
Member, School of Mathematics
April 30, 2020
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 theorem which derives the Eulerianity of certain Fourier coefficients from that of another coefficient. I will also discuss some applications of this theorem to Fourier coefficients of automorphic forms in minimal and next-to-minimal representations.

Local-global compatibility in the crystalline case

Ana Caraiani
Imperial College
April 16, 2020
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 compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems. I will discuss joint work with J.